Home > Nonlinear Analysis of Orthotropic Membrane and Shell Structures Including Fluid-Structure Interaction

PhD Thesis by

Jes��s Gerardo Vald��s V��zquez

Nowadays, fluid-structure interaction problems are a great challenge of different fields in engineering and applied sciences. In civil engineering applications, wind flow and structural motion may lead to aeroelastic instabilities on constructions such as long-span bridges, high-rise buildings and light-weight roof structures. On the other hand, biomechanical applications are interested in the study of hemodynamics, i.e. blood flow through large arteries, where large structural membrane deformations interact with incompressible fluids. In the structural part of this work, a new methodology for the analysis of geometrically nonlinear orthotropic membrane and rotation-free shell elements is developed based on the principal fiber orientation of the material. A direct consequence of the fiber orientation strategy is the possibility to analyze initially out-ofplane prestressed membrane and shell structures. Additionally, since conventional membrane theory allows compression stresses, a wrinkling algorithm based on modifying the constitutive equation is presented. The structure is modeled with finite elements emerging from the governing equations of elastodynamics. The fluid portion of this work is governed by the incompressible Navier-Stokes equations, which are modeled by stabilized equal-order interpolation finite elements. Since the monolithic solution for these equations has the disadvantage that take great computer effort to solve large algebraic system of equations, the fractional step methodology is used to take advantage of the computational efficiency given by the uncoupling of the pressure from the velocity field. In addition, the generalized-J time integration scheme for fluids is adapted to be used with the fractional step technique. The fluid-structure interaction problem is formulated as a three-field system: the structure, the fluid and the moving fluid mesh solver. Motion of the fluid domain is accounted for with the arbitrary Lagrangian-Eulerian formulation with two different mesh update strategies. The coupling between the fluid and the structure is performed with the strong coupling block Gauss-Seidel partitioned technique. Since the fluid-structure interaction problem is highly nonlinear, a relaxation technique based on Aitken��s method is implemented in the strong coupling formulation to accelerate the convergence. Finally several example problems are presented in each field to verify the robustness and efficiency of the overall algorithm, many of them validated with reference solutions.

UP C UPC

UNIVERSITAT POLITÈCNICA DE CATALUNYA

ESCOLA TÈCNICA SUPERIOR D��ENGINYERS DE CAMINS, CANALS I PORTS DEPARTAMENT DE RESISTÈNCIA DE MATERIALS I ESTRUCTURES A L��ENGINYERIA

UPC

Universitat Polit��cnica de Catalunya Escola T��cnica Superior d��Enginyers de Camins, Canals i Ports

Departament de Resist��ncia de Materials i Estructures a l��Enginyeria

PhD Thesis

NONLINEAR ANALYSIS OF ORTHOTROPIC MEMBRANE AND SHELL STRUCTURES INCLUDING FLUID-STRUCTURE INTERACTION

by

Jes�� s Gerardo Vald��s V��zquez u e a

Advisors:

Eugenio O˜ ate n Juan Miquel Canet

Barcelona, October 18th , 2007

�� ACTA DE QUALIFICACIO DE LA TESI DOCTORAL Reunit el tribunal integrat pels sota signants per jutjar la tesi doctoral: T�� de la tesi: Nonlinear Analysis of Orthotropic Membrane and Shell Structures ıtol Includind Fluid-Structure Interaction Autor de la tesi: Jes��s Gerardo Vald��s V��zquez u e a Acorda atorgar la qualiﬁcaci�� de: o No apte Aprovat Notable Excel·lent Excel·lent Cum Laude

Barcelona, . . . . . . . . . . . . de/d�� . . . . . . . . . . . . . . . . . . . . . . . . de . . . . . . . . . . . .

El President

El Secretari

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El vocal

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A mis padres Irma y Jes��s u A mi hermano Fernando

Agradecimientos

Agradezco a mis directores de tesis, Eugenio O˜ate y Juan Miquel Canet, por n la oportunidad y buena disposici��n para llevar a cabo este trabajo. Gracias a o ellos y especialmente a Salvador Botello he descubierto el maravilloso mundo de la mec��nica computacional, destacado por la pasi��n, dedicaci��n y coraje que se a o o necesita para lograr nuevas metas. Asimismo quiero agradecer a Carlos Arnold y Sergio Silva, profesores de la facultad de Ingenier�� Civil de la Universidad de Guanajuato por haber depositado en ıa mi su conﬁanza y apoyarme para conseguir la beca PROMEP. Tambi��n agradezco e al CIMNE por la ﬁnanciaci��n prestada que me permiti�� concluir este trabajo. o o A todas aquellas personas que han hecho que este trabajo sea posible les agradezco inﬁnitamente su buena disposici��n, en especial al grupo de ﬂuidos del CIMNE o dirigido por Ram��n Codina y a mis compa˜eros Herbert Owen y Javier Principe. o n Tambi��n agradezco a Sergio Oller, Fernando Flores, Francisco Z��rate, Michele e a Chiumenti, Miguel Cervera, Carlos Agelet, Juan Carlos Cante, Javier Oliver, Gabriel Bugeda y Robert L. Taylor, porque siempre me resolvieron m��s de una duda. a Asimismo agradezco al personal del CIMNE y del departamento de resistencia de materiales y estructuras en ingenier�� RMEE por su siempre buena disposici��n y ıa o amables atenciones, en especial a Rosa Mar�� Olea. Debo agradecer tambi��n a ıa e Mariano V��zquez y Guillaume Houzeaux por las correcciones y recomendaciones a hechas a este trabajo. Quiero agradecer profundamente a Dorian Luis Linero, Martha Liliana Carre˜o, n Sergio Blanco, Carlos Gonz��lez Ferrari, Guillermo D�� y Mabel Cristina Marua ıaz landa, mis compa˜eros del despacho 111, por su compa˜�� compresi��n, y gran n nıa, o amistad nacida durante estos a˜os. Asimismo, quiero agradecer a Noel Hern��ndez, n a Fernando Guzm��n, Gelacio Ju��rez, Humberto Esqueda e Ivo Dias por esos grandes a a momentos dedicados al estudio y reir siempre a la hora de la comida. No me puedo olvidar de Roberto Clemente, Fernando Rastellini y Jeovan Faleiro con quienes compart�� ideas y momentos dif�� ı ıciles durante el doctorado, as�� como a Pablo Mata, ı Jos�� Manuel Gonz��lez, Joaqu�� Hern��ndez, Riccardo Rossi, Pere Andreu, Santiago e a ın a Bad�� y Xavier Mart�� ıa ınez. A las personas que me ense˜ aron el camino a seguir en Barcelona cuando reci��n n e llegue para comenzar mis estudios les estar�� por siempre agradecido, en especial a e mis queridos e inolvidables amigos Quino, Ysabel y por supuesto Elizabeth, que m��s puedo decir, simplemente gracias por todo. Tambi��n agradezco a mis amigos a e de la generaci��n de salida con quienes compart�� muchos agradables momentos, a o ı

Guillaume, Ana, Eduardo y Esteban. Tambi��n agradezco a las nuevas generaciones e del doctorado por esos momentos que hemos podido compartir, a Pablo, Maritzabel, Christian, Juan Carlos, Flavia y Mar�� Laura. ıa Quiero agradecer a la gente del futbol de los viernes por la noche por los momentos inolvidables durante y despu��s del partido, en especial a Manolo, Lucho, e Antonio, Alex y Gigio. No puedo dejar de mencionar en este trabajo a mis grandes amigos de toda la vida, Luis Eugenio, Victor, Ismael, Carlos y Julio C��sar, asi como a mis compa˜eros e n y tambien grandes amigos de la universidad, Alberto, Heriberto, Miguel Eugenio, Julio, Lalo, Vero, Ana y en especial a Sandra, gracias por las palabras de animo en �� los momentos m��s dif�� a ıciles cuando no ve�� la salida. ıa Y por ultimo, quiero agradecer a toda mi familia, empezando por mi hermano �� Fernando y mis padres Irma y Jes��s por estar siempre ah�� apoy��ndome y d��ndome u ı, a a palabras de aliento que me dieron la fuerza necesaria para salir adelante. Gracias a mis primos, t�� y todos mis familiares, y muy especialmente quiero agradecer a mis ıos abuelos Jos��, Jes��s y Lolo porque de ellos proviene mi escencia y de quienes siempre e u me acuerdo. S�� que algunos de ellos me vieron partir pero no pudieron verme e regresar, pero saben que siempre los llevar�� en mi coraz��n y les estar�� eternamente e o e agradecido.

Abstract

Nowadays, ﬂuid-structure interaction problems are a great challenge of diﬀerent ﬁelds in engineering and applied sciences. In civil engineering applications, wind ﬂow and structural motion may lead to aeroelastic instabilities on constructions such as long-span bridges, high-rise buildings and light-weight roof structures. On the other hand, biomechanical applications are interested in the study of hemodynamics, i.e. blood ﬂow through large arteries, where large structural membrane deformations interact with incompressible ﬂuids. In the structural part of this work, a new methodology for the analysis of geometrically nonlinear orthotropic membrane and rotation-free shell elements is developed based on the principal ﬁber orientation of the material. A direct consequence of the ﬁber orientation strategy is the possibility to analyze initially out-ofplane prestressed membrane and shell structures. Additionally, since conventional membrane theory allows compression stresses, a wrinkling algorithm based on modifying the constitutive equation is presented. The structure is modeled with ﬁnite elements emerging from the governing equations of elastodynamics. The ﬂuid portion of this work is governed by the incompressible Navier-Stokes equations, which are modeled by stabilized equal-order interpolation ﬁnite elements. Since the monolithic solution for these equations has the disadvantage that take great computer eﬀort to solve large algebraic system of equations, the fractional step methodology is used to take advantage of the computational eﬃciency given by the uncoupling of the pressure from the velocity ﬁeld. In addition, the generalized�� time integration scheme for ﬂuids is adapted to be used with the fractional step technique. The ﬂuid-structure interaction problem is formulated as a three-ﬁeld system: the structure, the ﬂuid and the moving ﬂuid mesh solver. Motion of the ﬂuid domain is accounted for with the arbitrary Lagrangian-Eulerian formulation with two diﬀerent mesh update strategies. The coupling between the ﬂuid and the structure is performed with the strong coupling block Gauss-Seidel partitioned technique. Since the ﬂuid-structure interaction problem is highly nonlinear, a relaxation technique based on Aitken��s method is implemented in the strong coupling formulation to accelerate the convergence. Finally several example problems are presented in each ﬁeld to verify the robustness and eﬃciency of the overall algorithm, many of them validated with reference solutions.

Resumen

Problemas de interacci��n ﬂuido-estructura representan hoy en d�� un gran desaf�� o ıa ıo en diferentes ��reas de ingenier�� y ciencias aplicadas. Dentro de las aplicaciones a ıa en ingenier�� civil, el ﬂujo del viento y los movimientos estructurales pueden ocaıa sionar inestabilidades aeroel��sticas en construcciones tales como puentes de gran a luz, rascacielos y cubiertas estructurales ligeras. Por otro lado, aplicaciones en biomec��nica est��n interesadas en el estudio de hemodin��mica, por ejemplo: ﬂujo a a a sangu�� ıneo en arterias, donde grandes deformaciones de las venas interact��an con u ﬂuidos. En la parte estructural de este trabajo, una nueva metodolog�� para el an��lisis ıa a geom��tricamente no-lineal ort��tropo de membranas y l��minas sin grados de libertad e o a de rotaci��n es desarrollada bas��ndose en la orientaci��n de la ﬁbra principal del o a o material. Una consecuencia directa de la estrategia de orientaci��n de ﬁbras es la o posibilidad de analizar membranas y l��minas pretensadas cuya conﬁguraci��n inicial a o est�� fuera del plano. Por otra parte, ya que la teor�� convencional de membranas a ıa permite que existan tensiones de compresi��n, un modelo de arrugado basado en la o modiﬁcaci��n de la ecuaci��n constitutiva se presenta. El desarrollo estructural es o o modelado con elementos ﬁnitos provenientes de las ecuaciones de la elastodin��mica. a La parte de ﬂuidos de este trabajo est�� gobernada por las ecuaciones de Naviera Stokes para ﬂujos incompresibles, las cuales son modeladas por interpolaciones estabilizadas de elementos ﬁnitos. Ya que la soluci��n monol�� o ıtica de dichas ecuaciones tiene la desventaja que consumen mucho tiempo en la soluci��n de grandes o sistemas de ecuaciones, el m��todo de pasos fraccionados se usa para aprovechar e las ventajas computacionales que brinda gracias al desacoplamiento de la presi��n o del campo de las velocidades. Adem��s, el esquema ��-generalizado para integraci��n a o en el tiempo para ﬂuidos es adaptado para que se use con la t��cnica de los pasos e fraccionados. El problema de interacci��n ﬂuido-estructura es formulado como un sistema de o tres campos: la estructura, el ﬂuido y el movimiento de la malla. El movimiento del dominio del ﬂuido es tomado en cuenta mediante la formulaci��n arbitraria o Lagrangiana-Euleriana, para la cual se usan dos estrategias de movimiento de malla. Para el acoplamiento entre el ﬂuido y la estructura se usa un acoplamiento fuerte por bloques usando la t��cnica de Gauss-Seidel. Debido a que la interacci��n entre el e o ﬂuido y la estructura es altamente no-lineal, se implementa el m��todo de relajaci��n e o basado en la t��cnica de Aitken, la cual acelera la convergencia del problema. e Finalmente varios problemas se presentan en los diferentes campos, los cuales

veriﬁcan la eﬁciencia de los algoritmos implementados.

Table of Contents

1 Introduction 1.1 Motivation 1.2 Background 1.3 Objectives . 1.4 Outline . . 1 1 3 5 6 9 9 12 14 15 15 16 19 21 21 22 23 24 25 25 25 27 31 31 33 36 40 52 54 54 55 57

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2 Continuum Mechanics 2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Strain Measures . . . . . . . . . . . . . . . . . . . . . . . 2.3 Stress Measures . . . . . . . . . . . . . . . . . . . . . . . 2.4 Conservation Equations . . . . . . . . . . . . . . . . . . 2.4.1 Mass Conservation . . . . . . . . . . . . . . . . . 2.4.2 Conservation of Linear and Angular Momentum 2.4.3 Conservation of Energy . . . . . . . . . . . . . . 2.5 Constitutive Equations . . . . . . . . . . . . . . . . . . . 2.5.1 Linear Elasticity . . . . . . . . . . . . . . . . . . 2.5.2 Nonlinear Elasticity . . . . . . . . . . . . . . . . 2.5.3 Newtonian Fluid . . . . . . . . . . . . . . . . . . 2.6 Navier-Stokes Equation . . . . . . . . . . . . . . . . . . 3 Structural Dynamics 3.1 Preliminaries . . . . . . . . . . . . . 3.1.1 Total Lagrangian Weak Form 3.1.2 Finite Element Discretization 3.2 Membrane Elements . . . . . . . . . 3.2.1 Introduction . . . . . . . . . 3.2.2 Membrane Formulation . . . 3.2.3 Fiber Orientation . . . . . . . 3.2.4 Finite Element Discretization 3.2.5 Wrinkling . . . . . . . . . . . 3.3 Cable Elements . . . . . . . . . . . . 3.3.1 Cable Formulation . . . . . . 3.3.2 Finite Element Discretization 3.4 Shell Elements . . . . . . . . . . . . i

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Table of Contents

3.5

3.6

3.7 3.8

3.4.1 Introduction . . . . . . . . . . . . 3.4.2 Shell Formulation . . . . . . . . . . 3.4.3 Finite Element Discretization . . . Time Integration Schemes . . . . . . . . . 3.5.1 Newmark Method . . . . . . . . . 3.5.2 Hilber-Hughes-Taylor Method . . . 3.5.3 Bossak Method . . . . . . . . . . . 3.5.4 Generalized-�� Method . . . . . . . 3.5.5 Structural Damping . . . . . . . . 3.5.6 Quasi-static Solution . . . . . . . . Solution Strategies . . . . . . . . . . . . . 3.6.1 Newton-Raphson Iterative Method 3.6.2 Linearization . . . . . . . . . . . . Code Development . . . . . . . . . . . . . Example Problems . . . . . . . . . . . . . 3.8.1 Inﬂation of a Square Airbag . . . . 3.8.2 Prestressed Membrane . . . . . . . 3.8.3 Nonlinear Plate . . . . . . . . . . . 3.8.4 Hemispherical Shell with 18o Hole 3.8.5 Free Vibration Pendulum . . . . . 3.8.6 Inﬂation of a Parachute . . . . . . 3.8.7 Hyperbolic Paraboloid . . . . . . . 3.8.8 Orthotropic Spinnaker . . . . . . .

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57 59 61 68 69 71 71 72 73 74 74 74 75 79 80 80 82 83 85 87 90 97 99

4 Fluid Dynamics 4.1 Introduction . . . . . . . . . . . . . . . . . . 4.2 Governing Equations . . . . . . . . . . . . . 4.2.1 Weak Form . . . . . . . . . . . . . . 4.2.2 Finite Element Discretization . . . . 4.3 Time Integration Schemes . . . . . . . . . . 4.3.1 ��-Family Method . . . . . . . . . . . 4.3.2 Backward Diﬀerentiation Method . . 4.3.3 Generalized-�� Method . . . . . . . . 4.4 Pressure Segregation Methods . . . . . . . . 4.4.1 Fractional Step Method . . . . . . . 4.4.2 Predictor-Corrector Method . . . . . 4.5 Stabilization Methods . . . . . . . . . . . . 4.5.1 SUPG/PSPG Stabilization . . . . . 4.5.2 OSS Stabilization . . . . . . . . . . . 4.5.3 FIC Stabilization . . . . . . . . . . . 4.6 Solution Strategies . . . . . . . . . . . . . . 4.6.1 Picard Iteration Method . . . . . . . 4.7 Code Development . . . . . . . . . . . . . . 4.8 Example Problems . . . . . . . . . . . . . . 4.8.1 Wall-Driven Cavity Flow . . . . . . 4.8.2 Flow Past a Circular Cylinder in 2D

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105 . 105 . 105 . 106 . 108 . 112 . 112 . 113 . 114 . 115 . 115 . 117 . 118 . 120 . 121 . 122 . 124 . 124 . 125 . 125 . 125 . 128

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4.8.3

Flow Past a Circular Cylinder in 3D . . . . . . . . . . . . . . 133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 . 137 . 140 . 142 . 145 . 146 . 146 . 147 . 147 . 148 . 149 . 151 . 152 . 152 . 160 . 163 . 167 . 171

5 Fluid-Structure Interaction 5.1 Coupling Strategies . . . . . . . . . . . . . . . . . . . . . . 5.2 ALE Framework . . . . . . . . . . . . . . . . . . . . . . . 5.3 Governing Equations . . . . . . . . . . . . . . . . . . . . . 5.4 Partitioned Methods . . . . . . . . . . . . . . . . . . . . . 5.4.1 Block Jacobi Method . . . . . . . . . . . . . . . . . 5.4.2 Block Gauss-Seidel Method . . . . . . . . . . . . . 5.4.3 Inexact Block Newton Method . . . . . . . . . . . 5.4.4 Exact Block Newton Method . . . . . . . . . . . . 5.5 Mesh Movement Techniques . . . . . . . . . . . . . . . . . 5.6 Strong Coupling with Relaxation . . . . . . . . . . . . . . 5.7 Code Development . . . . . . . . . . . . . . . . . . . . . . 5.8 Example Problems . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Nonlinear Shell Vibration Excited by Fluid Vortex 5.8.2 Cavity with Flexible Bottom Membrane in 2D . . 5.8.3 Cavity with a Thin Bottom Shell in 3D . . . . . . 5.8.4 Channel with Flexible Wall . . . . . . . . . . . . . 5.8.5 Pressure Pulse in a Compliant Vessel . . . . . . . .

6 Conclusions 179 6.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 6.2 Possible Future Research Lines . . . . . . . . . . . . . . . . . . . . . 183 References Index 185 197

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Table of Contents

List of Figures

1.1 1.2 1.3 Membrane long span structure . . . . Prestressed membrane structure . . . . Aeroelastic instabilities of the Tacoma U.S.A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Narrows suspension bridge, . . . . . . . . . . . . . . . . . 1 2 3

2.1 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27

Conﬁgurations of a body . . . . . . . . . . . . . . . . . . . . . . . . . 10 Curvilinear coordinates for a surface . . . . . . . . . . . . . Covariant base vectors forming a tangent plane . . . . . . . Principal ﬁber direction and local base system . . . . . . . . Principal ﬁber direction for a ﬁnite element . . . . . . . . . Assignment of principal ﬁber direction to adjacent elements Local Cartesian base systems for tangent spaces . . . . . . . Transferring reference principal ﬁber direction . . . . . . . . Principal ﬁbers direction of a mesh . . . . . . . . . . . . . . Local ﬁber base system . . . . . . . . . . . . . . . . . . . . Mapping of surface elements . . . . . . . . . . . . . . . . . . Triangular three-node ﬁnite element . . . . . . . . . . . . . Cable two-node and three-node ﬁnite elements . . . . . . . Shell middle surface . . . . . . . . . . . . . . . . . . . . . . Shell patch . . . . . . . . . . . . . . . . . . . . . . . . . . . z-displacements of inﬂated airbag . . . . . . . . . . . . . . . Principal stresses with wrinkling . . . . . . . . . . . . . . . Principal stresses without wrinkling . . . . . . . . . . . . . Prestressed membrane geometry . . . . . . . . . . . . . . . Finite element meshes used for the analysis . . . . . . . . . Central plate displacement by uniform load . . . . . . . . . z-displacements of nonlinear plate . . . . . . . . . . . . . . . z-displacements of nonlinear plate . . . . . . . . . . . . . . . Structured hemispherical shell geometry . . . . . . . . . . . Unstructured hemispherical shell geometry . . . . . . . . . . Load-displacement comparison . . . . . . . . . . . . . . . . Deformed without magniﬁcation factor . . . . . . . . . . . . Pendulum mesh . . . . . . . . . . . . . . . . . . . . . . . . . v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 34 36 37 37 38 38 39 40 41 48 54 59 61 80 81 81 82 83 84 84 85 85 86 87 87 88

vi

List of Figures

3.28 3.29 3.30 3.31 3.32 3.33 3.34 3.35 3.36 3.37 3.38 3.39 3.40 3.41 3.42 3.43 3.44 3.45 3.46 3.47 3.48 3.49 3.50 3.51 3.52 3.53 3.54 3.55 3.56 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16

Time-displacement graph . . . . . . . . . . . . . . . . . . . . . Deformed cable at time 1.00 s. . . . . . . . . . . . . . . . . . . Deformed cable at time 1.50 s. . . . . . . . . . . . . . . . . . . Deformed cable at time 2.18 s. . . . . . . . . . . . . . . . . . . Deformed cable at time 2.54 s. . . . . . . . . . . . . . . . . . . Deformed cable at time 2.78 s. . . . . . . . . . . . . . . . . . . Initial conﬁguration of the parachute, top and side views [m] . Inﬂation process of the parachute from reference conﬁguration . Inﬂation process of the parachute until ﬁnal conﬁguration . . . Vertical displacements at ﬁnal conﬁguration . . . . . . . . . . . Horizontal displacements at ﬁnal conﬁguration . . . . . . . . . Maximum principal stresses . . . . . . . . . . . . . . . . . . . . Minimum principal stresses . . . . . . . . . . . . . . . . . . . . Vertical displacements at ﬁnal conﬁguration . . . . . . . . . . . Horizontal displacements at ﬁnal conﬁguration . . . . . . . . . Maximum principal stresses . . . . . . . . . . . . . . . . . . . . Minimum principal stresses . . . . . . . . . . . . . . . . . . . . Final conﬁguration: left with wrinkling, right without wrinkling Geometry and mesh for hyperbolic parabolid . . . . . . . . . . Comparison of 1st principal stress Si [M N/m2 ] . . . . . . . . . Comparison of selected nodes . . . . . . . . . . . . . . . . . . . Materials to build sails . . . . . . . . . . . . . . . . . . . . . . . Principal ﬁber orientation for a spinnaker . . . . . . . . . . . . Spinnaker manufacturing process . . . . . . . . . . . . . . . . . Principal ﬁber direction: optimal . . . . . . . . . . . . . . . . . Principal ﬁber direction: horizontal . . . . . . . . . . . . . . . . Second Piola Kircchoﬀ stresses: optimal ﬁber orientation . . . . Optimal ﬁber orientation and 1st principal stress direction . . . Second Piola Kircchoﬀ stresses: horizontal ﬁber orientation . . Cavity four-node ﬁnite element mesh . . . . . Pressure contours for Re=400 . . . . . . . . . Velocity vx (0.5, y) vs. y . . . . . . . . . . . . Velocity vy (x, 0.5) vs. x . . . . . . . . . . . . Finite element mesh and boundary conditions Pressure contours . . . . . . . . . . . . . . . . x-velocity contours . . . . . . . . . . . . . . . y-velocity contours . . . . . . . . . . . . . . . Time history of y-velocity component . . . . CD for a ﬂow around a cylinder at Re=100 . CL for a ﬂow around a cylinder at Re=100 . Fourier spectrum of the lift coeﬃcient . . . . CD for a ﬂow around a cylinder at Re=100 . CL for a ﬂow around a cylinder at Re=100 . Finite element mesh and boundary conditions Detail of the mesh around the cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88 89 89 89 89 90 90 91 91 92 92 93 93 94 94 95 95 96 97 97 98 99 99 100 100 101 102 103 104 126 126 127 127 128 128 129 129 130 130 131 131 132 132 133 134

List of Figures

vii

4.17 4.18 4.19 4.20 4.21 4.22 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29

Pressure contours . . . . . . . . . . . . . . . x-velocity contours . . . . . . . . . . . . . . y-velocity contours . . . . . . . . . . . . . . CD for a ﬂow around a cylinder at Re=190 CL for a ﬂow around a cylinder at Re=190 Fourier spectrum of the lift coeﬃcient . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

134 134 135 135 136 136

Geometry and boundary conditions [cm] . . . . . . . . . . . . . . . . 152 Reference ﬁnite element mesh . . . . . . . . . . . . . . . . . . . . . . 153 History of maximum vertical tip displacement . . . . . . . . . . . . . 154 Fourier spectrum of the lift forces . . . . . . . . . . . . . . . . . . . . 154 Detail of the ﬂuid mesh around the shell . . . . . . . . . . . . . . . . 154 Pressure ﬁelds for diﬀerent time steps . . . . . . . . . . . . . . . . . 155 Pressure ﬁelds for diﬀerent time steps (cont.) . . . . . . . . . . . . . 156 History of maximum vertical tip displacement . . . . . . . . . . . . . 157 Pressure ﬁelds for diﬀerent time steps . . . . . . . . . . . . . . . . . 158 Pressure ﬁelds for diﬀerent time steps (cont.) . . . . . . . . . . . . . 159 Geometry and boundary conditions [m] . . . . . . . . . . . . . . . . 160 Bottom midpoint vertical displacement . . . . . . . . . . . . . . . . . 161 Pressure ﬁelds for diﬀerent time instants . . . . . . . . . . . . . . . . 162 Geometry and boundary conditions [m] . . . . . . . . . . . . . . . . 163 Shell midpoint vertical displacement . . . . . . . . . . . . . . . . . . 164 Shell vertical displacement ﬁeld . . . . . . . . . . . . . . . . . . . . . 165 Shell vertical displacement ﬁeld (cont.) . . . . . . . . . . . . . . . . . 166 Channel geometry and boundary conditions [m] . . . . . . . . . . . . 167 Channel inﬂow x-velocity component . . . . . . . . . . . . . . . . . . 167 Time history for pressure . . . . . . . . . . . . . . . . . . . . . . . . 168 Time history for x-displacement component . . . . . . . . . . . . . . 168 Contour ﬁeld for x-velocity component and deformed geometry . . . 169 Contour ﬁeld for x-velocity component and deformed geometry (cont.)170 Wave propagation generated by pressure pulse at the inﬂow . . . . . 172 Wave propagation generated by pressure pulse at the inﬂow (cont.) . 173 Structure deformed conﬁguration. Displacements norm . . . . . . . . 174 Structure deformed conﬁguration. Displacements norm (cont.) . . . 175 Time history for radial displacements . . . . . . . . . . . . . . . . . . 176 Time history for pressure . . . . . . . . . . . . . . . . . . . . . . . . 176

viii

List of Figures

Chapter 1

Introduction

1.1 Motivation

Modeling of structural elements, such as membranes and thin shells, is widely used in many engineering ﬁelds. In civil engineering applications, their elegance, eﬀectiveness and optimal material usage make these light weight structures an ideal construction element. The introduction of new ﬁber materials, such as glass, carbon or aramide ﬁbers with orthotropic material behavior have motivated a deep study of such elements which are used to build membrane and thin shell structures. Moreover membrane and thin shell structures are characterized by their low

Figure 1.1

Membrane long span structure

1

2

1. Introduction

Figure 1.2

Prestressed membrane structure

ﬂexural stiﬀness. Consequently these elements should not resist any compression at all. Therefore the usage of such construction materials is performed by introducing a prestressed force to the structure. Figs. 1.1 and 1.2 show diﬀerent applications of membranes in civil engineering structures. Other examples where light weight structures can be found include aircraft and spacecraft applications, parachutes, automobile airbags, sails, windmills and human tissues. Since this kind of structures are highly ﬂexible systems, they are susceptible to wind loads applied to them. Wind ﬂow and structural motion may lead to aeroelastic instabilities which may cause important damage or even collapse of the structure. Maybe one of the most important examples of aeroelastic instabilities is the disaster of the Tacoma Narrows suspension bridge that took place in the U.S.A. on November 7, 1940. The collapse of the bridge was due to wind-induced vibrations that at the beginning produced large transverse and rotational oscillations, as can be seen in Fig. 1.3. In general many physical problems of diﬀerent ﬁelds belong to multiphysic problems. In particular, numerical simulation of ﬂuid-structure interaction problems have gained great interest from the industry community in order to reduce development time and cost in coupled systems. This kind of problems are complex because they consist of structural nonlinear boundary conditions imposed on ﬂuid moving domains where the position is part of the solution. Recently biomechanical applications are interested in the numerical simulation of hemodynamics, which study the blood ﬂow through veins and arteries. In this problem, large structural deformations of the arteries interact with viscous blood

1.2 Background

3

Figure 1.3

Aeroelastic instabilities of the Tacoma Narrows suspension bridge, U.S.A.

ﬂow as a consequence of each heart beat. Another ﬁeld where ﬂuid-structure interaction plays an important role is the aerospace engineering which study wind ﬂow around ﬂexible wings of aircrafts.

1.2

Background

In this work, membrane and shell structures with large deformations are studied. A numerical solution for membranes may be constructed using the ﬁnite element method, which solution for small deformations can be found in Zienkiewicz and Taylor (1989), Cook et al. (1989) or O˜ate (1992). n Theory for large deformations can proceed following the presentations of Simo and Fox (1989), Simo et al. (1990a), B��tchter et al. (1992) or Braun et al. (1994). u In particular a large displacement formulation of membrane elements composed by three-node triangular ﬁnite elements based on rectangular Cartesian coordinates is proposed by Taylor (2001). This last formulation together with the study of Lu et al. (2001) form the basis for the development of the membrane formulation given in this work, which includes orthotropic material behavior and prestressed forces. Several studies have been carried out to study the geometrically nonlinear behavior of shells, for example the works of Simo and Fox (1989), Simo et al. (1990a), Simo and Kennedy (1992), among many others. Since shell analysis requires a lot of memory and cpu-time to compute, several authors have tried to derive plate and shell elements with displacements as the only nodal variables. In this area, O˜ate and Cervera (1993) proposed a general procedure based on n ﬁnite volume concepts for deriving linear thin plate elements of triangular and quadrilateral shapes with the nodal deﬂections as the only degree of freedom. Brunet and Sabourin (1994) proposed a diﬀerent approach to compute the con-

4

1. Introduction

stant curvature ﬁeld within each triangle in terms of the six-node displacement of a macro-element. This triangular element was successfully applied to nonlinear shell analysis using an explicit dynamic approach. Z��rate (1996) continue with the study a of rotation free elements of O˜ate and Cervera (1993) developing new triangular n elements. This formulation applied to large deformations with an explicit dynamic procedure was presented by Cendoya (1996). Rojek et al. (1998) proposed the same element that Cendoya (1996) but applied to metal forming processes. As an alternative formulation for large strain plasticity, the BST shell element was introduced by Flores and O˜ate (2001). This formulation constitutes the startn ing point for the development of the rotation-free shell formulation developed in this work, which includes orthotropic material behavior and prestressed forces. Besides the structural developments, to perform a ﬂuid-structure interaction study the ﬂuid ﬂow for incompressible problems has to be implemented. Finite element analysis of ﬂuids present potential numerical instabilities that emerge for incompressible ﬂow problems. To circumvent these problems, diﬀerent stabilization techniques have been proposed. One of the stabilization techniques that has been extensively used is the streamline-upwind/Petrov-Galerkin SUPG method. Here numerical oscillations could be avoided by introducing numerical diﬀusion only along the streamlines as explained in the work of Hughes and Brooks (1979) for advection-diﬀusion equation. The use of the streamline diﬀusion in the context of weighted residual methods is given in Hughes and Brooks (1982). Another kind of stabilization is the pressure-stabilizing/Petrov-Galerkin PSPG method. In Tezduyar et al. (1990), the SUPG and PSPG stabilization methods are used together with equal-order interpolations. With the idea to better understand the origins of stabilized methods, which can be derived from a ﬁrm theoretical foundation and a precise deﬁnition of the intrinsic time scale parameter, Hughes (1995) developed the subgrid scale method. In the context of these methods, the orthogonal sub-scales OSS method was introduced by Codina (2000). The stabilization methods described require the addition of some artiﬁcial diﬀusion terms. However another technique where the stabilization terms emerge from the governing equations of the problem is the ﬁnite calculus FIC method given by O˜ate (1998). An application of the FIC for incompressible n viscous ﬂow problems can be found in O˜ate (2000). n The monolithic coupled equations for incompressible ﬂuid problems have the disadvantage that take great computer eﬀort to solve the algebraic system for each time step in a transient analysis. Since the original works of Chorin (1967) and Temam (1969), fractional step methods for the incompressible Navier-Stokes equations have earned widespread popularity because of the computational eﬃciency given by the uncoupling of the pressure from the velocity ﬁeld. A detailed stability analysis of fractional step methods for incompressible ﬂows is given in Codina (2001). In this work, the FIC and OSS stabilization techniques are implemented to study the coupled problem of large structural deformations and incompressible ﬂuids ﬂow using pressure segregation methods. With the structural and ﬂuid problem introduced, the remaining task to study is the coupled problem between both of them. The implementation of a coupled problem can be done using two diﬀerent global strategies, which are the monolithic

1.3 Objectives

5

methods and the partitioned methods. In partitioned methods application of existing appropriate and sophisticated solvers for either structural or ﬂuid subsystems will be used. Partitioned methods were introduced by Park and Felippa (1983). The key idea for these methods is clearly described in Felippa et al. (1998). Partitioned solutions with staggered coupling algorithms are developed by Farhat et al. (1997) to be used in aeroelastic wing problems. Applications of strongly partitioned algorithms to large displacements structural problems coupled to viscous incompressible ﬂuids are given by Wall and Ramm (1998) and Wall (1999). Other large displacements structural problems interacting with incompressible ﬂuids are detailed in Mok (2001), Mok and Wall (2001) and Tallec and Mouro (2001). More sophisticated developments on strong partitioned method for FSI problems can be found in Steindorf (2002), Matthies and Steindorf (2004) and Tezduyar et al. (2006). A study on strong coupling partitioned methods for ﬂuid-structure interaction problems applied to hemodynamic can be found in Nobile (2001), Causin et al. (2005), and Fern��ndez and Moubachir (2005). In this a work, strong coupling partitioned methods are used for ﬂuid-structure interaction problems.

1.3

Objectives

Two general objectives in this work are pursued. The ﬁrst of them is the improvement of prior developments made at CIMNE related to nonlinear membrane and shell analysis. The second general objective is the study of ﬂuid-structure interaction problems using ﬂuid ﬂow pressure segregation methods and stabilization techniques developed at CIMNE. The two general objectives must be implemented in one eﬃcient, robust and accurate computational tool that use ﬁnite element technology to solve the problem in study, which also must have possibilities to analyze each subproblem, i.e. the solid or ﬂuid part, independently as a highly developed software that exchange data for the solution of the coupled problem with other separate solvers. The following particular objectives belong to the structural part: • To review the state of the art for geometrically nonlinear membrane and shell ﬁnite elements. • To improve the membrane and rotation-free shell ﬁnite elements developed at CIMNE in prior studies. • To develop a new methodology to analyze orthotropic material behavior of membrane and shell ﬁnite elements, including the wrinkling phenomena to avoid compression stresses in membranes. • To propose a new strategy to study prestressed membrane elements. • To explore existing time integration schemes for structural dynamic problems and to work with the best choice for long time analysis periods.

6

1. Introduction

• To implement the membrane and rotation-free shell ﬁnite elements together with their new developments in the COMET program, which is a software for coupled contact, mechanical and thermal analysis built at CIMNE.

The particular objectives belonging to the ﬂuid-structure interaction problem part are: • To implement in the COMET program the ﬁnite element ﬂuid problem based on the incompressible Navier-Stokes equations and to use the stabilization techniques developed at CIMNE. • To implement the mesh update strategy developed at CIMNE for the motion of the ﬂuid domain. • To explore existing time integration schemes for ﬂuid dynamic problems and to work with the best choice for long time analysis periods. • To review the state of the art for ﬂuid-structure interaction problems. • To implement a general and simple methodology to perform the study of coupled problems using the COMET program. All these objectives are oriented to improve and to combine the developments made in-the-house in the solid mechanics ﬁeld, i.e. membrane and thin shell ﬁnite elements, and the ﬂuid dynamics ﬁeld, i.e. stabilization techniques and mesh moving algorithms, to perform ﬂuid-structure interaction of problems involving large structural displacements and incompressible ﬂuid ﬂows.

1.4

Outline

Since this work deals with diﬀerent topics, a detailed review of the state-of-the-art is provided at the beginning of each theme to be developed. The work presented is organized as follows: Chapter 2. Since the computational models emerged in this work are developed from the mechanics of a continuous medium, in this chapter the kinematics, stress and strain measures for solids and ﬂuids, conservation equations, and the constitutive and Navier-Stokes equations used in this work are given. The remainder of this work is referred to these equations. Chapter 3. In this chapter a review of the total Lagrangian formulation for geometrically nonlinear solid mechanics is made. From the continuum mechanics theory, the general formulation of each subject is presented which later is discretized with ﬁnite elements. Here the concept of principal ﬁber orientation is introduced. Next a new formulation for membrane elements is developed based on the ﬁber orientation, including orthotropic material behavior and initially out-of-plane prestressed conditions. Also a basic wrinkling algorithm to avoid compression stresses is studied.

1.4 Outline

7

Then the rotation-free shell elements are derived, using the ﬁber orientation to yield a new formulation to study prestressed and orthotropic material behavior of shells. Later a review of some time integration schemes for solids are addressed. The solution strategy used in this work and linearization of the semi-discrete equations of motion is explained. Finally some example problems are presented to validate the ﬁnite element implementations in the COMET code. Chapter 4. This chapter deals with the ﬂuid dynamic equations that solve the incompressible ﬂow problem. Next the governing equations which yield the weak form and the ﬁnite element discretization for ﬂuids are explained. Later a review of some time integration schemes for ﬂuids are studied. Two diﬀerent pressure segregation methods to solve the incompressible ﬂuid equations problem are presented. Then three stabilization techniques for the incompressible ﬂuid problem are introduced. The solution strategy used in this work to solve the nonlinear algebraic equations is shown. Finally some benchmark problems are solved and compared to validate the ﬁnite element implementation in the COMET program. Chapter 5. In this chapter the coupling strategies for ﬂuid-structure interaction problems are explained. The ﬁnite element formulation described in chapter 4 dealing with the incompressible ﬂow problem is extended to account for moving ﬂuid domains by means of the arbitrary Lagrangian-Eulerian formulation. Next the governing equations which yield the weak form of the coupled problem are obtained. Then diﬀerent partitioned methods to solve the ﬂuid-structure interaction problem are studied. Later two diﬀerent mesh update techniques are presented. A detailed algorithm for the strong coupling with relaxation used in this work is given. Finally some example problems are presented to validate the ﬁnite element implementations in the COMET code. Chapter 6. Here the conclusions and achievements of this work are presented. This thesis concludes with some suggestions for future research lines to be developed as a direct consequence of this work.

8

1. Introduction

Chapter 2

Continuum Mechanics

This work deals with the study of structures with large deformations interacting with incompressible ﬂuids. As mentioned in Batchelor (2000), the distinction between solids and ﬂuids is not a sharp one. While a solid has a deﬁnite shape which changes are small when there is a small change in the external conditions, a ﬂuid does not have a preferred shape and the relative positions of their elements change by an amount which is not small even tough the applied forces are small. The macroscopic behavior of ﬂuids, solids and structures is given by models emerged from the mechanics of a continuous medium. In this chapter a summary of the continuum mechanics used in this work for ﬂuid-structure interaction problems is presented. A general and more detailed description of continuum mechanics can be found in Malvern (1969), Gurtin (1981), Holzapfel (2000), Batchelor (2000), Kundu and Cohen (2002), and Spencer (1980) among many other well known books. This chapter begins with the kinematics that involved motion of a body. Next the concepts of stress and strain related to nonlinear solid mechanics and ﬂuids are described. Then the conservation equations, also known as balance equations, are presented. Finally the constitutive and Navier-Stokes equations are given.

