Peter Wriggers, Fadi Aldakheel, Blaž Hudobivnik

Virtual Element Methods in Engineering Sciences (eBook, PDF)

• Format: PDF

Produktbeschreibung

This book provides a comprehensive treatment of the virtual element method (VEM) for engineering applications, focusing on its application in solid mechanics. Starting with a continuum mechanics background, the book establishes the necessary foundation for understanding the subsequent chapters. It then delves into the VEM's Ansatz functions and projection techniques, both for solids and the Poisson equation, which are fundamental to the method. The book explores the virtual element formulation for elasticity problems, offering insights into its advantages and capabilities. Moving beyond elasticity, the VEM is extended to problems in dynamics, enabling the analysis of dynamic systems with accuracy and efficiency. The book also covers the virtual element formulation for finite plasticity, providing a framework for simulating the behavior of materials undergoing plastic deformation. Furthermore, the VEM is applied to thermo-mechanical problems, where it allows for the investigation of coupled thermal and mechanical effects. The book dedicates a significant portion to the virtual elements for fracture processes, presenting techniques to model and analyze fractures in engineering structures. It also addresses contact problems, showcasing the VEM's effectiveness in dealing with contact phenomena. The virtual element method's versatility is further demonstrated through its application in homogenization, offering a means to understand the effective behavior of composite materials and heterogeneous structures. Finally, the book concludes with the virtual elements for beams and plates, exploring their application in these specific structural elements. Throughout the book, the authors emphasize the advantages of the virtual element method over traditional finite element discretization schemes, highlighting its accuracy, flexibility, and computational efficiency in various engineering contexts.

Produktdetails

• Verlag: Springer International Publishing
• Erscheinungstermin: 28. Oktober 2023
• Englisch
• ISBN-13: 9783031392559
• Artikelnr.: 69232220

Autorenporträt

Professor Dr.-Ing. habil. P. Wriggers studied Civil Engineering at the University Hannover; he obtained his Dr.-Ing degree at the University of Hannover in 1980 on “Contact-impact problems.” Since April 2022, he is Emeritus Professor at Leibniz Universität Hannover. Peter Wriggers is Member of the “Braunschweigische Wissenschaftliche Gesellschaft,” the Academy of Science and Literature in Mainz, the German National Academy of Engineering “acatech” and the National Academy of Croatia. He was President of GAMM, President of GACM and Vice-President of IACM. Furthermore, he acts as Editor-in-Chief for the International Journal “Computational Mechanics” and “Computational Particle Mechanics.” He was awarded the Fellowship of IACM and received the “Computational Mechanics Award” and the “IACM Award” of IACM, the “Euler Medal” of ECCOMAS as well as three honorary degrees from the Universities of Poznan, ENS Cachan and TU Darmstadt.

Professor Dr.-Ing. habil. F. Aldakheel is since April 2023 professor for high performance computing at Leibniz Universität Hannover. After studying engineering in Aleppo, he initially worked at Alfurat University in Syria before moving to the Institute of Applied Mechanics at the University of Stuttgart for the master and Ph.D. studies and then the postdoc period. There he was course director for the international master's programme "Computational Mechanics of Materials and Structures" (COMMAS) as well as local director for the excellence programme "Erasmus Mundus Master of Science in Computational Mechanics". Most recently, he was Chief-Engineer/Group-Leader at the Institute for Continuum Mechanics at Leibniz Universität Hannover and Associate Professor (Honorary) at the Zienkiewicz Centre for Computational Engineering at Swansea University, UK. He has been awarded numerous awards, among them the Richard-von-Mises Prize of GAMM (Association of Applied Mathematics and Mechanics). His research interests are related to the modeling of material behaviors, variational principles, computational solid mechanics, structural mechanics, finite and virtual element methods, multiphysics and multi-scales problems, machine learning, energy transition and experimental validation.

Dr. Blaž Hudobivnik studied Civil Engineering at the University of Ljubljana. He was awarded his Doctoral degree in 2016 under the supervision of Prof. Jože Korelc. He worked as Young researcher/Researcher between 2011 and October 2016 at the University of Ljubljana and after that he was employed as Postdoctoral researcher until April 2023 at the Institute of Continuum Mechanics at the Leibniz Universität Hannover. Since April 2023 he is employed in industry as simulation expert in mechanical design of batteries. His primary research fields are efficient implementation of nonlinear coupled problems, the development of the virtual element method and its application to a wide range of engineering problems. This includes 2D and 3D applications for linear and nonlinear materials, for static and dynamic solids, plate and contact problems, coupled problems (thermo-hydro-mechanics), phase field methods, multi-scale and optimization problems. Alongside research, he advises other institute members in numerical implementations due to his expert knowledge of the Software-Tool AceGen/AceFEM, developed by his doctoral advisor Prof. Korelc.

Inhaltsangabe

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 History and recent developments of virtual elements . . . . . . . . . . . . . 2 1.2 Introductory examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 Virtual element formulation of a truss using a linear ansatz . 7 1.2.2 Quadratic ansatz for a one-dimensional virtual truss element 11 1.3 Contents of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Continuum mechanics background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.2 Balance laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Constitutive Eqations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.1 Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.2 Finite elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.3 Elasto-plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.1 Potential and weak form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.2 Incompressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3.3 Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3.4 Heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3 VEM Ansatz functions and projection for solids . . . . . . . . . . . . . . . . . . . 37 3.1 Two-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.1.1 General ansatz space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.2 Computation of the projection . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1.3 Equivalent projector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.1.4 Projection for a linear ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.1.5 Computation of the projection using symbolic software . . . . 52 3.1.6 Projection for a quadratic ansatz . . . . . . . . . . . . . . . . . . . . . . . . 54 3.1.7 Serendipity virtual element for a quadratic ansatz . . . . . . . . . 59 vii viii Contents 3.1.8 Computation of the second order projection using automatic di erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.1.9 Higher order ansatz for virtual elements . . . . . . . . . . . . . . . . . 65 3.2 Three-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.2.1 General ansatz space in three dimensions . . . . . . . . . . . . . . . . 67 3.2.2 Computation of the projection in three dimensions . . . . . . . . 69 3.2.3 Projection for linear ansatz in three dimensions . . . . . . . . . . . 70 4 VEM Ansatz functions and projection for the Poisson equation . . . . . . 77 4.1 Two-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.1.1 Computation of the projection . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.1.2 Projection for a linear ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.1.3 Projection for a quadratic ansatz . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2 Three-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5 Construction of the virtual element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.1 Consistency Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.1.1 Weak form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.1.2 Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.2 Stabilization techniques for virtual elements . . . . . . . . . . . . . . . . . . . . 90 5.2.1 Stabilization by a discrete bi-linear form . . . . . . . . . . . . . . . . . 90 5.2.2 Energy stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.3 Assembly to the global equation system . . . . . . . . . . . . . . . . . . . . . . . . 95 5.4 Numerical example for the Poisson equation . . . . . . . . . . . . . . . . . . . . 96 6 Virtual elements for elasticity problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.1 Linear elastic response of two-dimensional solids . . . . . . . . . . . . . . . . 104 6.1.1 Consistency term using Voigt notation. . . . . . . . . . . . . . . . . . . 105 6.1.2 Consistency term using tensor notation . . . . . . . . . . . . . . . . . . 108 6.1.3 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.1.4 Definition and labeling of di erent mesh types . . . . . . . . . . . . 116 6.1.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.2 Finite Strain: compressible elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.2.1 Consistency term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.2.2 Stability term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.2.3 Nonlinear virtual elements for three-dimensional problems in elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.2.4 General solution for nonlinear equations . . . . . . . . . . . . . . . . . 130 6.2.5 Numerical examples, compressible case . . . . . . . . . . . . . . . . . 132 6.3 Incompressible elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.3.1 Linear virtual element with constant pressure . . . . . . . . . . . . . 143 6.3.2 Quadratic serendipity virtual element with linear pressure . . 144 6.3.3 Nearly incompressible behaviour . . . . . . . . . . . . . . . . . . . . . . . 150 6.3.4 Numerical examples, incompressible case . . . . . . . . . . . . . . . . 151 6.4 Anisotropic elastic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Contents ix 6.4.1 Numerical examples, anisotropic case . . . . . . . . . . . . . . . . . . . 161 7 Virtual elements for problems in dynamics . . . . . . . . . . . . . . . . . . . . . . . . 167 7.1 Continuum formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 7.2 Mass matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 7.3 Solution algorithms for small strains . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 7.3.1 Matrix formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 7.3.2 Numerical integration in time, time stepping schemes . . . . . . 174 7.4 Solution algorithms for finite strains . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.5.1 Transversal Beam Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.5.2 Cook’s membrane problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.5.3 3D Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 8 Virtual element formulation for finite plasticity . . . . . . . . . . . . . . . . . . . . 189 8.1 Formulation of the virtual element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 8.1.1 Consistency part due to projection . . . . . . . . . . . . . . . . . . . . . . 189 8.1.2 Algorithmic treatment of finite strain elasto-plasticity. . . . . . 190 8.1.3 Energy stabilization of the virtual element for finite plasticity192 8.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 8.2.1 Necking of cylindrical bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 8.2.2 Taylor Anvil Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9 Virtual elements for thermo-mechanical problems . . . . . . . . . . . . . . . . . 203 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 9.