2.1

Kinematics

Kinematics is the study of motion and deformation of a body without regard to the forces responsible for such action. A body B can be imagined as a composition of a set of particles which are called material points. This body is in an initial state when time t = 0 as shown in Fig. 2.1. The domain of the body in this state is denoted by ��0 which occupies a region in space and is known as the initial conﬁguration. To describe the kinematics of a body another conﬁguration is needed where equations are referred to and is called the reference conﬁguration. Most of the times the initial conﬁguration is used as the referenced conﬁguration, unless we specify otherwise. Now we assume that the domain ��0 moves to a new region �� which is occupied by the body B for any subsecuent time t > 0. At this time the conﬁguration of the 9

10

2. Continuum Mechanics

I0

d

I

u X

Z

x

Y

X

Figure 2.1

Conﬁgurations of a body

body is called the current conﬁguration, also known as the deformed conﬁguration. The boundary of the domain, in this case for the current conﬁguration, is denoted by ��. The dimension of any model is denoted by ndime denoting the number of space dimensions of the body B. The position vector of a material point in the reference conﬁguration is deﬁned by X, where

ndime

X = Xi ei =

i=1

Xi ei

(2.1.1)

where Xi are the components of X in the reference conﬁguration and ei are the unit base vectors for a rectangular Cartesian coordinate system. The variable vectors X are called material coordinates or Lagrangian coordinates. The motion of the body B is given by x = ��(X, t) = x(X, t) where

ndime

(2.1.2)

x = xi ei =

i=1

xi ei

(2.1.3)

is the position of the material point X in the current conﬁguration. The variable vectors x are called spatial coordinates or Eulerian coordinates, and the function ��(X, t) is a mapping of the reference conﬁguration onto the current conﬁguration. When describing the kinematics of a continuum two approaches can be used. First, if we take material coordinates Xi and time t as the independent variables, the description is called material description or Lagrangian description. On the other hand, if the independent variables are the spatial coordinates xi and time

2.1 Kinematics

11

t, we are taking about a spatial description or Eulerian description. In general solid mechanics use Lagrangian descriptions while ﬂuid mechanics use Eulerian descriptions. The diﬀerence for a material point between its current and reference conﬁguration gives the displacement which in material description is u(X, t) = x − X Using Eq. (2.1.1) and Eq. (2.1.2) into Eq. (2.1.4) yields u(X, t) = ��(X, t) − ��(X, 0) = ��(X, t) − X (2.1.5) (2.1.4)

since for t = 0, x = ��(X, 0) = X which means that at reference conﬁguration x = X. Given (x, t) as the independent variables, the inverse mapping of the motion is deﬁned as X = ��−1 (x, t) = X(x, t) (2.1.6)

meaning that the material point X is associated with the place x at time t. For a material point the velocity is the rate of change, or derivative, of the position vector. When X is held constant then the derivative is called material time derivative or total time derivative. Using Eq. (2.1.2) and Eq. (2.1.5) the material velocity is given by ∂u(X, t) ∂x(X, t) ˙ = = u(X, t) (2.1.7) ∂t ∂t The material acceleration is the rate of change of the velocity, or the material time derivative of the velocity expressed by v(X, t) = ∂v(X, t) ˙ �� = v(X, t) = u(X, t) (2.1.8) ∂t For expressions given in spatial description, i.e. the velocity v(x, t) = v(x(X, t), t) where we use Eq. (2.1.2), its material time derivative can be found using a(X, t) = ∂vi (x, t) ∂vi (x, t) ∂xj (X, t) ∂vi ∂vi Dvi (x, t) · vj = + = + Dt ∂t ∂xj ∂t ∂t ∂xj

(2.1.9)

where ∂vi (x, t)/∂t is the spatial time derivative and the second term in the right hand side is the convective term, where ∂vi /∂xj is the right gradient of the velocity vector ﬁeld with respect to the spatial coordinates, which in indicial form is vi,j or in tensor notation is v∇. Using the inverse mapping of the motion, Eq. (2.1.6) to express the velocity in spatial description, Eq. (2.1.9) can be written as Dv(x, t) ∂v(x, t) = + v(x, t) · ∇v(x, t) (2.1.10) Dt ∂t where ∇v is the left gradient of the velocity vector ﬁeld with respect to the spatial coordinates, which in indicial form is ∂j vi . It is important to see that

12

2. Continuum Mechanics

Dv(x, t) ∂v(X, t) = (2.1.11) Dt ∂t In general the material time derivative of any function, vector or tensor given in spatial variables x and time t can be obtained with D(•) ∂(•) = + v · ∇(•) (2.1.12) Dt ∂t When a continuum body moves from the reference conﬁguration ��0 to the current conﬁguration ��, it changes its size and shape giving a deformation. A primary measure of deformation in nonlinear mechanics is the material deformation gradient tensor given by F= ∂x ∂X or Fij = ∂��i ∂xi = ∂Xj ∂Xj (2.1.13)

which relates a quantity in the reference conﬁguration to its corresponding quantity in the current conﬁguration. For example consider an inﬁnitesimal line segment d X in the reference conﬁguration, then using Eq. (2.1.13) the resulting line segment d x in the current conﬁguration is dx = F · dX or dxi = Fij dXj (2.1.14)

The deformation gradient F is also known as the Jacobian matrix . Another important quantity related to F is the Jacobian determinant given by J = det(F) (2.1.15)

The Jacobian determinant is useful to relate integrals in the reference conﬁguration to its counterpart in the current conﬁguration. Using x from Eq. (2.1.4) in Eq. (2.1.13), the deformation gradient tensor can be expressed by Fij = ∂ui ∂Xi ∂ui + = + ��ij ∂Xj ∂Xj ∂Xj (2.1.16)

where ∂ui /∂Xj is the material displacement gradient tensor and ��ij is the Kronecker delta which values are ��ij = 1 0 when i = j otherwise (2.1.17)

2.2

Strain Measures

In the behavior of materials, the strain measures the geometrical deformation caused by the forces applied on a continuum body B. The strain is computed as the change between the undeformed initial conﬁguration of the body and its

2.2 Strain Measures

13

ﬁnal deformed conﬁguration. Therefore, the strain expresses itself the motion and deformation of a body. There are many kinematic measures of strain in continuum mechanics. For Lagrangian descriptions the most essential strain is the Green-Lagrange strain tensor deﬁned by 1 1 T (2.2.1) or Eij = FT · F − I Fik Fkj − ��ij 2 2 which also can be expressed as a function of the displacement gradient tensor yielding E= Eij = 1 2 ∂uj ∂uk ∂uk ∂ui + + ∂Xj ∂Xi ∂Xi ∂Xj (2.2.2)

For linear strain problems, the inﬁnitesimal strain tensor can be found from Eq. (2.2.2) by neglecting the nonlinear terms giving ��ij = 1 2 ∂uj ∂ui + ∂Xj ∂Xi (2.2.3)

Now we deﬁne the spatial velocity gradient tensor by l= ∂v ∂x or lij = ∂vi ∂xj (2.2.4)

which can be decomposed into its symmetric and skew-symmetric parts using 1 1 l+lT + l−lT (2.2.5) 2 2 where the spatial rate of deformation tensor d, also known as the velocity strain tensor or strain rate tensor is given by the symmetric part of the velocity gradient according to l= d= 1 l+lT 2 or dij = 1 2 ∂vj ∂vi + ∂xj ∂xi (2.2.6)

The spatial rate of rotation tensor w, also known as the spin tensor is given by the skew-symmetric part of l yielding w= 1 l−lT 2 or wij = 1 2 ∂vj ∂vi − ∂xj ∂xi (2.2.7)

Taking the material time derivative of the deformation gradient tensor, Eq. (2.1.13), gives ∂v ˙ F= ∂X and now Eq. (2.2.4) can be written as ˙ l = F · F−1 or

−1 ˙ lij = Fik Fkj

or

∂vi ˙ Fij = ∂Xj

(2.2.8)

(2.2.9)

14

2. Continuum Mechanics

where we use the spatial deformation gradient tensor F−1 = ∂X ∂x or

−1 Fkj =

∂Xk ∂xj

(2.2.10)

If we take the material time derivative of the Green-Lagrange strain tensor, Eq. (2.2.1), we get 1 ˙ ˙T ˙ FT · F + F · F = FT · d · F E= 2 (2.2.11)

2.3

Stress Measures

The motion and deformation of a continuum body B gives rise to forces emerging from interactions between interior parts of the body or between the body and its environment. Physically, the stress measures a force per unit area within a body. Let P be a point on the boundary �� of the body, n the outward normal unit vector for P and d �� the part of the surface on the body where P is contained. Then dfs is the surface force acting at P that depends of n and d ��. Henceforth the surface traction t at point P on the surface with normal n is deﬁned by dfs (2.3.1) d����0 d�� where t does not necessarily coincide in direction with n. It is important to see that the surface traction has units of force per unit area. There exists a spatial tensor ﬁeld �� called the Cauchy stress tensor such that for each unit vector n t = t(n) = lim t = n · �� = ��T · n or ti = ��ji nj (2.3.2)

which is also known as the Cauchy��s theorem. Since the Cauchy stress tensor involves the normal to the current surface and the traction on the current surface too, this tensor is also known as the true physical stress tensor and has the property that is symmetric, see section 2.4.2. In the reference conﬁguration the counterpart of Eq. (2.3.2) is t0 = n0 · P or t0 = Pji n0 i j (2.3.3)

where P is the nominal stress tensor and t0 and n0 is the traction force and unit normal respectively in the reference conﬁguration. Unlike the Cauchy stress tensor, the nominal stress tensor is not symmetric and is important to see that the normal is to the left. The transpose of the nominal stress tensor is the ﬁrst Piola-Kirchhoﬀ stress tensor. The second Piola-Kirchhoﬀ stress tensor S is deﬁned by F−1 · t0 = n0 · S (2.3.4)

where the transformation of the forces by F−1 makes it a symmetric tensor. The transformation between these stresses is given by �� = J −1 F · P = J −1 F · S · FT (2.3.5)

2.4 Conservation Equations

15

P = JF−1 · �� = S · FT S = JF−1 · �� · F−T = P · F−T

(2.3.6) (2.3.7)

2.4

Conservation Equations

The conservation equations reﬂect some physical quantity for a continuum medium which always must be satisﬁed and that are not restricted in their application to any material. Applying the conservation equations to the domain �� of a body B leads to an integral relation. Since the integral relation must hold for any subdomain of the body, then the conservation equations can be expressed as partial diﬀerential equations. Before continuing with the conservation equations, the material time derivative of an integral relation for any spatial property is deﬁned by D Dt (•) =

�� ��

D(•) + (•)∇ · v d�� Dt

(2.4.1)

which is the Reynold��s transport theorem. The divergence ∇ · (•) taken respect to current coordinates can also be expressed as div(v) or in indicial form vi,i .

2.4.1

Mass Conservation

Consider the domain �� of a body B bounded by the surface �� which is ﬁlled with a constant material density ��(X, t). Then the mass of the body is given by m=

��

��(X, t)d�� =

��0

��(X, t)Jd��0 =

��0

��0 (X)d��0

(2.4.2)

where we use Eq. (2.1.15) to relate integrals in the reference and current conﬁgurations. Mass conservation requires that the mass of any material domain be constant. Consequently the material time derivative of the mass must be zero, giving Dm D = Dt Dt ��d�� = 0

��

(2.4.3)

which leads to the following integral relation using Eq. (2.4.1) Dm = Dt D�� + ��∇ · v d�� = 0 Dt (2.4.4)

��

where the quantities used are expressed in spatial coordinates. Since the above holds for any subdomain ��, the mass conservation yields the following ﬁrst-order partial diﬀerential equation D�� + ��∇ · v = 0 Dt or �� + ��vi,i = 0 ˙ (2.4.5)

16

2. Continuum Mechanics

The above equation is also known as the continuity equation. Using the deﬁnition of material time derivative Eq. (2.1.12) in ﬁrst term of Eq. (2.4.5), the continuity equation can be written as ∂�� (2.4.6) + (��vi ),i = 0 ∂t which is known as the conservative form of the mass conservation equation. When a material is said to be incompressible, the density keeps constant in time so that its material time derivative vanishes and the continuity equation becomes ∇·v=0 or vi,i = 0 (2.4.7)

This is the continuity equation used in this work for ﬂuid problems which always are considered as incompressible. For Lagrangian descriptions, the mass conservation equation Eq. (2.4.3) can be integrated in time to obtain an algebraic equation for the density in the form of Eq. (2.4.2) yielding ��(X, t)J = ��0 (X) which is the Lagrangian description for the mass conservation equation. (2.4.8)

2.4.2

Conservation of Linear and Angular Momentum

The conservation of linear momentum states that the rate of change of its linear momentum is equal to the total force applied to it. The conservation of linear momentum is also known as the balance of momentum principle or simply the momentum conservation principle. If we consider an arbitrary domain �� with boundary �� in the current conﬁguration subjected to body forces ��b and surface tractions t, where b is a force per unit mass, then the total force f is given by f(t) =

��

��b(x, t)d�� +

��

t(x, t)d��

(2.4.9)

The linear momentum is given by the product of the density �� and the velocity v over the domain �� in the form p(t) =

��

��v(x, t)d��

(2.4.10)

Hence the conservation of linear momentum is expressed by D Dt ��v(x, t)d�� =

�� ��

��b(x, t)d�� +

��

t(x, t)d��

(2.4.11)

Using Eq. (2.4.1) and Eq. (2.4.5) in Eq. (2.4.11), the rate of change of the linear momentum is found to be D Dt ��v(x, t)d�� =

�� ��

��

Dv(x, t) d�� Dt

(2.4.12)

2.4 Conservation Equations

17

The boundary integral in Eq. (2.4.11) can be transformed to a domain integral using Eq. (2.3.2) and Gauss�� divergence theorem yielding t(x, t)d�� =

�� ��

∇ · ��(x, t)d��

(2.4.13)

Substituting Eq. (2.4.12) and (2.4.13) into (2.4.11) gives ��

��

Dv − ��b − ∇ · �� d�� = 0 Dt

(2.4.14)

and since it holds for any arbitrary domain we ﬁnd �� Dv = ∇ · �� + ��b Dt or �� Dvi ∂��ij + ��bi = Dt ∂xj (2.4.15)

This is called the momentum equation. In an Eulerian description for the momentum equation, the material time derivative of the velocity in Eq. (2.4.15) is developed using Eq. (2.1.12) yielding ∂v + v · ∇v ∂t ∂vi + vj ∂j vi ∂t ∂��ij + ��bi (2.4.16) ∂xj

��

= ∇ · �� + ��b

or

��

=

where all the quantities are given is spatial coordinates. In this work, Eq. (2.4.16) is the one to be used in the ﬂuid mechanics problem. For ﬂuid ﬁnite elements this equation is called Eulerian formulation. The momentum equation Eq. (2.4.15) can also be written in a Lagrangian description where all the quantities are expressed in material coordinates, giving �� ∂v = ∇ · �� + ��b ∂t or �� ∂vi ∂��ij + ��bi = ∂t ∂xj (2.4.17)

Since Eq. (2.4.15) is in the current conﬁguration, the divergence term is taken respect to spatial coordinates and therefore ��(X, t) is expressed by �� ��−1 (x, t), t so that the spatial gradient of the stress ﬁeld can be evaluated. For nonlinear solid ﬁnite elements, Eq. (2.4.17) is called updated Lagrangian formulation. The conservation of angular momentum is obtained by taking the cross product of the current vector position x by each term of the linear momentum equation Eq. (2.4.11), yielding D Dt

x �� ��v(x, t)d�� =

�� ��

x �� ��b(x, t)d�� +

��

x �� t(x, t)d��

(2.4.18)

where it can be shown that leads to the following result �� = ��T (2.4.19)

The acceleration term in Eq. (2.4.15) can be neglected when the loads are applied slowly so that the inertial forces become insigniﬁcant. Then we can write

18

2. Continuum Mechanics

∇ · �� + ��b = 0

or

∂��ij + ��bi = 0 ∂xj

(2.4.20)

which is known as the equilibrium equation. Problems that use Eq. (2.4.20) are called static problems. The conservation of linear momentum can also be expressed in the reference conﬁguration. Consider an arbitrary domain ��0 with boundary ��0 in the reference conﬁguration subjected to body forces ��0 b and surface tractions t0 , then the total force f is given by f(t) =

��0

��0 b(X, t)d��0 +

��0

t0 (X, t)d��0

(2.4.21)

The linear momentum is given by p(t) =

��0

��0 v(X, t)d��0

(2.4.22)

Hence the conservation of linear momentum is expressed by d dt ��0 v(X, t)d��0 =

��0 ��0

��0 b(X, t)d��0 +

��0

t0 (X, t)d��0

(2.4.23)

The boundary integral in Eq. (2.4.23) can be transformed to a domain integral using Eq. (2.3.3) and Gauss�� divergence theorem yielding t0 (X, t)d��0 =

��0 ��0

∇0 · P(X, t)d��0

(2.4.24)

where ∇0 · (•) is the divergence taken respect to material coordinates. Leaving the derivation of the conditions which follow from Eq. (2.4.23), the conservation of linear momentum in reference conﬁguration for Lagrangian coordinates is ��0 ∂v = ∇ 0 · P + ��0 b ∂t or ��0 ∂vi ∂Pji = + �� 0 bi ∂t ∂Xj (2.4.25)

This is called the Lagrangian form of the momentum equation. For nonlinear solid ﬁnite elements, Eq. (2.4.25) is called total Lagrangian formulation. The corresponding equilibrium equation for this description is ∇ 0 · P + ��0 b = 0 or ∂Pji + �� 0 bi = 0 ∂Xj (2.4.26)

As a consequence of the conservation of angular momentum Eq. (2.4.18) and Eq. (2.4.19) the nominal stress tensor yields F · P = PT · FT (2.4.27)

which is in general not symmetric. The number of conditions imposed by angular momentum conservation are usually imposed directly on the constitutive equation.

2.4 Conservation Equations

19

Using Eq. (2.3.6) in Eq. (2.4.27) we obtain that for the second Piola-Kirchhoﬀ stress S = ST is a symmetric tensor. (2.4.28)

2.4.3

Conservation of Energy

The kinetic energy of a material is given by E kin =

��

1 ��v · vd�� 2

(2.4.29)

which for a continuum body B is only part of the total energy. The remainder energy is called the internal energy that is expressed by wint per unit mass. The internal energy per unit volume is denoted by E int =

��

��wint d��

(2.4.30)

The total energy is then expressed by E tot = E int + E kin . Then the conservation of energy requires that the power of the total energy equals the power of the applied forces plus the power at which other energy enters in the domain. The other energy may take diﬀerent forms, but the most important is the energy due to heat sources and heat ﬂux across B. Other energy sources arise from radiation, chemical changes, electromagnetic ﬁelds, etc. We consider thermomechanical processes only. The power of the total energy is given by P tot = P int + P kin = D Dt ��wint d�� +

��

D Dt

��

1 ��v · vd�� 2

(2.4.31)

while the power of the applied forces is expressed by P ext =

��

v · ��bd�� +

��

v · td��

(2.4.32)

The power supplied by heat sources s and the heat ﬂux q is P heat =

��

��sd�� −

��

n · qd��

(2.4.33)

The conservation of energy states that P tot = P ext + P heat (2.4.34)

which is known as the ﬁrst law of thermodynamics. Replacing Eqs. (2.4.31)-(2.4.33) into Eq. (2.4.34) yields the equation of conservation of energy

20

2. Continuum Mechanics

D Dt

��wint d�� +

��

D Dt

��

1 ��v · vd�� = 2 ��sd�� −

��

v · ��bd��+

��

(2.4.35)

v · td�� +

�� ��

n · qd��

The equation which emerges from the above integral form leads to the following Eulerian partial diﬀerential equation of energy conservation �� Dw int = �� : d − ∇ · q + ��s Dt (2.4.36)

For a purely mechanical process, the above equation becomes �� Dw int =��:d Dt (2.4.37)

which is no longer a partial diﬀerential equation. As a consequence of Eq. (2.4.37) we can say that the Cauchy stress tensor �� and the rate of deformation tensor d are conjugate in power. The conservation of energy can also be expressed in Lagrangian coordinates and in the reference conﬁguration, where the counterpart of Eq. (2.4.35) gives d dt 1 ��0 wint + ��0 v · v d��0 = 2 v · t0 d��0 +

��0 ��0

v · ��0 bd��0 +

��0

��0

(2.4.38) n0 · qd��0

��0 sd��0 −

��0

which gives the Lagrangian partial diﬀerential equation of energy conservation ˙T ��0 wint = P : F − ∇0 · q + ��0 s ˙ For a purely mechanical process, the Lagrangian energy conservation is ˙T ˙ ��0 wint = P : F (2.4.40) (2.4.39)

showing that the nominal stress tensor is conjugate in power to the material time derivative of the deformation gradient tensor. Using Eq. (2.3.6) in Eq. (2.4.40) we obtain the energy conservation equation in terms of the second Piola-Kirchhoﬀ stress tensor ˙ ˙ ��0 wint = S : E (2.4.41)

which shows that the second Piola-Kirchhoﬀ stress tensor is conjugate in power to the rate of the Green-Lagrange strain tensor.

2.5 Constitutive Equations

21

2.5

Constitutive Equations

The equations given so far are still insuﬃcient to describe the mechanical behavior of any material. Therefore we need additional equations called constitutive equations which complete the set of equations specifying the mechanical properties of a material. For a purely mechanical process, the constitutive equation of a material speciﬁes the dependence of the stress tensor in terms of kinematic variables such as the strain tensor.

2.5.1

Linear Elasticity

Engineering materials such as metals or concrete usually undergo very small changes of shape when they are subjected to the forces which they are exposed. They also have an initial shape to which they will return if the forces applied are removed. Since the changes of shape are very small, there is no diﬀerence between the reference and current conﬁguration. The linear elasticity theory gives an excellent model for the behavior of such materials. The inﬁnitesimal strain tensor �� is used to measure strains while the Cauchy stress tensor �� measures the stresses. For the linear elasticity theory the energy conservation equation takes the form ˙ ˙ ˙ ��0 wint = �� : �� = ��ij ��ij (2.5.1)

˙ where �� and �� are conjugate in power. It is conventional to denote the internal energy per unit volume ��0 wint by W int which is called the strain energy function. For a linear elastic material the strain energy function depends only of the components ��ij and is a quadratic function of the form 1 1 or W int = �� : C : �� (2.5.2) ijkl ��ij ��kl 2 2 where ijkl are called elastic constants. Since the elastic constants possess symmetry of the form W int =

ijkl

=

jikl

=

ijlk

=

klij

(2.5.3)

then for an isotropic material , its properties are the same in all directions. Since W int depends only on ��ij , the material time derivative of Eq. (2.5.2) gives ∂W int ∂��ij ∂W int ∂W int ˙ ��ij = = ∂t ∂��ij ∂t ∂��ij (2.5.4)

where the symmetry of the material has been used. Substituting Eq. (2.5.4) into Eq. (2.5.1) gives ��ij = ∂W int ∂��ij (2.5.5)

However from Eq. (2.5.2) and Eq. (2.5.3)

22

2. Continuum Mechanics

∂W int = ∂��ij

ijkl ��kl

(2.5.6)

Substituting Eq. (2.5.6) into Eq. (2.5.5) yields ��ij =

ijkl ��kl

or

��=C:��

(2.5.7)

which is the constitutive equation that relates stresses and strains. The constitutive equation complete the equations to describe the mechanical behavior of linear elastic materials. For an isotropic material ijkl takes the form = �˦�ij ��kl + �� (��ik ��jl + ��il ��jk ) or C = ��I ⊗ I + 2��I (2.5.8)

ijkl

where only two constants �� and �� of the original 81 of the fourth-order tensor survived after the restrictions of material isotropy and stress symmetry. The two independent material constants �� and �� are called the Lam�� constants, I is the e second-order identity tensor and I is the fourth-order symmetric identity tensor given by 1 (��ik ��jl + ��il ��jk ). The constitutive equation Eq. (2.5.7) becomes 2 ��ij = �˦�kk ��ij + 2�̦�ij where tr(��) is the trace of �� = ��kk . or �� = ��tr(��)I + 2�̦� (2.5.9)

2.5.2

Nonlinear Elasticity

Engineering applications also involved small strains and large deformations, where these eﬀects arise from large displacements and large rotations of the structure. The response of such materials may be modeled with a Saint Venant-Kirchhoﬀ material which is a generalization of the linear theory to large deformations giving the nonlinear elasticity theory. The strain energy function for a nonlinear elastic material is a generalization of Eq. (2.5.2) and is given by W int = where the stress is Sij = ∂W int ∂Eij (2.5.11) 1 2

ijkl Eij Ekl

or

W int =

1 E:C:E 2

(2.5.10)

The counterpart of Eq. (2.5.7) in the nonlinear theory yields Sij =

ijkl Ekl

or

S=C:E

(2.5.12)

where ijkl is given by Eq. (2.5.8). Finally, the constitutive equation for nonlinear elastic materials is Sij = ��Ekk ��ij + 2��Eij or S = ��tr(E)I + 2��E (2.5.13)

2.5 Constitutive Equations

23

The Lam�� constants �� and �� can be expressed in terms of other physical meae surements given by ��= ��= E 2(1 + ��) (2.5.14) (2.5.15)

��E (1 + ��)(1 − 2��)

2 K =��+ �� (2.5.16) 3 where E is the Young��s modulus, �� is the Poisson��s ratio and K is the bulk modulus.

2.5.3

Newtonian Fluid

An equation that linearly relates the stress tensor to the rate of strain tensor in a ﬂuid medium is called the constitutive equation for Newtonian ﬂuids. In a static ﬂuid there are only normal components of the stress tensor on a boundary, so the stress tensor for a ﬂuid at rest is isotropic and takes the form ��ij = −p��ij (2.5.17)

where p is the thermodynamic pressure related to the density �� and the temperature T. A moving ﬂuid develops additional components of stress due to viscosity yielding ��ij = −p��ij + ��dev ij (2.5.18)

where the deviatoric stress tensor ��dev is linearly related to the strain rate tensor ij by ��dev = ij

ijkl dkl

(2.5.19)

Substituting Eq. (2.2.6) and Eq. (2.5.8) into Eq. (2.5.19) and this new equation in Eq. (2.5.18), the resulting equation is ��ij = −p��ij + ��dkk ��ij + 2��dij (2.5.20)

where dkk = ∇·v is the volumetric strain rate. If the Stokes assumption, ��+ 2 �� = 0, 3 is used in Eq. (2.5.20) to relate �� and �� the new equation is 2 ��ij = − p + ��∇ · v ��ij +2��dij 3 2 �� = − p + ��∇ · v I+2��d (2.5.21) 3

or

which is the constitutive equation for Newtonian ﬂuids. For incompressible ﬂuids the continuity equation Eq.(2.4.7) is substituted into Eq. (2.5.21) and the constitutive equation for incompressible ﬂuids takes the simple form ��ij = −p��ij + 2��dij or �� = −pI + 2��d (2.5.22)

where p is the pressure. For incompressible ﬂuids, p is called mechanical pressure.

24

2. Continuum Mechanics

2.6

Navier-Stokes Equation

The equation of motion for a Newtonian ﬂuid is obtained by substituting the constitutive equation for Newtonian ﬂuids Eq. (2.5.21) into the momentum equation in Eulerian description Eq. (2.4.16) to obtain ∂vi + vj ∂j vi ∂t ∂p ∂ + ��bi + ∂xi ∂xj 2 2��dij − ��(∇ · v)��ij 3

��

=−

(2.6.1)

Equation (2.6.1) is the general form of the Navier-Stokes equation. If �� is taken as a constant, the derivative in the right hand side term can be written as ∂ ∂xj 2 2��dij − ��(∇ · v)��ij 3 = �� ∇2 vi + 1 ∂ (∇ · v) 3 ∂xi (2.6.2)

where ∇2 vi is the Laplacian 1 of vi . For incompressible ﬂuids the continuity equation Eq.(2.4.7) is substituted into Eq. (2.6.2) and the Navier-Stokes equation reduces to �� ∂vi + vj ∂j vi ∂t =− ∂p + ��bi + ��∇2 vi ∂xi (2.6.3)

If viscous eﬀects are negligible, which may occur far from boundaries of the ﬂow ﬁeld, �� ∂vi + vj ∂j vi ∂t =− ∂p + ��bi ∂xi (2.6.4)

and the Euler equation is obtained. Now consider given a characteristic velocity scale vc and a characteristic length scale lc , then the Reynolds number is deﬁned as Re = ��vc lc /��. When the Reynolds number for the ﬂow is very low, the convective term can be neglected yielding �� ∂p ∂vi + − ��∇2 vi = ��bi ∂t ∂xi (2.6.5)

which is known as the Stokes ﬂow. In the literature, it is common to express Eq. (2.6.5) without the inertial term.

1∇ 2v

i

=

∂ ∂vi ∂xj ∂xj

=

∂ 2 vi ∂x2 1

+

∂ 2 vi ∂x2 2

+

∂ 2 vi ∂x2 3

Chapter 3

Structural Dynamics

Structural elements such as membranes and thin shells are widely used in modern technology in many engineering ﬁelds. Their elegance, eﬀectiveness and optimal material usage make these light weight structures an ideal construction element for structural and decorative purposes. This chapter begins with a review of the standard total Lagrangian formulation. Then membrane elements are developed for isotropic and orthotropic material behavior and prestressed ﬁelds. Next cable elements are given as a particular case of the membrane theory. Rotation-free shell elements are developed for isotropic and orthotropic materials too. Finally time integration schemes and solution strategies for these structural elements are presented.

3.1

Preliminaries

Before developing membrane and shell elements, the principle of virtual work is developed for standard elements of the total Lagrangian formulation. The principle of virtual work emerges as a consequence of the strong form of the momentum equation.

3.1.1

Total Lagrangian Weak Form

The strong form consists of the momentum equation, the displacement boundary conditions ui on the Dirichlet boundary ��D and the traction boundary conditions ¯ ¯ ti on the Neumann boundary ��N . See for example Hughes (1987), Zienkiewicz and Taylor (1989) and Bathe (1996). In particular for total Lagrangian formulations, the momentum equation used is Eq. (2.4.25). A complete deduction can be found in Bonet and Wood (1997) and Belytschko et al. (2000). To develop the weak form, test function ��ui (X) and trial functions ui (X, t) are require. The space of the test functions is deﬁned as ��ui (X) �� U0 , U0 = ��ui |��ui �� C 0 (X), ��ui = 0 on ��D 25 (3.1.1)

26

3. Structural Dynamics

where C 0 describes the continuity of the function and the boundary ��0 is deﬁned by ��0 = ��D �� ��N . In general, a function is C n if the nth derivative is a continuous function. Eq. (3.1.1) means that for the weak form, the integral over the kinematic boundary is neglected and remains only the integral over the traction boundary. Since the principal of virtual work is to be developed, the space of the trial functions for the displacements is given by ui (X, t) �� U, U = ui |ui �� C 0 (X), ui = ui on ��D ¯ (3.1.2)

Note that the spaces of test and trial functions are similar except that the test displacements vanish wherever the trial displacement are prescribed. The development of a Galerkin-type weak form consists of taking the product of the momentum equation Eq. (2.4.25) by the test function ��ui and integrating over the reference conﬁguration giving ��ui

��0

∂Pji + �� 0 bi − �� 0 u i �� ∂Xj

d��0 = 0

(3.1.3)

This weak form is useless because the space of trial functions for the displacements needs to be C 1 . To solve this problem, the underlined term of Eq. (3.1.3) is integrated by parts1 leading to (��Fij Pji − ��ui ��0 bi + ��ui ��0 ui ) d��0 − ��

��0

��N 0

¯i ��ui t0 d��0 = 0

(3.1.4)

which is the weak form of the momentum equation together with the traction ¯ boundary conditions ti . Note that the spaces of test and trial functions are C 0 . Substituting Eq. (2.3.6) into Eq. (3.1.4), the resulting equation takes the form

(��Fij Sjk Fik − ��ui ��0 bi + ��ui ��0 ui ) d��0 − ��

��0

��N 0

¯i ��ui t0 d��0 = 0

(3.1.5)

where the second Piola-Kirchhoﬀ stress tensor has been replaced by the nominal stress tensor. Eq. (3.1.4) and Eq. (3.1.5) are the principle of virtual work which can be written as ��W int − ��W ext + ��W kin = 0 where (3.1.6)

��W int =

��0

��Fij Pji d��0 =

��0

��Fij Sjk Fik d��0 ��F : S · F d��0

T T ��0

or (3.1.7)

��W

1

int

=

��0

��F : Pd��0 =

R udv = uv − R

T

R

d(uv) =

R

udv +

R

vdu,

vdu

3.1 Preliminaries

27

��W ext =

��0

��ui ��0 bi d��0 +

��N 0

¯i ��ui t0 d��0 ��u · ¯0 d��0 t

or (3.1.8)

��W

ext

=

��0

��0 ��u · bd��0 +

��N 0

��W kin =

��0

��ui ��0 ui d��0 ��

or

��W kin =

��0

�� ��0 ��u · ud��0

(3.1.9)

are the virtual internal work, the virtual external work and the virtual kinetic work respectively. From the virtual internal work the following identity emerges ��FT : P = tr (P · ��F) = tr S · FT · ��F = tr FT · ��F · S = = ��FT · F : S = = 1 ��FT · F + FT · ��F : S = 2 (3.1.10)

1 �� FT · F − I : S = ��E : S 2

and the virtual internal work becomes ��W int =

��0

��Eij Sij d��0

or

��W int =

��0

��E : Sd��0

(3.1.11)

as it was expected from Eq. (2.4.41). Substituting Eq. (3.1.10) into Eq. (3.1.4) yields (��Eij Sij − ��ui ��0 bi + ��ui ��0 ui ) d��0 − ��

��0

��N 0

¯i ��ui t0 d��0 = 0

(3.1.12)

which is another form to express the principle of virtual work in the reference conﬁguration.

3.1.2

Finite Element Discretization

In this section the ﬁnite element discretization for total Lagrangian formulations is described . It is assumed that the reference domain ��0 is discretized by a ﬁnite number of elements that conform the ﬁnite element mesh. For each ﬁnite element of the mesh, the equation of motion is approximated by

nnode

xh (X, t) i

=

I=1

NI (X)xiI (t)

∀i = 1, ndime

(3.1.13)

where NI (X) are the shape functions of each node, nnode is the number of nodes for the ﬁnite element and xiI (t) are the nodal values of the motion at node I with direction i. The displacements are given by

28

3. Structural Dynamics

nnode

uh (X, t) i

=

I=1

NI (X)uiI (t)

∀i = 1, ndime

(3.1.14)

The corresponding velocity and acceleration is

nnode

uh (X, t) ˙i

=

I=1 nnode

NI (X)uiI (t) ˙

∀i = 1, ndime

(3.1.15)

uh (X, t) = ��i

I=1

NI (X)��iI (t) u

∀i = 1, ndime

(3.1.16)

Substituting Eq. (3.1.13) into Eq. (2.1.13) the material deformation gradient tensor is found to be

0 Fij = xiI BjI

or

F = x · BT 0

(3.1.17)

where

0 BjI =

∂NI ∂Xj

(3.1.18)

is the strain-displacement tensor. The space of the test functions is time independent and its discretization yields

nnode

��uh (X) i

=

I=1

NI (X)��uiI

∀i = 1, ndime

(3.1.19)

Since XI is a constant vector, the variation of xI gives ��xiI = ��uiI Consequently, the variation of the material deformation gradient tensor is

0 ��Fij = ��uiI BjI

(3.1.20)

or

��F = ��u · BT 0

(3.1.21)

Recalling that work can be obtained by a force multiplied by a distance, the internal forces emerge from the virtual internal work, Eq. (3.1.7), as

int ��W int = ��uiI fiI =

��Fij Pji d��0 = ��uiI

��0 ��0

0 BjI Pji d��0

(3.1.22)

Since the variations of the displacements ��uiI are arbitrary, the internal forces are expressed by

int fiI = 0 BjI Pji d��0

or

f int =

��0

BT · Pd��0 0

(3.1.23)

��0

where the internal forces are given in terms of the nominal stress tensor. Since the nominal stress tensor is not symmetric, it is more convenient to express the internal

3.1 Preliminaries

29

forces as a function of the second Piola-Kirchhoﬀ stress tensor S. Substituting Eq. (2.3.6) into Eq. (3.1.23) yields

int fiI = 0 BjI Sjk Fik d��0 = 0 BjI Fik Sjk d��0

(3.1.24)

��0

��0

where the second Piola-Kirchhoﬀ stress tensor S is a symmetric tensor in (j, k) and deﬁning the strain-displacement tensor B as

0 BijkI = sym(j,k) BjI Fik

(3.1.25)

the internal forces are expressed by

int fiI =

BijkI Sjk d��0

��0

(3.1.26)

At this point, it is convenient to use the Voigt notation to express Eq. (3.1.26) in the form

int fa =

��0

T Bab Sb d��0

or

int fa

f int =

��0

BT {S}d��0

or

fIint =

��0

BT {S}d��0 I (3.1.27)

where the internal forces

=

int fiI

and its positions are given by (3.1.28)

a = (I − 1)ndime + i

The second Piola-Kirchhoﬀ stress components are transformed by the kinetic Voigt rule as shown in table 3.1. Sij i 1 2 3 2 1 1

Table 3.1

j 1 2 3 3 3 2

Sa a 1 2 3 4 5 6

Voigt rule for stresses

The strain-displacement tensor BI in Voigt notation for 3D problems is

30

3. Structural Dynamics

⎡

⎥ ⎥ ⎥ h h ∂NI ∂x2 ∂NI ∂x3 ⎥ ⎥ ∂X2 ∂X2 ∂X2 ∂X2 ⎥ ⎥ h h ⎥ ∂NI ∂x2 ∂NI ∂x3 ⎥ ∂X3 ∂X3 ∂X3 ∂X3 ⎥ ⎥ ⎥ h h h h ∂NI ∂x2 ∂N ∂x ∂NI ∂x3 ∂N ∂x ⎥ + ∂XI ∂X2 + ∂XI ∂X3 ⎥ ∂X2 ∂X3 ∂X2 ∂X3 3 2 3 2⎥ ⎥ ⎥ ∂xh ∂xh ∂xh ∂xh ⎥ ∂NI ∂N ∂NI ∂N 2 3 + ∂XI ∂X2 + ∂XI ∂X3 ⎥ ∂X1 ∂X3 ∂X1 ∂X3 3 1 3 1 ⎥ ⎦ h h h h ∂NI ∂x2 ∂NI ∂x2 ∂NI ∂x3 ∂NI ∂x3 ∂X1 ∂X2 + ∂X2 ∂X1 ∂X1 ∂X2 + ∂X2 ∂X1 (3.1.29) Assuming that body forces and surface tractions are given, the external forces come from the virtual external work by substituting Eq. (3.1.19) into Eq. (3.1.8) yielding

⎢ ⎢ ⎢ h ∂NI ∂x1 ⎢ ⎢ ∂X2 ∂X2 ⎢ ⎢ h ⎢ ∂NI ∂x1 ⎢ ∂X3 ∂X3 ⎢ BI = ⎢ ⎢ h ⎢ ∂NI ∂xh ∂N ∂x ⎢ ∂X2 ∂X1 + ∂XI ∂X1 3 3 2 ⎢ ⎢ ⎢ ⎢ ∂NI ∂xh + ∂NI ∂xh 1 1 ⎢ ∂X1 ∂X3 ∂X3 ∂X1 ⎢ ⎣ h h ∂NI ∂x1 ∂NI ∂x1 ∂X1 ∂X2 + ∂X2 ∂X1

h ∂NI ∂x1 ∂X1 ∂X1

h ∂NI ∂x2 ∂X1 ∂X1

h ∂NI ∂x3 ∂X1 ∂X1

⎤

ext ��W ext = ��uiI fiI = ��uiI

NI ��0 bi d��0 +

��0

��N 0

¯i NI t0 d��0

(3.1.30)

where the external forces are found to be

ext fiI =

NI ��0 bi d��0 +

��0

��N 0

¯i NI t0 d��0

(3.1.31)

The kinetic forces are a consequence of the virtual kinetic work, which are expressed by the relation

kin ��W kin = ��uiI fiI =

��0

��uh ��0 uh d��0 = ��uiI ��i i

NI ��0 NJ uiJ d��0 ��

��0

(3.1.32)

where Eq. (3.1.16) and Eq. (3.1.19) have been substituted into Eq. (3.1.32), and the kinetic forces are given by

kin fiI =

NI ��0 NJ uiJ d��0 = ��

��0 ��0

NI ��0 NJ d��0 uiJ ��

(3.1.33)

However, it is common to express the kinetic forces as the product of the mass matrix and the accelerations. From Eq. (3.1.33) the mass matrix is deﬁned by MijIJ = ��ij

��0

��0 NI NJ d��0

(3.1.34)

and the kinetic forces become

3.2 Membrane Elements

31

kin fiI = MijIJ ujJ = MijIJ ajJ ��

(3.1.35)

The kinetic forces are also known as the inertial forces. All the above equations have not been discretized in time, so instead of ﬁnite element discretization, sometimes these equations are called ﬁnite element semidiscretization. Finally the equations of motion are given by

int ext fiI + MijIJ ajJ = fiI

or

f int + Ma = f ext

(3.1.36)

3.2

3.2.1

Membrane Elements

Introduction

Membrane structures are used for many purposes because they are built with very light materials which are optimally used since the structures are subjected to membrane tension stresses. Examples include aircraft and spacecraft applications, parachutes, automobile airbags, sails, windmills, human tissues and long span structures. A membrane is essentially a thin shell with no ﬂexural stiﬀness, consequently a membrane should not resist any compression at all. In such a theory only the inplane stress resultants are included. The position of points on the two-dimensional surface in the Euclidean space gives the deformation state for a membrane. A numerical solution for membranes may be constructed using the ﬁnite element method, which solution for small deformations can be found in Zienkiewicz and Taylor (1989), Cook et al. (1989) or O˜ate (1992). Theory for large deforman tions can proceed following the presentations of Simo and Fox (1989), Simo et al. (1990a), B��tchter et al. (1992) or Braun et al. (1994). A general formulation for u membranes based on curvilinear coordinates is given by Bonet et al. (2000) and Lu et al. (2001). Taylor (2001) proposed a large displacement formulation of a membrane composed of three-node triangular elements based on rectangular Cartesian coordinates, where details of the various terms involved are given in Vald��s (2002). e This work has been generalized for diﬀerent ﬁnite elements by Rossi (2005). Some membrane structures have a very low ﬂexural stiﬀness that can support a small amount of compressive stress before buckling appears. In order to avoid compression stresses, membranes are prestressed. Levy and Spillers (1995), Raible (2003) and Gil (2003) use a prestressed method to analyze membranes which are initially ﬂat in the Euclidean space. An approach that include curved pre-stressed membranes using a projection scheme can be found in Bletzinger and W��chner u (2001). In the present work, analysis of initially curved pre-stressed membranes is performed using the ﬁber orientation strategy, which is an extension of the work of Vald��s et al. (2004). Also the ﬁber orientation allow to analyze orthotropic meme branes, where other possibilities are studied in Raible (2003) and W��chner and u Bletzinger (2005).

32

3. Structural Dynamics

When a membrane is subjected to compression in one principal direction and tension in the other principal direction, it will buckle and many narrow wrinkles will form with crests and troughs roughly parallel to the tensile direction. As the ﬂexural stiﬀness decreases, so do the critical buckling stress and the distance between the crests. When a ﬂexural stiﬀness vanishes, so does the critical buckling stress and there would be an inﬁnite number of wrinkles exactly parallel to the tensile direction, as mentioned by Libai and Simmonds (1998). However conventional membrane theory can resist compression without wrinkling although its ﬂexural stiﬀness vanishes. Therefore one diﬃculty in modelling membranes is to account for wrinkling phenomena that are not predicted by normal membrane theory. A membrane theory which accounts for wrinkling does not allow any negative stress to appear. When a negative stress is about to appear the membrane will wrinkle. The modelling of wrinkled membranes was started by Wagner (1929). He tried to explain the behavior of thin metal webs and spars carrying a shear load in excess of the initial buckling value. Many authors contributed to the linear analysis of wrinkles like Reissner (1938), Kondo et al. (1955), Mansﬁled (1970) or Mansﬁled (1977). Since then, many signiﬁcant contributions to analyze wrinkling models of membranes have been studied extensively with diﬀerent approaches. In one approach, the constitutive relation of the membrane is modiﬁed to simulate wrinkling. For example, Contri and Schreﬂer (1988) use a no-compression material model that was carried out in a two step procedure that allows the folds and the mean deformed position of the wrinkled surface to be obtained. Liu et al. (2001) propose a penalty parameter modiﬁed material model with a constant parameter, which rotates the constitutive equation to the direction of principal strains. Then the values of the constitutive equation related to the direction of the second principal strains are penalize almost to zero and then the constitutive equation is rotated back to its original position. Rossi et al. (2003) and Rossi et al. (2005) use the same procedure that Liu et al. (2001) but with one extra parameters that keeps the convergence properties of the element. Inspired in the work of Liu et al. (2001) and Rossi et al. (2003), Jetteur (2005) proposed a material model with small resistance in compression in order to have a good convergence in a static scheme. In the present work, following Vald��s et al. (2005) a modiﬁed material model for e orthotropic materials is presented. An advantage of this kind of procedure is that since commercial ﬁnite element codes do not usually support tension ﬁeld models, the method can be input to the code through the user-deﬁned material model port. A second approach is based on modifying the deformation gradient tensor without changing the constitutive relation. Wu and Canﬁeld (1981) presented a model describing the wrinkling of membranes in ﬁnite plane-stress theory. They modiﬁed the deformation gradient tensor by introducing an extra parameter. The value of this parameter was determined by the condition that the stress in wrinkling direction is zero. The modiﬁcation of the deformation gradient tensor was chosen in a way that the principal Cauchy directions did not change because of the wrinkling, which is only true when the material is isotropic. Another model capable of dealing with anisotropy was introduced by Roddeman et al. (1987a) and Roddeman et al. (1987b). They also introduced the correct criterion to judge the state of the membrane at a point. Due to the complexity of the formulation, explicit expressions for

3.2 Membrane Elements

33

the nodal forces and stiﬀness matrix lead to lengthy derivations. Later Roddeman (1991) presented a much more simple element derivation and instead of deriving explicit expression for the nodal forces and the stiﬀness matrix, the equivalent nodal forces are obtained numerically. The tangent stiﬀness matrix follows from numerical diﬀerentiation of the nodal forces. Muttin (1996) has generalized the wrinkling theory of Roddeman (1991) for curved membranes using curvilinear coordinates, which also uses numerical diﬀerentiation to calculate the internal forces and the tangent stiﬀness matrix. Based on the wrinkling condition of Roddeman et al. (1987a), Ziegler (2001) and Ziegler et al. (2003) develop an algorithm in analogy to the small strain elasto-plasticity model to calculate the wrinkling strains, which was applied to isotropic materials only. The wrinkling theory of Roddeman et al. (1987a) is formulated by Lu et al. (2001) using curvilinear coordinates, and a robust scheme to ﬁnd the wrinkling direction is derived with concise explicit formulas for the internal forces and the tangent stiﬀness matrix. An alternative formulation of Roddeman et al. (1987a) is given by Schoop et al. (2002) where a reference conﬁguration methodology results in a simpler formulation. Another approach similar to the second one, decompose the strain tensor in its principal directions and then using a new variable, the second principal direction is modiﬁed until the compression stresses disappear. In the works of Raible (2003) and L��hnert et al. (2003) this approach is used, where explicit formulas for the o internal forces are derived.

3.2.2

Membrane Formulation

Until now a rectangular Cartesian coordinate system have been employed as the basis for the representation of vectors and tensors. However for the membrane theory, a curvilinear coordinate system based on diﬀerential geometry of surfaces will be used, as can be found in Farrashkhalvat and Miles (2003), Lu et al. (2001) and W��chner and Bletzinger (2005). Here Greek indices on membrane mid-surface u take on values of 1 and 2 in a plane stress state in Euclidean space.

X3 e3 e2 X

x

1

2

x

X2

e1 X1

Figure 3.1

Curvilinear coordinates for a surface

The position vector X on the surface in the reference conﬁguration ��0 is deﬁned

34

3. Structural Dynamics

by two independent curvilinear coordinates �� 1 and �� 2 , shown in Fig. 3.1, as X = X(�� 1 , �� 2 ) (3.2.1)

It is assumed that there is an invertible relationship between Eq. (2.1.1) and Eq. (3.2.1). The position vector x on the surface in the current conﬁguration �� is given by x = x(�� 1 , �� 2 , t) (3.2.2)

An invertible relationship between Eq. (2.1.2) and Eq. (3.2.2) is assumed. The convected covariant base vectors of the curvilinear coordinate system on ��0 and �� are deﬁned respectively as G�� = ∂X , ∂�� �� g�� = ∂x ∂�� �� (3.2.3)

Note that the covariant base vectors G�� and g�� form the tangent space TX B to the membrane surface and in general they are neither unit vector nor orthogonal to each other, as can be seen in Fig. 3.2.

G2

G1

x

2

T

X

B

B

x

1

Figure 3.2

Covariant base vectors forming a tangent plane

Therefore the surface normals are determined by G3 = G1 �� G2 , G3 , G3 g3 = g 1 �� g 2 , g3 g3

N=

n=

(3.2.4)

in reference and current conﬁguration respectively. The normals are normalized given a unit vector. The covariant components of the metric tensors are deﬁned by G���� = G�� · G�� , g���� = g�� · g�� (3.2.5)

for the reference and current conﬁgurations respectively. The convected contravariant base vectors are given for ��0 and �� respectively by

3.2 Membrane Elements

35

G�� = G���� · G�� ,

−1

g�� = g ���� · g��

−1

(3.2.6)

where the contravariant components of the metric tensors are obtained from G���� = G���� , g ���� = g���� (3.2.7)

for the corresponding conﬁgurations. For the case when the contravariant base vectors are given, the covariant base vector can be obtained from G�� = G���� · G�� , g�� = g���� · g�� (3.2.8)

for reference and current conﬁgurations respectively. The covariant and contravariant base vectors deﬁne the scalar product identities

�� G �� · G �� = �Ħ� , �� g �� · g�� = �� ��

(3.2.9)

where the Kronecker delta is given by

�� �Ħ� =

1 when �� = �� 0 otherwise

(3.2.10)

The deformation gradient tensor F in curvilinear coordinates is given by F = g�� ⊗ G�� , FT = G�� ⊗ g�� , F−1 = G�� ⊗ g�� , F−T = g�� ⊗ G�� (3.2.11)

and substituting Eq. (3.2.11) into the Green-Lagrange strain tensor, Eq. (2.2.1), yields 1 1 FT · F − I = G�� ⊗ g�� · g�� ⊗ G�� − G���� G�� ⊗ G�� 2 2

E=

(3.2.12)

which components for the membrane surface in a plane stress state are given by 1 (3.2.13) (g���� − G���� ) 2 In Eq. (3.2.12) the identity tensor I was written as the product of F−1 · F and Eq. (3.2.8) was used to expressed I = G���� G�� ⊗ G�� . Using the appropriate constitutive equation to relate the second Piola-Kirchhoﬀ stress tensor and the Green-Lagrange strain tensor in curvilinear coordinates, the components of the stress tensor are deﬁned as E = E���� G�� ⊗ G�� , E���� = S = S ���� G�� ⊗ G�� (3.2.14)

Finally the virtual internal work, Eq. (3.1.11), can be expressed in curvilinear coordinates as ��W int =

��0

��E���� S ���� d��0

(3.2.15)

36

3. Structural Dynamics

3.2.2.1

Pressure Follower Forces

Although the most common example of a body force is gravity loading and a wide variety of surface traction forces exists, the most important case for geometrically nonlinear membrane elements is the case of uniform normal pressure follower forces, that change their direction each time the normal to the surface changes in the current conﬁguration. Consider a membrane element with an applied uniform pressure p acting on the current conﬁguration having a pointwise normal n. Then the traction force vector t is expressed as pn, and the corresponding virtual external work in the current conﬁguration is ��W ext =

��

��u · pnd��

(3.2.16)

3.2.3

Fiber Orientation

The idea for the ﬁber orientation comes from the manufacturing process of membrane structures, which can be built with orthotropic or composite materials and a reference principal ﬁber direction is needed to perform correctly an analysis with ﬁnite elements. Even for isotropic materials, the reference principal ﬁber direction is needed if the membrane has an initial pre-stressed ﬁeld. With the methodology proposed, a pre-stressed ﬁeld for orthotropic materials is also possible. Another important aspect of the ﬁber orientation comes from postprocessing the strain and stress ﬁeld in-plane on the membrane surface. Other possibilities for postprocessing these values are given in O˜ate (1992). n To build the ﬁber orientation for a membrane structure, ﬁrst a ﬁnite element mesh is needed and for each element the following methodology is applied. A local Cartesian base system is obtained from the covariant base vectors as G1 , G1 G1 ⊗ G 2 , G1 ⊗ G 2

ed

eloc = 1

eloc = N = 3

eloc = eloc ⊗ eloc 2 3 1

(3.2.17)

ed ed ed ed ed

e2

loc

ed

e1

loc

Figure 3.3

Principal ﬁber direction and local base system

3.2 Membrane Elements

37

If a curved membrane structure is meshed with ﬁnite elements, the local Cartesian axes of each element generally have diﬀerent orientations, even for structured meshes as can be seen in Fig. 3.3. If dashed-lines are the reference principal ﬁbers direction or the orthotropy direction for the material, then an angle �� is needed to rotate each local Cartesian base system eloc in order to apply the material orthotropy correctly (or the prei stress ﬁeld). Suppose that ed is a given vector that deﬁnes the principal ﬁber direction for a ﬁnite element and lies in the tangent space of the element, as shown in Fig. 3.4. This principal ﬁber direction is always known at least for one ﬁnite element and is given by the manufacture process of the structure.

eloc 2

S

loc e1

ed

Figure 3.4

Principal ﬁber direction for a ﬁnite element

To assign correctly the principal ﬁber direction from a known element (or source element) to the whole mesh, ﬁrst the adjacent elements of the source element must be identiﬁed, see Fig. 3.5(a). Next build the tangent space TX BS for the source element together with its normal that will be called NS . For one adjacent element its tangent space TX BN is also built and its normal will be called NN . The intersection line to both tangent spaces will be a common vector between both of them.

(a)

(b)

(c)

Figure 3.5

Assignment of principal ﬁber direction to adjacent elements

A new Cartesian base system for TX BS is constructed. The Cross product between the normal NS and the vector XS , given by the intersection line between both tangent spaces, yields YS . This new vector YS must have a direction pointing inside of the source element, as can be seen in Fig. 3.6. In a similar way, a new Cartesian base system for the adjacent element will be built. The Cross product between the normal NN and the vector given by the intersection line XN yields YN , where XN = −XS . This new vector YN must point inside the adjacent element. Remark that vectors XS and YS belong to

38

3. Structural Dynamics

NS

XS NN

YS

YN XN

TXB

S

TBN X

Intersection line

Figure 3.6

Local Cartesian base systems for tangent spaces

the tangent space of the source element while vectors XN and YN belong to the tangent space of the adjacent element. Once both local Cartesian base systems are deﬁned, the next step is to choose an arbitrary point a on axis XS . Now over point a the principal ﬁber direction vector ed is passed and its intersection over axis YS will be the point b, as shown in Fig. 3.7.

XS

ed YS

TX B

S

a o c d YN

TX B

N

b

XN

Figure 3.7 Transferring reference principal ﬁber direction

Finally to transfer the principal ﬁber direction to the adjacent element, the distance from the origin o to the point a must be the same that the distance from the origin o to the point c located on axis XN . Also the distance from the origin o to the point b must be the same that from the origin o to the point d over axis YN . Then build the vector from point c to point d which will be the reference principal ﬁber direction of the neighbor element, as can be seen in Fig. 3.7. This procedure of transferring the principal ﬁber direction from a source element to one of its adjacent elements is repeated for all adjacent elements of the source element, as in Fig. 3.5. Once each adjacent element has a principal ﬁber direction transferred, the procedure is repeated for the whole ﬁnite element mesh until every

3.2 Membrane Elements

39

element has a known principal ﬁber direction ed . A ﬁnite element mesh with all its elements with a principal ﬁber direction assigned is shown in Fig. 3.8.

Figure 3.8

Principal ﬁbers direction of a mesh

With every ﬁnite element of the mesh with a principal ﬁber direction vector ed known, the angle �� for every element that is needed to rotate each local Cartesian base system, as shown in Fig. 3.4, is given by sin �� = −ed · eloc , 2 and �� = tan−1 sin �� cos �� (3.2.19) cos �� = ed · eloc 1 (3.2.18)

The local Cartesian base system with the ﬁber orientation is denominated local ﬁber Cartesian base system, and is found with the equation ef iber = R(��, N) · eloc i i (3.2.20)

where R(��, N) is the Rodrigues�� rotation formula that rotates an angle �� about a ﬁxed axis speciﬁed by a unit vector, in this case the unit normal vector to the surface element N, and is given by

2 − cos��) 6 cos�� + 6 6 6 6N3 sin�� + N1 N2 (1 − cos��) 6 6 4 N1 N3 (1 − cos��) − N2 sin��

2 N1 (1

3 N1 N2 (1 − cos��) − N3 sin��

2 cos�� + N2 (1 − cos��)

R(��, N) =

N1 sin�� + N2 N3 (1 − cos��)

N2 sin�� + N1 N3 (1 − cos��)7 7 7 7 N2 N3 (1 − cos��) − N1 sin�� 7 7 7 5 2 cos�� + N3 (1 − cos��)

(3.2.21) The ﬁnite element mesh for the analysis with the ﬁber orientation is shown in Fig. 3.9. A fast algorithm to ﬁnd the adjacent elements in a ﬁnite element mesh is given in L��hner (2001), which is an important step to be applied in this methodology. o

40

3. Structural Dynamics

ed

ed ed ed ed ed

e2

fiber

ed

e1

fiber

Figure 3.9

Local ﬁber base system

3.2.4

Finite Element Discretization for Membranes

This section contains diﬀerent subsections that begins with the general discretization for membrane elements expressed in curvilinear coordinates. Next this discretization is transformed to local Cartesian coordinates that simplify the implementation and allow the analysis of isotropic membranes without any pre-stressed ﬁelds. To be able to add a pre-stress ﬁeld or an orthotropic material to the internal forces, the discretization with ﬁber orientation is presented. Finally the implementation of a triangular three-node ﬁnite element is particularized. 3.2.4.1 Discretization in Curvilinear Coordinates

The discretization is established for the total Lagrangian formulation. The ﬁnite element discretization is developed with shape functions expressed in terms of the so called parent element coordinates or master element coordinates. The most common parent element coordinates are the isoparametric coordinates. The parent element coordinates are denoted by �� �� for each element with a parent domain �� . The shape of the parent domain depends on the type of element and the problem dimension. In Fig. 3.10 it is shown that the �� 1 and �� 2 surface coordinates of the parent element can be mapped to curvilinear coordinates when plotted in local rectangular Cartesian coordinates for a given ﬁnite element. Therefore these coordinates are used to develop membrane elements. Consider a membrane element with parent element coordinates �� 1 and �� 2 as the curvilinear coordinate system. Then Eq. (3.2.1) can be approximated by

nnode nnode

X (��) =

I=1

h

NI (��)XI

or

h Xi (��)

=

I=1

NI (��)XiI

∀i = 1, ndime

(3.2.22) where NI (��) are the parent element shape functions. The equation of motion Eq. (3.2.2) is given by

3.2 Membrane Elements

41

Y

2

d

(-1,1)

Y

2 (1,1) 1

Y

X2

1

Y

(-1,-1) (1,-1)

X1

d

Y

2

Y Y

1 (0,1)

2

X2

(0,0)

(1,0)

Y

1

X1

Figure 3.10

Mapping of surface elements

nnode

xh (��, t) i

=

I=1

NI (��)xiI (t)

nnode

∀i = 1, ndime (3.2.23) NI (��)xI (t)

or and the displacements are

nnode

x (��, t) =

I=1

h

uh (��, t) i

=

I=1

NI (��)uiI (t)

nnode

∀i = 1, ndime (3.2.24) NI (��)uI (t)

or

uh (��, t) =

I=1

Substituting Eq. (3.2.22) into the covariant base vectors of the curvilinear coordinates in the reference conﬁguration ��0 of Eq. (3.2.3) yields ∂ G�� = �� ∂�� where

nnode nnode

NI (��)XI

I=1

=

I=1

NI,�� XI

(3.2.25)

42

3. Structural Dynamics

NI,�� =

∂NI (��) ∂�� ��

(3.2.26)

Following the same procedure, Eq. (3.2.23) is substituted into the covariant base vectors of the curvilinear coordinates in the current conﬁguration ��, Eq. (3.2.3), to express

nnode

g�� =

I=1

NI,�� xI (t)

(3.2.27)

With these quantities known, the covariant components of the metric tensor G���� and g���� given in Eq. (3.2.5) are found. Then components of the Green-Lagrange strain tensor are computed with E���� = The variation ��E���� becomes ��E���� = and ��g���� = ��g�� · g�� + g�� · ��g�� (3.2.30) 1 1 ��(g���� − G���� ) = ��g���� 2 2 (3.2.29) 1 (g���� − G���� ) 2 (3.2.28)

where Eq. (3.2.5) has been used. The variation of the covariant base vectors in current conﬁguration results from Eq. (3.2.27) yielding

nnode nnode

��g�� =

I=1

NI,�� ��xI =

I=1

NI,�� ��uI

(3.2.31)

Substituting Eq. (3.2.27) and Eq. (3.2.31) into Eq. (3.2.30), the resulting equation is

nnode nnode nnode nnode

��g���� =

I=1

NI,�� ��uiI ·

J=1

NJ,�� xiJ +

J=1

NJ,�� xiJ ·

I=1

NI,�� ��uiI

(3.2.32)

Substituting Eq. (3.2.32) into Eq. (3.2.29), the variation of the Green-Lagrange strain tensor becomes

nnode nnode nnode nnode

2 ��E���� =

I=1

NI,�� ��uiI ·

J=1

NJ,�� xiJ +

J=1

NJ,�� xiJ ·

I=1

NI,�� ��uiI

(3.2.33)

and the virtual internal work, Eq. (3.2.15), yields

3.2 Membrane Elements

43

nnode

nnode

2 ��W

int

=

��0 I=1 nnode

NI,�� ��uiI ·

J=1 nnode

NJ,�� xiJ S ���� + (3.2.34) NI,�� ��uiI S ���� d��0

I=1

NJ,�� xiJ ·

J=1

From Eq. (3.1.22), the virtual internal work was expressed as

nnode

��W int =

I=1

int ��uiI fiI

∀i = 1, ndime

(3.2.35)

Substituting Eq. (3.2.34) into Eq. (3.2.35), the internal forces for a particular direction i and node I can be expressed as

int fiI =

��0

1 2

nnode

(NI,�� NJ,�� + NJ,�� NI,�� ) xiJ S ���� d��0

J=1

(3.2.36)

where the strain-displacement tensor in curvilinear coordinates is given by

cur B����iI

1 = 2

nnode

(NI,�� NJ,�� + NJ,�� NI,�� ) xiJ

J=1

(3.2.37)

and the internal forces, Eq. (3.1.26), for membrane elements in curvilinear coordinates can be written as

int fiI = cur B����iI S ���� d��0

(3.2.38)

��0

Eq. (3.2.37) can be expressed in the simple form

cur B����iI =

1 NI,�� xh + NI,�� xh i,�� i,�� 2

(3.2.39)

where the equation

nnode

xh i,��

=

J=1

NJ,�� xiJ

(3.2.40)

has been used. Using the Voigt notation to express the internal forces in curvilinear coordinates, Eq. (3.2.38), yields

int fa =

��0

T [Bab ]cur {S b }cur d��0

or

fIint =

��0

[BT ]cur {S}cur d��0 I

(3.2.41)

where the strain-displacement matrix Bcur is given by I

44

3. Structural Dynamics

⎡ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣

h ∂NI ∂x1 ∂�� 1 ∂�� 1 h ∂NI ∂x1 2 ∂�� 2 ∂�� h ∂NI ∂x1 ∂�� 1 ∂�� 2 h ∂NI ∂x1 ∂�� 2 ∂�� 1

h ∂NI ∂x2 ∂�� 1 ∂�� 1 h ∂NI ∂x2 2 ∂�� 2 ∂�� h ∂NI ∂x2 ∂�� 1 ∂�� 2 h ∂NI ∂x2 ∂�� 2 ∂�� 1

h ∂NI ∂x3 ∂�� 1 ∂�� 1 h ∂NI ∂x3 2 ∂�� 2 ∂�� h ∂NI ∂x3 ∂�� 1 ∂�� 2 h ∂NI ∂x3 ∂�� 2 ∂�� 1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Bcur I

+

+

+

(3.2.42) In Voigt notation, the virtual internal work Eq. (3.1.22) is written as

T T ��W int = {��Eb }cur {S b }cur = {��ua }T [Bab ]cur {S b }cur

(3.2.43)

where the variation of the Green-Lagrange strain tensor in curvilinear coordinates and Voigt notation is given by

cur cur ��Eb = Bba ��ua

(3.2.44)

3.2.4.2

Discretization in Cartesian Coordinates

The Green-Lagrange strain tensor E expressed in local Cartesian coordinates can be found using the following transformation equation, see i.e. Crisﬁeld (1991), ¯ ¯T Ecur = J�� Eloc J�� (3.2.45)

¯ where the Jacobian transformation tensor J�� in the reference conﬁguration is deﬁned as G1 · eloc 1 ¯ J�� = G1 · eloc 2 G2 · eloc J11 1 loc = 0 G2 · e 2 J12 J22 (3.2.46)

Note that the product G1 · eloc = 0 because they are orthogonal vectors. From Eq. 2 (3.2.45) the Green-Lagrange strain tensor Eloc is found by Eloc = TT Ecur T�� �� ¯ where T�� is used to denote the inverse of J�� by T ¯ −1 T�� = J�� = 11 0 where T11 = 1 , J11 T12 = −J12 , J11 J22 T22 = 1 J22 (3.2.49) T12 T22 (3.2.48) (3.2.47)

Eq. (3.2.47) can be written in Voigt notation yielding {E}loc = [Q]{E}cur or

loc cur Ec = Qcb Eb

(3.2.50)

From Eq. (3.2.47), the variation of the Green-Lagrange strain tensor is

3.2 Membrane Elements

45

��Eloc = TT ��Ecur T�� �� which in Voigt notation can be written as {��E}loc = [Q]{��E}cur or

loc cur ��Ec = Qcb ��Eb

(3.2.51)

(3.2.52)

where the transformation matrix Q = Q(T�� ) gives the transformation from curvilinear coordinates to local Cartesian coordinates and is deﬁned by ⎤ ⎡ 2 0 0 T11 2 2 T22 T22 T12 ⎦ (3.2.53) Q = ⎣ T12 2T11 T12 0 T11 T22 The virtual internal work, Eq. (3.2.43), written in local Cartesian coordinates yields

T loc T loc T loc ��W int = {��Ec }loc Sc = {��ua }T [Bac ]loc Sc = {��ua }T [Bab ]cur QT Sc bc

(3.2.54)

where the strain-displacement tensor in local Cartesian coordinates and in Voigt notation is deﬁned by

loc cur Bca = Qcb Bba

or

Bloc = QBcur

(3.2.55)

The internal forces in Voigt notation and local Cartesian coordinates are given by

int fa =

��0

T loc [Bab ]cur QT Sc d��0 bc

or

fIint =

��0

[BT ]cur [QT ]{S}loc d��0 (3.2.56) I

This equation is deﬁned only for isotropic materials without any pre-stressed ﬁeld. To add a pre-stressed ﬁeld or an orthotropic material to the internal forces, the ﬁber orientation has to be used. 3.2.4.3 Discretization with Fiber Orientation

The following relations to rotate the stress S and strain E tensor in Voigt notation are needed, see i.e. Decolon (2000), {S}loc = [T�� ]{S}f ib and {E}loc = [T�� ]{E}f ib (3.2.58) (3.2.57)

where the rotation matrix for the stresses T�� and the strains T�� are given respectively by

46

3. Structural Dynamics

cos2 �� T�� = ⎣ sin2 �� sin��cos�� and cos2 �� T�� = ⎣ sin2 �� 2sin��cos�� ⎡

⎡

sin2 �� cos2 �� −sin��cos��

⎤ −2sin��cos�� 2sin��cos�� ⎦ cos2 �� − sin2 �� ⎤ −sin��cos�� sin��cos�� ⎦ cos2 �� − sin2 ��

(3.2.59)

sin2 �� cos2 �� −2sin��cos��

(3.2.60)

where �� is given by Eq. (3.2.19). The inverse relationship of Eq. (3.2.57) and Eq. (3.2.58) is given respectively by {S}f ib = [TT ]{S}loc �� and {E}f ib = [TT ]{E}loc �� (3.2.62) (3.2.61)

The variation of the Green-Lagrange strain tensor in Voigt notation with the ﬁber orientation yields {��E}f ib = [TT ]{��E}loc = [TT ][Q]{��E}cur = [TT ][Q][B]cur {��u} �� �� �� (3.2.63)

where the strain-displacement matrix with the ﬁber orientation is given by the following equation

f ib T cur Bea = [Tec ]�� Qcb Bba

or

Bf ib = TT QBcur ��

(3.2.64)

Finally the internal forces with the ﬁber orientation and in Cartesian coordinates are given by

fIint =

��0

[BT ]cur [QT ][T�� ]{S}f ib d��0 = I

��0

[BT ]f ib {S}f ib d��0 I

(3.2.65)

or in indicial notation

faint =

��0

T [Bab ]cur [QT ][Tce ]�� {Se }f ib d��0 = bc

��0

T [Bae ]f ib {Se }f ib d��0

(3.2.66)

Note that if the angle �� = 0 when no ﬁber orientation is performed, the rotation matrix T�� = I3��3 and the classical membrane element is recovered. To add a prestressed ﬁeld to the membrane structure once the ﬁber orientation is performed, simple add the desire prestressed value, yielding fIint = [BT ]f ib {S}f ib + {S}prestressed d��0 I (3.2.67)

��0

3.2 Membrane Elements

47

or in indicial notation faint =

T [Bae ]f ib {Se }f ib + {Se }prestressed d��0

(3.2.68)

��0

Also an orthotropic analysis can be performed once the ﬁber orientation is done, as shown next. 3.2.4.4 Plane Stress Constitutive Equation

The constitutive equation used in this work is a Saint Venant-Kirchhoﬀ material model that express the stresses {S}f ib from the strains {E}f ib as {S}f ib = [C]{E}f ib ⎡ ⎤ 0 ⎦ 0 (1 − ��)/2 (3.2.69)

where the constitutive matrix [C] for plane stress isotropic materials is given by 1 E ⎣ �� C= 1 − ��2 0 �� 1 0

(3.2.70)

where E is the elastic or Young��s modulus, and �� is the Poisson��s ratio. For orthotropic materials the constitutive matrix is ⎡ C= 1 1 − ��xy ��yx Ex ⎣��xy Ey 0 ��yx Ex Ey 0 ⎤ 0 ⎦ 0 (1 − ��xy ��yx )Gxy

(3.2.71)

with Ex ��yx = Ey ��xy and Gxy being the shear modulus. The Green-Lagrange strain tensor in Voigt notation {E}f ib is found substituting Eq. (3.2.50) into Eq. (3.2.62) yielding {E}f ib = [TT ][Q]{E}cur �� 3.2.4.5 (3.2.72)

Implementation of Triangular Three-Node Finite Elements

Rotation-free thin shells give excellent solutions to many structural problems when the ﬁnite element used for the discretization is a three-node triangle. Since any shell element is composed of a bending part and a membrane part, a membrane element for three-node triangles is developed next, which will be used as the membrane part of the rotation-free shell element described in section 3.4. Consider the triangular element of Fig. 3.11. To clarify the notation, components of the vector (x1 , x2 , x3 ) are change to (x, y, z), while the components of (�� 1 , �� 2 ) are now (��, ��). The shape functions for this elements are given by N1 = 1 − �� − ��, Derivatives of the shape functions are N2 = ��, N3 = �� (3.2.73)

48

3. Structural Dynamics

R 3

1

Z

2

Y X

Y

Figure 3.11

Triangular three-node ﬁnite element

∂N1 = −1, ∂�� ∂N2 = 1, ∂�� ∂N3 = 0, ∂��

∂N1 = −1 ∂�� ∂N2 =0 ∂�� ∂N3 =1 ∂��

(3.2.74)

From Eq. (3.2.42) the following equation is derived ∂xh ∂xh 1 = = ∂�� 1 ∂��

nnode =3 J=1

∂N1 ∂N2 ∂N3 ∂NJ x = x + x + x J J=1 J=2 ∂�� ∂�� ∂�� ∂�� J=3

(3.2.75)

where Eq. (3.2.40) has been used, and the subindexes naming the element nodes J = 1, J = 2 and J = 3 are used to clarify the notation. Substituting the corresponding values from Eq. (3.2.74) into Eq. (3.2.75) yields ∂xh = (−1)x + (1)x + (0)x = x − x J=1 J=2 J=3 J=2 J=1 ∂�� Similarly for the other derivatives of Eq. (3.2.42) respect to �� ∂y h ∂xh 2 = = ∂�� 1 ∂��

3

(3.2.76)

J=1

∂NJ y = yJ=2 − yJ=1 ∂�� J

(3.2.77)

3.2 Membrane Elements

49

∂z h ∂xh 3 = = ∂�� 1 ∂��

3

J=1

∂NJ z = zJ=2 − zJ=1 ∂�� J

(3.2.78)

Derivatives of Eq. (3.2.42) respect to �� yield ∂xh ∂xh 1 = = ∂�� 2 ∂��

3

J=1

∂NJ ∂N1 ∂N2 ∂N3 x = x + x + x J J=1 J=2 ∂�� ∂�� ∂�� ∂�� J=3

(3.2.79)

∂xh = (−1)x + (0)x + (1)x = x − x J=1 J=2 J=3 J=3 J=1 ∂�� and ∂y h ∂xh 2 = = ∂�� 2 ∂�� ∂z h ∂xh 3 = = ∂�� 2 ∂��

3

(3.2.80)

J=1 3

∂NJ y = yJ=3 − yJ=1 ∂�� J ∂NJ z = zJ=3 − zJ=1 ∂�� J

(3.2.81)

(3.2.82)

J=1

The strain-displacement matrix for this element is composed of Bcur = [Bcur , Bcur , Bcur ] 1 2 3 (3.2.83)

where each submatrix is found substituting Eqs. (3.2.74)-(3.2.82) into Eq. (3.2.42) to yield x −x J=1 J=2 ⎣ x −x = J=1 J=3 2x − x − x J=1 J=2 J=3 ⎡ ⎡ yJ=1 − yJ=2 yJ=1 − yJ=3 2yJ=1 − yJ=2 − yJ=3 ⎤ zJ=1 − zJ=2 ⎦ zJ=1 − zJ=3 2zJ=1 − zJ=2 − zJ=3 (3.2.84)

Bcur 1

or ⎤ T − (x2 − x1 ) ⎢ ⎥ T =⎣ − (x3 − x1 ) ⎦ T T − (x2 − x1 ) − (x3 − x1 )

Bcur 1 Similarly ⎡ x −x J=2 J=1 0 =⎣ x −x J=3 J=1

(3.2.85)

Bcur 2 and

yJ=2 − yJ=1 0 yJ=3 − yJ=1

⎤ ⎡ T⎤ zJ=2 − zJ=1 (x2 − x1 ) ⎦=⎣ ⎦ 0 0 T zJ=3 − zJ=1 (x3 − x1 )

(3.2.86)

50

3. Structural Dynamics

⎡ Bcur = ⎣x J=3 3 x J=2

0 −x J=1 −x J=1

yJ=3 yJ=2

0 − yJ=1 − yJ=1

zJ=3 zJ=2

⎤ ⎡ ⎤ 0 0 T − zJ=1 ⎦ = ⎣(x3 − x1 ) ⎦ T − zJ=1 (x2 − x1 )

(3.2.87)

The complete strain-displacement matrix for the three-node triangular ﬁnite element is given by − (x2 − x1 ) ⎢ T =⎣ − (x3 − x1 ) T T − (x2 − x1 ) − (x3 − x1 ) ⎡

T

Bcur

(x2 − x1 ) 0 T (x3 − x1 )

T

⎤ 0 T⎥ (x3 − x1 ) ⎦ T (x2 − x1 )

(3.2.88)

The internal forces for this element in Cartesian coordinates are given by

int f(9��1) =

��0

[BT ]cur [QT ](3��3) [T�� ](3��3) {S}f ib d��0 (9��3) (3��1)

(3.2.89)

Usually at this point numerical integration is performed. However, for this element exact integration can be found using the Cross product A0 = 1 (X2 − X1 ) �� (X3 − X1 ) 2 (3.2.90)

giving the area A0 of the triangular membrane element. Therefore reference domain integral for the internal forces can be written as f int = A0 h[BT ]cur [QT ][T�� ]{S}f ib or in indicial notation

T faint = A0 h[Bab ]cur [QT ][Tce ]�� {Se }f ib bc

(3.2.91)

(3.2.92)

where h is the membrane thickness in the reference conﬁguration. To ﬁnd the components of the strain tensor, discretization of the covariant base vectors for this element in the reference conﬁguration, Eq. (3.2.25), yields

nnode =3

G1 =

I=1

∂NI XI = X2 − X1 ∂��

(3.2.93)

and

nnode =3

G2 =

I=1

∂NI XI = X3 − X1 ∂��

(3.2.94)

The corresponding discretization in the current conﬁguration for this element is given by Eq. (3.2.27) yielding

3.2 Membrane Elements

51

nnode =3

g1 =

I=1

∂NI xI = x2 − x1 ∂��

(3.2.95)

and

nnode =3

g2 =

I=1

∂NI xI = x3 − x1 ∂��

(3.2.96)

Substituting Eqs. (3.2.93)-(3.2.96) into Eq. (3.2.5) gives the covariant components of the metric tensor G���� and g���� , which substituted into Eq. (3.2.13) yields the components of the strain tensor Ecur . The transformation matrix Q is a function of the components of the tensor J�� given by Eq. (3.2.46) where G1 · eloc = G1 · 1 G1 G1 (3.2.97)

Here Eq. (3.2.17) has been substituted. Using the identity G1 = (G1 · G1 )

1/2

(3.2.98)

and the covariant base vector G1 of Eq. (3.2.93), Eq. (3.2.97) can be written as G1 · eloc = X2 − X1 1 (3.2.99)

¯ The remaining components of the Jacobian tensor J�� are found using the same methodology yielding ⎤ ⎡ (X2 −X1 )·(X3 −X1 ) X2 − X1 X2 −X1 ⎥ ⎢ ¯ J�� = ⎣ (3.2.100) ⎦ (X2 −X1 )��(X3 −X1 ) 0 X2 −X1 Components of Eq. (3.2.100) are used to build the transformation matrix Q giving by Eq. (3.2.53) and Eq. (3.2.49). Note that this matrix is always evaluated in the reference conﬁguration. An important advantage of this element is that all the expression can be obtained with the vectors X2 − X1 and X3 − X1 of the reference conﬁguration which are evaluated just once, and the vectors x2 − x1 and x3 − x1 of the current conﬁguration. 3.2.4.6 Pressure Follower Forces Discretization

From the virtual external work for pressure follower forces given by Eq. (3.2.16), the external forces emerge for the three-node triangular element with constant pressure as f ext = 1 Apn 3 (3.2.101)

52

3. Structural Dynamics

where A is the area of the triangular membrane element in current conﬁguration given by A= 1 (x2 − x1 ) �� (x3 − x1 ) 2 (3.2.102)

and n is the normal to the current surface, which for the three-node triangle is given by n= (x2 − x1 ) �� (x3 − x1 ) (x2 − x1 ) �� (x3 − x1 ) (3.2.103)

Substituting Eqs. (3.2.102)-(3.2.103) into Eq. (3.2.101), the external forces can be written as f ext = p (x2 − x1 ) �� (x3 − x1 ) 6 (3.2.104)

3.2.5

Wrinkling

Here a wrinkling algorithm based on a modiﬁed material model is developed. The idea of this algorithm is to present a simple algorithm that allow to solve the wrinkling phenomena that are not predicted by normal membrane theory even its ﬂexural stiﬀness vanishes. This algorithm is not intended for time-history analysis, consequently only the ﬁnal solution is accurate. At any point on its surface, a membrane must be in one of three states. In a slack state, the membrane is not stretched in any direction. In a taut state, the membrane is in tension in all directions. If the membrane is neither taut not slack, it is in a wrinkle state corresponding to uniaxial tension. In a slack or wrinkled criterion the real conﬁguration of the membrane is undeﬁned. To avoid this, the slack or wrinkled region can be replaced with an average smoothed pseudo-surface where material points on the real wrinkled surface are projected onto the pseudosurface. To predict the real conﬁguration of a wrinkled membrane, a very dense ﬁnite element mesh is needed to perform the analysis, which sometimes needs an initial perturbation, see Tessler et al. (2003). However from an engineering point of view, the shape of the wrinkles are not as important as the membrane stresses. Therefore diﬀerent wrinkling algorithms are developed. In this work, the wrinkling criterion based on principal stresses and principal strains introduced by Roddeman et al. (1987a) is used. Its main objective is the diﬀerentiation of a membrane state for isotropic and orthotropic materials. Table 3.2 shows this classiﬁcation. First the membrane state is determined. If the membrane is taut, the constitutive matrix [C] does not need to be modiﬁed. Then [CCON ] = [C] and the state stress is computed as {S}f ib = [CCON ]{E}f ib (3.2.105)

3.2 Membrane Elements

53

Stress and strain SII > 0 EI > 0 and SII �� 0 EI �� 0

Table 3.2

Wrinkling No One axial Two axial

Membrane Taut Wrinkled Slack

Wrinkling criterion

If the membrane is slack, components of the constitutive matrix [C] are modiﬁed yielding [CCON ] = 0 and the stress ﬁeld is simply {S}f ib = 0 (3.2.106)

Finally if the membrane is wrinkled, the angle �Ȧ� of principal stresses is found. With this angle the rotation matrix for stresses given by Eq. (3.2.59) and the rotation matrix for strains given by Eq. (3.2.60) are built, where the angle �� is replaced by the angle �Ȧ� . Then the constitutive matrix is rotated with the following equation [C] = [TT ][C][T�� ] �� ⎡ ⎤ (3.2.108) (3.2.107)

This new rotated constitutive matrix is modiﬁed as shown next yielding ⎢ ⎢ [CM OD ] = ⎢ 0 ⎢ ⎣ C1,1 0 0 0 C1,3 ⎥ ⎥ 0 ⎥ ⎥ ⎦

C3,1

C3,3

Finally this modiﬁed constitutive matrix is rotated back to its original position with the following equation giving [CCON ] = [T�� ][CM OD ][TT ] �� Now the stress ﬁeld for the wrinkled state is computed using {S}f ib = [CCON ]{E}f ib (3.2.110) (3.2.109)

Since a geometrically non-linear problem is being solved, it is common to use a Newton-Raphson scheme with an appropriate linearization. Usually each time step ti is solved within 3 to 5 Newton-Raphson iterations. If the wrinkling algorithm presented is applied at each iteration, then the number of iterations may be increased signiﬁcantly. To avoid this problem, the wrinkling algorithm is applied only for the ﬁrst two iterations of the time step. After the second iteration the constitutive matrix [CCON ] is kept constant until convergence for the time step is reached. Therefore the model perform 4 to 6 Newton-Raphson iterations per time step.

54

3. Structural Dynamics

An important aspect to be accounted for orthotropic materials is to ﬁnd the correct wrinkling direction, as given in Jarasjarungkiat et al. (2007). A limitation of this work is that the principal stress direction is taken as the wrinkling direction, which is not true for orthotropic materials. However for the studied cases, the solution is quite similar to the reported by other investigations.

3.3

Cable Elements

Cable elements are modeled as geometrically non-linear unidimensional truss elements in Euclidean space. Here a cable element is developed as a particular case of the membrane theory.

3.3.1

Cable Formulation

The position vector X for the cable element in the reference conﬁguration ��0 is deﬁned by one independent curvilinear coordinate ��, shown in Fig. 3.12, as

2 3 Y Y 1

Z

2 1

Y X

Figure 3.12

Cable two-node and three-node ﬁnite elements

X = X(��) The position vector x of the current conﬁguration �� is given by x = x(��, t)

(3.3.1)

(3.3.2)

The covariant base vector of the curvilinear coordinate system for this element on ��0 and �� are deﬁned respectively by G1 = ∂X , ∂�� g1 = ∂x ∂�� (3.3.3)

The covariant components of the metric tensors are given by G11 = G1 · G1 g11 = g1 · g1 (3.3.4)

3.3 Cable Elements

55

for the reference and current conﬁgurations respectively. The contravariant base vectors are given for ��0 and �� respectively by G1 = G11 · G1 g1 = g 11 · g1 (3.3.5)

where the contravariant components of the metric tensors are G11 = 1 G11 g 11 = 1 g11 (3.3.6)

for the corresponding conﬁgurations. The Green-Lagrange strain tensor for the cable element is given by

cur E11 =

1 (g11 − G11 ) 2

(3.3.7)

Using the appropriate constitutive equation to relate the second Piola-Kirchhoﬀ stress tensor and the Green-Lagrange strain tensor in curvilinear coordinates, the 11 stress tensor Scur is found. cur From Eq. (3.3.4) and Eq. (3.3.7), the variation ��E11 yields

cur ��E11 =

1 ��g11 = g1 · ��g1 2

(3.3.8)

Finally the virtual internal work, Eq. (3.1.11), can be expressed in curvilinear coordinates for the cable element as ��W int =

��0 cur 11 ��E11 Scur d��0

(3.3.9)

3.3.2

Finite Element Discretization for Cables

The discretization is established for the total Lagrangian formulation. Here a twonode cable element is developed. Consequently Eq. (3.3.1) is approximated by

nnode nnode

X (��) =

I=1

h

NI (��)XI

or

h Xi (��)

=

I=1

NI (��)XiI

∀i = 1, ndime (3.3.10)

where the shape functions NI (��) are given by N1 = 1 (1 − ��) , 2 N2 = 1 (1 + ��) 2 (3.3.11)

Derivatives of the shape functions yield 1 ∂N1 = − ��, ∂�� 2 ∂N2 1 = �� ∂�� 2 (3.3.12)

The equation of motion, Eq. (3.3.2), is given by

56

3. Structural Dynamics

nnode

xh (��, t) i

=

I=1

NI (��)xiI (t)

nnode

∀i = 1, ndime (3.3.13) NI (��)xI (t)

or

xh (��, t) =

I=1

The covariant base vector in the reference conﬁguration ��0 is expressed by

nnode

G1 =

I=1

∂NI 1 XI = (X2 − X1 ) ∂�� 2

(3.3.14)

In the current conﬁguration �� the covariant base vector is

nnode

g1 =

I=1

1 ∂NI xI = (x2 − x1 ) ∂�� 2

(3.3.15)

which variation ��g1 is expressed as ��g1 = 1 (��u2 − ��u1 ) 2 (3.3.16)

Covariant components of the metric tensor, Eq. (3.3.4), yield 1 1 (x2 − x1 ) · (x2 − x1 ) = l 2 4 4 (3.3.17) where L is the length of the element in the reference conﬁguration and l is the length of the current conﬁguration. Substituting Eq. (3.3.17) into Eq. (3.3.7) the Green-Lagrange strain tensor yields G11 = g11 = 1 2 l − L2 (3.3.18) 8 Transformation from curvilinear coordinates to Cartesian coordinates requires

cur E11 =

1 1 (X2 − X1 ) · (X2 − X1 ) = L2 , 4 4

J11 = G1 · eloc = 1 and

1 L 2

(3.3.19)

cur loc 2 loc E11 = J11 E11 J11 = J11 Ex

(3.3.20)

From Eq. (3.3.20)

loc Ex =

1 cur 2 E J11 11

(3.3.21)

and

3.4 Shell Elements

57

loc Ex =

l 2 − L2 2L2

(3.3.22)

cur The discretization of ��E11 , given by Eq. (3.3.8) yields

1 (x2 − x1 ) · (��u2 − ��u1 ) 4 which can be written in matrix notation as

cur ��E11 = cur ��E11 =

(3.3.23)

1 T − (x2 − x1 ) 4

(x2 − x1 )

T

��u1 ��u2

(3.3.24)

where the strain-displacement matrix for the cable element is given by Bcur = (1��6) 1 T − (x2 − x1 ) 4 (x2 − x1 )

T

(3.3.25)

From Eq. (3.3.21), transformation of Bcur from curvilinear coordinates to Cartesian coordinates yields 1 T T (x2 − x1 ) − (x2 − x1 ) L2 Finally, the internal forces for a two-node cable element are

loc B(1��6) = int loc f(6��1) = A0 L [BT ](6��1) {Sx }loc (1��1)

(3.3.26)

(3.3.27)

where A0 is the cross-sectional area of the element, L is the length in the reference loc loc conﬁguration and Sx = E Ex with E being the Young��s modulus.

3.4

3.4.1

Shell Elements

Introduction

Shells are used in wide structural ﬁelds such as civil, mechanical, naval and airspace engineering. A formulation based on thin-shell theory can be used to analyze several applications of shell structures. Since shell analysis demand a lot of computer memory and cpu time to be computed, a formulation using displacements as the only degrees of freedom is developed. The basic idea of the shell theory is to take into account the particular threedimension model and by integration over the thickness get a two-dimensional model in the Euclidean space formulated on the middle surface of the shell. The pioneers of this kind of derivation were Kirchhoﬀ (1876) and Love (1934). The main assumption of this theory of shells (plates) is to preserve the orthogonality of the normal to the middle-surface. The well-known problems to derive conforming C 1 continuous thin shell elements motivated a number of authors to explore the possibilities of Reissner (1945) and Mindlin (1951) theory. This theory relaxes the normal orthogonality condition, introducing the shear deformation stress that is important for thick shell

58

3. Structural Dynamics

simulations. Unfortunately Reissner-Mindlin plate and shell elements suﬀer the socalled shear locking in the thin limit. This diﬃculty wrecked the full success of Reissner-Mindlin plate and shell elements for practical engineering analysis. The ﬁrst developments of plate and shell elements using Kirchhoﬀ-Love theory were made by Melosh (1961), Zienkiewics and Cheung (1964), Clough and Tocher (1965) among many others. On the other hand, Hughes et al. (1977), O˜ate et al. n (1979), Bathe and Dvorkin (1985) and Zienkiewics et al. (1990) proposed the ﬁrst plate and shell elements based on Reissner-Mindlin theory. A complete state of the art for linear plate elements can be found in O˜ate (1992). n Nonlinear behavior of shell elements is continuously increasing because of its important aspects in which large rotations and large deformations are involved as well as nonlinear material properties. Several studies have been carried out for example by Simo and Fox (1989), Simo et al. (1990a), Simo and Kennedy (1992), among many others. Since shell analysis requires a lot of memory and cpu-time to compute, several authors have tried to derive plate and shell elements with displacements as the only nodal variables. The idea of using the deﬂection as the only nodal variables, i.e. rotation-free, is not new and many diﬀerent procedures are based on this approach, i.e. Ugural (1981). The well-known diﬃculties of ﬁnite diﬀerence analysis with boundary conditions and the problems for dealing with non-orthogonal or unstructured grids limited their progress. One of the ﬁrst attempts to use ﬁnite elements was due to Nay and Utku (1972) who derived a rotation free thin plate triangle using a leastsquare quadratic approximation to describe the deﬂection ﬁeld within the patch surrounding a node in terms of the deﬂections of the patch nodes. Later, Barnes (1977) proposed a method for deriving a three-node triangle with the nodal deﬂections as the only degrees of freedom based on the computation of the curvatures in terms of the nodal rotations at the middle-side points from the nodal deﬂections of adjacent elements. This method was exploited by Hampshire et al. (1992) assuming that the elements are hinged together at their common boundaries and the bending stiﬀness is represented by torsional springs resisting rotations about the hinge line. O˜ate and Cervera (1993) proposed a general procedure based on ﬁnite volume conn cepts for deriving thin plate elements of triangular and quadrilateral shapes with the nodal deﬂections as the only degree of freedom. Brunet and Sabourin (1994) proposed a diﬀerent approach to compute the constant curvature ﬁeld within each triangle in terms of the six-node displacement of a macro-element. This triangular element was successfully applied to nonlinear shell analysis using an explicit dynamic approach. Z��rate (1996) continue with the study of rotation free elements of a O˜ate and Cervera (1993) developing new triangular elements. This formulation n applied to large deformations with an explicit dynamic procedure was presented by Cendoya (1996). Rojek et al. (1998) proposed the same element that Cendoya (1996) but applied to metal forming processes. As an alternative formulation for large strain plasticity, the BST shell element was introduced by Flores and O˜ate n (2001). A new perspective using a subdivision of surfaces for thin-shell analysis was introduced by Cirak et al. (2000) for small strains, and Cirak and Ortiz (2001) for large deformations. An improvement of the BST shell element using an assumed strain approach is given by Flores and O˜ate (2005). n

3.4 Shell Elements

59

3.4.2

Shell Formulation

A ﬁnite deformation shell formulation is presented. Here Greek indices take on values of 1 and 2 while lower latin indices take on values of 1, 2 and 3. The position vector R on the middle surface in the reference conﬁguration is deﬁned by the indepent curvilinear coordinates �� 1 , �� 2 and �� as R �� 1 , �� 2 , �� = X �� 1 , �� 2 + ��N �� 1 , �� 2 where N is the normal to the middle surface on ��0 and − h0 �� �� �� 2 being the shell thickness in the reference conﬁguration, see Fig. 3.13.

h0 2

(3.4.1) with h0

N

Y2 h0

Y1

Figure 3.13

Shell middle surface

The position vector r on the current conﬁguration is given by r �� 1 , �� 2 , ��, t = x �� 1 , �� 2 , t + �Ʀ� �� 1 , �� 2 , t n �� 1 , �� 2 , t (3.4.2)

where n is the normal to the middle surface on �� and �� is the thickness stretch that relates the thickness h of the deformed shell to the thickness h0 of the undeformed shell. Finite thickness stretch measured by �� play an important role in problems involving ﬁnite membrane strains or contact, as given in Simo et al. (1990b). However, in this work a Saint Venant-Kirchhoﬀ material model is used and therefore only small strains are assumed. Consequently, the �� term is not considered. The convected covariant base vectors of the curvilinear coordinates system on ��0 are deﬁned by ∂R ∂X ∂N = �� + �� �� = G�� + ��N,�� �� ∂�� ∂�� ∂��

G�� =

(3.4.3)

∂R G3 = =N ∂��

60

3. Structural Dynamics

Here G�� are the middle surface basis vectors in the reference conﬁguration. The convected covariant base vectors on the current conﬁguration �� are given by ∂r ∂x ∂n = �� + �� �� = g�� + ��n,�� ∂�� �� ∂�� ∂�� ∂r =n g3 = ∂��

g�� =

(3.4.4)

where g�� are the middle surface basis vectors in the current conﬁguration. The convected contravariant base vectors follow from the relations G · Gj = ��ji ,

i

g i · gj = ��ji

(3.4.5)

where ��ji is the Kronecker delta. The covariant metric tensors in both conﬁgurations follow as Gij = Gi · Gj , gij = gi · gj (3.4.6)

Components of the Green-Lagrange strain tensor are given as the diﬀerence between the covariant metric tensors on the current and reference conﬁgurations of the shell yielding 1 (3.4.7) gij − Gij 2 The Green-Lagrange strain tensor can be extended to be written in the form of Eij = Eij =

ij

+ �Ʀ�ij + �� 2 ��ij

(3.4.8)

where the non-zero components of the above expression are given by 1 g · g − G �� · G�� , 2 �� �� 1 = (n · n − N · N) 2 = 1 (g · n − G�� · N) 2 ��

���� 33

��3

=

(3.4.9)

�ʦ��� = g�� · n,�� − G�� · N,�� �æ��� =

(3.4.10)

1 (3.4.11) (n,�� · n,�� − N,�� · N,�� ) 2 This work is intended for the Kirchhoﬀ-Love theory of thin shells, consequently the deformed director n coincides with the unit normal to the current middle surface. Therefore the values ��3 and 33 vanish identically and values for thin shells of �� 2 can be neglected. This constraint yields the components of the Green-Lagrange strain tensor to be deduced from the deformation of the middle surface of the shell as

3.4 Shell Elements

61

E���� = where

����

+ �Ʀʦ��� = E����

memb

+ ��E����

bend

(3.4.12)

����

=

1 (g���� − G���� ) 2

(3.4.13)

is identical to Eq. (3.2.13) and measure membrane strains. For convenience of the discretization given in section 3.4.3, the bending strains are written as2 �ʦ��� = G��,�� · N − g��,�� · n = K���� − k���� (3.4.14)

The variation of the Green-Lagrange strain tensor is given by the variation of Eq. (3.4.12) yielding ��E���� = ��E����

memb

+ �Ʀ�E����

bend

(3.4.15)

With an appropriate constitutive equation to relate stresses and strains, the virtual internal work is expressed as ��W

int

=

��0

+h 2 −h 2

��E���� S ���� d��d��0

(3.4.16)

where ��0 is the middle surface domain.

3.4.3

Finite Element Discretization for Shells

The discretization used for the thin shell element is given for the total Lagrangian formulation. Discretization only for bending strains is presented since membrane strains given in Eq. (3.4.13) are developed in section 3.2.4. In this work the rotation free triangle element presented by Flores and O˜ate n (2001) is extended to account for the principal ﬁber direction and perform the analysis with initially pre-stressed ﬁelds or orthotropic materials.

n2

3 5 2 M 1 3 6 1 2

n1

4

n3

Shell patch

Figure 3.14

2 (g ��

· n) ,�� = g�� · n,�� + g��,�� · n = 0 and (G�� · N) ,�� = G�� · N,�� + G��,�� · N = 0

62

3. Structural Dynamics

Bending eﬀect for this rotation-free triangle element is given by the displacement ﬁeld of one element and all nodes of immediately adjacent elements, as shown in Fig. 3.14. The path description is as follows: • • • • Element number is inside a circle. Nodes of the main element (M) are numbered locally as 1, 2 and 3. Sides of the main element are deﬁned by its local node opposite to the side. Adjacent elements are numbered with the number associated to the common side 1, 2 and 3. • The remaining nodes of the patch are numbered locally as 4, 5, and 6 corresponding to nodes on adjacent elements (1), (2) and (3) respectively. Conectivities for the path element are deﬁned in table 3.3. Element (M) (1) (2) (3) Node 1 4 5 6 Conectivities 1 Node 2 Node 3 2 3 3 2 1 3 2 1

Table 3.3

Patch conectivities

A local coordinate system must be deﬁned for the patch. In the original work of Flores and O˜ate (2001), the following deﬁnition was proposed: the unit vector n loc e1 was directed from node 1 to node 2. Unit vector eloc was given by the normal 3 of the main element (M). The last unit vector was obtained by eloc = eloc �� eloc . 2 3 1 In this work, the local coordinate system is given by the local ﬁber Cartesian base system, deﬁned in section 3.2.3. Then the base system for each ﬁnite element is given by the unit vectors ef ib , ef ib and the normal ef ib . The choice for this local 1 2 3 system allow us to compute shells with a prestressed ﬁeld as well as orthotropic material deﬁnitions. The bending part of Eq. (3.4.16) is given by, ��W

bend

int

=

+h 2 −h 2 ��0

���� ��E���� Sbend d��0 d��

bend

(3.4.17)

and the values of E���� and ��E���� are needed to evaluate the internal forces. The bending strains for the current conﬁguration are expressed by k���� = g��,�� · n which can be written in the form k���� = 1 A0 g��,�� d��0 · n (3.4.19) (3.4.18)

bend

��0

3.4 Shell Elements

63

and then applying the divergence theorem3 yields k���� = 1 A0 n�� g�� d��0 · n ¯ (3.4.20)

��0

where n�� are the components of the normal to the boundary, in-plane with the unit ¯ base vectors ef ib and ef ib , of the main element (M), as shown in Fig. 3.14. Since 1 2 the boundary integral of the three-node triangular main element can be computed explicitly, then Eq. (3.4.20) is expressed as k���� 1 = A0

nsides

lJ nJ g�� · n ¯��

J=1

(3.4.21)

where nsides = 3, J is the number of the side in the main element, lJ is the length of side J and nJ are the components of the normal to the boundary of side J of ¯�� the main element. At this point it is convenient to change from curvilinear coordinates to the local ﬁber Cartesian base system, so the whole patch is with the same local coordinate system. Discretization of g�� is expressed in Eq. (3.2.27), which can be written explicitly as g1 = g2

nnode I=1

⎡ ∂NI ⎤ ⎣

∂�� ∂NI ∂��

⎦ xI (t)

(3.4.22)

Since the formulation is to be carried out in the local ﬁber system, the derivatives of the shape functions need to be transformed. The Jacobian transformation tensor J�� in the current conﬁguration is deﬁned as g1 · ef ib 1 J�� = g1 · ef ib 2 and the Cartesian derivatives become ⎡ ∂NI ⎤ ⎣

∂x ∂NI ∂y ��

g2 · ef ib 1 g2 · ef ib 2 ⎡ ∂NI ⎤

∂��

(3.4.23)

⎦ = J−T ⎣

⎦

(3.4.24)

∂NI ∂��

Then Eq. (3.4.22) can be written in Cartesian coordinates as xh ,1 xh ,2

nnode

⎡ ∂NI ⎤ ⎣

∂x ∂NI ∂y

=

I=1

⎦ xI (t)

(3.4.25)

Using Voigt notation, Eq. (3.4.21) can be written in local ﬁber Cartesian coordinates as

3

R

��

∂u d�� ∂xi

=

R

��

ni ud�� ¯

64

3. Structural Dynamics

⎡

⎤ k11 ⎣k22 ⎦ = 1 A0 k12

nsides J=1

nJ ¯1 J ⎣ 0 l nJ ¯2

⎡

⎤ 0 xh · n nJ ⎦ ,1 ¯2 xh · n ,2 nJ ¯1

(3.4.26)

The constraints imposed by the Kirchhoﬀ-Love theory of thin shells make the product xh ·n = 0. This problem is solved taking for xh the average value between ,�� ,�� the main triangle and each one of the adjacent elements yielding ⎡ ⎤ ⎤ ⎡ J ⎤ ⎡1 M J nsides n1 0 ¯ k11 2 x,1 + x,1 · n ⎣k22 ⎦ = 1 ⎦ ¯2 (3.4.27) l J ⎣ 0 nJ ⎦ ⎣ A0 1 J J M J k12 n2 n1 ¯ ¯ J=1 2 x,2 + x,2 · n which can be simpliﬁed to ⎡ ⎤ k11 ⎣k22 ⎦ = 1 2A0 k12 ⎡ J n1 ¯ J ⎣ 0 l nJ ¯2 ⎤⎡ J ⎤ x,1 · n 0 ⎦ nJ ⎦ ⎣ ¯2 J J n1 ¯ x,2 · n

nsides J=1

(3.4.28)

since xM · n = 0 and where xJ is given by ,�� ,��

⎡ J⎤ ⎡ J⎤ ∂NI nnode x,1 ⎢ ∂x ⎥ J ⎣ ⎦= ⎣ ⎦ xI J J ∂NI x,2 I=1

∂y

(3.4.29)

The same methodology is used to obtain the conﬁguration yielding ⎤ ⎡ J ⎡ nsides n1 ¯ K11 ⎣K22 ⎦ = 1 LJ ⎣ 0 2A0 K12 nJ ¯2 J=1

bending strains for the reference ⎤⎡ J ⎤ X,1 · N 0 ⎦ nJ ⎦ ⎣ ¯2 J J n1 ¯ X,2 · N

(3.4.30)

The bending strain tensor in Voigt notation is then given by ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ K11 k11 ��11 bend {E} = ⎣��22 ⎦ = ⎣K22 ⎦ − ⎣k22 ⎦ ��12 K12 k12 The variation of the bending strain tensor in Voigt notation yields ⎤ ⎡ ⎤ ⎡ ��k11 �Ħ�11 bend �� {E} = ⎣�Ħ�22 ⎦ = − ⎣��k22 ⎦ �Ħ�12 ��k12 where ⎡ ⎤ ��k11 ⎣��k22 ⎦ = 1 2A0 ��k12

nsides J=1

(3.4.31)

(3.4.32)

nJ ¯1 J ⎣ 0 l nJ ¯2

⎡

⎤⎡ ⎤ �� xJ · n 0 ,1 ⎦ nJ ⎦ ⎣ ¯2 J J n1 ¯ �� x,2 · n

(3.4.33)

The last term of the right hand side of Eq. (3.4.33) can be expanded to become

3.4 Shell Elements

65

�� xJ · n = ��xJ · n + xJ · ��n ,�� ,�� ,�� The variation ��xJ yields ,�� ⎡ J⎤ ⎡ J⎤ ∂NI nnode ��x,1 ⎢ ∂x ⎥ J ⎦= ⎣ ⎣ ⎦ ��uI J J ∂NI ��x,2 I=1

∂y

(3.4.34)

(3.4.35)

From Eq. (3.2.4), the normal n is given as a function of g3 which can be written as n= xh �� xh g �� g2 g3 ,1 ,2 = 1 = g3 g1 �� g 2 xh �� xh ,1 ,2 1 ��g · g − g3 3 33 g3 g3 (3.4.36)

Therefore the normal variation yields ��n = �� g3 g3 = ��g3 (3.4.37)

which components of the convective covariant base vectors xh give ,�� ��n1 = xh · ��n = xh · ��g3 ,1 ,1 ��n2 = xh ,2 · ��n = xh ,2 · ��g3 1 ��g · g − xh · g3 3 33 ,1 g3 g3 1 ��g · g − xh · g3 3 33 ,2 g3 g3

(3.4.38)

where xh · g3 = 0. The variation of g3 is ,�� ��g3 = �� xh �� xh = ��xh �� xh + xh �� ��xh ,1 ,2 ,1 ,2 ,1 ,2 (3.4.39)

Instead of the Cross product, it is convenient to express Eq. (3.4.39) in a matrix form yielding x,2 ,1 ˆ ,1 ,2 ��g3 = −ˆ h ��xh + xh ��xh where ⎡ ˆ ,�� xh = ⎣ x3,�� −x2,�� 0 −x3,�� 0 x1,�� ⎤ x2,�� −x1,�� ⎦ 0 (3.4.41) (3.4.40)

Substituting Eq. (3.4.40) into Eq. (3.4.38) leads to 1 g3 1 ��n2 = g3 −xh · xh ��xh + xh · xh ��xh ,1 ˆ ,2 ,1 ,1 ˆ ,1 ,2 (3.4.42) −xh ,2 · ˆ ,2 ,1 xh ��xh + xh ,2 · ˆ ,1 ,2 xh ��xh

��n1 =

66

3. Structural Dynamics

However xh · xh ��xh = xh · xh ��xh = 0, yielding ,1 ˆ ,1 ,2 ,2 ˆ ,2 ,1 ��n1 = 1 g3 1 ��n2 = g3 −xh �� xh · ��xh = −n · ��xh ,1 ,2 ,1 ,1 (3.4.43) xh ,2 �� xh ,1 · ��xh ,2 = −n · ��xh ,2

and Eq. (3.4.37) can be written as ˜ ,1 ˜ ,2 ��n = ��n1 xh + ��n2 xh (3.4.44)

˜ ,�� where xh are the convective contravariant base vectors given by Eq. (3.2.6) or simply by ˜ ,1 xh = 1 xh �� n , g3 ,2 ˜ ,2 xh = − 1 xh �� n g3 ,1 (3.4.45)

From Eq. (3.4.25) the variation ��xh is ,�� ��xh ,1 ��xh ,2

nnode

⎡ ∂NI ⎤ ⎣

∂x ∂NI ∂y

=

I=1

⎦ ��uI

(3.4.46)

Substituting Eq. (3.4.46) into Eq. (3.4.43), and the resulting equation into Eq. (3.4.44) yields

nnode

��n = −

I=1

∂NI h ∂NI h ˜ ˜ x,1 + x n · ��uI ∂x ∂y ,2

(3.4.47)

Substituting Eq. (3.4.35) and Eq. (3.4.47) into Eq. (3.4.34) leads to ⎡ J⎤ ⎡ ⎡ ⎤ ∂NI ∂NI J J ˜h nnode nnode �� x,1 · n ∂x x,1 · x,1 + ∂x ⎢ ⎥ ⎢ J ⎣ ⎦= ⎣ ⎦ n · ��uI − ⎣ J J ∂NI J ∂NI I=1 I=1 �� x,2 · n ˜h ∂x x,2 · x,1 +

∂y ∂NI J ∂y x,1 ∂NI J ∂y x,2

˜ ,2 · xh ˜ ,2 · xh

⎤ ⎥ ⎦ n · ��uI (3.4.48)

Now substituting Eq. (3.4.48) into Eq. (3.4.33) yields ⎤ ⎡ ��k11 ⎣��k22 ⎦ = 1 2A0 ��k12 ⎡ nJ ¯1 ⎤ ⎡

nnode

J ∂NI ∂x

⎤

lJ ⎣ 0 nJ ¯2 J=1 ⎡ J nsides n1 ¯ 1 J ⎣ 0 l − 2A0 nJ ¯2 J=1

nsides

0 ⎢ ⎥ J nJ ⎦ ¯2 ⎣ ⎦ n · ��uI + J nJ I=1 ∂NI ¯1 ∂y ⎡ ⎤ ⎤ ∂N ∂N ˜h ˜h 0 nnode ∂xI xJ · x,1 + ∂yI xJ · x,2 ,1 ,1 ⎢ ⎥ nJ ⎦ ¯2 ⎣ ⎦ n · ��uI J h h ∂NI J ∂NI J n1 I=1 ¯ ˜ ˜ ∂x x,2 · x,1 + ∂y x,2 · x,2 (3.4.49)

3.4 Shell Elements

67

Finally substituting Eq. (3.4.49) into Eq. (3.4.32), the variation of the bending strain tensor in Voigt notation can be written explicitly as ⎡ J nsides n1 ¯ 1 bend = lJ ⎣ 0 �� {E} 2A0 nJ ¯2 J=1 ⎡ J nsides n1 ¯ 1 lJ ⎣ 0 − 2A0 nJ ¯2 J=1 ⎤ ⎡

∂NI J ∂x x,1

0 nnode ⎢ nJ ⎦ ¯2 ⎣ nJ I=1 ∂NI xJ · xh + ¯1 ˜ ,1 ∂x ,2 ⎡ J⎤ ⎤ ∂N 0 nnode ∂xI ⎢ ⎥ J nJ ⎦ ¯2 ⎣ ⎦ n · ��uI J nJ I=1 ∂NI ¯1

∂y

˜ ,1 · xh +

∂NI J ∂y x,1 ∂NI J ∂y x,2

˜ ,2 · xh ˜ ,2 · xh

⎤ ⎥ ⎦ n · ��uI +

(3.4.50) The variation of the bending strain tensor can be expressed in compact form as �� {E}

bend

= [B]

main

��uI + [B]

adj

��uJ I

main

(3.4.51)

for the main element where the expression of the strain-displacement matrix [B] adj and [B] for the adjacent element is taken from Eq. (3.4.50). The complete straindisplacement matrix for bending is given by [B] 3.4.3.1

bend

= [B]

main

+ [B]

adj

(3.4.52)

Stress resultants and internal forces

The membrane and bending stress tensors may be given in a direct mechanistic interpretation as force and moment resultants . The constitutive equation given by Eq. (3.2.69) can be written for shell elements as {S} = [C] · {E} = [C] · {E}

memb

+ �� {E}

bend

(3.4.53)

and the virtual internal work given by Eq. (3.4.16) for shells is expressed by

h 2

��W

int

=

��0

−h 2

�� {E}

memb

+ �Ʀ� {E}

bend

· [C] {E}

memb

+ �� {E}

bend

d��d��0 (3.4.54)

which can be split to yield

h 2

��W

int

=

��0

−h 2 h 2 −h 2

�� {E}

memb

· [C] {E}

memb

d��d��0 + (3.4.55) d��d��0

�� 2 �� {E}

bend

· [C] {E}

bend

��0

68

3. Structural Dynamics

In this work a linear material model is used and integration of the virtual internal work becomes ��W int =A0 h�� {E} A0 where the forces resultant {N} as

memb

· [C] {E}

memb

+ (3.4.56)

res

h3 bend bend · [C] {E} �� {E} 12 and the moments resultant {M}

res

are obtained

{N} {M}

res res

= h [C] {E} =

memb

h3 bend [C] {E} 12

(3.4.57)

Finally the internal forces for the rotation-free shell element are expressed by f int = A0 BT

memb

{N}

res

+ A0 BT

bend

{M}

res

(3.4.58)

where all expressions related to the membrane part are given directly in section 3.2.4. 3.4.3.2 Boundary conditions

The main diﬀerence between classical formulations for shells and rotation-free formulations for shells is that the boundary conditions for rotation-free elements become a part of the formulation that also needs to be implemented into the ﬁnite element code. It is important to give special considerations to the boundaries where there might be one or two missing adjacent elements. In this work, the boundary conditions are treated as given by Flores and O˜ate (2001). n

3.5

Time Integration Schemes

Time integration schemes can be classiﬁed as explicit solution and implicit solution. In an explicit scheme the lumped mass is commonly used and consists in ﬁnding the position vector x at time tn+1 using the known value of x at time tn . However x is deﬁned as x = X + u, and the only variable to ﬁnd reduces to the displacement vector u at time tn+1 . Therefore in an explicit method, the time integration of the discrete momentum equations does not require the solution of any equations. An explicit solution is conditionally stable and requires that the time step size ��t be less or equal than a critical time step size ��tcrit for all time steps. The critical time step depends on element size and the maximum wave speed for the element material. The resulting time increment is often too small for practical considerations in computer eﬀort and for the response necessary to model slowly varying loads.

3.5 Time Integration Schemes

69

Implicit schemes are based on ﬁnding the position vector x at time tn+1 using not only the known value of x at time tn but also the values at tn+1 . Therefore in an implicit method, the time integration of the discrete momentum equations requires the solution of algebraic equations. An implicit solution is generally unconditionally stable, i.e. see Hughes (1987), and can be one or two orders of magnitude larger than the time step used in an explicit scheme. However the accuracy of the implicit schemes deteriorates as the time step size increases relative to the period of response of the system. In this work only implicit schemes are solved for structural dynamic problems. The semi-discrete equations of motion to be solved are given by u f int(un+1 ) + M�� n+1 = f ext(un+1 ) (3.5.1)

�� where the acceleration vector un+1 has to be integrated in time to solve the algebraic equations for un+1 from the second-order diﬀerential equations. Among the several numerical integration methods available to integrate second order equations, the most popular used in structural dynamics is the method developed by Newmark (1959). A major drawback of the Newmark integrator is the tendency for high frequency noise to persist in the solution. On the other hand, when linear damping or artiﬁcial viscosity is added, the accuracy is markedly degraded. Therefore other integration schemes are utilized with minor modiﬁcations as the ones given by Hilber et al. (1977), Wood et al. (1980) and Chung and Hulbert (1993). These methods improve numerical dissipation for high frequencies without degrading the accuracy as much. Details of these methods can also be found in Adams and Wood (1983) and Barbat and Canet (1994).

3.5.1

Newmark Method

In the Newmark method it is supposed that the solution at time step tn is known ˙ �� for the displacements un and its time derivatives un and un . The semi-discrete equations of motion to be solved at time tn+1 are given by f int(un+1 ) + M�� n+1 = f ext(un+1 ) u (3.5.2)

and the displacements un+1 and its time derivative are approximated according to ˙ �� �� ˙ un+1 = un + ��t (1 − ��) un + ��t�� un+1 ˙ un+1 = un + ��tun + ��t2 1 �� �� − �� un + ��t2 �� un+1 2 (3.5.3) (3.5.4)

where ��t is the time step size, and ��, �� are the parameters that determine the stability and accuracy of the scheme. The diﬀerent values for the parameters �� and �� originate the Newmark family methods. Stability conditions for the Newmark method are given for implicit schemes by Unconditional 2�� �� �� �� 1 2 (3.5.5)

70

3. Structural Dynamics

Conditional with

1 �� , ��< 2 2 �ئ�t �� ��crit �á�

(3.5.6)

where ��crit is the stability condition and �� is the natural frequency and must be satisﬁed for each mode in the system. Second-order accuracy is achieved if and only if �� = 1 , and viscous damping has no eﬀect on stability. When �� > 1 the 2 2 eﬀect of viscous damping is to increase the critical time step of conditionally stable Newmark methods. Two of the methods from the Newmark family for implicit schemes with second-order of accuracy are given in table 3.4. Other methods can be found, i.e. Hughes (1987). Method Trapezoidal rule Linear acceleration

Table 3.4

�� parameter 1/2 1/2

�� parameter 1/4 1/6

Stability condition Unconditional �� ��crit = 2 3

Most commons members of the implicit Newmark family

To solve the semi-discrete equations of motion with algebraic equations, the �� value of un+1 is obtained from Eq. (3.5.4) yielding �� un+1 = 1 1 ˙ (un+1 − un ) − un − �¦�t2 �¦�t 1 �� − 1 un 2�� (3.5.7)

˙ The value of un+1 is obtained from the resulting equation of substituting Eq. (3.5.7) into Eq. (3.5.3) yielding �� (un+1 − un ) − �¦�t �� ˙ − 1 un − ��t �� �� �� − 1 un 2��

˙ un+1 =

(3.5.8)

When the system has some kind of damping, i.e. viscous damping as described ˙ in section 3.5.5, the internal forces are function not only of un+1 but also of un+1 int ˙ and are written as f (un+1 , un+1 ). Now substituting Eq. (3.5.7), and Eq. (3.5.8) when necessary, into Eq. (3.5.2) yields f int(un+1 ) + 1 M un+1 − f ext(un+1 ) = �¦�t2 1 1 ˙ u + M un + 2 n �¦�t �¦�t

1 �� − 1 un 2��

(3.5.9)

and the algebraic equations can be solved for un+1 since all values at time tn are ˙ �� known. With un+1 solved, the values for un+1 and un+1 are updated with Eqs. (3.5.8) and (3.5.7) respectevely. The solution strategy used in this work to solve Eq. (3.5.9) is given ahead in section 3.6.

3.5 Time Integration Schemes

71

3.5.2

Hilber-Hughes-Taylor Method

Hilber, Hughes and Taylor introduced the ��-method as given in Hilber et al. (1977). This method introduces numerical dissipation for high frequencies without degrading the order of accuracy. In the ��-method, the Newmark formulas given by Eqs. (3.5.3) and (3.5.4) are retained, whereas the semi-discrete equations of motion are modiﬁed as follows u f int(un+1+��H ) + M�� n+1 = f ext(un+1+��H ) where the only change is given by un+1+��H = (1 + ��H ) un+1 − ��H un (3.5.11) (3.5.10)

If ��H = 0 this method reduces to the Newmark method. The method is unconditionally stable for linear systems when the parameters are selected such that 1 ��H �� − , 0 , 3 ��= 1 − 2��H , 2 ��= (1 − ��H )2 4 (3.5.12)

Any value given here for ��H results in a second-order accurate scheme. If ��H = 0, the method correspond to the Newmark trapezoidal rule method. There are no general stability results for this method for nonlinear problems. Now substituting Eq. (3.5.7), and Eq. (3.5.8) when necessary, into Eq. (3.5.10) yields f int(un+1+��H ) + 1 M un+1 − f ext(un+1+��H ) = 2 �¦�t 1 1 1 ˙ �� un + − 1 un u + M 2 n �¦�t �¦�t 2��

(3.5.13)

and the algebraic equations can be solved for un+1 . Note that the expression f int(un+1+��H ) is not equal to (1 − ��H ) f int(un+1 ) − ��H f int(un ) since the problem is nonlinear. The solution strategy used in this work to solve Eq. (3.5.13) is given ahead in section 3.6.

3.5.3

Bossak Method

An extension of the Newmark method was proposed by Bossak as given in Wood et al. (1980). This method deﬁned by the Newmark formulas given by Eqs. (3.5.3) and (3.5.4), and the semi-discrete equations of motion modiﬁed as follows �� f int(un+1 ) + M un+1−��B = f ext(un+1 ) where the only change is given by �� �� �� un+1−��B = (1 − ��B ) un+1 + ��B un (3.5.15) (3.5.14)

The method is unconditionally stable for linear systems when the parameters are selected such that

72

3. Structural Dynamics

1 ��B �� − , 0 , 3

��=

1 − 2��B , 2

��=

(1 − ��B )2 4

(3.5.16)

Again, any value given here for ��B results in a second-order accurate scheme. If ��B = 0, the method correspond to the Newmark trapezoidal rule method. A comparison of Bossak��s method and the Hilber-Hughes-Taylor ��-method is presented in Adams and Wood (1983). Both methods posses the same high-frequency dissipation properties. Howerver it is easier to implement in a ﬁnite element code the Bossak��s method. Now substituting Eq. (3.5.7), Eq. (3.5.15) , and Eq. (3.5.8) when necessary, into Eq. (3.5.14) yields f int(un+1 ) + 1 − ��B M un+1 − f ext(un+1 ) = �¦�t2 1 − ��B 1 − ��B ˙ un + un + M 2 �¦�t �¦�t

1 − ��B �� − 1 un 2��

(3.5.17)

and the algebraic equations can be solved for un+1 . The solution strategy used in this work to solve Eq. (3.5.17) is given ahead in section 3.6.

3.5.4

Generalized-�� Method

The generalized-�� method was introduced by Chung and Hulbert (1993), which achieves high-frequency dissipation while minimizing unwanted low-frequency dissipation. This method is a combination of all the above methods presented. In the generalized-�� method, the Newmark formulas given by Eqs. (3.5.3) and (3.5.4) are retained, whereas the semi-discrete equations of motion are modiﬁed as follows

s s u f int un+��f + M�� n+��s = f ext un+��f m

(3.5.18)

where the changes are given by

s s s un+��f = 1 − ��f un + ��f un+1 s s �� �� �� un+��s = (1 − ��m ) un + ��m un+1 m

(3.5.19) (3.5.20)

Low frequency dissipation is optimal with ��s �� [0, 1], ��

s ��f =

1 , 1 + ��s ��

s ��m =

2 − ��s �� 1 + ��s ��

(3.5.21)

and the method is second-order accurate and posses high frequency dissipation when 1 1 s s s s 2 + ��m − ��f , 1 + ��m − ��f ��= (3.5.22) 2 4 s s Other possibilities for ��f and ��m , as given in Chung and Hulbert (1993), result in s s the methods of Newmark, Bossak or ��-method. For example if ��f = 1 and ��m = 1, ��=

3.5 Time Integration Schemes

73

the method correspond to the Newmark trapezoidal rule method. The algebraic equations to be solved for un+1 are given by

s f int un+��f +

s ��m s M un+1 − f ext un+��f = 2 �¦�t s s ��s ��m ��m ˙ �� un + m un + − 1 un M �¦�t2 �¦�t 2��

(3.5.23)

The solution strategy used in this work to solve Eq. (3.5.23) is given ahead in section 3.6.

3.5.5

Structural Damping

The purpose of structural damping is to dissipate strain energy in a structure that is stored due to deformation processes. The energy dissipation within a structure due to material and structural damping depends on many factors such as the structural material or the magnitude of deformations experienced. In a dynamic analysis, this energy dissipation usually is accounted for by specifying an amount of viscoelastic damping that would result in energy dissipation in the analytical model equivalent to that expected to occur as a result of material and structural damping in the real structure. In this work, a viscoelastic damping is taken in account for geometrically nonlinear structures and consists of an extension of the viscoelastic damping given in Oller (2001) for geometrically linear structures with Kelvin��s model. In this model, the viscoelastic strain tensor is given as a function of the GreenLagrange strain tensor by ˙ Evis = E + �� E (3.5.24)

where �� is the retardation time. For linear problems this model yields a damping term which is a function of the stiﬀness matrix K in the form of D = ��K (3.5.25)

where D is the damping matrix. A very popular damping scheme in structural dynamics is given by the Rayleigh damping, which is a linear combination of the stiﬀness and mass matrices as D = ��M + ��K (3.5.26)

where �� and �� are the mass and stiﬀness damping constants respectively. As demonstrated in Oller (2001), when �� = 0, the Rayleigh and viscoelastic damping are equivalent, that is D = ��K = �� K (3.5.27)

74

3. Structural Dynamics

and the constant �� = ��. Note that the damping matrix is only proportional to the stiﬀness matrix. For nonlinear analysis, the damping is introduced directly only in the internal forces, which are written as ˙ f int = f int (un+1 , un+1 ) (3.5.28)

This way of introducing the viscoelastic strain yields in a natural form to add the damping, and not like a trick as in the Rayleigh case. The only diﬀerence in the solution between both damping schemes is given by the stress ﬁeld. While in the Rayleigh damping the stresses are given by S = CE, ˙ the stresses for the viscoelastic model are given by S = C E + �� E .

3.5.6

Quasi-static Solution

The methods described before are used to analyze dynamic problems since the inertial forces are computed. However if a static analysis is required, problems may be encountered when the membrane or shell problem has coplanar nodes yielding a singular matrix. To solve this inconvenient, a quasi-static solution can be employed. The quasi-static problem is expressed as a dynamic problem, only that the inertial terms are neglected and a damping matrix is added explicitly. This damping matrix is taken equal to the mass matrix. The problems solved in this work with the quasistatic analysis give a good solution with this methodology.

3.6

Solution Strategies

The numerical procedures used to solve nonlinear algebraic equations, given i.e. by Eq. (3.5.9) for the Newmark method, are iterative. Among the many diﬀerent procedures available to solve nonlinear problems, in this work only the Newton-Raphson method, see i.e. Reddy (2004), is outlined for structural dynamic problems.

3.6.1

Newton-Raphson Iterative Method

In the Newton-Raphson method, Eq. (3.5.9) is written in a residual form given by 1 M un+1 − f int(un+1 ) + �¦�t2 1 1 ˙ u + un + M 2 n �¦�t �¦�t

Rn+1 = f ext(un+1 ) −

1 �� − 1 un 2��

(3.6.1)

where it is supposed that the solution for iteration i−1 is known and the solution for iteration i is been searched. If the residual is expanded about the known solution ui−1 in Taylor��s series yields n+1 Ri n+1 �� Ri−1 n+1 ∂Rn+1 + ∂un+1

i−1

· ��ui n+1 = 0

(3.6.2)

3.6 Solution Strategies

75

where higher-order terms have been neglected. If the tangent stiﬀness matrix is deﬁned as KT = ∂R ∂u (3.6.3)

then Eq. (3.6.2) can be written in the form KT

i−1 n+1 i−1 · ��ui n+1 = −Rn+1

(3.6.4)

which is an algebraic system for ��ui . If the procedure converges, the residual n+1 is gradually reduced to zero and the solution at the ith iteration is given by

i−1 i ui n+1 = un+1 + ��un+1

(3.6.5)

The iteration procedure is continued until a certain convergence criterion is satisﬁed for the system. Then the value for the current position vector is given by xn+1 = X + un+1 . For the ﬁrst iteration i = 1 of the time step n + 1, the predictor for ui−1 is n+1 taken from the previous converged solution at time step n, that is u0 = u n n+1 (3.6.6)

From the residual equation given by Eq. (3.6.1), the tangent stiﬀness matrix is evaluated yielding KT = K T − K T − KT ext kin int (3.6.7)

where the internal tangent matrix KT comes from the internal forces, the kinetic int tangent matrix KT is associated to the mass matrix and and the external tangent kin matrix KT comes from the external forces. Here the external tangent matrix is ext diﬀerent from zero only when pressure follower forces exist or in the presence of any other non-conservative load types.

3.6.2

Linearization

A diﬃcult task in the solution of implicit systems for the semi-discrete equations of motion is the linearization of the governing equations. Here the expressions for the tangent stiﬀness matrix are derived. This is done with the continuum tangent moduli which does not account for the actual constitutive update algorithm and is the one used in this work because it is intended only for linear materials. Other constitutive equations may need an algorithmic tangent moduli, which gives rise to the consistent tangent stiﬀness matrix. The linearization depends on the type of formulation for the diﬀerent elements studied. Here only membrane, cable and shell linearization is taken into account.

76

3. Structural Dynamics

3.6.2.1

Membrane Linearization

The internal tangent stiﬀness matrix for a membrane element is computed by taking the derivative of the internal forces in the displacements direction. The general expression for the internal membrane forces is given in Eq. (3.2.38) yielding the following derivative (KijIJ )int =

T int ∂fiI = ∂ujJ cur B����iI cur ∂B����iI ���� ∂S ���� d��0 + S ∂ujJ ∂ujJ

(3.6.8)

��0

Expressing the second Piola-Kirchhoﬀ stress tensor as a function of the GreenLagrange strain tensor, then S ���� = C ���¦� E�� and since in this work the components of the constitutive tensor are constant, the ﬁrst term in the integral is expressed by

cur B����iI

∂S ���� ∂E�� cur = B����iI C ���¦� ∂ujJ ∂ujJ 1 ∂g�� ∂E�� = ∂ujJ 2 ∂ujJ

(3.6.9)

Now substituting Eq. (3.2.5) in the derivative term yields (3.6.10)

which can be expanded leading to ∂g�� ∂g ∂g�� = · g + g�� · ∂ujJ ∂ujJ ∂ujJ and

nnode ∂g�� ∂NI ∂xI = ∂ujJ ∂�� �� ∂ujJ I=1

(3.6.11)

(3.6.12)

However, the last term can be expressed as ∂xI ∂uiI = ei = ��ij ��IJ ei = ��IJ ej ∂ujJ ∂ujJ ∂g�� ∂NJ = ej ∂ujJ ∂�� �� If the covariant base vector is discretized by

nnode

(3.6.13)

Substituting Eq. (3.6.13) into Eq. (3.6.12) the following expression is obtained (3.6.14)

g =

I=1

∂NI xiI ei ∂��

(3.6.15)

then the ﬁrst product of Eq. (3.6.11) yields

nnode ∂g�� ∂NJ ∂NI ·g = xjI ∂ujJ ∂�� �� ∂�� I=1

(3.6.16)

3.6 Solution Strategies

77

This last equation allow us to expressed Eq. (3.6.10) in the form of ∂E�� 1 = ∂ujJ 2

nnode

(NJ,�� NI, + NI,�� NJ, ) xjI

I=1

(3.6.17)

which is equal to the expression for the strain-displacement tensor given in Eq. (3.2.37) and then

cur B�� jJ =

∂E�� ∂ujJ

(3.6.18)

Then the tangent material stiﬀness is deﬁned as

mat KijIJ = cur cur B����iI C ���¦� B�� jJ d��0

(3.6.19)

��0

or in Voigt notation Kmat = IJ [BT ]f ib [C] [BJ ]f ib d��0 I (3.6.20)

��0

With Eq. (3.6.18), the internal forces can be written in a more useful equation instead of Eq. (3.2.38) yielding

int fiI =

��0

∂E���� ���� S d��0 ∂uiI

(3.6.21)

which derivative is given by

int ∂fiI = ∂ujJ

(KijIJ )int =

T

��0

∂E���� ∂S ���� ∂ 2 E���� ���� d��0 + S ∂uiI ∂ujJ ∂uiI ∂ujJ

(3.6.22)

This is another way to express Eq. (3.6.8). To compute the second derivative on the right hand side of Eq. (3.6.22), the term 1 ∂ 2 g���� ∂ 2 E���� = ∂uiI ∂ujJ 2 ∂uiI ∂ujJ where ∂ 2 g�� ∂ 2 g���� ∂ 2 g�� ∂g�� ∂g�� ∂g�� ∂g�� = · g�� + · + · + g�� · (3.6.24) ∂uiI ∂ujJ ∂uiI ∂ujJ ∂uiI ∂ujJ ∂ujJ ∂uiI ∂uiI ∂ujJ From Eq. (3.6.14), the second derivative term in Eq. (3.6.24) yields ∂ 2 g�� ∂NJ ∂ej = =0 ∂uiI ∂ujJ ∂�� �� ∂uiI Now substituting Eq. (3.6.14) into Eq. (3.6.24), we ﬁnd that (3.6.25) (3.6.23)

78

3. Structural Dynamics

1 ∂ 2 E���� = (NI,�� NJ,�� + NI,�� NJ,�� ) ��ij ∂uiI ∂ujJ 2

(3.6.26)

This equation and the second Piola-Kirchhoﬀ stress tensor give rise to the tangent geometrical stiﬀness tensor, which is given by

geo KijIJ =

��0

∂ 2 E���� ���� S d��0 ∂uiI ∂ujJ

(3.6.27)

or in Cartesian tensor notation as Kgeo = I(3��3) IJ BT S B0J d��0 0I

��0

(3.6.28)

0 where the components of B0I (2��1) are given by BjI in Eq. (3.1.18) for j = 1, 2 and S is expressed in tensorial notation for the membrane element. Finally, the internal tangent matrix is deﬁned by

KT = Kmat + Kgeo int

(3.6.29)

Another expression that is needed to compute the tangent stiﬀness matrix is the kinetic tangent matrix that is obtained directly from Eq. (3.6.1) by taking the derivative of the mass term respect to the displacements, which yields KT = kin 1 M �¦�t2 (3.6.30)

The last term in the tangent stiﬀness matrix comes from the derivative of the external forces respect to the displacements. Only when the external forces are functions of the displacements, an external tangent matrix diﬀerent from zero is found. Since in this work pressure follower forces depend of the current displacements at each time step, this value can be computed from Eq. (3.2.104) giving ∂ f ext p ∂ = (x2 − x1 ) �� (x3 − x1 ) ∂u 6 ∂u The resulting equation is given in an explicit form of KT = ext

2

(3.6.31)

KT ext

p = 6

(3.6.32) A general expression for the external tangent matrix for pressure follower forces can be found in Bonet and Wood (1997) or Belytschko et al. (2000). With all these expressions the tangent stiﬀness matrix is built with K T = KT − K T − KT − KT ext mat geo kin and the algebraic equations given by Eq. (3.6.4) can be solve iteratively. (3.6.33)

0 6 6 6z − z 2 6 3 4 y2 − y3

z2 − z3 0 x3 − x2

y3 − y2 x2 − x3 0

0 z1 − z3 y3 − y1

z3 − z1 0 x 1 − x3

y1 − y3 x3 − x1 0

0 z2 − z1 y1 − y2

z1 − z2 0 x2 − x1

7 7 x1 − x2 7 7 5 0

y2 − y1

3

3.7 Code Development

79

3.6.2.2

Cable Linearization

Following the same methodology given for the tangent stiﬀness matrices for membranes, the tangent stiﬀness matrices for cables are found. From Eq. (3.6.20) we can conclude that the material tangent stiﬀness matrix is Kmat = A0 L[BT ]loc [C] [B]loc (3.6.34)

where [B]loc is given by Eq. (3.3.26). The geometrical tangent stiﬀness matrix is found with the same procedure that for membrane elements, yielding Kgeo = A0 {Sx } L

loc

I(3��3) I(3��3)

I(3��3) I(3��3)

(3.6.35)

The kinetic tangent stiﬀness matrix gives the same expression that Eq. (3.6.30), but with the corresponding mass matrix for two-node cable elements. 3.6.2.3 Shell Linearization

As explained in section 3.4.2, a shell has a membrane and a bending part. Consequently its linearization is formed by a membrane tangent stiﬀness matrix and a bending tangent stiﬀness matrix. The membrane tangent stiﬀness matrix is detailed in section 3.6.2.1 and is exactly the same to be used in shell elements. However the missing part corresponds to the bending tangent stiﬀness matrix. For the rotatinal-free shell element used in this work, this matrix is very complicated to derive, as can be found in Flores and O˜ate (2001), and has no practical advantages if it is incorporated in a ﬁnite n element program. Therefore as explained in Flores and O˜ate (2005), the bending n stiﬀness tangent matrix can be avoided and only the membrane tangent stiﬀness matrix is used in this work.

3.7

Code Development

In this work the membrane, cable and shell formulation have been implemented in the ﬁnite element program COMET (2007). The dynamic analysis with Bossak method, Hilber-Hughes-Taylor method and Generalized-�� method have also been added to the code. This software is developed at the International Center for Numerical Methods in Engineering (CIMNE) and is used for coupled contact, mechanical and thermal analysis using the ﬁnite element method. The ﬁnite element meshes and input data used in this work where generated using the program GiD (2007). This software is also developed at the International Center for Numerical Methods in Engineering (CIMNE) which is a preprocessor and postprocessor for ﬁnite element programs.

80

3. Structural Dynamics

3.8

3.8.1

Example Problems

Inﬂation of a Square Airbag

A square membrane is inﬂated by internal pressure to simulate an airbag, as given by Contri and Schreﬂer (1988) and Lu et al. (2001). The internal pressure is 2 0.5 lb/ft . The side length of the airbag is 1 f t with a membrane thickness of 2 0.0001 f t. The Young��s modulus is E = 4.32 �� 106 lb/ft and the Poisson ratio 3 is �� = 0.3. The density used for the quasi-static analysis is �� = 100 lb/ft , with 200 steps and a pseudo-time step ��t = 0.0001. The wrinkling algorithm is used for the analysis to avoid unwanted compression stresses. Because of the symmetry, only one-eighth of the airbag is modelled. A structured mesh of 200 three-node triangular membrane elements is used. The origin O is placed at the center of the airbag. Symmetry boundary conditions with ﬁxed y-displacements are applied on side OA, ﬁxed x-displacements are applied on side OC and ﬁxed z-displacements are applied on side AB and BC. The inﬂated airbag analyzed with the wrinkling algorithm is shown in Fig. 3.15 without ampliﬁcation factor. The maximum transverse displacement is presented at the origin O in the z-direction with 0.249 f t, while the references give a value of 0.252 f t with nine-node quadrilateral elements. The maximum in-plane displacement is 0.142 f t symmetric for x- and y-direction, while the references give a value of 0.144 f t.

0.25 0.20 0.15 0.10 0.05

0.00

[ft]

0.25

Figure 3.15

z-displacements of inﬂated airbag

3.8 Example Problems

81

The maximum principal stress is at the origin O with 4105 lb/ft , and no compression stresses appear in the solution since the wrinkling algorithm is used, see Fig. 2 3.16. The maximum principal stress given by the references is about 4000 lb/ft .

2

Figure 3.16

Principal stresses with wrinkling

A completely diﬀerent solution is found with a conventional membrane analysis (no wrinkling algorithm is used), where high compression stresses appear in the solution, see blue vectors in Fig. 3.17. These negative stresses do not allow to inﬂate the airbag with such ﬁnite element discretization. In this analysis, the maximum transverse displacement is presented at the origin O in the z-direction with 0.097 f t, while the references give a value of 0.100 f t with nine-node quadrilateral elements. The maximum in-plane displacement is 0.066 f t symmetric for x- and y-direction, while the references give a value of 0.077 f t.

Figure 3.17

Principal stresses without wrinkling

2

The maximum principal tension stress is about 13000 lb/ft , and the maxi-

82

3. Structural Dynamics

mum principal compression stress is −35000 lb/ft . This example demonstrates the importance in membrane analysis of using an eﬀective wrinkling algorithm to eliminate compression stresses. When no wrinkling algorithm is available for the membrane analysis, then a very ﬁne mesh is needed in order to obtain a reasonable good solution.

2

3.8.2

Prestressed Membrane

An initially prestressed membrane is loaded by a transversal point load in its middle domain, as given by Levy and Spillers (1995) and Gil (2003). The point load is −10000 lb and the membrane side length is 240 in with a thickness of 0.004167 in. The Young��s modulus is E = 30.0 �� 106 psi and the Poisson ratio is �� = 0.3. The 3 density used for the quasi-static analysis is �� = 0.06 lb/in , with 40 steps and a pseudo-time step ��t = 0.5. The prestressing eﬀect is considered to be ��xx = 80000 psi and ��yy = 80000 psi in the whole domain. The sides of the membrane are ﬁxed for the analysis. In Fig. 3.18 the mesh used in the analysis is shown, together with control nodes and control elements surrounded by a circle. The mesh has 32 three-node triangular membrane elements and 25 nodes.

1 3 4 11 9 8

z

y x

Figure 3.18

Prestressed membrane geometry

A comparison of the displacements at control nodes is shown in Table 3.5.

Node 4 8 9 Levy x-disp 0.015 0.000 0.000 and Spillers y-disp z-disp -0.015 -1.431 -0.017 -2.605 0.000 -6.642 x-disp 0.014 0.000 0.000 Gil y-disp -0.014 -0.017 0.000 z-disp -1.423 -2.600 -6.626 Present work x-disp y-disp z-disp 0.014 -0.014 -1.429 0.000 -0.017 -2.600 0.000 0.000 -6.626

Table 3.5

Membrane displacements [in]

It can be seen that the displacements agree well with both referenced analysis using the same mesh. The major diﬀerence is at node 9 in the z-displacement, where Levy and Spillers (1995) give a greater value.

3.8 Example Problems

83

Elem 1 3 11

Levy ��xx 97377 83510 144691

and Spillers ��yy ��xy 85212 -2801 96859 -8657 97831 -15616

��xx 97300 83502 144471

Gil ��yy 85164 96830 97849

��xy -2797 -8631 -15582

Present work ��xx ��yy ��xy 97328 85139 -2794 83503 96839 -8677 144812 97649 -15711

Table 3.6

Membrane Cauchy stresses [psi]

A comparison of the Cauchy stresses at control elements is shown in Table 3.6, where it can be seen that the stresses are very similar between the referenced analysis. In this example the ﬁber orientation is performed in order to give the elements the same orientation and then assign the correct prestressed values to the mesh. However in this special case where an initially ﬂat membrane is analyzed, a simpler method can be used. Just to compare the maximum z-displacement, the membrane is analyzed without the prestressed force which results in a displacement of -9.242 in compared to -6.626 in with the prestressed load. This show us how important can be the prestressed force in membrane elements.

3.8.3

Nonlinear Plate

A plate with uniform load is analyzed. The analysis is assumed to be geometrically nonlinear, as presented by Zienkiewicz and Taylor (1989) and Clemente (2007). The same geometry and material properties of Clemente are used in this analysis to simplify the comparison of the diﬀerent solutions. The only diﬀerence is that Clemente used 8-node three-dimensional brick elements while in this work the 3node rotation-free shell elements are employed. The side length of the plate is L = 2a = 20 m with a thickness of 1 m. The Young��s modulus is taken as E = 12 P a and the Poisson ratio is �� = 0.0. The density used for the quasi-static analysis is �� = 1.0 kg/m3 , with 10 steps and a pseudo-time step ��t = 0.1. Because of the symmetry, only a quarter of the plate is analyzed.

Figure 3.19

Finite element meshes used for the analysis

84

3. Structural Dynamics

Two meshes are used for the analysis as shown in Fig. 3.19. The structured mesh has 800 elements and 441 nodes, while the unstructured mesh has 816 elements with 447 nodes. A variable uniform load q is applied to the plate, with values from 0.00 to −0.04 P a. The adimensional solution of the problem is plotted in Fig. 3.20. The transversal displacement at the central point of the plate w is normalized by the thickness t in the horizontal axis, while the load q is normalized by Dt/a4 in the vertical axis, where the value D = Et3 /12.

400 350 300 250

Present work structured Vald��s structured Present work unstructured Vald��s unstructured

Zienkiewicz/T aylor Clemente Linear Analysis

qa 4/Dt

200 150 100 50 0 0.0

0.5

1.0 w/t

1.5

2.0

2.5

Figure 3.20

Central plate displacement by uniform load

The deformed without ampliﬁcation factor is shown in Fig. 3.21 and Fig. 3.22 for two diﬀerent views, where the reference mesh is drawn in gray color.

z

0.00 y [m] x -2.27

Figure 3.21

z-displacements of nonlinear plate

We can conclude that our shell analysis gives the same solution that using threedimensional brick elements. Less computational eﬀort is required in our analysis since the degrees of freedom are drastically reduced to those used by Clemente, where his analysis needed 1600 elements and 2205 nodes with four layers over the

3.8 Example Problems

85

y z 0.00

[m]

x

-2.27

Figure 3.22

z-displacements of nonlinear plate

thickness. From Fig. 3.20 we can also see that the solution of the linear analysis becomes an important aspect to take care of when the load is incremented and the behavior of the linear and nonlinear analysis give great diﬀerences.

3.8.4

Hemispherical Shell with 18o Hole

A popular benchmark problem for linear shell analysis is the pinched hemisphere which is concerned with the nearly inextensional deformation of a hemispherical shell with an 18o hole at the top under the action of two inward and two outward forces 90o apart. Symmetry conditions are used in this problem and only onequarter needs to be modelled, as shown in Fig. 3.23.

100

Free Sym.

y Free x z Sym.

100

Figure 3.23

Structured hemispherical shell geometry

86

3. Structural Dynamics

To illustrate the large deformation capabilities of the rotation-free formulation, the forces of the linear problem are incremented 100 times to obtain deﬂections of nearly 60% of the initial radius. This problem is compared with the solution obtained by Simo et al. (1990a) and Flores and O˜ate (2001). The problem geometry n consists of a sphere of radius R = 10 with an 18o hole and thickness of 0.04. The material properties are E = 6.825 �� 107 and �� = 0.30. The forces and boundary conditions are shown in Fig. 3.23.

100

Free Sym.

y Free x z Sym.

100

Figure 3.24

Unstructured hemispherical shell geometry

This problem was solved by Simo et al. (1990a) using a structured mesh of 256 quadrilateral elements with 1632 d.o.f. To compare the benchmark, in this work two meshes are studied, a structured mesh of 640 elements with 1036 d.o.f., as shown in Fig. 3.23, and an unstructured mesh of 640 elements with 1032 d.o.f. as shown in Fig. 3.24. A summary of the maximum displacements in x- and y-direction is given in Table 3.7. Author Sim�� o Present work Present work Elements 256 640 640

Table 3.7

d.o.f. 1632 1036 1032

Mesh structured structured unstructured

u disp 3.380 3.319 3.324

v disp -5.875 -5.878 -5.929

Maximum displacements

A plot diﬀerent of load-displacement values is given in Fig. 3.25.

3.8 Example Problems

87

6

Vald��s unstructured u Present work unstructured u

5

Displacement

4 3 2 1 0 0

Vald��s unstructured v Present work unstructured v Vald��s structured u Present work structured u Present work structured v Vald��s structured v Simo u Simo v

20

40 Load

60

80

100

Figure 3.25

Load-displacement comparison

The complete deformed structured mesh without magniﬁcation factor is given in Fig. 3.26, where the norm of displacements is shown.

y x z 0.00 5.87

Figure 3.26

Deformed without magniﬁcation factor

We can conclude that the rotation-free shell formulation has an excellent behavior for problems involving large deformations with both structured and unstructured meshes, the former with uniform element size distribution.

3.8.5

Free Vibration Pendulum

The objective of the free vibration pendulum problem is to show the importance of the time integration scheme used in structural problems since sometimes high

88

3. Structural Dynamics

frequency noise persists in the solution for large deformations analysis. The pendulum problem is also analyzed by Rossi (2005) using quadrilateral elements for the whole problem, and showing a graph of instability given by the modulus of reaction. The mesh used in this analysis is shown in Fig. 3.27.

y x

z

Figure 3.27

Pendulum mesh

The cable length is 1.5 m with a cross sectional area of 0.0005 m2 . The circular end mass has a radio of 0.15 m with a thickness of 1.0 m. The Young��s modulus for both the cable and mass is taken as E = 2.1 �� 1011 P a and the Poisson ratio is �� = 0.3. The density used for the dynamic analysis is �� = 7800 kg/m3 for the whole problem. The mesh used in the analysis has 20 two-node cable elements and 184 three-node triangular plane stress elements with a total of 131 nodes. The time step size is ��t = 0.01 s. Gravity forces are applied to the structure. Fig. 3.28 show the time vs. y-displacement graph, where the Generalized-�� method is stable for the time studied. However the Newmark algorithm is stable only at the beginning of the analysis.

0.0

Generalized-J Newmark

-0.4

y-displacement [m]

-0.8

-1.2

-1.6 0 2 4 Time [s] 6 8 10

Figure 3.28

Time-displacement graph

In Figs. 3.29-3.33 various time instants are plotted comparing the deformed shape of the pendulum between the Generalized-�� method and the Newmark algorithm. Fig. 3.31 shows that at time instant 2.18 s the high frequencies begin to appear in

3.8 Example Problems

89

Generalized-J

Newmark

Figure 3.29

Generalized-J

Deformed cable at time 1.00 s.

Newmark

Figure 3.30

Generalized-J

Deformed cable at time 1.50 s.

Newmark

Figure 3.31

Generalized-J

Deformed cable at time 2.18 s.

Newmark

Figure 3.32

Deformed cable at time 2.54 s.

90

3. Structural Dynamics

Generalized-J

Newmark

Figure 3.33

Deformed cable at time 2.78 s.

the Newmark analysis. Later in Fig. 3.32 the high frequencies become more evident until that in Fig. 3.33 the high frequencies spoil the solution for the Newmark algorithm. However the Generalized-�� method keeps its convergence properties for the time interval studied. Also the Hilber-Hughes-Taylor and Bossak��s methods show the same accuracy and stability that the Generalized-�� method. The Newmark method give the same solution that the former methods with a time step size of ��t = 0.001 s.

3.8.6

Inﬂation of a Parachute

In this example an initially highly folded parachute is inﬂated. Since high compression stresses are presented in the solution, the wrinkling algorithm is used in order to avoid an inappropriate deformed conﬁguration. The initial conﬁguration of the parachute is shown in Fig. 3.34. Diﬀerent inﬂation processes of parachutes are studied by Lu et al. (2001) and Tezduyar et al. (2006).

0.76 0.76

3.05

6.10

Figure 3.34

Initial conﬁguration of the parachute, top and side views [m]

The mesh used for the example has 1664 membrane elements and 16 cable elements, with a total of 882 nodes. The internal pressure is taken as 5 P a. Parachute material properties are used with the Young��s modulus as E = 2.07 �� 108 P a, the Poisson ratio 0.3 and the material density �� = 9.61 kg/m3 . The thickness is taken as 0.00003 m. The cable properties are Young��s modulus E = 2.07 �� 109 P a and

3.8 Example Problems

91

the cross-sectional area of 0.00013 m2 . The initial conﬁguration of the problem is shown in Fig. 3.35 together with an intermediate deformation step.

Figure 3.35

Inﬂation process of the parachute from reference conﬁguration

The problem uses the wrinkling algorithm with a quasi-static analysis which stabilizes the solution. The number of steps are 3000 with a pseudo-time of 0.0001. Fig. 3.36 shows diﬀerent instants of the inﬂation process of the parachute.

Figure 3.36

Inﬂation process of the parachute until ﬁnal conﬁguration

The same parachute is analyzed dynamically with the wrinkling algorithm using 3000 steps and a time step size ��t = 0.0001 s. In this case stiﬀness proportional damping was applied to stabilized the solution of the problem. Both analysis, quasi-static and dynamic lead to the same solution.

92

3. Structural Dynamics

Fig. 3.37 shows the vertical displacements while Fig. 3.38 shows the horizontal displacements where it can be seen that the problem leads to a symmetric solution.

0.00

[m]

-1.58

Figure 3.37

Vertical displacements at ﬁnal conﬁguration

1.42

[m]

-1.42

Figure 3.38

Horizontal displacements at ﬁnal conﬁguration

3.8 Example Problems

93

Fig. 3.39 shows the maximum principal stresses while Fig. 3.40 shows the minimum principal stresses. In both ﬁgures it can be seen that there are no compression stresses due to the use of the wrinkling algorithm.

2.6 x 10 6

[Pa]

75.6

Figure 3.39

Maximum principal stresses

3.2 x 10 5

[Pa]

0.0

Figure 3.40

Minimum principal stresses

94

3. Structural Dynamics

The same parachute is analyzed dynamically without the wrinkling algorithm. Fig. 3.41 shows the vertical displacements while Fig. 3.42 shows the horizontal displacements.

0.00

[m]

-1.03

Figure 3.41

Vertical displacements at ﬁnal conﬁguration

1.31

[m]

-1.31

Figure 3.42

Horizontal displacements at ﬁnal conﬁguration

3.8 Example Problems

95

Fig. 3.43 shows the maximum principal stresses while Fig. 3.44 shows the minimum principal stresses. In both ﬁgures it can be seen that there are compression stresses due to the use of conventional membrane theory that do not include any wrinkling algorithm.

1.1 x 10 6

[Pa]

-1.2 x 10 5

Figure 3.43

Maximum principal stresses

8.2 x 10 5

[Pa]

-7.7 x 10 5

Figure 3.44

Minimum principal stresses

96

3. Structural Dynamics

Fig. 3.45 shows two ﬁnal conﬁgurations of the parachute. One includes the wrinkling algorithm while the other one does not include it.

Figure 3.45

Final conﬁguration: left with wrinkling, right without wrinkling

It can be seen that the shape of the inﬂated parachute is completely diﬀerent if the wrinkling algorithm is used or not. Also the maximum principal stresses are very diﬀerent. In the analysis with the wrinkling algorithm there are no compression stresses, just as it really happens. However conventional membrane analysis that do not include any wrinkling algorithm lead to an unreal ﬁnal conﬁguration due to the presence of compression stresses.

3.8 Example Problems

97

3.8.7

Hyperbolic Paraboloid

Ziegler (2001) and Raible (2003) have studied hyperbolic paraboloids under vertical �� loading. Geometry parameters are given in Fig. 3.46, where a = 12 2 m, b = 2.88 m with a membrane thickness of 0.3 mm. Membrane boundary is ﬁxed. The Young��s modulus is E = 21000 kN/cm2 and the Poisson ratio is �� = 0.3. A vertical load of 10 kN/m2 is applied in various steps.

M

b

B

b

A

a

a

Figure 3.46

Geometry and mesh for hyperbolic parabolid

Also in Fig. 3.46 control points M, A and B are given for comparison purposes. A quasi-static analysis is performed using the wrinkling algorithm to avoid unwanted compression stresses. A structured three-node membrane triangular mesh of 1600 elements and 841 nodes is used for the analysis.

Raible Present work

117.2

Figure 3.47

1298.0

179.1

1316.8

Comparison of 1st principal stress Si [M N/m2 ]

98

3. Structural Dynamics

In Fig. 3.47 the 1st principal stress Si is compared with the one obtained by Raible (2003). The tendency of both works is the same, where little diﬀerences are found. Also maximum and minimum values for Si are very similar in both cases. In this ﬁgure, the geometry of the hyperbolic paraboloid is indicated as follows: black circles denote the lower corner nodes while white circles mean upper ones. At point A, Ziegler (2001) seems to indicate a slack membrane state, while Raible (2003) indicate a wrinkled membrane state. In this work, at point A the slack membrane state is found. Finally a plot of the vertical displacements at the control points is given in Fig. 3.48, where solutions of Ziegler (2001) and Raible (2003) are also included. In the work of Raible (2003) two wrinkling algorithms are studied, and here only the corrected approach solution is included since it delivers the most reliable results. In these three works, vertical displacement of control point M is very similar and in this work a value of 36.4 cm is found. A closer vertical displacement between these three works is found for control point B with a value of 26.3 cm. However, control point A delivers some diﬀerences between the vertical displacements of the compared cases. The solution presented in this work lies between the works of Ziegler (2001) and Raible (2003), but it can be seen that the present solution is closer to the former one, which in this work correspond to 24.6 cm.

Ziegler

Raible

Present work

vertical displacement [cm]

load [kN/m 2]

Figure 3.48 Comparison of selected nodes

3.8 Example Problems

99

3.8.8

Orthotropic Spinnaker

Usually sails are built with composite materials, which sometimes are modeled with isotropic or orthotropic materials. Fig. 3.49 shows diﬀerent materials for sails, where the material on the left picture can be modeled as an isotropic material while the material on the right picture can be modeled as an orthotropic material.

Figure 3.49

Materials to build sails

A sail is built as an assembling process of single parts of the cutting pattern which are sewed and glued together. Each one of the parts has ﬁbers of principal reinforcement direction that for optimization of the sail take diﬀerent forms. In the case of a spinnaker, their orientations are shown in Fig. 3.50.

Figure 3.50

Principal ﬁber orientation for a spinnaker

100

3. Structural Dynamics

The objective of this example is to show how a membrane structure build with an orthotropic material can be analyzed, including diﬀerent local orientations for the material. Fig. 3.51 shows the manufacturing process for a spinnaker.

Figure 3.51

Spinnaker manufacturing process

Following the methodology given in this work for the ﬁber orientation, Figs. 3.52 - 3.53 show vectors of local x-direction for two conﬁgurations.

Figure 3.52

Principal ﬁber direction: optimal

3.8 Example Problems

101

Figure 3.53

Principal ﬁber direction: horizontal

Fig. 3.52 shows the optimal ﬁber orientation for a spinnaker, while Fig. 3.53 shows an horizontal orientation. Material orthotropic properties are taken with a Young modulus Ex = 1100 N/mm2 , Ey = 385 N/mm2 , Poisson ratio ��xy = 035, ��yx = 0.1225 and shear modulus Gxy = 220 N/mm2 . A uniform pressure follower load of 20 N/m2 is applied for the analysis. The bottom dimension is 6.5 m, with a high of 14.0 m and thickness of 0.1 mm. The structure is discretized with 686 three-node triangular membrane elements and 384 nodes. Fig. 3.54 shows values of the second Piola-Kircchoﬀ stress tensor S. On the left part of the ﬁgure, values for Sxx are plotted, where the local direction for xx is given by the optimal orientation of the ﬁbers, as illustrated in Fig. 3.52. The corresponding local values for Sxx are maximum with 25.87 M P a and minimum with 0.64 M pa. The right part of Fig. 3.54 shows values for the 1st principal component Si of the second Piola-Kircchoﬀ stress tensor, corresponding to a maximum of 25.92 M pa and minimum with 0.68 M pa. In this case, both solutions are almost identical because the orientation for the principal ﬁber of the material follows the vectors of the 1st principal component of the stress tensor, as shown in Fig. 3.55. On the contrary, Fig. 3.56 gives values of the second Piola-Kircchoﬀ stress tensor S. On the left part of the ﬁgure, values for Sxx are plotted, where the local

102

3. Structural Dynamics

direction for xx is given by the horizontal orientation of the ﬁbers, as illustrated in Fig. 3.53. The corresponding local values for Sxx are maximum with 8.22 M P a and minimum with −0.57 M pa. The right part of Fig. 3.56 shows values for the 1st principal component Si of the second Piola-Kircchoﬀ stress tensor, corresponding to a maximum of 24.46 M pa and minimum with 0.72 M pa. In this case, both solutions are very diﬀerent because the orientation for the principal ﬁber of the material is horizontal, which is completely diﬀerent from the vectors of the 1st principal component of the stress tensor that give an optimal material usage.

-0.57

[MPa]

25.92

Figure 3.54

Second Piola Kircchoﬀ stresses: optimal ﬁber orientation

3.8 Example Problems

103

Figure 3.55

Optimal ﬁber orientation and 1st principal stress direction

104

3. Structural Dynamics

-0.57

[MPa]

25.92

Figure 3.56

Second Piola Kircchoﬀ stresses: horizontal ﬁber orientation

Chapter 4

Fluid Dynamics

In this chapter, the ﬂuid dynamics equations used in this work to solve incompressible ﬂow problems are presented. The objective here is only to explain the tools implemented to solve the ﬂuid equations that later will be used as an application in the ﬂuid-structure interaction problem.

4.1

Introduction

Fluid mechanics deals with the ﬂow of ﬂuids. The main diﬀerence between a ﬂuid and a solid is that while solids have very strong intermolecular attractive forces, ﬂuids are characterized by the relative mobility of its molecules. It is common that the stress in a solid is proportional to the strain, while the stress in a ﬂuid is proportional to the rate of strain. This proportional parameter in ﬂuids is known as the viscosity. Fluid mechanics is usually divided into smaller areas based on characteristics of the ﬂuid properties. An inviscid ﬂuid is one where the viscosity is assumed to be zero. An incompressible ﬂuid is one which density variations compared to a reference density are negligible. The motion of a ﬂuid is governed by the laws of conservation of mass, momentum (resulting in the Navier-Stokes equation) and energy, all of them described in chapter 2. When temperature eﬀects are not important, as assumed in this work, the energy equation is uncoupled and only the Navier-Stokes equation and the continuity equation are solved. With the help of computers, predictions of ﬂuid ﬂow based on the governing equations can be done, and the science responsible for this is called computational ﬂuid dynamics (CFD).

4.2

Governing Equations

In ﬂuid mechanics problems, Lagrangian methods are totally inappropriate. Lagrangian elements become severely distorted when the material is similar deformed since they follow the material. On the other hand, in ﬂuid mechanics problems the 105

106

4. Fluid Dynamics

interest is focused on a particular spatial subdomain, therefore this type of problem is more suited to Eulerian elements. This kind of elements are ﬁxed in space, thus undergo no distortion. However other kind of problems emerge as a consequence of the convection of the material through the elements.

4.2.1

Weak Form

The strong for consists of the Navier-Stokes and continuity equation, the velocity boundary conditions vi on the Dirichlet boundary ��D and the traction boundary ¯ ¯i on the Neumann boundary ��N , where the boundary ��0 is deﬁned by conditions t ��0 = ��D �� ��N . To develop the weak form of the Navier-Stokes equation, the space of the test functions for the Eulerian description is deﬁned as ��vi (x) �� V0 , V0 = ��vi |��vi �� H 1 (��), ��vi = 0 on ��D (4.2.1)

where H 1 (��) consists of vector functions whose components and their derivatives of order 1 are square-integrable. The space of the trial functions for the velocities is given by vi (x, t) �� V V = vi |vi �� H 1 (��), vi = vi on ��D ¯ (4.2.2)

The development of a Galerkin-type weak form consists in taking the product of the general form of the Navier-Stokes equation, Eq. (2.6.1), by the test function ��vi , integrating over the current domain and using appropriate boundary conditions, yielding

��vi

��

��

∂vi + vj ∂j vi ∂t

+

∂p 1 ∂ − ��bi − �� ∇2 vi + (∇ · v) ∂xi 3 ∂xi

d�� = 0

(4.2.3) where Eq. (2.6.2) has been substituted into Eq. (2.6.1) and �� is the dynamic viscosity . Integrating by parts1 the underlined terms of Eq. (4.2.3), leads to

∂vi ∂��vi ∂vi − ��bi d�� − p d�� + + ��vj ∂t ∂xj �� �� ∂xi ∂vi ∂��vi 1 ∂��vi �� d�� + d�� = �� (∇ · v) ∂xi �� ∂xj ∂xj �� 3 ∂vi 1 �̦�vi nj (∇ · v) ��ij d�� − �̦�vi nj d�� + ��vi nj p��ij d�� ∂xj ��N ��N 3 ��N ��vi �� The boundary integrals of this equation can be developed as detailed next

1

(4.2.4)

R

d(uv) =

R

udv +

R

vdu,

R

udv = uv −

R

vdu

4.2 Governing Equations

107

��N

��vi nj −p��ij + �� ��vi nj ��vi nj

��N

��N

∂vi 1 ∂vj + �� ��ij d�� = ∂xj 3 ∂xj ∂vi ∂vj 2 ∂vj −p��ij + �� d�� = +�� − �� ∂xj ∂xi 3 ∂xi 2 −p��ij + 2��dij − �� (∇ · v) ��ij d�� 3

(4.2.5)

From Eq. (2.5.21), we can see that the term inside the parenthesis of Eq. (4.2.5) is equal to the stress tensor for Newtonian ﬂuids. This allow us to express the boundary integral as ��vi nj ��ij d�� = ��vi ti d�� (4.2.6)

��N

��N

For simplicity, in this work the homogeneous no-slip Dirichlet boundary condition is taken, therefore the Neumann boundary, Eq. (4.2.6), vanishes. Moreover, if the incompressibility condition given by Eq. (2.4.7) is used, i.e. ∇ · v = 0, which is widely acceptable in the application range of civil engineering and other areas2 , then Eq. (4.2.4) yields ∂vi ∂vi + ��vj ∂t ∂xj ∂��vi d�� + ∂xi ∂vi ∂��vi d�� = ∂xj ∂xj

��vi ��

��

d�� −

��

p

��

��

��vi ��bi d��

��

(4.2.7) These equations are known as the weak form for the incompressible Navier-Stokes equations. To develop the weak form of the continuity equation, their space of test function is deﬁned as ��p(x) �� Q, Q= ��p|��p �� L2 (��),

��

��pd�� = 0

(4.2.8)

where L2 (��) consists of square-integrable functions. The space of the trial function for the pressure is given by p(x, t) �� Q (4.2.9)

The development of a Galerkin-type weak form consists in taking the product of the continuity equation, Eq. (2.4.7), by the test function ��p, integrating over the current domain and using appropriate boundary conditions, yielding ��p

��

∂vj d�� = 0 ∂xj

(4.2.10)

which is known as the weak form of the continuity equation for incompressible materials. These weak forms can be written in a simple form, ﬁrst dividing Eq.

2 For

Mach number < 0.3, air can be considered incompressible

108

4. Fluid Dynamics

(4.2.7) by the density and next using the following mathematical deﬁnitions. The standard L2 inner product for functions belonging to L2 (��) is given by (u, v ) =

��

u · v d��

(4.2.11)

Also the following bilinear forms are deﬁned a(u, v ) = ��

��

∇u : ∇v d��

∀u, v �� H 1 (��)

(4.2.12) (4.2.13)

b(q, v ) =

��

q ∇ · v d��

∀v �� H 1 (��) and ∀q �� L2 (��)

and the trilinear form c(u, v , w ) =

��

u · ∇v · w d��

∀u, v , w �� H 1 (��)

(4.2.14)

With these deﬁnitions, the incompressible ﬂow equations given by Eq. (4.2.7) and Eq. (4.2.10) can be simplify to yield respectively (∂t v, ��v) + c(v, v, ��v) − b(p, ��v) + a(v, ��v) = (b, ��v) b(��p, v) = 0 (4.2.15) (4.2.16)

In this work, these equations are going to be called the compact form for the incompressible Navier-Stokes equations. More details of this mathematical formulation can be found in Gunzburger (1989). In Eq. (4.2.12), the term �� is the kinematic viscosity which is deﬁned by �� = ��/��. In Eq. (4.2.15), the density has been absorbed into the pressure yielding a kinematic pressure.

4.2.2

Finite Element Discretization

Finite elements discretization of the incompressible ﬂow equations is presented in this section for the Galekin-type weak form. The velocities are approximated by

nnode h vi (x, t)

=

I=1

NI (x)viI (t)

∀i = 1, ndime

(4.2.17)

where NI (x) are the shape functions in Eulerian coordinates and viI (t) are the nodal values of the velocity ﬁeld. The space of the test functions for the momentum equations is time independent and its discretization yields

nnode h ��vi (x)

=

I=1

NI (x)��viI

∀i = 1, ndime

(4.2.18)

The material time derivative of the velocity gives the acceleration, which is approximated by

4.2 Governing Equations

109

h ∂vi (x, t) = ∂t

nnode

NI (x)viI (t) ˙

I=1

∀i = 1, ndime

(4.2.19)

Gradients of the velocity and gradients of the test functions of the momentum equations yield the following Cartesian derivatives

h ∂vi (x, t) = ∂xj nnode I=1

∂NI (x) viI (t) ∂xj ∂NI (x) ��viI ∂xj

∀i, j = 1, ndime

(4.2.20)

h ∂��vi (x) = ∂xj

nnode I=1

∀i, j = 1, ndime

(4.2.21)

The divergence of the test and trial functions are given by

h ∂��vi (x) = ∂xi h ∂vi (x, t) = ∂xi nnode ndime I=1 i=1

∂NI (x) ��viI ∂xi

(4.2.22)

nnode ndime I=1 i=1

∂NI (x, t) viI (t) ∂xi

(4.2.23)

The pressure is approximated by

nnode

p(x, t) =

I=1

NI (x)pI (t)

(4.2.24)

and the test function for the continuity condition gives

nnode

��p(x) =

I=1

NI (x)��pI

(4.2.25)

These equations allow to express the incompressible Navier-Stokes equations, Eq. (4.2.7), in the form given ahead. The kinetic term can be discretized as

h ∂vi d�� = ��viI ∂t

��

h ��vi ��

NI ��NJ viJ d�� ˙

��

= ��viI

��

NI ��NJ ��ij d�� vjJ ˙

(4.2.26)

= ��viI MijIJ vjJ ˙ where the mass matrix for the eulerian description is given by MijIJ = ��ij

��

��NI NJ d��

(4.2.27)

Discretization of the convective term yields

110

4. Fluid Dynamics

��

h h ��vi ��vj

h ∂vi d�� = ��viI ∂xj

= ��viI

∂NJ viJ d�� ∂xj �� ∂NJ NI �� vh ��ij d�� vjJ ∂xj �� NI �� vh

(4.2.28)

c = ��viI KijIJ vjJ

where vh is the discretized velocity vector given by Eq. (4.2.17) and the convective stiﬀness matrix is deﬁned by

c KijIJ = ��ij

��

��NI vh

∂NJ d�� ∂xj

(4.2.29)

The pressure term can be discretized in the form of

h ∂��vi ∂NI d�� = ��viI NJ pJ d�� ��xi �� ∂xi = ��viI GiIJ pJ

p

��

(4.2.30)

where the pressure matrix is expressed as GiIJ =

��

∂NI NJ d�� ∂xi

(4.2.31)

The viscous term can be discretized following the same procedure to obtain

h h ∂vi ∂��vi d�� = ��viI ∂xj ∂xj

��

��

��

= ��viI =

�� v ��viI KijIJ vjJ

∂NI ∂NJ �� viJ d�� ∂xj ∂xj ∂NI ∂NJ �� ��ij d�� vjJ ∂xj ∂xj

(4.2.32)

where the viscous stiﬀness matrix is given by

v KijIJ = ��ij ��

��

∂NI ∂NJ d�� ∂xj ∂xj

(4.2.33)

Since the variation ��viI is arbitrary, then the forces yielding from the Navier-Stokes equations can be written in the following discretized matrix form ˙ Mv + K(v)v − Gp = f ext (4.2.34)

where the stiﬀness matrix is given by K(v) = Kc (v) + Kv . The compact form for these equations can be written as (∂t vh , wh ) + c(vh , vh , wh ) − b(ph , wh ) + a(vh , wh ) = (bh , wh ) (4.2.35)

4.2 Governing Equations

111

where the discretized test functions for the velocity are given by

nnode

wh (x) =

I=1

NI (x)

(4.2.36)

The incompressibility condition given by Eq. (4.2.10) can be discretized and the divergence term leads to

h ∂vj ��p d�� = ��pI ∂xj ��

NI

=

�� ��pI GT vjJ jIJ

∂NJ vjJ d�� ∂xj

(4.2.37)

where divergence matrix is expressed by GT = jIJ NI

��

∂NJ d�� ∂xj

(4.2.38)

Since the variation ��pI is arbitrary, then the continuity equation can be written in the following discretized matrix form GT v = 0 which can be written in compact form as b(qh , vh ) = 0 where the discretized test function for the pressure is given by

nnode

(4.2.39)

(4.2.40)

qh (x) =

I=1

NI (x)

(4.2.41)

Note that the discretized incompressible ﬂow equations are coupled, and the problem is expressed by ˙ Mv + K(v)v − Gp = f ext GT v = 0

(4.2.42)

which are used to ﬁnd the velocity ﬁeld and pressure of the problem monolithically. These equations can also be expressed by ˙ (vh , wh ) + c(vh , vh , wh ) − b(ph , wh ) + a(vh , wh ) = (bh , wh ) b(qh , vh ) = 0

(4.2.43)

which is the compact form to express the coupled discretized incompressible ﬂow equations. Eq. (4.2.43) has the advantage that no matrices need to be built to later be multiplied by a vector, which results in a more optimal code.

112

4. Fluid Dynamics

4.3

Time Integration Schemes

In this part, the same ideas of section 3.5 for time integration schemes of solids are followed. Therefore only implicit schemes are solved for incompressible ﬂow problems. The semi-discrete equations to be solved are given by

ext ˙ Mvn+1 + K(vn+1 )vn+1 − Gpn+1 = fn+1

GT vn+1 = 0

(4.3.1)

˙ where the acceleration vector vn+1 has to be integrated in time to solve the algebraic equations for vn+1 , pn+1 from the ﬁrst-order diﬀerential equations. The compact form of the above equations yields

n+1 n+1 n+1 n+1 n+1 ˙ n+1 (vh , wh ) + c(vh , vh , wh ) − b(ph , wh ) + a(vh , wh ) = (bh , wh ) n+1 b(qh , vh ) = 0

(4.3.2)

Among the diﬀerent numerical integration methods available to integrate ﬁrst-order equations, the simplest choice in ﬂuid dynamics problems is the �� f -family of approximation. Multistep schemes of the type of backward diﬀerentiation are also used in ﬂuid dynamics problems. Both of these methods are better described in Gunzburger (1989). The tendency of these methods for high frequency noise to persist in the solution for long time periods can lead to non-physical instabilities. To avoid this problem, Jansen et al. (2000) have developed the generalized-�� method for ﬂuid dynamics, with the same dissipation properties that the generalized-�� method for structural dynamics problems. An analysis of time integration algorithms for incompressible ﬂuids ﬂow is detailed in Dettmer and Peric (2003).

4.3.1

��-Family Method

In this section, the �� f -family method for implicit schemes is presented. The solution at time step tn is assumed to be known for vn , pn . In this method, the time derivative of the velocity is approximated for two consecutive time steps by linear interpolation, yielding vn+1 − vn ˙ ˙ ˙ vn+��f = (1 − �� f )vn + �� f vn+1 ∼ = ��t (4.3.3)

with 1 �� �� f �� 1 for unconditionally stable implicit schemes and ��t = tn+1 − tn . 2 The velocity ﬁeld is approximated according to vn+1 = which is usually written as vn+��f = (1 − �� f )vn + �� f vn+1 (4.3.5) 1 f v f − (1 − �� )vn �� f n+�� (4.3.4)

4.3 Time Integration Schemes

113

For �� f = 1, the method is ﬁrst-order accurate and is known as the backward Euler method. For �� f = 1 , the method is known as the Crank-Nicolson method. Only 2 when �� f = 1 the method is second-order accurate. The semi-discrete equations to 2 be solved for the �� f -family method are then given by 1 ext (vn+1 − vn ) + K(vn+��f )vn+��f − Gpn+1 = fn+��f ��t GT vn+��f = 0

M

(4.3.6)

and the unknowns for the algebraic equations to be solved are vn+1 , pn+1 since all values at time tn are known. Also we can write

n+�� n+�� n+�� n+1 n+�� ˙ n+�� , wh ) + c(vh (vh , vh , wh ) − b(ph , wh ) + a(vh , wh ) = (bh , wh ) n+�� b(qh , vh )=0

f f f f f f

(4.3.7) which is the compact form for incompressible ﬂuids.

4.3.2

Backward Diﬀerentiation Method

Backward diﬀerentiation methods are also known as Gear schemes. The backward diﬀerentiation method used in this work are of ﬁrst and second-order of accuracy. The simplest prototype of backward diﬀerentiation is the backward Euler, which coincides with the �� f -family method when �� f = 1. Then the time derivative for the velocity can be approximated by ˙ vn+1 = vn+1 − vn ��t (4.3.8)

and the semi-discrete equations to be solved for the BDF1 (Backward DiFferentiation method order 1) method are then given by 1 ext (vn+1 − vn ) + K(vn+1 )vn+1 − Gpn+1 = fn+1 ��t GT vn+1 = 0

M

(4.3.9)

and the unknowns for the algebraic equations to be solved are vn+1 , pn+1 since all values at time tn are known. For second-order accuracy, the method is known as BDF2 and the velocity is expressed as ˙ vn+1 = 3vn+1 − 4vn + vn−1 2��t (4.3.10)

and the semi-discrete equations to be solved are then given by

114

4. Fluid Dynamics

M

1 ext (3vn+1 − 4vn + vn−1 ) + K(vn+1 )vn+1 − Gpn+1 = fn+1 2��t GT vn+1 = 0

(4.3.11)

and the unknowns for the algebraic equations to be solved are vn+1 , pn+1 . One advantage of these methods is that they do not require an initial guess for the pressure, as reported in Gunzburger (1989). One disadvantage for the BDF2 is that it requires more starting values that just v0 . But this problem can be avoided if always the ﬁrst iteration is perform with the BDF1 method and then for the second iteration change to the BDF2 method.

4.3.3

Generalized-�� Method

The generalized-�� method for ﬁrst-order systems was introduced by Jansen et al. (2000). The idea is to achieve high-frequency dissipation while minimizing unwanted low-frequency dissipation for analysis with long time periods. This method was motivated from the excellent results of the same method for second-order systems given by Chung and Hulbert (1993). In the generalized-�� method, the semidiscrete equations for incompressible ﬂuids ﬂow are given by

ext ˙ Mvn+��f + K(vn+��f )vn+��f − Gpn+1 = fn+1 f f m

GT vn+1 = 0 where the changes are given by ˙ ˙ vn+1 = vn + ��t 1 − �� f vn + ��t �� f vn+1

f f vn+��f = 1 − ��f vn + ��f vn+1 f f f ˙ ˙ ˙ vn+��m = 1 − ��m vn + ��m vn+1 f

(4.3.12)

(4.3.13) (4.3.14) (4.3.15)

f f where ��f , ��m and �� f are integration parameters deﬁned ahead. From Eq. (4.3.13) we can obtain

˙ vn+1 =

1 (vn+1 − vn ) − ��t�� f

1 ˙ − 1 vn ��f

(4.3.16)

which will allow us to update the acceleration at time tn+1 when the velocities vn+1 are found. Substituting Eq. (4.3.16) into Eq. (4.3.15) yields ˙ vn+��f = m

f ��m ��f (vn+1 − vn ) + 1 − m ��t �� f ��f

˙ vn

(4.3.17)

This last equation together with Eq. (4.3.14) are the ones that are substituted into Eq. (4.3.12) and allow us to ﬁnd the unknowns vn+1 , pn+1 for the algebraic equations since all values at time tn are known.

4.4 Pressure Segregation Methods

115

The method is second-order accurate and posses high frequency dissipation when the integration parameter �� f is selected such that 1 f f + ��m − ��f 2 Low frequency dissipation is optimal when ��f = ��f �� [0, 1], ��

f ��f =

(4.3.18)

1 , 1 + ��f ��

f ��m =

1 3 − ��f �� 2 1 + ��f ��

(4.3.19)

For ��f = 1 the method is identical to the Crank-Nicolson method given by in the �� �� f -family. Numerical damping of the method increases with smaller values of ��f . ��

4.4

Pressure Segregation Methods

The monolithic coupled equations for incompressible ﬂuid problems have the disadvantage that take great computer eﬀort to solve the algebraic system for each time step in a transient analysis. Since the original works of Chorin (1967) and Temam (1969), fractional step methods for the incompressible Navier-Stokes equations have earned widespread popularity because of the computational eﬃciency given by the uncoupling of the pressure from the velocity ﬁeld. A detailed stability analysis of fractional step methods for incompressible ﬂows is given in Codina (2001). Another proposal in the context of fractional step methods is the predictormulticorrector algorithm born from the ideas of Brooks and Hughes (1982). Development of this method can be found in Blasco et al. (1998), and Codina and Soto (2004). Stability analysis for these methods are given by Codina and Badia (2006).

4.4.1

Fractional Step Method

The easiest form to understand the development of the fractional step method is to part from the incompressible Navier-Stokes equations given by 1 ext (vn+1 − vn ) + K(vn+1 )vn+1 − Gpn+1 = fn+1 ��t GT vn+1 = 0

M

(4.4.1) (4.4.2)

where Eq. (4.4.1) can be split and yield the equivalent incompressible ﬂow equations 1 ext v v (˜ n+1 − vn ) + K(˜ n+1 )˜ n+1 − Gpn = fn+1 v ��t 1 ˜ M (vn+1 − vn+1 ) − G(pn+1 − pn ) = 0 ��t GT vn+1 = 0

M

(4.4.3) (4.4.4) (4.4.5)

116

4. Fluid Dynamics

˜ where vn+1 is an auxiliary velocity variable and the essential approximation v v K(vn+1 )vn+1 �� K(˜ n+1 )˜ n+1 (4.4.6)

˜ has been taken. From Eq. (4.4.4), vn+1 can be expressed in terms of vn+1 yielding ˜ vn+1 = vn+1 + ��t M−1 G(pn+1 − pn ) Substituting this last equation into Eq. (4.4.5) yields ˜ −��t GT M−1 G(pn+1 − pn ) = GT vn+1 (4.4.8) (4.4.7)

Now observe that GT M−1 G represent an approximation to the Laplacian operator, as mentioned in Codina (2001), given by LIJ = �� (∇NI , ∇NJ ). Finally the incompressible ﬂuid ﬂow equations Eq. (4.4.3)-(4.4.5) to be solved using the �� f -family integration scheme are expressed as 1 ext v v (˜ n+1 − vn ) + K(˜ n+��f )˜ n+��f − Gpn = fn+1 v ��t ˜ −��t L(pn+1 − pn ) = GT vn+1 1 ˜ M (vn+1 − vn+1 ) − G(pn+1 − pn ) = 0 ��t

M

(4.4.9) (4.4.10) (4.4.11)

with 1 �� �� f �� 1 for unconditionally stable implicit schemes. Remember that for 2 �� f = 1, the method coincides with the backward diﬀerentiation BDF1 method. These equation are expressed in a diﬀerent form when the integration scheme used is the BDF2, yielding 1 ext v v (3˜ n+1 − 4vn + vn−1 ) + K(˜ n+1 )˜ n+1 − Gpn = fn+1 v 2��t 2 ˜ − ��t L(pn+1 − pn ) = GT vn+1 3 1 v (3vn+1 − 3˜ n+1 ) − G(pn+1 − pn ) = 0 M 2��t

M

(4.4.12) (4.4.13) (4.4.14)

In this work we use a diﬀerent time integration scheme that those ﬁnd in the literature for the fractional step method. This scheme is the generalized-�� method, used in the monolithic problem, that for fractional steps take the form of

f ��f ��m ext (˜ n+1 − vn ) + K(˜ n+��f )˜ n+��f − Gpn = fn+1 − M 1 − m v v f v f ��t �� f ��f

M

˙ vn (4.4.15)

− M

��t �� f ˜ L(pn+1 − pn ) = GT vn+1 f ��m

(4.4.16) (4.4.17)

f ��m ˜ (vn+1 − vn+1 ) − G(pn+1 − pn ) = 0 ��t �� f

4.4 Pressure Segregation Methods

117

The compact form for these equations is given by

f n+��f n+��f ��m n n ˜ (˜h vn+1 − vh , wh ) + c(˜h f , vh f , wh ) − b(ph , wh ) + v ��t �� f

a(˜h v

n+��f

f

n+1 , wh ) = (bh , wh ) − 1 −

f ��m ��f

˙n (vh , wh ) (4.4.18)

−

��t �� f n+1 n ˜n+1 (∇(ph − ph ), ∇qh ) = b(qh , vh ) f ��m

(4.4.19) (4.4.20)

f ��m n+1 n ˜n+1 (vn+1 − vh , wh ) − b(ph − ph , wh ) = 0 ��t �� f h

As mention before, these equations have the advantage that no matrices need to be built to later be multiplied by a vector, which results in a more optimal code.

4.4.2

Predictor-Corrector Method

Here the predictor multicorrector scheme emerging from Eqs. (4.4.9)-(4.4.11) proposed by Codina and Soto (2004) is used. The goal of this method is to converge to the monolithic problem. The linearized algebraic system for this method using the �� f -family for integration in time is taken directly from Codina and Badia (2006), yielding 1 ext (vn+1,i − vn ) + K(vn+��f ,i−1 )vn+��f ,i − Gpn+��f ,i−1 = fn+��f ��t −��t L(pn+��f ,i − pn+��f ,i−1 ) = GT vn+��f ,i

M

(4.4.21) (4.4.22)

which is an iterative algebraic system where the pressure is uncoupled from the velocity ﬁeld. If the BDF2 time integration method is used, then Eqs. (4.4.21)(4.4.22) are written as 1 ext (3vn+1,i − 4vn + vn−1 ) + K(vn+1,i−1 )vn+1,i − Gpn+1,i−1 = fn+1 2��t (4.4.23) 2 − ��t L(pn+1,i − pn+1,i−1 ) = GT vn+1,i 3 (4.4.24)

M

Note that for the ﬁrst iteration, i = 1, a predictor for the pressure and the velocity is required. The ﬁrst order predictor for the pressure and velocity are taken as pn+1,0 = pn , while the second order predictor yields vn+1,0 = vn (4.4.25)

118

4. Fluid Dynamics

pn+1,0 = 2pn − pn−1 ,

vn+1,0 = 2vn − vn−1

(4.4.26)

Again in this work we use a diﬀerent time integration scheme that those ﬁnd in the literature for the predictor-corrector method. This scheme is the generalized-�� method, that for the predictor-corrector algorithm take the form of

f ��m (vn+1,i − vn ) + K(vn+��f ,i−1 )vn+��f ,i − Gpn+��f ,i−1 = f f f ��t �� f ��f ext ˙ fn+��f − M 1 − m vn f ��f

M

(4.4.27) (4.4.28)

−

��t �� f L(pn+��f ,i − pn+��f ,i−1 ) = GT vn+��f ,i f f f f ��m

The compact form for these equations is given by

f n+��f ,i−1 n+��f ,i n+��f ,i−1 ��m n+1,i n (vh − vh , wh ) + c(vh f , vh f , wh ) − b(ph f , wh ) + ��t �� f n+��f ,i n+��f ��f ˙n a(vh f , wh ) = (bh f , wh ) − 1 − m (vh , wh ) (4.4.29) ��f

−

n+��f ,i n+��f ,i−1 n+��f ,i ��t �� f (∇(ph f − ph f ), ∇qh ) = b(qh , vh f ) f ��m

(4.4.30)

This mathematical form is the one that is used in the code implemented for solving the computational ﬂuid dynamics equations. We note that for ��f = 1 the method �� is identical to the Crank-Nicolson method given by Eqs. (4.4.21)-(4.4.22) when �� f = 1. 2

4.5

Stabilization Methods

It is well known that two sources of potential numerical instabilities emerge for incompressible ﬂow problems. The ﬁrst is presented when the Reynolds number for the ﬂow is high, and the convective term tend to dominate the ﬂow equations which yield oscillations in the velocity ﬁeld. These oscillations can be avoided with a reﬁnement of the ﬁnite element mesh for the problem. Unfortunately this is not a solution to the problem due to the high computational cost for very ﬁne meshes. The other source of instabilities is due to the incompressibility constraint which has several inconveniences due to the zero divergence condition for the velocity ﬁeld. When the standard Galerkin formulation is used, compatible spaces for the pressure and the velocity ﬁeld have to be used and must satisfy the Babu��ka-Brezzi s (BB) stability condition, see i.e. Babuska (1973), Brezzi (1974). To circumvent these problems, several stabilization techniques have been proposed. Essentially, the stabilizing terms added to the original Galerkin formulation

4.5 Stabilization Methods

119

involve the residual of the momentum equation as a factor. The stabilization techniques are consistent in the sense that when an exact solution is substituted into the stabilized formulation, these added terms vanish and the exact solution is satisﬁed. One of the stabilization techniques that has been extensively used in convention dominated problems is the streamline-upwind/Petrov-Galerkin method, which acronym is SUPG. The original SUPG method was designed for the steady state version of the advection-diﬀusion equation to avoid the numerical oscillations found using the standard Galerkin formulation. This misbehavior could be avoided by introducing numerical diﬀusion only along the streamlines as explained in the work of Hughes and Brooks (1979). The ﬁnal step was the use of the streamline diﬀusion in the context of weighted residual methods, as given in Hughes and Brooks (1982), that satisfy the BB stability condition. As demonstrated in Hughes et al. (1986), elements that do not satisfy the BB condition still can achieve a proper stabilization for the Petrov-Galerkin method with equal-order interpolations. Another kind of stabilization is the pressurestabilizing/Petrov-Galerkin method, which acronym is PSPG. The PSPG proposed by Tezduyar et al. (1990) and Tezduyar et al. (1992) consists of a generalization of the Petrov-Galerkin stabilization term proposed in Hughes et al. (1986) for Stokes ﬂows. Coeﬃcients of the PSPG stabilization terms vary with the Reynolds number just as the SUPG terms do. In the zero Reynolds number limit, the PSPG stabilization term reduces to the one proposed in Hughes et al. (1986). In Tezduyar et al. (1990), the SUPG and PSPG stabilization methods are used together with equal-order interpolations. Because the PSPG stabilizing terms involve the residual of the momentum equation as a factor, the stabilized formulation is consistent. A more general stabilization approach that includes in essence the SUPG and PSPG methods for stabilization was found by Hughes et al. (1989). This idea leads to the well known formulation of the Galerkin least-square method, which acronym is GLS. For time-dependent problems, the GLS stabilization needs ﬁnite element discretization in both space and time, and therefore leads to a space-time ﬁnite element formulation of the problem. Since the stabilizing terms added to the GLS method are obtained by minimizing the sum of the square residual of the momentum equation as a factor, the stabilized formulation is consistent. With the idea to better understand the origins of stabilized methods, which can be derived from a ﬁrm theoretical foundation and a precise deﬁnition of the intrinsic time scale parameter, Hughes (1995) developed the subgrid scale method. This technique is known as the SGS method and consists in splitting the unknown into a part which can be represented by the ﬁnite element mesh and another part that accounts for the unresolvable scales that cannot be reproduced because of the mesh size. In fact the SGS is a family of stabilization techniques where various methods can be recovered as a particular case. In the context of these methods, the orthogonal sub-scales method was introduced by Codina (2000). The acronym used for this method is OSS. The main idea here is to assume that the subgrid components are orthogonal to the ﬁnite element space. The stabilization of both, convection dominated ﬂows and pressure with transient problems can be found in Codina (2002). So far all the stabilization methods described require the addition of some ar-

120

4. Fluid Dynamics

tiﬁcial diﬀusion terms. However another technique where the stabilization terms emerge from the governing equations of the problem is the ﬁnite calculus method given by O˜ate (1998), which acronym is FIC. In this method the concept of ﬂow n equilibrium is taken over a ﬁnite domain that allows to reinterpret the stabilization terms as an intrinsic and natural contribution to the original diﬀerential equations, instead of a correction term introduced at the discrete level. With this natural stabilization concept, the standard forms of the balancing terms appearing in many well known stabilized schemes are easily recognized and can be reinterpreted in a more physical manner. An application for incompressible viscous ﬂows can be found in O˜ate (2000). n Other well known stabilization methods are the characteristic Galerkin method (CG) developed by Douglas and Russell (1982), and the Taylor-Galerkin method (TG) introduced by Donea (1984). A detailed comparison of the most popular stabilization techniques is given by Codina (1998). An interesting comparison between the GLS and the SUPG/PSPG methods can be found in Tezduyar (1992). These stabilization techniques are also explained in Donea and Huerta (2003). In this work the stabilizations used are the SUPG/PSPG, OSS and FIC techniques. They are only implemented in our code to be used for ﬂuid-structure interaction problems.

4.5.1

SUPG/PSPG Stabilization

The stabilization procedure introduced in this section can be found in Tezduyar (1992). In order to account for the linear ﬁnite element interpolation chosen for this kind of stabilization, we refer to the work of Dettmer and Peric (2003), where the ﬁnal outcome consists of adding the discretized term ˙ ˜ (vh

n+��f m n+��f n+��f n+��f

˜ + vh

f

· ∇˜h v

f

˜ + ∇pn+1 , ��v vh h

f

· ∇wh + ��p ∇qh )

(4.5.1)

to the compact form of the Galerkin method for the fractional step with the generalized-�� time discretization. The contribution to the momentum equation is obtained when the pressure test function qh = 0, whereas the contribution to the continuity equation is found when wh = 0. The ﬁnal result for the fractional step method are the stabilized equations ˙ ˜ (vh

n+��f m n+��f n+��f n+��f

, wh ) + c(˜h v

f

f

˜ , vh

f

n , wh ) − b(ph , wh ) + a(˜h v n+��f f h

f

, wh ) +

˙ n+�� ˜ ˜ ��v (vh m + v −

n+��f f h

· ∇˜ v

˜ + ∇pn , v h

n+��f f h

n+1 · ∇wh ) = (bh , wh ) (4.5.2)

n+��f n+��f f ��t �� f n+1 n ˙ n+�� ˜ ˜ (∇(ph − ph ), ∇qh ) − ��p (vh m + vh f · ∇˜h f + ∇pn+1 , ∇qh ) v h f ��m

˜n+1 = b(qh , vh ) (4.5.3)

f ��m n+1 n ˜n+1 (vn+1 − vh , wh ) − b(ph − ph , wh ) = 0 ��t �� f h

(4.5.4)

4.5 Stabilization Methods

121

where ��v and ��p are the intrinsic time parameters which multiply the advection term and the pressure term respectively. In this work, the stabilization parameters are the ones chosen by Dettmer and Peric (2003), which are he 2 vh �� 1 1+

3 Reh 2

��v =

(4.5.5)

��p =

he 2 vh ��

��1 1+

3��2 Reh 2

(4.5.6)

where Reh is the element Reynold number given by vh he �� 2��

Reh =

(4.5.7)

Here the characteristic element size he is deﬁned as the diameter of the circle, the area of which correspond to the ﬁnite element e in two dimensions. For 3D problems, he is taken as given by Codina (2000). All numerical examples with this stabilization take ��1 = 30 and ��2 = 100 in this work. Observe that the pressure in the ﬁrst equation is treated explicitly, in order to keep the uncoupling of the velocity and pressure calculations. Only the ﬁrst equation is a nonlinear system, while the second is a linear system. The third equation is simply an update of the velocity.

4.5.2

OSS Stabilization

The orthogonal sub-scales stabilization procedure described in this section is given by Codina (2000). In order to account for the stabilized fractional step method, we refer to the work of Codina (2001), where the ﬁnal outcome consists in adding

n+��f n+��f n+��f

��(˜h v

f

· ∇˜h v

f

n ˜ + ∇pn+1 − ��h , vh h

f

· ∇wh + ∇qh )

(4.5.8)

to the compact form of the Galerkin method for the fractional step with the generalized-�� time discretization, where ��h is a projection explained ahead. The contribution to the momentum equation is obtained when the pressure test function qh = 0, whereas the contribution to the continuity equation is found when wh = 0. The ﬁnal result for the fractional step method are the stabilized equations

122

4. Fluid Dynamics

˙ ˜ (vh

n+��f m

, wh ) + c(˜h v ��(˜h v

n+��f

f

f

˜ , vh

n+��f

f

n , wh ) − b(ph , wh ) + a(˜h v n ˜ + ∇pn − ��h , vh h n+��f

f

n+��f

f

, wh ) + (4.5.9)

n+��f

· ∇˜h v

n+��f

f

n+1 · ∇wh ) = (bh , wh )

−

n+��f n+��f ��t �� f n+1 n n (∇(ph − ph ), ∇qh ) − ��(˜h f · ∇˜h f + ∇pn+1 − ��h , ∇qh ) v v h f ��m

˜n+1 = b(qh , vh ) (4.5.10)

f ��m n+1 n ˜n+1 (vn+1 − vh , wh ) − b(ph − ph , wh ) = 0 ��t �� f h n+1 v (��h , ��h ) − (˜h n+��f

f

(4.5.11) (4.5.12)

· ∇˜h v

n+��f

f

+ ∇pn+1 , ��h ) = 0 h

Observe that in the last equation, ��h has the meaning of being the projection of ˜ v vh · ∇˜h + ∇ph onto the ﬁnite element space. This last equation is just an update of the orthogonal projection, just as the third equation is simply an update of the velocity. The stabilization parameter �� is given in Codina (2001), which is ��= 2 vh 4�� + 2 he he

−1

(4.5.13)

This equation is also used to estimate the critical time step size for explicit schemes, where ��tcrit is substituted instead of �� in Eq. (4.5.13).

4.5.3

FIC Stabilization

The FIC stabilization introduced by O˜ate (2000) for incompressible ﬂow equan tions is described in this section. The idea behind the method consists in applying the standard conservation laws expressing balance of momentum and mass over a control domain. Assuming that control domain has ﬁnite dimensions and representing the variation of mass and momentum over the domain using Taylor series expansions of one order higher than those used in standard inﬁnitesimal theory, the following expressions for the momentum and mass balance respectively are found 1 ∂rmi =0 rmi − hj 2 ∂xj 1 ∂rd rd − hj =0 2 ∂xj (4.5.14) (4.5.15)

where rmi is the residual of the momentum equation obtained from Eq. (4.2.35) and rd is the residual of the divergence of the velocity given by Eq. (4.2.40). In O˜ate et al. (2006), the following assumption is obtained n ∂rd hj ∂rmi �� ∂xj 2ai ∂xi where (4.5.16)

4.5 Stabilization Methods

123

ai =

vi hi 2�� + 3 2

(4.5.17)

and the following alternative expression for the stabilized mass balance equation is found rd − ��i with ��i = 8�� 2vi 2 + h 3hi i

−1

∂rmi =0 ∂xi

(4.5.18)

(4.5.19)

��i are the intrinsic time parameters. After integrating by parts the resulting momentum and mass balance equations is weak form are ∂vi ∂vi + vj ∂t ∂xj ∂vi ∂��vi d�� �� �� ∂xj ∂xj hj ∂��ui − ��vi bi d�� = rmi d�� �� �� 2 ∂xj p �� ��i

��

��vi

��

d�� −

∂��vi d�� + ∂xi

(4.5.20)

��p

��

∂vi d�� = ∂xi

∂��p rm d�� ∂xi i

(4.5.21)

where the only new terms involving the residual rmi appear on the right hand side of both equations. The computation of the residual terms can be simpliﬁed if we introduce the convective projection ci and the pressure gradient projection ��i ˆ deﬁned respectively as ˆ rmi = ci − vj rmi = ��i − ∂vi =0 ∂xj (4.5.22)

∂p =0 ∂xi

(4.5.23)

Now rmi can be expressed in Eqs. (4.5.20) and (4.5.21) in terms of ci and ��i respecˆ tively which then become additional variables. The system of integral equations is now augmented in the necessary number of additional equations by imposing that the residual vanished for both forms given by Eqs. (4.5.22) and (4.5.23). This gives the ﬁnal system of governing equations as ∂vi ∂vi + vj ∂t ∂xj ∂��vi ∂vi ∂��vi d�� + �� d�� ∂xi �� ∂xj ∂xj hj ∂��ui ∂vi vj − ci d�� = ˆ 2 ∂xj ∂xj

��vi

��

d�� −

��

p

+

��

��vi bi d�� (4.5.24)

��

124

4. Fluid Dynamics

��p

��

∂vi d�� + ∂xi

��i

��

∂��p ∂xi ∂vi ∂xj

∂p − ��i d�� = 0 ∂xi d�� = 0 d�� = 0

(4.5.25) (4.5.26) (4.5.27)

��ˆi ci − vj c ˆ

��

�Ħ�i ��i −

��

∂p ∂xi

If the fractional step method is applied to Eqs. (4.5.24)-(4.5.27) together with the generalized-�� method, the ﬁnal result for the stabilized equations in compact form yields ˙ ˜ (vh

n+��f m

, wh ) + c(˜h v

n+��f

f

n ˜ , vh f , wh ) − b(ph , wh ) + a(˜h f , wh ) + v f n+��f 1 n+��f n+1 ˆh · ∇˜h f − cn , h · ∇wh ) = (bh , wh ) (4.5.28) v (˜h v 2

n+��f

n+��f

−

��t �� f n+1 n n ˜n+1 (∇(ph − ph ), ∇qh ) − ��(∇pn+1 − ��h , ∇qh ) = b(qh , vh ) (4.5.29) h f ��m f ��m n+1 n ˜n+1 (vn+1 − vh , wh ) − b(ph − ph , wh ) = 0 (4.5.30) ��t �� f h (ˆn+1 , ��h ) − (˜h ch v (��h

n+1 n+��f

f

· ∇˜h v

n+��f

f

, ��h ) = 0 , ��h ) = 0

(4.5.31) (4.5.32)

, ��h ) − (∇ph

n+1

Observe that the third equation is simply an update of the velocity, just as the fourth and ﬁfth equations are the updates for the convective projection and pressure projections terms respectively. Advances of the FIC formulation with applications to ﬂuid-structure interaction is been developed by Lynga (2005).

4.6

Solution Strategies

The numerical procedures used to solve nonlinear algebraic equations are iterative, given i.e. by Eqs. (4.5.2), (4.5.9) or (4.5.28), for the only nonlinear terms in fractional steps or predictor corrector schemes. Among the many diﬀerent procedures available to solve nonlinear problems, in this work only the Picard iteration method, see i.e. Reddy (2004), is outlined for computational ﬂuid dynamic problems.

4.6.1

Picard Iteration Method

The Picard iteration method of successive substitution, also known as the direct iteration technique, is the simplest method used in nonlinear problems. In the Picard method, the solution at the ith iteration is obtained from K(vi−1 ) vi = f (4.6.1)

4.7 Code Development

125

where the nonlinear matrix K(v) is evaluated using the known solution from the i−1 iteration. It is assumed that the initial or guess vector v0 satisﬁes the speciﬁed boundary conditions of the problem and the nonlinear matrix is invertible. The Picard iteration method converges when the nonlinearity is mild, and it diverges if the nonlinearity is severe. In this last case, other initial conditions should be taken or a diﬀerent iterative procedure can be used. The linearized algebraic system for the fractional step method using the generalized-�� method for integration in time, applied to Eqs. (4.5.2), (4.5.9) and (4.5.28) yields ˙ ˜ (vh

n+��f ,i m

, wh ) + c(˜h v

f

n+��f ,i−1

f f

˜ , vh

n+��f ,i

f f

n , wh ) − b(ph , wh ) + a(˜h v

n+��f ,i

f

, wh ) +

˙ n+�� ,i ˜ ˜ ��v (vh m + vh

n+��f ,i−1

· ∇˜h v

n+��f ,i

˜ + ∇pn , vh h

n+��f ,i−1

f

n+1 · ∇wh ) = (bh , wh ) (4.6.2)

˙ ˜ (vh

n+��f ,i m

, wh ) + c(˜h v

n+��f f h ,i−1

n+��f ,i−1

f

˜ , vh

,i

n+��f ,i

f

n , wh ) − b(ph , wh ) + a(˜h v n+��f f h ,i−1

n+��f ,i

f

, wh ) +

��(˜ v

· ∇˜ v

n+��f f h

n ˜ + ∇pn − ��h , v h

n+1 · ∇wh ) = (bh , wh ) (4.6.3)

˙ ˜ (vh

n+��f ,i m

, wh ) + c(˜h v

n+��f ,i−1

f

˜ , vh

n+��f ,i

f

n , wh ) − b(ph , wh ) + a(˜h v n+��f f h ,i

n+��f ,i

f

, wh ) +

1 (˜ v 2

n+��f f h

,i−1

· ∇˜ v

n+1 ˆh − cn , h · ∇wh ) = (bh , wh )

(4.6.4) which are the nonlinear terms of the SUPG/PSPG, OSS and FIC stabilization techniques respectively.

4.7

Code Development

In this work the ﬂuid problem with the presented stabilizations have been implemented in the ﬁnite element program COMET (2007). The dynamic analysis with the ��-family method, backward diﬀerencing method and Generalized-�� method for ﬂuids have also been added to the code. This software is developed at the International Center for Numerical Methods in Engineering (CIMNE) and is used for coupled contact, mechanical and thermal analysis using the ﬁnite element method.

4.8

4.8.1

Example Problems

Wall-Driven Cavity Flow

The wall-driven cavity ﬂow problem is often used as a benchmark problem since it has been extensively studied by analytical, numerical and experimental methods.

126

4. Fluid Dynamics

vx=1, vy=0

vx=0, vy=0

vx=0, vy=0

p=0

vx=0, vy=0

Figure 4.1

Cavity four-node ﬁnite element mesh

Assuming a unit square and that the velocity of the top wall is unity, the problem is solved with a 40��40 quadrilateral mesh which has been reﬁned near the boundaries, as shown in Fig. 4.1. In the literature this problem is solved for various Reynolds numbers, which are given by Re = ��vc lc /��. In this problem the characteristic velocity vc = 1 and characteristic length lc = 1. If the viscosity is kept constant as �� = 1, then the Reynolds number is given by Re = ��. In this example the wall-driven cavity ﬂow for a Reynold number of 400 is studied. Consequently, the density for the problem is taken as �� = 400. The generalized-�� time integration method has been used with ��f = 0.9. �� Singularities for this problem exists at each corner where the moving lid of the top meets the side ﬁxed walls. Here it is assumed that vx = 1, vy = 0 at singularity points. In this example, the FIC stabilization technique has been used with a mesh of 1600 elements, 1681 nodes and a time step size ��t = 0.05. The pressure contours reached at the steady state for the problem are shown in Fig. 4.2. The accuracy of the numerical results have been compared to those presented

Figure 4.2

Pressure contours for Re=400

4.8 Example Problems

127

by Ghia et al. (1982), which were obtained with ﬁnite diﬀerence solutions and using a very ﬁne grid that made them become a standard reference. The comparison of the x-velocity proﬁle along the cavity mid-section for x = 0.5 versus the vertical distance is given in Fig. 4.3.

1.0 0.8 0.6

y

0.4 0.2 0.0 -0.50

Ghia et al. Present work

-0.25

0.0

0.25 x-velocity

0.50

0.75

1.0

Figure 4.3

Velocity vx (0.5, y) vs. y

Again the solution of this work is compared to the numerical results presented by Ghia et al. (1982) for the y-velocity proﬁle along the cavity mid-section for y = 0.5 versus the horizontal distance, as shown in Fig. 4.4.

0.5 0.3

Ghia et al. Present work

y-velocity

0.1 -0.1 -0.3 -0.5 0.0 0.25 0.50 x 0.75 1.0

Figure 4.4

Velocity vy (x, 0.5) vs. x

Figs. 4.3-4.4 show that the implementation of the computational ﬂuid dynamics methods used in this work together with the FIC stabilization described in section 4.5 lead to the expected solution for this problem.

128

4. Fluid Dynamics

4.8.2

Flow Past a Circular Cylinder in 2D

The transient ﬂow past a circular cylinder is another widely solved benchmark problem. The interest of this problem is the periodic ﬂow patterns that develops for values of the Reynolds number Re > 40. In this case vortex shedding are generated periodically and alternately from each side of the cylinder, leading to the well known von Karman vortex street.

vx=1, vy=0

vx=1, vy=0

p=0

vx=1, vy=0

Figure 4.5

Finite element mesh and boundary conditions

To describe the geometry of the problem, a circular cylinder of unit diameter is placed inside the domain �� = {−8 �� x �� 25, −8 �� y �� 8}. The center of the cylinder is placed at the point (0,0). The boundary conditions are given by a value of 1.0 for the x-velocity at the inﬂow, top and bottom boundaries. At these boundaries, the y-velocity is set to zero. Additionally, the pressure is set to zero at the outﬂow boundary, as shown in Fig. 4.5.

Figure 4.6

Pressure contours

The material properties are density �� = 0.01 and viscosity �� = 0.0001, yielding a Reynolds number Re = 100. The ﬁnite element mesh used for this problem is

4.8 Example Problems

129

shown in Fig. 4.5 and consists of 2880 four-node quadrilateral elements and 3000 nodes. Second order time integration is employed with the Generalized-�� scheme with ��f = 0.9. The number of time steps studied are 1500 with a time step �� size ��t = 0.20. In this problem, the FIC, OSS and SUPG/PSPG stabilization techniques have been employed and compared. Fig. 4.6 shows pressure contours for a time t = 154.2.

Figure 4.7

x-velocity contours

Fig. 4.7 shows x-velocity contours for a time t = 154.2, while Fig. 4.8 shows y-velocity contours at the same time.

Figure 4.8

y-velocity contours

Fig. 4.9 shows a typical plot of the time history of the velocity component vy at the point (x, y) = (2, 0) given by the SUPG/PSPG stabilization technique. The forces around the cylinder are used to compute the evolution of the drag CD and lift CL coeﬃcients given respectively by

130

4. Fluid Dynamics

0.8 0.6 0.4

y-velocity

0.2 0.0 -0.2 -0.4 -0.6 -0.8 0 50 100 time 150 200

Figure 4.9

Time history of y-velocity component

CD = CL =

fD 1 2 2 ��v A fL 1 2 2 ��v A

(4.8.1) (4.8.2)

where fD is the drag force measured in the direction of the ﬂow while fL is the lift force perpendicular to the direction of ﬂow. The solution using the three methods in this example is plotted in the temporal evolution of the drag coeﬃcient CD , as well as the evolution of the lift coeﬃcient CL as shown in Figs. 4.10 and 4.11.

1.45

FIC OSS SUPG/ S U P G /P S P G PSPG

F IC OSS

1.44

CD

1.43

1.42

1.41 200

202

204

206 time

208

210

212

214

Figure 4.10

CD for a ﬂow around a cylinder at Re=100

For this example, the lift coeﬃcient is almost the same for the three diﬀerent methods, while in the case of the drag coeﬃcient, the FIC stabilization technique shows a little bit more diﬀusive behavior than the OSS and SUPG/PSPG stabilization

4.8 Example Problems

131

0.5

FIC FIC

OSS OSS

0.3 0.1

SUPG/PSPG SUPG/PSPG

CL

-0.1 -0.3 -0.5 200

202

204

206 time

208

210

212

214

Figure 4.11

CL for a ﬂow around a cylinder at Re=100

techniques. However the amplitude between the three methods is practically the same. For the drag coeﬃcient, the amplitude is about 0.027, while a value of 0.865 is obtained for the lift coeﬃcient. These values are in good agreement with those presented in the numerical comparison of CBS and SGS stabilization techniques presented by Codina et al. (2006). The Strouhal number or adimensional frequency of the solution is one of the most studied quantities and describes the oscillating ﬂow mechanisms, which is given by St = f req · lc vc (4.8.3)

where St is the dimensionless Strouhal number, f req is the frequency of vortex shedding, lc is the characteristic length (in this example taken as the cylinder diameter) and vc is the characteristic velocity of the ﬂuid.

0.5

Fourier coefficient

0.4 0.3 0.2 0.1 0.0 0 0.2 0.4 frequency 0.6 0.8

Figure 4.12

Fourier spectrum of the lift coeﬃcient

132

4. Fluid Dynamics

A Fourier analysis of the lift coeﬃcient CL is performed within the time range [120,300] in order to ﬁnd the dominant frequency of the solution. In Fig. 4.12, the Fourier spectrum obtained is shown, leading to a frequency f req = 0.16602 Hz, equivalently to a period of 6.02 s. This value gives a dimensionless Strouhal number St = 0.16602 which is in good agreement with the experimental result St = 0.166 reported by Hammache and Gharib (1991).

1.48 1.47 1.46

F IC _ FIC G f in e OSS_ OSS G f in e S U P G _ G f in e SUPG/PSPG

CD

1.45 1.44 1.43 200 202 204 206 time 208 210 212 214

Figure 4.13

CD for a ﬂow around a cylinder at Re=100

This problem is solved again using a ﬁner mesh, consisting of 32000 four-node quadrilateral elements and 32400 nodes. The same second order time integration is employed. The number of time steps studied are 1500 with a time step size ��t = 0.20. Again, the FIC, OSS and SUPG/PSPG stabilization techniques have been used. Drag and lift coeﬃcients for this mesh are given in Figs. 4.13 and 4.14 respectively. For the ﬁner mesh, the lift coeﬃcient is practically the same for the three

0.6 0.4 0.2

F IC FIC_ G f in e OSS OSS_ G f in e S U P G _ G f in e SUPG/PSPG

CL

0.0 -0.2 -0.4 -0.6 200 202 204 206 time 208 210 212 214

Figure 4.14

CL for a ﬂow around a cylinder at Re=100

4.8 Example Problems

133

diﬀerent methods, while in the case of the drag coeﬃcient some little diﬀerences are presented. The amplitude between the three methods for the drag coeﬃcient is about 0.04, while a value of 1.0 is obtained for the lift coeﬃcient. It can be seen that when the mesh is reﬁned, all methods lead to the same solution. Very small frequency discrepancies between these methods are found, but all of them give a period about 6.0 s. In this problem, it is shown that the stabilization techniques of these three methods are in fact very similar. The numerical comparison yields no conclusive remarks, but shows that a ﬁner mesh leads to close numerical solutions.

4.8.3

Flow Past a Circular Cylinder in 3D

Here the transient ﬂow past a circular cylinder in 3D is solved using a high quality mesh constructed by Rainald L��hner. The objective of the example is to verify the o behavior of the formulations implemented in this work, using a large example and comparing it with a code specialized for ﬂuids. To describe the geometry of the problem, a circular cylinder of unit diameter is placed inside the domain �� = {−4 �� x �� 15, −4 �� y �� 4, 0 �� z �� 0.2}. The center of the cylinder is placed at the point (0,0,0). The boundary conditions are given by a value of 1.0 for the x-velocity at the inﬂow, and free x- and z-velocity at top and bottom boundaries. At these boundaries, the y-velocity is set to zero. The right and left boundaries have the z-velocity component set to zero while the x- and y-velocity components are free. Additionally, the pressure is set to zero at the outﬂow boundary, as shown in Fig. 4.15.

v y=0

vx=1 vy=0 vz=0

p=0

v y=0

Figure 4.15

Finite element mesh and boundary conditions

The material properties are density �� = 1.0 and viscosity �� = 0.005263, yielding a Reynolds number Re = 190. The ﬁnite element mesh used for this problem is shown in Fig. 4.15 and consists of 108,147 four-node tetrahedral elements and 30,000 nodes. A detail of the mesh around the cylinder is shown in Fig. 4.16. Second order time integration is employed with the Generalized-�� scheme and f �ѡ� = 0.9. The number of time steps studied are 2000 with a time step size

134

4. Fluid Dynamics

y

z

x

Figure 4.16

Detail of the mesh around the cylinder

��t = 0.05, beginning from an initially stable conﬁguration. In this problem, the OSS stabilization techniques is been employed and the solution is compared with the one obtained by Coppola-Owen (2006) using the same stabilization and the BDF2 time integration scheme. Fig. 4.17 shows pressure contours for time t = 100.

Figure 4.17

Pressure contours

Fig. 4.18 shows x-velocity contours for a time t = 100, while Fig. 4.19 shows y-velocity contours at the same time.

Figure 4.18

x-velocity contours

4.8 Example Problems

135

Figure 4.19

y-velocity contours

The forces around the cylinder are used to compute the evolution of the drag CD and lift CL coeﬃcients given respectively by Figs. 4.20 and 4.21.

1.58 1.56 1.54

M y D ra g O th e r D ra g

Present work Coppola-Owen

CD

1.52 1.50 1.48 1.46 90

92

94 time

96

98

100

Figure 4.20

CD for a ﬂow around a cylinder at Re=190

In this case, the drag coeﬃcient has an amplitude of 0.10, while a value of 1.45 is obtained for the lift coeﬃcient. A Fourier analysis of the lift coeﬃcient CL is performed within the range [65,100] in order to ﬁnd the dominant frequency of the solution. In Fig. 4.22, the Fourier spectrum obtained is shown, leading to a frequency f req = 0.19531 Hz. The Strouhal number for this example yields a dimensionless value of 0.19531 and a period of 5.12 s. The solution found by Coppola-Owen (2006) for this example is the same as the one presented in this work.

136

4. Fluid Dynamics

0.8 0.6 0.4 0.2

M y L ift O th e r L ift

Present work Coppola-Owen

CL

0.0 -0.2 -0.4 -0.6 -0.8 90 92 94 time 96 98 100

Figure 4.21

CL for a ﬂow around a cylinder at Re=190

0.6

Fourier coefficient

0.5

0.4 0.3 0.2 0.1 0.0 0 0.2 0.4 0.6 frequency 0.8 1.0 1.2

Figure 4.22

Fourier spectrum of the lift coeﬃcient

Chapter 5

Fluid-Structure Interaction

Many physical problems of diﬀerent ﬁelds in engineering and applied sciences belong to multiphysic problems. Recently numerical simulation of Fluid-Structure Interaction (FSI) problems have gained great interest from the computational mechanics community in order to reduce development time and cost in coupled systems. In case of civil engineering applications, the interaction of wind ﬂow and structural motion may lead to aeroelastic instabilities which may cause the collapse of the structure. This problem can occur on constructions such as long-span bridges, high-rise buildings and light weight roof structures. Maybe the most analyzed FSI problem is found in aerospace applications, which are concerned with aeroelastic instabilities for the study of wind ﬂow around ﬂexible wings of aircrafts. Recently, on the other hand, biomechanical applications are interested in the study of hemodynamics, i.e. blood ﬂow through large arteries, which are simulated using numerical FSI of large structural membrane deformations interacting with viscous ﬂows. FSI problems are complex because they consist of structural nonlinear boundary conditions imposed on ﬂuid moving boundaries where the position is part of the solution. Because the moving position of the structure prescribes part of the ﬂuid boundary, it becomes necessary to perform the integration of the Eulerian ﬂuid equations on a moving mesh. Among the several methods proposed for this problem, in this work it is used the Arbitrary Lagrangian-Eulerian (ALE) formulation. This kind of FSI problems usually are viewed as a two-ﬁeld coupled problem, however the moving mesh can be viewed as another structural problem, and therefore the complete coupled problem can be formulated as a three-ﬁeld system: the structure, the ﬂuid and the moving mesh.

5.1

Coupling Strategies

The implementation of a coupled problem can be done using two diﬀerent global strategies, which are the monolithic methods and the partitioned methods. In 137

138

5. Fluid-Structure Interaction

monolithic methods, the discretized ﬂuid-structure system is solved together with the mesh movement system in a single iteration loop, leading to a very large system of nonlinear equations to be solved simultaneously. Some advantages of this method are that it ensures stability and convergence of the whole coupled problem. On the contrary, in simultaneous solution procedures the time step has to be equal for all subsystems, which may be ineﬃcient if diﬀerent time scales are presented for the problem. An important disadvantage is the considerably high computing time eﬀort required to solve each algebraic system and sometimes the necessity to develop new software and solution methods for the coupling method. A monolithic approach to FSI is presented by H��bner et al. (2004). u In partitioned methods application of existing appropriate and sophisticated solvers for either structural or ﬂuid subsystems will continue to be used. These methods enjoy great popularity due to the simpliﬁed coupling procedure in many cases. Partitioned methods are divided into weak or loose coupling algorithms and strong or implicit coupling schemes. Weak algorithms are also known as staggered or explicit schemes. The major drawbacks of partitioned methods are lack of accuracy and stability problems, which sometimes may diverge from the solution. Partitioned methods were introduced by Park and Felippa (1983). The key idea for these methods is clearly described in Felippa et al. (1998). Partitioned solutions with staggered coupling algorithms are developed by Farhat et al. (1997) to be used in aeroelastic wing problems. Strong coupling of partitioned algorithms are applied to large displacements 2D structural problems coupled to viscous incompressible ﬂuids by Wall and Ramm (1998) and Wall (1999). They also applied the same method for a coupled ﬂuid structure environment with an initially ﬂat three-dimensional shell model as given in Wall and Ramm (2000). Other large displacements structural problems interacting with incompressible ﬂuids are detailed in Mok (2001), Mok and Wall (2001) and Tallec and Mouro (2001). FSI with large displacements applied to wind problems is developed by Rossi (2005), W��chner u and Bletzinger (2005), Badia (2006) and W��chner (2006). u More sophisticated developments on strong partitioned method for FSI problems can be found in Steindorf (2002), Matthies and Steindorf (2004), Matthies and Steindorf (2005), Matthies and Steindorf (2006) and Tezduyar et al. (2006). A study on strong coupling partitioned methods for FSI applied to hemodynamic problems can be found in Nobile (2001), Causin et al. (2005), and Fern��ndez and a Moubachir (2005). Strong coupling of ﬂuid-structure interaction including free surfaces is studied in Dettmer (2004) and Wall et al. (2007). Recently, a new approach based on Robin transmissions conditions for ﬂuid-structure interaction problems is given in Badia et al. (2007). These concepts are better explained next. A simple example for a two-ﬁeld x and y scalar problem, as given in Felippa (2004), is assumed to be governed by the diﬀerential equations 2x(t) + 4y(t) = f (t) y(t) − 3y(t) + 5x(t) = g(t) ˙

(5.1.1)

5.1 Coupling Strategies

139

in which f (t) and g(t) are the external forces. Using the backward Euler time integration leads to yn+1 − yn (5.1.2) ��t where ��t is the time step size. Assuming known all values at time step n, and substituting Eq. (5.1.2) into Eq. (5.1.1) yields the following algebraic system yn+1 = ˙ 2 5��t 4 1 − 3��t fn+1 xn+1 = yn+1 ��t gn+1 + yn (5.1.3)

which can be solved simultaneously for xn+1 and yn+1 as a monolithic approach, where a general multi-ﬁeld solver has to be used or developed. However if we want to use our existing codes that solve independently xn+1 with the F ﬁeld solver, and yn+1 with the S ﬁeld solver, then a partitioned approach has to be used. It is assumed that the solvers F and S are two separate but communicating programs. A simple partitioned solution procedure is obtained with the following steps for a staggered partition. 1) Make a prediction for yn+1 , yn+1 = y p (5.1.4)

where the usual choices for the prediction are y p = yn or y p = yn + ��t yn . Send ˙ this information to the F ﬁeld solver as an external force. 2) With the F ﬁeld solver, ﬁnd xn+1 from the partitioned system extracting the ﬁrst equation from the algebraic system in Eq. (5.1.3) yielding xn+1 = 1 (fn+1 − 4y p ) 2 (5.1.5)

3) Send xn+1 to the S ﬁeld solver. 4) Perform a correction with the S ﬁeld solver for yn+1 from the partitioned system extracting the second equation from the algebraic system in Eq. (5.1.3) giving yn+1 = 1 (��t gn+1 + yn − 5��t xn+1 ) 1 − 3��t (5.1.6)

5) Now the time step is advanced and the process is repeated for the next time step. This coupling procedure is known as the staggered partitioned scheme or weak coupling scheme. It is common to use this kind of coupling method for explicit ﬁeld solvers where little time steps are used and no convergence of the coupling is checked. For implicit ﬁeld solvers, the time step is usually 10 to 100 times larger than the critical time step used for explicit ﬁeld solvers. In this case, convergence of the coupling after step 4) is required. If the convergence is not reached, step 5) is not performed and the procedure is repeated within each time step, i.e. steps 2 to 4, where y p of step 2) is substituted by yn+1 of step 4). These iterations turn

140

5. Fluid-Structure Interaction

the staggered approach over a strong coupling partitioned procedure which tends to the solution of the monolithic problem. For ﬂuid-structure interaction problems, the system to solve is nonlinear and adds diﬃculty to the problem being studied, which sometimes diverge. However the basic idea for solving the problem keeps.

5.2

ALE Framework

Solid and structural formulations are commonly based on Lagrangian descriptions since they deform with the material. On the contrary, ﬂuids are usually formulated on Eulerian descriptions since they involve ﬁxed spatial domains where interest is focused on a particular part of the ﬁxed mesh, i.e. the ﬂow around a solid. In FSI problems, the Eulerian mesh of the ﬂuid is not ﬁxed in space anymore since the motion of the structural mesh moves the ﬂuid mesh and consequently the Eulerian ﬂuid equations must be formulated using a mesh movement technique. Hybrid techniques such as the ALE descriptions combine the advantages of Lagrangian and Eulerian methods while minimizing the disadvantages. Since ALE descriptions are an arbitrary combination of the Lagrangian and Eulerian descriptions, the user must select an adequate mesh motion as a result of the combination of both descriptions. Equations describing the ALE methods are very similar to those used for the Eulerian formulations. In fact, Eulerian methods can be considered as a special case of ALE methods. A complete description and state of the art for ALE methods can be found in Belytschko et al. (2000) and Donea and Huerta (2003). Until now for Lagrangian and Eulerian descriptions two conﬁgurations have been used: the reference conﬁguration and the deformed conﬁguration. In ALE description, another conﬁguration is needed which is the ALE conﬁguration. The ˆ domain for an ALE conﬁguration is denoted by ��, which coordinates X are called ALE coordinates. Two diﬀerent motions are described in ALE methods: material motion and mesh motion. Material motion is described just as Eq. (2.1.2) yielding x = ��(X, t) (5.2.1)

The corresponding velocity and acceleration is given in Eq. (2.1.7) and Eq. (2.1.8) respectively. The mesh motion is described by ˆ x = ��(X , t) (5.2.2)

where x represents the deformed or spatial conﬁguration. ALE coordinates can be obtained from Eq. (5.2.2) leading to ˆ X = ��−1 (x, t) (5.2.3)

Substituting Eq. (5.2.1) into Eq. (5.2.3) gives the relation between material and ALE coordinates yielding ˆ X = ��−1 (��(X, t), t) = ��(X, t) (5.2.4)

5.2 ALE Framework

141

ˆ where �� represents the composition of functions ��−1 ◦��. With all these equations, the material motion can also be expressed as a composition of functions given by ˆ x = ��(��(X, t), t) ˆ and it is concluded that �� = �� ◦ ��. The mesh displacement is deﬁned by ˆ ˆ u = x − X = ��(X , t) − X (5.2.6) (5.2.5)

The corresponding mesh velocity and mesh acceleration is given respectively by ˆ ˆ ∂ u(X , t) ∂ ��(X , t) = ∂t ∂t ˆ ∂ v(X , t) ∂t

ˆ v=

(5.2.7)

ˆ a=

(5.2.8)

The material time derivative can be obtained similar to the Eulerian description. Therefore, the material time derivative of a function f (X , t) yields ∂f ∂f ∂xj ∂Xi ∂f ∂f Df (X , t) cj = + = + Dt ∂t ∂xj ∂Xi ∂t ∂t ∂xj (5.2.9)

where cj is the convective velocity given by the diﬀerence between the material and ˆ mesh velocities c = v − v = v − vmesh . In general the material time derivative of any function, vector or tensor given in ALE variables X and time t can be obtained with ∂(•) D(•) = + c · ∇(•) Dt ∂t (5.2.10)

It is obvious to think that the conservation equations, expressed before in Lagrangian and Eulerian coordinates, must now be expressed in ALE coordinates. Since we are dealing with incompressible ﬂuid equations, the only equation that we need to describe is the momentum equation, yielding

��

∂v + c · ∇v ∂t

= ∇ · �� + ��b

or

��

∂vi + cj ∂j vi ∂t

=

∂��ij + ��bi (5.2.11) ∂xj

where the material time derivative of the velocity has been employed. The resulting equations for the fractional step method using the OSS stabilization in ALE conﬁguration yields

142

5. Fluid-Structure Interaction

˙ ˜ (vh

n+��f m

, wh ) + c(˜h c ��(˜ c

n+��f

f

˜ , vh

n+��f

f

n , wh ) − b(ph , wh ) + a(˜h v n ˜ + ∇pn − ��h , c h n+��f f h

n+��f

f

, wh ) +

n+��f f h

· ∇˜ v

n+��f f h

n+1 · ∇wh ) = (bh , wh )

−

n+��f n+��f ��t �� f n+1 n n (∇(ph − ph ), ∇qh ) − ��(˜h f · ∇˜h f + ∇pn+1 − ��h , ∇qh ) c v h f ��m

˜n+1 = b(qh , vh )

f ��m n+1 n ˜n+1 (vn+1 − vh , wh ) − b(ph − ph , wh ) = 0 ��t �� f h n+1 (��h , ��h ) − (˜h c n+��f

f

(5.2.12)

· ∇˜h v

n+��f

f

+ ∇pn+1 , ��h ) = 0 h

The ﬁnal equations for the fractional step method using the FIC stabilization in ALE conﬁguration are expressed by

n+��f m n+��f n+��f n+��f

˙ ˜ (vh

, wh ) + c(˜h c

f

n ˜ , vh f , wh ) − b(ph , wh ) + a(˜h f , wh ) + v f n+��f 1 n+��f n+1 ˆh · ∇˜h f − cn , h · ∇wh ) = (bh , wh ) v (˜h c 2

−

��t �� f n+1 n n ˜n+1 (∇(ph − ph ), ∇qh ) − ��(∇pn+1 − ��h , ∇qh ) = b(qh , vh ) h f ��m (5.2.13) f ��m n+1 n ˜n+1 (vn+1 − vh , wh ) − b(ph − ph , wh ) = 0 ��t �� f h c (ˆn+1 , ��h ) − (˜h ch (��h

n+1 n+��f

f

· ∇˜h v

n+��f

f

, ��h ) = 0 , ��h ) = 0

, ��h ) − (∇ph

n+1

From the OSS and FIC stabilizations described in ALE coordinates, it can be shown that when the mesh velocity vmesh = 0, the convective velocity c = v and the Eulerian methods described in chapter 4 are recovered.

5.3

Governing Equations

The governing equations for the couple incompressible ﬂuid-structure problem consist of the momentum equations together with the continuity equation. However the ﬂuid and the structural parts of the domain must be treated diﬀerently. Then the problem is split into the ﬂuid test functions over the ﬂuid domain and the solid test function operating over the structural part. For convenience, the boundary of the coupled problem is divided into the Dirichlet boundary for the ﬂuid ��D and the solid ��D , the Neumann boundary for the s f ﬂuid ��N and the solid ��N , and a common interface boundary ��I between the ﬂuid s f and the solid. Then the boundary of the couple problem is �� = ��D �� ��N �� ��I , where ��D = ��D �� ��D and ��N = ��N �� ��N . s s f f

5.3 Governing Equations

143

Also the coupled problem domain is divided into a solid part �� s and a ﬂuid part �� f , where �� = �� s �� �� f . In particular, the solid motion xs sets the solid displacements us yielding us (X, t) = xs − X (5.3.1)

s where X �� ��0 . The ﬂuid mesh motion xf is deﬁned as a function of the solid interface displacements xs �� ��I as an extension over the ﬂuid domain �� f , where the ﬂuid (mesh) displacements are given by

ˆ u = uf = Ext(us��I )

(5.3.2)

where diﬀerent forms to make this extension are presented ahead in section 5.5. Since our ﬂuid domain is moving, the ALE formulation is needed. The corresponding mesh velocity is found substituting Eq. (5.3.2) into Eq. (5.2.7) yielding ˆ vmesh = v(uf ). Before writing the continuous formulation of the couple ﬂuid-structure interaction problem, the subspace test functions for the ﬂuid, with a homogeneous Dirichlet boundary condition, are deﬁned by W0 = ��wi �� H 1 (��f ), ��wi = 0 on ��D �� ��I , ��wi = 0 on ��s f (5.3.3) and the subspace test functions for the structure are expressed as ��wi �� W0 , W0 = ��wi �� H 1 (��s ), ��wi = 0 on ��D , ��wi = Ext(��wi|��I ) on ��f s (5.3.4) Note that the test functions for the ﬂuid vanish at the interface and inside the solid subdomain, while the solid test functions are nonzero on the interface and extend inside the ﬂuid subdomain. In this way, the kinematic continuity for the displacement and velocity ﬁeld is imposed as Dirichlet boundary conditions on the ﬂuid by the interface, and kinetic continuity for the traction is given as Neumann boundary conditions on the structure at the interface. Since the continuity equation is the same that for CFD problems, then the ﬂuid problem is obtained by substituting into the momentum equation the corresponding test functions, which vanish in the solid part, yielding after integrating by parts ∂vi ∂vi + ��cj ∂t ∂xj ∂��wi d�� = ∂xi ��wi �� W0 ,

��f

��wi ��

d�� +

��f

��ij

��f

��wi ��bi d��

(5.3.5)

This equation must be complemented by adequate boundary conditions dictated by the ﬂuid space test functions. Since ﬂuid the test functions vanish at the interface, then the displacement and velocity ﬁeld can only be governed by imposing the kinematic compatibility given by the Dirichlet condition uf��I = us��I (5.3.6)

144

5. Fluid-Structure Interaction

vf��I = vs��I

(5.3.7)

Similarly, the structural problem is obtained by considering the space of the solid test functions on the momentum equations, yielding

��s 0

��wi ��0 ui d��0 + ��

��s 0

��Eij Sij d��0 =

��s 0

��wi ��0 bi d��0 +

��N 0;s

¯i ��wi t0 d��0 + L(��wi|��I )

(5.3.8) where L is the interface load corresponding to the part of the momentum equation which is integrated on the ﬂuid domain, and is given by the choice of the solid test functions in Eq. (5.3.4). Then L can be written as ∂vi ∂vi + ��cj ∂t ∂xj

L(��wi|��I ) =

��f

Ext(��wi|��I )��bi d�� −

��f

Ext(��wi|��I ) ��

d�� (5.3.9)

∂Ext(��wi|��I ) − ��ij d�� ∂xi ��f

which is the residual of the ﬂuid equations evaluated with the extension of the interface test functions Ext(��wi|��I ), as given in Tallec and Mouro (2001). Remembering that the ﬂuid is considered with a homogeneous Dirichlet boundary condition, the last term in Eq. (5.3.9) is integrated by parts yielding ∂Ext(��wi|��I ) d�� = ∂xi ∂��ij d�� ∂xj

��f

��ij

��I

��wi nf ��ij d�� − j

��f

Ext(��wi|��I )

(5.3.10)

Substituting Eq. (5.3.10) into Eq. (5.3.9) leads to ∂vi ∂vi + ��cj ∂t ∂xj

L(��wi|��I ) =

��f

Ext(��wi|��I )��bi d�� −

��f

Ext(��wi|��I ) ��

d�� (5.3.11)

∂��ij + Ext(��wi|��I ) d�� − ∂xj ��f

��I

��wi nf ��ij d�� j

Since the ﬁrst three terms in the above equation are the solution of the ﬂuid part, Eq. (5.3.11) is ﬁnally expressed as L(��wi|��I ) = −

f ��wi nf ��ij d�� = j f ��wi ns ��ij d�� j

(5.3.12)

��I

��I

which shows the interface traction obtained directly from the momentum conservation equation and not considered as an additional independent equation. The continuity equation for the ﬂuid part remains the same that for the CFD problem, which is given by Eq. (4.2.10). The stabilization terms involve are expressed by Eq. (5.2.12) or Eq. (5.2.13).

5.4 Partitioned Methods

145

The ﬂuid solver F for the couple problem is given by the following weak formulation ∂vi ∂vi + ��cj ∂t ∂xj

��f

��f

��wi �� − +

��I

d�� − ��p

p

��f

∂��wi d�� + ∂xi

��f

��

��f

∂vi ∂��wi d�� ∂xj ∂xj (5.3.13)

��f

��wi ��bi d�� −

∂vj d�� + ∂xj

��I

m ��wi uf − Ext(us��I ) d��

u ��wi uf − us d�� +

v ��wi vf − vs d�� + Stabilization = 0

where the ALE framework for the momentum equation, with its corresponding stabilization technique, has been considered together with the incompressibility constraint. Besides the mesh movement in the ﬂuid subdomain and the kinematic compatibility at the interface are included in the formulation. The corresponding solid solver S for the couple problem is given by the weak formulation shown next ��Eij Sij d��0 −

f ��wi ns ��ij d�� j

��s 0

��wi ��0 ui d��0 + �� −

��s 0

��s 0

��wi ��0 bi d��0 (5.3.14)

��N s

¯ ��wi ti d�� −

=0

��I

where the momentum equation is complemented with the traction ﬂuid forces of the couple problem at the interface.

5.4

Partitioned Methods

With the developed equations for the ﬂuid solver F and the solid solver S given in the last section, the problem consist in ﬁnding the appropriate partitioned method to solve the ﬂuid-structure interaction problem. Numerical simulation of FSI problems is not only diﬃcult because of the problems associated with the ﬂuid or structure solution, but also because of the coupling interface between these two ﬁelds which sometimes represents another challenge. These diﬃculties depend strongly on the added mass eﬀect introduced by the ﬂuid over the structure, as given in Causin et al. (2005). When the structure density �� s is much larger than the ﬂuid density �� f , the added mass eﬀect of the ﬂuid is not signiﬁcant and the problem can be solved with staggered partitioned schemes or strong coupling partitioned techniques as in the case of aeroelasticity. However when the structure and ﬂuid densities are of the same order, as in the case of hemodynamics, the added mass eﬀect of the ﬂuid over the structure is very important and the coupling algorithm to be used must be a strong coupling partitioned scheme with relaxation, or even better techniques as the exact Newton or inexact Newton method for strong coupling problems, or the recently partitioned procedures based on Robin transmissions conditions given by Badia et al. (2007).

146

5. Fluid-Structure Interaction

To explain these methods, assume that the ﬂuid unknowns are grouped together in the vector x and that the nonlinear iterative ﬂuid solver is written as xi = F(xi−1 , y) (5.4.1)

where y represent the displacements of the structure, which also deﬁne the current conﬁguration of the interface. The nonlinear iterative solid solver is given by yi = S(yi−1 , x) (5.4.2)

where x deﬁnes the nodal ﬂuid velocities and pressure including the traction ﬂuid forces at the interface for the structure. With these equations, the following strong coupling schemes are described within a time step.

5.4.1

Block Jacobi Method

The easiest method to perform an iterative strong coupling scheme between diﬀerent ﬁeld solvers is the Block Jacobi method. This method consist in iterating the ﬂuid and solid solvers independently as shown next xi = Fp (xi−1 , yi−1 ) yi = Sq (yi−1 , xi−1 ) (5.4.3) (5.4.4)

where p and q are the number of times that the solvers F and S are repeated 0 respectively. Also for i = 1, x0 n+1 = xn and yn+1 = yn . Once the tolerance or maximum number of iterations is reached, the time step is advanced and the process is repeated.

5.4.2

Block Gauss-Seidel Method

With a simple modiﬁcation to Eqs. (5.4.3)-(5.4.4), the Block Gauss-Seidel method is obtained. As before, this method consist in iterating the ﬂuid and solid solvers independently yielding xi = Fp (xi−1 , yi ) yi = Sq (yi−1 , xi ) (5.4.5) (5.4.6)

where in Eq. (5.4.5) a predicted value for y1 is needed when i = 1, as explained in step 1) of section 5.1 for the staggered partition. When the tolerance or maximum number of iterations is reached, the time step is advanced and the process is repeated. This method is the most common to ﬁnd in ﬂuid-structure interaction problems with strong coupling schemes, because existing and sophisticated solvers for each ﬁeld can be used with minor modiﬁcations.

5.4 Partitioned Methods

147

Even though the iterations within each time step lead to the monolithic solution, sometimes relaxation techniques are combined with this method when important values for the added mass eﬀect are presented to ensure a convergence of the problem, or simply to accelerate the convergence of the coupling scheme when the added mass eﬀect is negligible. Such technique is described in the works of Mok (2001), Mok and Wall (2001), Tallec and Mouro (2001), W��chner and Bletzinger (2005), u Causin et al. (2005), W��chner (2006), and Wall et al. (2007). The relaxation u technique used in this work is detailed in section 5.6.

5.4.3

Inexact Block Newton Method

The inexact Block Newton method, or quasi-Newton method, is derived from the monolithic Newton-Raphson procedure, which for a two-ﬁeld couple problem can be written as ∂x F − I ∂x S ∂y F ∂y S − I xi − xi−1 x−F i i−1 = y−S y −y (5.4.7)

xi−1 ,yi−1

xi−1 ,yi−1

where ∂x and ∂y are the partial derivatives with respect x and y respectively. Since a strong coupling partitioned procedure is being developed, then Eq. (5.4.7) is split into two separate equations. However the problem emerges when the cross Jacobians ∂y F and ∂x S, which represents the sensitivity of the ﬂuid state with respect to the structure and vice versa, need to be evaluated. This is due to the fact that existing solvers are used and we do not have direct access to the cross Jacobians of Eq. (5.4.7). Approximation of the cross Jacobians is developed in the work of Steindorf (2002) and Matthies and Steindorf (2004) where ﬁnite diﬀerences are used with an auxiliary vector z and a small time step-size h yielding 1 F(x, y + hz) − F(x, y) (5.4.8) h More details can also be found in Matthies and Steindorf (2005) and Matthies and Steindorf (2006). Another method that evaluate approximately the cross Jacobians is given by Tezduyar et al. (2006), where a mixed analytical/numerical elementvector based technique is introduced. ∂y F(x, y)z =

5.4.4

Exact Block Newton Method

The exact Block Newton method, or simply Newton method, is derived from the monolithic Newton-Raphson procedure, just as the inexact block Newton method does, where the Eq. (5.4.7) is used again. In the exact block Newton method, the cross Jacobians are derived exactly using the existing solvers with some more advanced ideas, as given in Dettmer (2004), Fern��ndez and Moubachir (2005) and Dettmer and Peric (2006). a The exact block Newton method shows superior convergence properties that any of the above methods mentioned with just a few iterations. While sometimes the Gauss-Seidel method with relaxation or the quasi-Newton fail to converge, it is

148

5. Fluid-Structure Interaction

demonstrated in Fern��ndez and Moubachir (2005) that the Newton method does a converge. However the cost of each Newton and quasi-Newton iteration is much higher that the Gauss-Seidel method. Consequently, Newton based methods do not imply an overall reduction of computational cost. Besides the computational implementation is harder than the Gauss-Seidel method. These reasons make the block Gauss-Seidel partitioned method still be an attractive method in the ﬂuidstructure interaction community.

5.5

Mesh Movement Techniques

Since the interface of the ﬂuid-structure interaction problem follows the solid displacements, it is necessary to use mesh movement techniques, also known as mesh update equations, to update the internal nodes of the ﬁnite element mesh inside the ﬂuid subdomain. The mesh moving algorithm updates the nodal coordinates of the ﬂuid mesh in response to the deforming boundaries at the interface, and maintaining freeze the rest of the boundary. Among the wide possibilities to update meshes in this ﬁeld, the most common are mentioned next. The simplest one is the Laplacian method which can be found in Belytschko et al. (2000). Other methods are based on the pseudo-structural system, which can be done through the elastic spring analogy, i.e. see Farhat (1997) and Degand and Farhat (2002), or by solving the elasticity equation as a pseudo-elastic system, i.e. see Johnson and Tezduyar (1994), Belytschko et al. (2000) and Chiandussi et al. (2000). In this way, the mesh is considered as another system. Therefore the ﬂuid-structure interaction problem yields a couple three-ﬁeld system. The Laplacian method, used in this work, updates the position of the nodes by solving the Laplace equation, where the contours solution are approximately orthogonal. This method in strong form consists in ﬁnding the mesh displacements such that ˆ ∇2 u = 0 in ��f ˆ u=u ˆ u=0 on on ��I ��D f (5.5.1)

ˆ where ∇2 is the Laplacian of the mesh displacements u, and u are the displacements on the interface given by the structural deformation. This method has the advantage that it can be uncoupled for the displacements ˆ and ﬁnd independently ux , uy and uz , which represents a faster solution of the ˆ ˆ mesh displacements. Since it is desirable to retain the original shape of the elements in the reﬁned areas, this method can be improved by assigning diﬀerent element stiﬀness to small and large elements by simply dividing (instead of multiplying) the Jacobian determinant during element integration. On the other hand, in the pseudo-elastic structural technique, the mesh is assumed to be an elastic body, and a standard linear problem is solved. This method in strong form consists in ﬁnding the mesh displacements such that

5.6 Strong Coupling with Relaxation

149

Kˆ = 0 in ��f u ˆ u=u ˆ u=0 on on ��I ��D f (5.5.2)

ˆ where K is the stiﬀness matrix for the elastostatic system, u are the unknown mesh displacements, and u are the displacements on the interface given by the structural problem. For small displacements on the interface, the linear elastic system is suﬃcient, however in the presence of large deformations at the interface, a nonlinear model for the mesh may be more accurate. For example, Dettmer and Peric (2006) use a nonlinear hyperelastic Neo-Hookean model, which after a few iterations yield a mesh with a uniform element distortion. In this work the pseudo-structural system of Chiandussi et al. (2000) is used, which is simple to implement and instead of solving an iterative nonlinear structural problem, a linear-elastic problem is solved twice. Of course this method is more expensive than the Laplacian method, but for large displacements of the interface this method yields a better mesh distributions of the internal ﬂuid nodes. The method is described next. First, assign an isotropic homogeneous material with constant Young properties, i.e. E = 1 and �� = 0, to each element. Then solved the pseudo-structural problem with the appropriate boundary conditions at the interface. After the solution of the ﬁrst analysis pseudo-structural problem is found, the strain ﬁeld is computed together with its principal strains ��i . Then the new Young modulus for each element is found with 1 E= 2 �� ndime ¯

ndime

��2 i

i=1

(5.5.3)

where �� is the uniform strain. Now the second pseudo-structural problem is solved ¯ with the appropriate boundary conditions at the interface. In this second analysis, a uniform mesh distortion is obtained. Further details on this method including other alternatives for evaluating the Young modulus and measurements of the mesh quality using a mesh quality indicator are given in Chiandussi et al. (2000). Since this method is more expensive than the Laplacian method, applying only a few iterations when using iterative solvers, instead of iterating until convergence, gives a good alternative to move nodes inside the ﬂuid subdomain.

5.6

Strong Coupling with Relaxation

In this work, the strong coupling block Gauss-Seidel partitioned method with relaxation has been implemented. The structural solver S given by Eq. (5.3.14) is now referred to as the computational solid dynamic (CSD) solver. The ﬂuid solver F given by Eq. (5.3.13) solves the ﬂuid equations plus the movement of the mesh, yielding in a high-cost task from the computational point of view. Therefore the

150

5. Fluid-Structure Interaction

ﬂuid solver F is split into the computational mesh dynamic (CMD) solver that moves the interior nodes of the ﬁnite element mesh of the ﬂuid subdomain with the methods given in section 5.5, and the computational ﬂuid dynamic (CFD) solver that uses the fractional step method described in chapter 4 and the ALE technique incorporated in the momentum equation as given in section 5.2. All solvers of each ﬁeld use an implicit scheme. There are several methods in the literature to accelerate the solution of the problem by means of relaxation, i.e. Mok (2001). In this work the Aitken method is implemented, i.e. Irons and Tuck (1969), which yields excellent solutions with simple modiﬁcations to the code. In order to compute the coupled ﬂuid-structure interaction problem, a uniﬁed algorithmic framework for the whole procedure is presented next. Considering known all values of solid, ﬂuid and mesh at time step tn , the new step tn+1 is found following the simple steps given ahead: 1. Advance time step: tn+1 = tn + ��t 2. Set iteration i = 1 3. Compute interface predictor displacement with one of the following methods ˆ |��I ,i (a) Structural predictor: Solve the CSD problem to ﬁnd un+1 at the interface from un+1 with a predicted external force given by: |��s ,i i. Pressure: · Order 1: pn+1 = pn · Order 2: pn+1 = 2pn − pn−1 , (for n �� 2)

f f ii. Force tf = ns ��|��I : (where ��|��I ,0 = 0) f · Order 1: tf n+1 = tn f f · Order 2: tf n+1 = 2tn − tn−1 , (for n �� 2)

ˆ |��I ,i (b) Interface displacement predictor: Set u1 I ,1 = 0 and ﬁnd directly un+1 |�� at the interface with a prediction of the form ˆ |��I ,i ˆ |�� i. Order 0: un+1 = un I ˆ |��I ,i ˆ |�� ii. Order 1: un+1 = un I + ��t ˆ n I v|�� ˆ |��I ,i ˆ |�� iii. Order 2: un+1 = un I + ��t 4. Iterate the coupled FSI problem (a) CMD solver: Move Mesh with Eq. (5.5.1) or Eq. (5.5.2) ˆ |��I ,i i. Transfer un+1 to the mesh solver ˆi ˆ |��f ,i ii. Solve the CMD problem and ﬁnd un+1 = un+1 ˆi iii. Compute mesh velocities vn+1 (b) CFD solver: Solve ﬂuid with Eq. (5.2.12) or Eq. (5.2.13) ˆi i. Transfer vn+1 to the ﬂuid solver ii. Solve the CFD problem and ﬁnd vn+1 , pn+1 i i f iii. Compute ﬂuid stress tensor at interface ��|��I

3 n ˆ 2 v|��I

ˆ − 1 vn−1 , (for n �� 2) 2 |��I

5.7 Code Development

151

(c) CSD solver: Solve structure with Eq. (5.3.14) f i. Transfer ��|��I to the solid solver and compute structure forces tf ii. Solve the CSD problem and ﬁnd un+1 i+1 (d) Relaxation phase i. Compute optimal relaxation parameter wi via Aitken method ¯ ii. Relaxation of predicted interface position with ˆ |��I ,i+1 un+1 = (1 − wi )ˆ n+1 + wi un+1 ¯ u|��I ,i ¯ |��I ,i+1 (e) Advance iteration: i = i + 1 (f) Check convergence. If reached, go to 5, else go to 4 5. Check time step. If end of time not reached, go to 1, else end of calculation The Aitken method of relaxation is based on Aitken��s acceleration method for vectors. The method can be easily implemented in any code which then ensures converge of the coupled problem for adequate time step parameters. The Aitken relaxation parameter is computed with the following algorithm, as given in Wall et al. (2007). 1. For the ﬁrst time step and i = 1, ��n+1 = 0 and w1 = 1 ¯1 ¯ ˆ |��I ,0 ˆ |�� 2. For i = 1, ��n+1 = ��n ¯0 ¯imax and un+1 = un I ,imax 3. Compute Aitken optimal relaxation parameter wi ¯ (a) Compute the diﬀerence between previous and actual interface solution ˆ |��I ,i−1 ��ˆ n+1 = un+1 − un+1 ui |��I ,i ˆ |��I ,i ��ˆ n+1 = un+1 − un+1 u i+1 |��I ,i+1 (b) Compute Aitken factor

ui u i+1 u i+1 (��ˆ n+1 −��ˆ n+1 )·��ˆ n+1 2 ui u i+1 (��ˆ n+1 −��ˆ n+1 ) (c) Compute Aitken optimal relaxation parameter wi = 1 − ��n+1 ¯ ¯i

¯i−1 ��i−1 ��n+1 = ��n+1 + (¯n+1 − 1) ¯i

More sophisticated and computationally expensive methods, such as the gradient method, lead to solutions as good as the Aitken method for ﬂuid-structure interaction problems. Additional computational cost for this technique is insigniﬁcant since only vector operations over the interface nodes are performed.

5.7

Code Development

In this work the ﬂuid-structure interaction problem has been implemented in the ﬁnite element program COMET (2007) following the algorithm given in section 5.6. The dynamic analysis with the ��-family method, backward diﬀerencing method and Generalized-�� method for ﬂuids have also been added to the code. This software is developed at the International Center for Numerical Methods in Engineering (CIMNE) and is used for coupled contact, mechanical and thermal analysis using the ﬁnite element method.

152

5. Fluid-Structure Interaction

5.8

5.8.1

Example Problems

Nonlinear Shell Vibration Excited by Fluid Vortex

This model problem was originally proposed by Wall and Ramm (1998) to demonstrate the ability of their FSI formulation to deal with complex ﬂow-ﬂexible structure interactions exhibiting large deformations. Later Steindorf (2002) and recently Dettmer (2004) test their numerical simulations with this problem. Also, with a modiﬁcation at the inﬂow velocity, H��bner et al. (2004) have used this problem to u test their FSI monolithic code. The problem consists of a thin elastic nonlinear shell attached to a ﬁxed square rigid body, which are submerged in an incompressible ﬂuid ﬂow. Vortices separating from the corners of the rigid body generate oscillating forces on the shell. Geometry and boundary conditions are given in Fig. 5.1.

vy=0

vx=v vy=0

1.0 1.0 4.0

0.06

p=0

12.0

vy=0

5.5

15.5

Figure 5.1

Geometry and boundary conditions [cm]

5.8.1.1

H�� bner��s problem u

The easiest couple problem involving ﬂuid and structural large deformations was proposed by H��bner et al. (2004), where the material properties are given for u the structure with a density ��s = 2.0 g/cm3 , Young��s modulus E = 2.0 �� 106 g/(cm s2 ) and Poisson��s ratio �� = 0.35. The ﬂuid material properties are density ��f = 1.18��10−3 g/cm3 and viscosity �� = 1.82��10−4 g/(cm s). The inﬂow velocity is taken as v = 31.5 cm/s, leading to a Reynolds number Re = 204 if the length of ¯ the square rigid body is taken as the characteristic length. The ﬁnite element mesh used for this problem is shown in Fig. 5.2 and consists of 14218 three-node triangular elements and 7278 nodes for the ﬂuid part, and 20 nine-node quadrilateral elements in one layer over the thickness with 123 nodes for

5.8 Example Problems

153

the structural part using plane stress conditions. The ﬂuid part uses the second order backward diﬀerencing BDF2 time integration scheme, while the generalized�� time integration method with ��s = 0.8 is used for the structural part. The �� time interval studied is of 25 s using a time step size ��t = 0.005. Since the density ��s >> ��f , a staggered coupling technique is used together with a structural predictor of 1st order.

Figure 5.2

Reference ﬁnite element mesh

For this example, the FIC stabilization technique has been employed. Mesh movement is obtained by employing the pseudo-elastic structural technique presented in section 5.5. Initially the ﬂuid and the structure are at rest, and at t = 0 the inﬂow velocity is applied instantaneously. Fig. 5.3 shows the maximum vertical tip displacement of the structure, that oscillates mainly in the second mode and it is compared with the work of H��bner et al. (2004). u The ﬁrst two natural frequencies of the structural part in its initial conﬁguration s s become f req1 = 0.607 Hz and f req2 = 4.087 Hz. The dominant frequency of the ﬂuid ﬁeld yields f req f = 3.71 Hz. A Fourier analysis of the coupled problem for lift forces fL is performed in order to ﬁnd the dominant frequency of the solution. Fig. 5.4 shows the Fourier spectrum obtained, leading to a frequency for the couple problem f req c = 3.22 Hz. This value is in good agreement with the result reported by H��bner et al. (2004) with a u frequency of the couple problem of 3.10 Hz. It can be seen that even the dominant frequency of the ﬂuid ﬁeld and the second natural frequency of the structural part are similar, the coupled problem dominant frequency is lower that both of them. This may be though as a consequence of the inﬂuence of the ﬁrst frequency of the structure over the couple problem. A close-up of the ﬂuid mesh around the shell is given in Fig. 5.5 for a time instant when the structure has a maximum tip displacement. There can also be observed that the structure displacements oscillates in its second mode of vibration. Figs. 5.6-5.7 show diﬀerent time instants of the pressure ﬁeld and its corresponding structural displacements.

154

5. Fluid-Structure Interaction

Present work 1.0 0.8 0.6

H��bner

y-displacement [cm]

0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 0 5 10 time [s] 15 20 25

Figure 5.3

0.0018 0.0016 0.0014 0.0012 0.0010 0.0008 0.0006 0.0004 0.0002 0.0000

History of maximum vertical tip displacement

Fourier coefficient

0

5

10 frequency

15

20

Figure 5.4

Fourier spectrum of the lift forces

Figure 5.5

Detail of the ﬂuid mesh around the shell

5.8 Example Problems

155

t = 22.575 s

t = 22.600 s

t = 22.675 s

-2.32

[g/(cm s2)]

1.38

Figure 5.6

Pressure ﬁelds for diﬀerent time steps

156

5. Fluid-Structure Interaction

t = 22.750 s

t = 22.775 s

t = 22.825 s

-2.32

[g/(cm s2)]

1.38

Figure 5.7

Pressure ﬁelds for diﬀerent time steps (cont.)

5.8 Example Problems

157

5.8.1.2

Wall��s problem

H��bner��s problem is solved again, but with the material properties and boundary u conditions proposed by Wall and Ramm (1998). This problem is more complicated to solve that the one proposed by H��bner et al. (2004). The material properties u are given for the structure with a density ��s = 0.1 g/cm3 , Young��s modulus E = 2.5 �� 106 g/(cm s2 ) and Poisson��s ratio �� = 0.35. The ﬂuid material properties are density ��f = 1.18 �� 10−3 g/cm3 and viscosity �� = 1.82 �� 10−4 g/(cm s). The inﬂow velocity is taken as v = 51.3 cm/s, leading to a Reynolds number Re = 333 if the ¯ length of the square rigid body is taken as the characteristic length. The ﬁnite element mesh used for the coupled problem is the same that the one used for the former problem and it is shown in Fig. 5.2. The ﬂuid part uses the generalized-�� time integration scheme with ��f = 0.8 while the generalized-�� �� time integration method with ��s = 0.5 is used for the structural part. The time �� interval studied is of 10 s using a time step size ��t = 0.005. In this example, the OSS stabilization technique has been employed. After 100 initial steps for the ﬂuid with the inﬂow velocity applied instantaneously, the coupling between the ﬂuid and the structure is performed. For this problem, a strongly coupling procedure is used with a structural predictor of 1st order and without relaxation. Fig. 5.8 shows the maximum vertical tip displacement of the structure, that oscillates mainly in the ﬁrst mode and it is compared with the work of Steindorf (2002).

Present work 1.5 Steindorf

1.0

y-displacement [cm]

0.5

0.0

-0.5

-1.0

-1.5

1.0

1.5

2.0

2.5 time [s]

3.0

3.5

4.0

Figure 5.8

History of maximum vertical tip displacement

The ﬁrst two natural frequencies of the structural part in its initial conﬁguration s s become f req1 = 3.033 Hz and f req2 = 19.023 Hz. The dominant frequency of the ﬂuid ﬁeld yields f req f = 5.761 Hz. A Fourier analysis of the coupled problem for

158

5. Fluid-Structure Interaction

lift forces fL is performed in order to ﬁnd the dominant frequency of the solution yielding a frequency for the couple problem f req c = 3.125 Hz. In this case, there are no similarities between the dominant frequency of the ﬂuid and the structural part. However the couple dominant frequency is higher than the ﬁrst frequency of the structure, maybe because of the little inﬂuence of the ﬂuid dominant frequency over the couple interaction problem. Figs. 5.9-5.10 show diﬀerent time instants of the pressure ﬁeld and its corresponding structural displacements, corresponding mainly to the oscillation of the ﬁrst structural mode.

t = 7.85 s

t = 7.88 s

-4.13

[g/(cm s2)]

2.33

Figure 5.9

Pressure ﬁelds for diﬀerent time steps

5.8 Example Problems

159

t = 7.92 s

t = 8.01 s

-4.13

[g/(cm s2)]

2.33

Figure 5.10

Pressure ﬁelds for diﬀerent time steps (cont.)

160

5. Fluid-Structure Interaction

5.8.2

Cavity with Flexible Bottom Membrane in 2D

This example is taken from the work of Mok (2001) which consists of a modiﬁcation of the well known wall-driven cavity ﬂow problem used in ﬂuid dynamics. In this problem the originally constant top velocity is change for an oscillatory top velocity, including the corner nodes, given by v = 1 − cos ¯ 2��t 5 (5.8.1)

yielding a velocity interval between vx = 0 m/s and vx = 2 m/s. The cavity bottom originally ﬁxed is changed for a ﬂexible bottom membrane. Also the ﬂuid inﬂow and outﬂow is allowed near the top face which allow the volume to change in time. Geometry and boundary conditions for this problem are shown in Fig. 5.11.

vx=v vx=v, vy=0 p=0 0.125

vx=0 vy=0

vx=0 vy=0

0.875

vx=0, vy=0 0.002 bottom membrane 1.0

Figure 5.11

Geometry and boundary conditions [m]

The ﬂuid mesh consists of a four-node quadrilateral mesh of 32 �� 32 divisions and the structural part is constructed with 16 nine-node quadrilateral plane-stress elements. The ﬂuid properties are taken with a density ��f = 1.0 Kg/m3 and a viscosity �� = 0.01 m2 /s. For the structure, the density is taken as ��s = 500 Kg/m3 , Young��s modulus E = 250.0 N/m2 and a Poisson ratio of �� = 0.0. The time interval analyzed is t �� [0.0, 70.0], using a time-step size of ��t = 0.01 s. Time integration for the ﬂuid is made with the generalized-�� method with ��f = 0.9 and �� the structural time integration uses the generalized-�� technique with ��s = 0.8. �� As shown in Mok (2001), this example can be solved without any relaxation method. In this case, the number of Gauss-Seidel coupling iterations needed to reach a tolerance of 10−6 varies between 12 and 20 iterations. If the relaxation method is used with a ﬁxed value w = 0.825, the range of iterations to reach the tolerance is reduced between the interval 8 to 10 iterations. Finally if the Aitken optimal relaxation parameter is computed automatically, the number of

5.8 Example Problems

161

iterations to reach the tolerance is reduced to an interval between 6 and 8 iterations. Consequently, the Aitken relaxation method shows an excellent behavior that is preferable to use instead of the standard block Gauss-Seidel coupling technique. Fig. 5.12 shows the vertical displacement of the membrane midpoint. It can be noted that the present solution is indeed diﬀerent from the solution presented by Mok. The reason is that not all the boundary conditions used by Mok to solve the problem are available in the reference cited. The amplitude obtained by Mok is of 7.3 cm while the amplitude obtained in this work is of 7.4 cm. The period of both solutions is the same with a value of 5.12 s. The main diﬀerence between both works is that while Mok yield a mean vertical displacement of 18.1 cm, the present solution has a mean vertical displacement of 23.5 cm. However the general tendency of the coupled problem is the same. Also Rossi (2005) ﬁnd this general tendency using cable elements in the bottom of the cavity instead of nonlinear plane-stress elements as used in this work.

Present work 0.3 Mok

vertical displacement [m]

0.2

0.1

0.0

-0.1 0 10 20 30 time [s] 40 50 60 70

Figure 5.12

Bottom midpoint vertical displacement

In this example the structural density is two orders of magnitude greater than the ﬂuid density and it can be thought that also the faster staggered coupling technique can be employed in the problem solution. However Mok (2001) showed that staggered schemes failed to obtain the problem solution even if any predictor is used and only strong coupling iterative methods yield a stable long-time solution. Fig. 5.13 shows diﬀerent time instants of the pressure ﬁeld together with their deformation associated to the bottom membrane displacements. In this ﬁgure, when time t = 2.54 s the vertical displacement is just beginning. After time t = 17.04 s, the deformation cycle is repeated with a period of 5.12 s. From Fig. 5.12, it can be seen that for time t = 17.04 s, the minimum displacement is presented and has a value of 19.8 cm. Graphically this deformation is shown in Fig. 5.13. This value is repeated every 5.12 s. For time t = 19.54 s, the maximum displacement is given with a value of 27.2 cm, and it is repeated every 5.12 s.

162

5. Fluid-Structure Interaction

t = 2.54 s

t = 17.04 s

t = 19.54 s

-0.394

[N/m2]

0.898

Figure 5.13

Pressure ﬁelds for diﬀerent time instants

5.8 Example Problems

163

5.8.3

Cavity with a Thin Bottom Shell in 3D

This example is also taken from the work of Mok (2001) which consists of an extension to 3D of the well known wall-driven cavity ﬂow problem used in ﬂuid dynamics. In this problem the originally constant top velocity is change for an oscillatory top velocity, including the corner edges, given by v = 1 − cos ¯ 2��t 5 (5.8.2)

yielding a velocity interval between vx = 0 m/s and vx = 2 m/s. The cavity bottom originally ﬁxed is changed for a thin bottom shell. Also the ﬂuid inﬂow and outﬂow is allowed near the top face which allow the volume to change in time, everywhere else the velocity is ﬁxed to zero in all directions. Geometry and boundary conditions for this problem are shown in Fig. 5.14.

vx=v vx=v p= 0 0.125

vx= 0 vx= 0

0.875

1.0

1.0

Bottom Shell

Figure 5.14

Geometry and boundary conditions [m]

The ﬂuid mesh consists of an eight-node hexahedral mesh of 24��24��24 divisions and the structural part is constructed with 1152 three-node triangular rotation-free shell elements with 625 nodes. The ﬂuid properties are taken with a density ��f = 1.0 Kg/m3 and a viscosity �� = 0.01 m2 /s. For the structure, the density is taken as ��s = 500 Kg/m3 , Young��s modulus E = 250.0 N/m2 , Poisson ratio of �� = 0.0 and the thickness is 0.002 m. The time interval analyzed is t �� [0.0, 70.0], using a time-step size of ��t = 0.1 s. Time integration for the ﬂuid is made with the generalized-�� method with ��f = 0.9 and the structural time integration uses the �� generalized-�� technique with ��s = 0.9. ��

164

5. Fluid-Structure Interaction

As shown by Mok (2001), this problem can be solved with the staggered coupling scheme using a displacement predictor of order zero. If a displacement predictor of order 1st or 2nd is used, the coupling scheme fails. If the Gauss-Seidel coupling technique is used, the method converges yielding a stable long-time solutions. In this example the iterative coupling scheme is used together with the automatic Aitken relaxation method for faster convergence. The number of iteration needed to reach a tolerance of 10−6 lies between 3 and 4 per time step with the Aitken method. Fig.5.15 shows the vertical displacement of the shell middle point. As in the previous problem, the solution obtained is diﬀerent from the solution presented by Mok. Again the reason is that not all boundary conditions used by Mok to solve this problem are available in the reference cited. The amplitude obtained by Mok is of 6.0 cm while the amplitude obtained in this work is of 8.8 cm. The period of both solutions is the same with a value of 5.12 s. Another diﬀerence between both works is that while Mok yield a mean vertical displacement of 11.0 cm, the present solution has a mean vertical displacement of 20.1 cm. However the general tendency of the coupled problem is the same.

0.3

vertical displacement [m]

0.2

0.1

0.0

-0.1 0 10 20 30 time [s] 40 50 60 70

Figure 5.15

Shell midpoint vertical displacement

Figs. 5.16-5.17 show diﬀerent time instants of the bottom shell deformation together with their displacement ﬁeld. Fig. 5.16 shows the vertical displacement for time t = 4.0 s and t = 6.0 s where the deformation process has just began. After time t = 21.8 s the deformation process become cyclical with a period of 5.12 s. From Fig. 5.15, it can be seen that for time t = 21.8 s, the minimum vertical displacement of the cycle is 16.3 cm, which is plotted in Fig. 5.17 and is repeated each 5.12 s. For time t = 24.3 s, the maximum vertical displacement is given with a value of 25.3 cm, which is repeated every 5.12 s.

5.8 Example Problems

165

t = 4.0 s

t = 6.0 s

-0.004

[m]

0.253

Figure 5.16

Shell vertical displacement ﬁeld

166

5. Fluid-Structure Interaction

t = 21.8 s

t = 24.3 s

-0.004

[m]

0.253

Figure 5.17

Shell vertical displacement ﬁeld (cont.)

5.8 Example Problems

167

5.8.4

Channel with Flexible Wall

This model problem was originally proposed by Mok (2001) and Wall and Mok (2001) as a challenging test for coupled problems involving similar densities between the ﬂuid and the structure. Obviously, if the Gauss-Seidel strong coupling method is used, this kind of problems can only be solved if a relaxation technique is employed. Standard Gauss-Seidel methods fail to solve problems with similar densities. Geometry and boundary conditions of the problem are given in Fig. 5.18.

vy = 0

0.20

A 0.005 B vx = 0, vy = 0 Flexible Wall 1.25

vx = 0, vy = 0 0.50

Figure 5.18

0.25

0.25

vx=v

p=0

Channel geometry and boundary conditions [m]

The material properties are taken for the structure with a density ��s = 1500.0 kg/m3 , Young��s modulus E = 2.3 �� 106 N/m2 and Poisson��s ratio �� = 0.45. The ﬂuid material properties are density ��f = 956.0 Kg/m3 and viscosity �� = 0.145 kg/(m s). The inﬂow velocity is parabolic and is applied slowly with the function shown in Fig. 5.19, where after 10 s the x-velocity component is kept constant with a value of 0.06067 m/s.

t v = 0.06067 1 - cos 2 10

(

)

25

[m/s]

0.06067 0 0 5 10 [s] 15 20

Figure 5.19

Channel inﬂow x-velocity component

The ﬁnite element mesh used for this problem has 6008 three-node triangular elements and 3218 nodes for the ﬂuid part, and 6 nine-node quadrilateral elements in one layer over the thickness with 39 nodes for the structural part using geometrically nonlinear plane stress conditions. The ﬂuid part uses the generalized-�� time integration method with ��f = 0.9, while the generalized-�� time integration �� method for second-order equations with ��s = 0.6 is used for the structural part. �� The time interval studied is 25 s using a time step size ��t = 0.1. The FIC stabi-

168

5. Fluid-Structure Interaction

lization technique has been employed in this example. Mesh movement is obtained by using the pseudo-elastic structural technique. Fig. 5.20 shows a comparison between the pressure history solution obtained by Mok (2001) and the present work. At point B, the solution of both works is practically the same, while point A present little variations. However the solution of point A is the same that the solution given by Rossi (2005).

25 Present work, A 20 B Present work, B Mok, A Mok, B 10 A 5 0 0 5 10 time [s] 15 20 25

pressure [N/m 2]

15

Figure 5.20

Time history for pressure

Fig. 5.21 shows a comparison between the x-velocity component time history solution. The tendency between both works is the same, presenting little changes probably because of the mesh and geometry studied.

10.0 Present work, A Present work, B

x-displacement [cm ]

7.5

A

5.0

Mok, A Mok, B

2.5 B 0.0 0 5 10 time [s] 15 20 25

Figure 5.21

Time history for x-displacement component

Figs. 5.22-5.23 show diﬀerent time instant of ﬂuid-structure interaction for the x-velocity component together with its associated deformation. This solution is in accordance with the solution presented by Mok (2001) and Wall and Mok (2001).

5.8 Example Problems

169

t = 2.5 s

t = 5.0 s

t = 7.5 s

-4.23

[m/s]

12.75

Figure 5.22

Contour ﬁeld for x-velocity component and deformed geometry

170

5. Fluid-Structure Interaction

t = 10.0 s

t = 15.0 s

t = 25.0 s

-4.23

[m/s]

12.75

Figure 5.23

Contour ﬁeld for x-velocity component and deformed geometry (cont.)

5.8 Example Problems

171

5.8.5

Pressure Pulse in a Compliant Vessel

This model problem was originally proposed by Nobile (2001) and Formaggia et al. (2001) to consider the ﬂuid-structure interaction arising in the modelling of blood ﬂow in large arteries. Recently Fern��ndez and Moubachir (2005) has made a variaa tion at the inlet data using the same geometry and physical parameters to demonstrate the quadratic convergence of their FSI formulation. The problem consists of a thin elastic vessel, that here is modelled with rotationfree shell elements, conveying the blood ﬂow modelled by an incompressible ﬂuid using the Navier-Stokes equations . The geometry used for the computation has been made with a straight cylinder of radius r0 = 0.5 cm and length L = 5 cm. The ﬂuid physical parameters are taken as those given by Formaggia et al. (2001) with a density ��f = 1 g/cm3 and viscosity �� = 0.03 g/(cm s). The surrounding solid parameters are density ��s = 1.2 g/cm3 , Young��s modulus E = 3.0 �� 106 g/(cm s2 ), Poisson ratio �� = 0.3 and a thickness of 0.1 cm. The structure is clamped at the inlet and outlet. The ﬂuid is initially at rest and an overpressure of 1.3332 �� 104 g/(cm s2 ) (10 mmHg) has been imposed at the inlet boundary for 0.003 s. To introduce this Neumann boundary condition, Eq. (4.2.6) has to be added to the momentum equation. The ﬂuid mesh consists of 1600 hexahedral eight-node elements and 1869 nodes, while the structure mesh has 1280 three-node triangular rotation-free shell elements and 656 nodes. The structure has also been modelled with 640 hexahedral eightnode elements with two layers over the thickness and 1008 nodes for comparison purposes. The ﬂuid solver uses the generalized-�� time integration method with ��f = 0.9, while the solid solver uses the generalized-�� time integration method �� with ��s = 0.8. The time interval studied is of 0.01 s using a time step size of �� ��t = 0.0001 s. The FIC stabilization technique has been employed in this example. Mesh movement is obtained by using the Laplacian method. The Gauss-Seidel strong coupling technique is used together with the automatic Aitken relaxation method. The number of iterations needed to reach a tolerance of 10−6 lies between 12 and 15 per time step. Figs. 5.24-5.25 shows a pressure wave propagation together with the ﬂuid deformed conﬁguration for diﬀerent time steps. A reﬂection wave is expected due to the numerical side-eﬀect of the outlet boundary condition. As can be seen, these results are similar to those given by Formaggia et al. (2001) and Fern��ndez and a Moubachir (2005). Fig. 5.26-5.27 shows the half solid deformed conﬁguration of the rotational free shell structure compared with the solid deformed conﬁguration of the hexahedral structure. It can be observed that the solution obtained with the rotation-free shell elements is practically the same that the solution found with the 3D solid elements. But the advantage of using the shell elements is the faster solution in the CSD solver. It is not the purpose of this example to study realistic physiological simulation of blood-arteries interaction, but to show the capabilities of the coupling formulation implemented to deal with this kind of problems. Displacements of all the ﬁgures in this example have been magniﬁed by a factor of 20 for clarity. Fig. 5.28 shows the time history analysis for radial displacements at diﬀerent

172

5. Fluid-Structure Interaction

t = 0.0025 s

-9300

[g/cm s2]

15500

t = 0.005 s

-9300

[g/cm s2]

15500

Figure 5.24

Wave propagation generated by pressure pulse at the inﬂow

5.8 Example Problems

173

t = 0.0075 s

-9300

[g/cm s2]

15500

t = 0.01 s

-9300

[g/cm s2]

15500

Figure 5.25

Wave propagation generated by pressure pulse at the inﬂow (cont.)

174

5. Fluid-Structure Interaction

Shell

t = 0.0025 s

Solid

0.0

[cm]

0.013

Shell

t = 0.005 s

Solid

0.0

[cm]

0.013

Figure 5.26

Structure deformed conﬁguration. Displacements norm

5.8 Example Problems

175

Shell

t = 0.0075 s

Solid

0.0

[cm]

0.013

Shell

t = 0.01 s

Solid

0.0

[cm]

0.013

Figure 5.27

Structure deformed conﬁguration. Displacements norm (cont.)

176

5. Fluid-Structure Interaction

0.015

L=5 l=

0.010

0.25 L 0.50 L L=15 0.75 L l=

L=10 l=

radial displacements [cm]

0.005

0

-0.005

-0.010

-0.015

0

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

time [s]

Figure 5.28

Time history for radial displacements

lengths from the inlet pressure: l = 0.25L cm, l = 0.50L cm and l = 0.75L cm. Since the outlet pressure is set to zero, it is expected that the maximum radial displacements become smaller when the distance to the outﬂow is closer.

15000

L=5 l=

10000

0.25 L L L=15 0.75 L l=

L=10 0.50 l=

pressure [g/cm s 2]

5000

0

-5000

-10000

-15000

0

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

time [s]

Figure 5.29

Time history for pressure

Fig. 5.29 shows the time history analysis for the pressure at diﬀerent lengths

5.8 Example Problems

177

from the inlet pressure. Also the eﬀect of the reﬂection wave can be observed in Figs. 5.28-5.29 due to the numerical side-eﬀect of the outlet boundary condition that develop negative pressures near this boundary. It is important to remark that the ﬂuid part of this example was solved using the predictor-corrector scheme because the fractional step method lead to some oscillations in the solution. A way to avoid these oscillations is the use of the fractional step method twice per time step. In this form, oscillations are drastically reduced, but still the predictor-corrector method gives a smoother solution.

178

5. Fluid-Structure Interaction

Chapter 6

Conclusions

This thesis deals with the analysis of structures with large deformations, incompressible ﬂuid ﬂows and ﬂuid-structure interaction problems, which objectives pursued were introduced in section 1.3 and have been fulﬁlled satisfactorily. An algorithm to solve these three ﬁeld problems has been developed and implemented in the context of the ﬁnite element method. Structures. The structural part of this work is concerned with the geometrically nonlinear analysis of membrane and rotation-free shell structures. Improvement to the formulations already existing is began with the continuum expression of the virtual internal work in curvilinear coordinates. Then a new formulation based on the principal ﬁber orientation is performed. The idea to use the ﬁber orientation of the material for our analysis comes from the manufacturing process where a direction for the principal ﬁber of the material is needed to build the structure. The ﬁber orientation is mesh dependent and has great advantages, which can be stated as follow: • It is based on an idea which is very simple to understand. • A suitable implementation as a preprocess in a ﬁnite element code makes it a true alternative to take into account for structural analysis, since a wider range of problems can be solved. • Allows adequate postprocess of the problem being analyzed since any number of patches that conform the structure can be postprocessed with diﬀerent local orientations. • It allows to perform analysis with orthotropic and composite materials including structures having an initially out-of-plane conﬁguration. Following a detailed explanation of the classical membrane theory, the inclusion of the ﬁber orientation allows to improve the membrane formulation, where the classical membrane theory is considered as a particular case of this new and general formulation. Even though the new ﬁnite element discretization for the membrane formulation is expressed in a general context, in particular the triangular three-node ﬁnite 179

180

6. Conclusions

element is implemented. This particularity is done with the intention to be used as a part of the rotation-free shell formulation considered in this work. An important aspect of the new formulation is the possibility to add prestressed forces to the membrane structure in a simple and almost direct way. This is possible because the new formulation of the three dimensional membrane structure has its manufacturing ﬁbers already oriented and then the prestressed forces can be added directly to the internal forces. Since conventional membrane theory allows compression stresses, a wrinkling algorithm based on modifying the constitutive equation has been developed. Without this wrinkling algorithm, ﬁnite element analysis of membranes would lead to inexistent compression stresses that would yield a ﬁnal conﬁguration of the membrane structure completely diﬀerent to the real one. On the other hand, conventional shell formulations require a lot of memory and computing time for their analysis. Therefore shell formulations with displacements as the only degrees of freedom have been investigated by several authors. In this work, the Kirchhoﬀ-Love theory of thin shells is studied. Since the virtual internal work for this kind of shells is built with a membrane and a bending part, the new membrane formulation is used to form the membrane part of these elements. For the bending part, the new formulation presented is based on the ﬁber orientation technique. The choice to use triangular three-node ﬁnite elements for the rotationfree shell discretization is due to the higher approximation of the solution compared to other type of ﬁnite elements used in these kind of shell formulations. The basic shell triangle formulation of Flores and O˜ate (2001) has been reforn mulated using the ﬁber orientation technique. The resulting new formulation has improved its range of applications for engineering problems. As in the case of membranes, the new rotation-free shell formulation can be used to analyze orthotropic and prestressed shells. Usually the Newmark time integration scheme is used in structural dynamic problems. However geometrically nonlinear analysis for long periods of time introduce spurious high frequencies that spoil the solution. In order to minimize unwanted low frequency dissipation while achieving high frequency dissipation without regarding the order of accuracy, other time integration schemes are studied. The easiest method to implement in a ﬁnite element code is the Bossak method. Another widely used method is the Hilber-Hughes-Taylor time integration technique. These last two methods produce very similar solutions for large deformation structural problems. However the generalized-�� strategy is the best choice for this kind of problems, since it can reproduce the solution of the last two methods mentioned before, and provide a third option for better results. The structural problems solved with the improved membrane and shell formulations have demonstrated their robustness and accuracy for the diﬀerent variety of applications presented. Fluids. The ﬂuid part of this work is concerned with the solution of the NavierStokes and continuity equation for incompressible ﬂow problems. The monolithic solution of these equations take great computer eﬀort to solve the nonlinear system of equations, specially for 3D problems. Instead of using a monolithic solu-

181

tion for incompressible ﬂow problems, pressure segregation methods have earned a widespread popularity because of the computational eﬃciency given by the uncoupling of the pressure from the velocity ﬁeld. In this work, the fractional step and predictor-corrector methods are explained. Three well known stabilization techniques have been implemented in the ﬁnite element code used in this work. They are the SUPG/PSPG, OSS and FIC stabilization methods with equal-order interpolation ﬁnite elements. Until now, the SUPG/PSPG stabilization method have been used with monolithic solvers for incompressible ﬂow problems. In this work, the SUPG/PSPG scheme is developed to be used with fractional step and predictor-corrector methods following the ideas that solve the OSS and FIC techniques with pressure-segregation methods. Additionally, the generalized-�� time integration scheme for ﬂuids is adapted to these three stabilization techniques with pressure segregation methods, which has only been used for monolithic solutions. As in the case of structural problems, the generalized-�� technique for ﬂuids seeks to minimize unwanted low frequency dissipation while achieving high frequency dissipation without regarding the order of accuracy. Fluid problems extensively studied by analytical, numerical and experimental methods are often used as benchmark problems to validate numerical simulations. In this work, some well-known example problems have validated and demonstrated the good performance of the ﬂuid formulations implemented. Coupling. From the available types for coupling ﬂuid and structural problems, the strong coupling block Gauss-Seidel partitioned technique is implemented in this work. The ﬂuid-structure interaction problem is built with the structure and ﬂuid contributions of this work. A very important aspect to take into account for ﬂuid-structure interaction problems with large structural deformations is the time integration scheme used for the simulation. Usually this kind of problem are studied for long periods of time, which introduced spurious high frequencies that spoil the solution. One way to avoid this problem is to reduce the time step size used for the analysis that sometimes is unacceptable. The best way to solve this problem is to used time integration schemes that minimize unwanted low frequency dissipation while achieving high frequency dissipation without regarding the order of accuracy. In this work, the method suggested to use is the generalized-�� technique for both, the structural and the ﬂuid parts. Many ﬂuid-structure interaction problems are solved with staggered coupling techniques, which are only acceptable for problems where the added mass eﬀect do not have inﬂuence in the structure. However this kind of problem may become unstable for long time periods of study. To avoid this problem, strong coupling partitioned schemes are advised. However when the added mass eﬀect plays an important role in the structure, partitioned methods with block Newton schemes are an excellent choice. Another option is the block Jacobi or block Gauss-Seidel method with relaxation techniques. These last two methods are useless if they do not include the relaxation technique. In this work, the strong coupling block Gauss-Seidel partitioned method with a re-

182

6. Conclusions

laxation technique based on Aitken��s method is implemented, yielding an excellent solution to all the examples presented. Finally, a detailed algorithm for the ﬂuid-structure interaction problem using the strong coupling block Gauss-Seidel partitioned method is presented. In there, a relaxation technique with Aitken��s method is explained. However, other relaxation techniques can be added with minor modiﬁcations.

6.1

Achievements

The achievements made in this work are summarized as follows: • A methodology for the ﬁber orientation of materials used for the construction of light-weight structures is developed based on the ﬁnite element mesh of the discretized problem. • Development of an improved membrane formulation based on the ﬁber orientation, where the classical membrane formulation can be considered as a particular case of this new formulation. • The new membrane formulation allows the analysis of orthotropic membrane structures. The initial conﬁguration of the membrane can be ﬂat or curved. • The inclusion of prestressed forces is accounted for in the new membrane formulation, since usually membranes have a prestressed ﬁeld that give the design shape to the structure. • Development of an improved rotation-free shell formulation based on the ﬁber orientation. • The new rotation-free shell formulation allows to analyze orthotropic and prestressed structured. • Local postprocess for membrane and shell structures with diﬀerent principal ﬁber orientations. • Development of the incompressible ﬂow formulation with the SUPG/PSPG stabilization technique in the context of pressure segregation methods. Specially for fractional step and predictor-corrector methods. • Adaptation of the generalized-�� time integration scheme for ﬂuids to fractional step and predictor-corrector methods, using the SUPG/PSPG, OSS and FIC stabilization techniques. • Coupling of large deformation membrane and shell structures with incompressible ﬂuid ﬂow formulations. Membrane and shell new formulation developed in this work are used in the structural part. Fluid improvements with the generalized-�� technique for pressure segregation methods are used in the ﬂuid part.

6.2 Possible Future Research Lines

183

6.2

Possible Future Research Lines

The following research lines to continue the development of the structure, ﬂuid and ﬂuid-structure interaction problems of this work are suggested: • To develop a constitutive equation for membrane and shell ﬁnite elements capable of dealing with composite materials for the study of recent structural applications. • Improving the computational ﬂuid dynamic solver to account for turbulence models and wall-functions will allow the solution of high mean ﬂow Reynolds numbers problems. • When excessive large displacements of the structure are transferred to the moving mesh algorithm, it may fail. Then it is necessary to remesh the ﬂuid domain. Therefore the implementation of an internal remesh solver in the actual code will be needed. • Implementation of strong coupling block Newton partitioned methods for ﬂuid-structure interaction problems. • Parallelization of the code to save computer time for complex analysis. • Develop a form-ﬁnding algorithm.

184

6. Conclusions

References

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Index

��-method time integration scheme, 71 ��-method time integration scheme, 112 acceleration material, 11 added mass eﬀect, 145 aeroelasticity, 137, 145 ALE formulation, 137 Backward diﬀerentiation method time integration scheme, 113 balance of momentum, 16 base vectors contracovariant, 34 covariant, 34, 41 Bossak method time integration scheme, 71 boundary Dirichlet, 25 Neumann, 25 cable linearization, 79 cable element formulation, 54 semidiscretization, 55 conﬁguration current, 10 deformed, 10 initial, 9 reference, 9 conjugate in power, 20 conservation equations, 15 angular momentum, 17 energy, 19 linear momentum, 16 mass, 15 constitutive equation plane stress isotropic material, 47 orthotropic material, 47 constitutive equations, 21 linear elasticity, 22 Newtonian ﬂuids, 23 nonlinear elasticity, 22 continuity equation, 16 contravariant base vectors, 34, 60 convection, 106 convective term, 11 convective velocity, 141 coordinate system curvilinear, 33 rectangular Cartesian, 10 coupled incompressible ﬂow equations, 111 coupling strategies monolithic methods, 137 partitioned methods, 137 covariant base vectors, 34, 41, 59 damping structural, 73 viscoelastic, 73 deformation, 12 gradient tensor, 12, 14, 35 density, 15 Dirichlet boundary, 25 displacement, 11 gradient tensor, 12 divergence, 15, 18 drag coeﬃcient, 129 dynamic viscosity, 106 197

198

Index

elastic constants, 21 elasticity linear theory, 21 nonlinear theory, 22 energy conservation, 19 internal, 19 kinetic, 19 equations of motion, 31 equilibrium equation, 18 Euler equation, 24 Eulerian coordinates, 10 description, 11 Eulerian elements, 106 Eulerian formulation, 17 external forces, 30, 51 ﬁber orientation, 36 ﬁnite element discretization for Eulerian formulations, 108 for total Lagrangian formulations, 27, 40, 61 ﬁnite element semidiscretization, 27, 31, 40, 61 ﬁrst law of thermodynamics, 19 ﬂuid incompressible, 105 inviscid, 105 ﬂuid-structure interaction, 137 governing equations, 142 staggered coupling, 139 strong coupling, 149 force resultants, 67 surface, 14 forces body, 16 pressure follower, 36 fractional step method, 115 function continuity, 26 Galerkin-type weak form, 26, 106, 107 Generalized-�� method time integration scheme, 72, 114 gradient

left, 11 right, 11 Green-Lagrange strain tensor, 13, 35, 42, 60 hemodynamics, 145 homogeneous Dirichlet boundary condition, 107 incompressible ﬂow equations, 111 ﬂuids, 23 material, 16 inertial forces, 31 integration by parts, 26 internal energy, 19 internal forces, 28, 29, 43, 45, 46, 50, 57, 68 isoparametric coordinates, 40 isotropic material, 21, 47 Jacobian determinant, 12 matrix, 12 Jacobian transformation tensor, 44, 51, 63 kinematic pressure, 108 viscosity, 108 kinematics, 9 kinetic energy, 19 kinetic forces, 30 Kirchhoﬀ-Love theory, 60 Kronecker delta, 12, 35 Lagrangian coordinates, 10 description, 10 Lagrangian formulation total, 18 updated, 17 Laplacian, 24 Laplacian operator, 116 lift coeﬃcient, 129

Index

199

linearization cable, 79 membrane, 76 shell, 79 Mach number, 107 mass, 15 conservation, 15 conservative form, 16 mass matrix, 30 master element coordinates, 40 material coordinates, 10 deformation gradient tensor, 12 description, 10 displacement gradient tensor, 12 isotropic, 47 orthotropic, 47 point, 9, 10 time derivative, 11 material motion, 140 membrane ﬁnite element semidiscretization, 40 formulation, 33 linearization, 76 slack state, 52 state-of-the-art, 31 structures, 31 taut state, 52 thickness, 50 wrinkle state, 52 wrinkling algorithm, 52 membrane analysis isotropic, 36 orthotropic, 36 membrane element discretization, 40 ﬁber orientation, 36 implementation, 47 theory, 33 mesh movement techniques, 148 update equations, 148 mesh motion, 140 metric tensor, 34, 42, 60 moment resultants, 67

momentum equation, 17 Eulerian description, 17 Lagrangian description, 17 weak form, 26 motion material, 140 mesh, 140 Navier-Stokes equation, 24 Neumann boundary, 25 Newmark time integration scheme, 69 Newton-Raphson method, 74 Newtonian ﬂuids, 23 nonlinear solution methods Newton-Raphson, 74 Picard, 124 normal, 14 orthotropic material, 47 parent element coordinates, 40 partial diﬀerential equations, 15 partitioned methods staggered coupling, 138, 139 strong coupling, 138 weak coupling, 138 Picard iteration method, 124 postprocess strain and stress ﬁeld, 36 predictor-corrector method, 117 pressure kinematic, 108 thermodynamic, 23 pressure follower forces, 36, 51 pressure segregation methods fractional step, 115 predictor-corrector, 115, 117 prestressed forces, 36, 46, 61 principle of virtual work, 25, 26 quasi-static analysis, 74 rate of deformation tensor, 13 rate of rotation tensor, 13 Reynolds number, 24 Rodrigues�� rotation formula, 39

200

Index

rotation formula, 39 rotation matrix strains, 45 stresses, 45 rotation-free shell triangle element, 62

SGS, 119 SUPG, 119 SUPG/PSPG, 120 TG, 120 staggered partitioned coupling, 139 static problems, 18 Stokes ﬂow, 24 Saint Venant-Kirchhoﬀ material, 22, 47, strain 59 energy function, 21 shape functions, 27, 40, 108 rate tensor, 13 three-node triangle, 47 strain tensor two-node linear, 55 Green-Lagrange, 13, 35, 42 shell inﬁnitesimal, 13 ﬁnite element semidiscretization, 61 strain-displacement matrix, 43, 45, 46, Kirchhoﬀ-Love theory, 57, 60 49, 57, 67 linearization, 79 strain-displacement tensor, 28, 43 Reissner-Mindlin theory, 57 stress rotation-free, 58 prestressed ﬁeld, 36 state-of-the-art, 57 resultants, 67 shell analysis stress tensor isotropic, 61 Cauchy, 14 orthotropic, 61 ﬁrst Piola-Kirchhoﬀ, 14 shell element nominal, 14 discretization, 61 second Piola-Kirchhoﬀ, 14, 35 formulation, 59 true physical, 14 theory, 57 Strong coupling solution strategies Block Gauss-Seidel, 146 Newton-Raphson method, 74 Block Jacobi, 146 Picard iteration method, 124 exact Block Newton, 147 space inexact Block Newton, 147 of the test functions, 25 strong form, 25 of the trial functions, 26 Strouhal number, 131 spatial structural damping, 73 coordinates, 10 surface deformation gradient tensor, 14 force, 14 description, 11 traction, 14 rate of deformation tensor, 13 rate of rotation tensor, 13 tangent space, 34, 37 velocity gradient tensor, 13 test functions spin tensor, 13 space, 25, 106, 107 stabilization techniques time derivative CG, 120 material, 11 FIC, 120, 122 spatial, 11 GLS, 119 total, 11 OSS, 119, 121 time integration schemes, 68 PSPG, 119 for ﬂuids

Index

201

��-family method, 112 Backward diﬀerentiation, 113 Gear methods, 113 Generalized-�� method, 114 for solids ��-method, 71 Bossak method, 71 Generalized-�� method, 72 Newmark method, 69 total Lagrangian elements, 27, 40, 61 traction surface, 14 transformation Jacobian tensor, 44 matrix, 45 trial functions space, 26, 106, 107 triangular three-node membrane element, 47 shape functions, 47 variation, 28 velocity gradient tensor, 13 material, 11 spatial, 11 strain tensor, 13 virtual external work, 36 internal work, 35, 43, 45, 61, 67 bending, 62 viscoelastic damping, 73 viscosity, 105 dynamic, 106 kinematic, 108 Voigt notation, 29 von Karman vortex, 128 weak form, 25, 106 Galerkin-type, 26, 106, 107 of the continuity equation, 107 of the momentum equation, 26 of the Navier-Stokes equation, 106 of the Navier-Stokes equations, 107 wrinkling algorithm for membranes, 52

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