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 9.2.1 Energetic and dissipative response functions . . . . . . . . . . . . . . 205 9.2.2 Global constitutive equations . . . . . . . . . . . . . . . . . . . . . . . . . . 208 9.2.3 Weak form and pseudo-potential energy function . . . . . . . . . . 208 9.3 Virtual element discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 9.4 Representative numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 9.4.1 Forming of a steel bolt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 10 Virtual elements for fracture processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 10.1 Brittle crack-propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 10.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 10.1.2 Equations of brittle crack propagation . . . . . . . . . . . . . . . . . . . 220 10.1.3 Modeling crack propagation with virtual elements . . . . . . . . . 221 10.1.4 Computation of stress intensity factors . . . . . . . . . . . . . . . . . . 221 10.1.5 Propagation criteria: Maximum circumferential stress criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 10.1.6 Stress intensity factor analysis using virtual elements . . . . . . 223 10.1.7 Cutting Technique and crack update algorithm . . . . . . . . . . . . 227 10.1.8 Crack propagation simulations based on the cutting technique231 10.2 Phase field methods for brittle fracture using virtual elements . . . . . . 235 10.2.1 Governing equations for elasticity . . . . . . . . . . . . . . . . . . . . . . 235 x Contents 10.2.2 Regularization of a sharp crack topology . . . . . . . . . . . . . . . . 235 10.2.3 Variational formulation to brittle fracture . . . . . . . . . . . . . . . . 238 10.2.4 Formulation of the virtual element method . . . . . . . . . . . . . . . 241 10.2.5 VEM for phase field brittle fracture simulations . . . . . . . . . . . 244 10.3 Phase field methods for ductile fracture using virtual elements . . . . . 246 10.3.1 Governing equations for phase field ductile fracture . . . . . . . 248 10.3.2 Formulation of the virtual element method . . . . . . . . . . . . . . . 251 10.3.3 VEM for phase field ductile fracture simulations . . . . . . . . . . 251 10.4 An adaptive scheme to follow crack paths combining phase field and cutting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 10.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 10.4.2 Modeling crack propagation using VEM . . . . . . . . . . . . . . . . 258 10.4.3 A discontinuous crack propagation using phase field . . . . . . 258 10.4.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 10.5 Fracturing analysis using damage mechanics . . . . . . . . . . . . . . . . . . . . 268 10.5.1 Governing equations for isotropic damage model . . . . . . . . . . 268 10.5.2 Virtual element formulation for damage . . . . . . . . . . . . . . . . . 271 10.5.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 11 Virtual element formulation for contact. . . . . . . . . . . . . . . . . . . . . . . . . . . 281 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 11.2 Theoretical background for contact of solids . . . . . . . . . . . . . . . . . . . . 283 11.2.1 Local contact kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 11.2.2 Constitutive relations for contact . . . . . . . . . . . . . . . . . . . . . . . 286 11.2.3 Potential form for solids in contact . . . . . . . . . . . . . . . . . . . . . . 288 11.3 Contact discretization using VEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 11.3.1 Node insertion algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 11.3.2 Inserted node and gap in the two-dimensional case . . . . . . . . 292 11.3.3 Discretization of the contact interface in 2d . . . . . . . . . . . . . . 294 11.3.4 Penalty formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 11.3.5 Augmented Lagrangian multiplier formulation . . . . . . . . . . . . 301 11.4 Stabilization of VEM in case of contact . . . . . . . . . . . . . . . . . . . . . . . . 303 11.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 11.5.1 Patch test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 11.5.2 Hertzian contact problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 11.5.3 Contacting Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 11.5.4 Hertz contact for large deformations . . . . . . . . . . . . . . . . . . . . 311 11.5.5 Ironing problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 11.5.6 Wall mounting of a bolt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 12 Virtual elements for homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Contents xi 13 Virtual elements for beams and plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 13.1 Virtual element formulations for Euler-Bernoulli beams . . . . . . . . . . 322 13.1.1 Fourth order ansatz for a one-dimensional virtual beam element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 13.2 Virtual element formulations for Kirchho -Love plates . . . . . . . . . . . 331 13.2.1 Mathematical model of the plate and constitutive relations . . 331 13.3 Formulation of the virtual element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 13.3.1 General notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 13.3.2 Ansatz and projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 13.3.3 Ansatz function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 13.3.4 Plate element with constant curvature . . . . . . . . . . . . . . . . . . . 339 13.3.5 Plate element with linear curvature . . . . . . . . . . . . . . . . . . . . . 342 13.3.6 Residual and sti ness matrix of the virtual plate element . . . 345 13.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 13.4.1 Notation used in the examples . . . . . . . . . . . . . . . . . . . . . . . . . . 347 13.4.2 Clamped plate under uniform load . . . . . . . . . . . . . . . . . . . . . . 348 13.4.3 Rhombic plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 13.4.4 Rectangular orthotropic plate . . . . . . . . . . . . . . . . . . . . . . . . . . 354 13.4.5 Plate with anisotropic material . . . . . . . . . . . . . . . . . . . . . . . . . 355 13.5 ⇠1 -continuous virtual elements for FEM codes . . . . . . . . . . . . . . . . . . 357 13.5.1 Clamped plate under uniform load . . . . . . . . . . . . . . . . . . . . . . 358 13.5.2 Clamped plate under point load . . . . . . . . . . . . . . . . . . . . . . . . 359 13.5.3 L-shaped plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 A Formulae in virtual element formulations . . . . . . . . . . . . . . . . . . . . . . . . . 363 A.1 Integration over polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 A.2 Computation of volume by surface integrals . . . . . . . . . . . . . . . . . . . . 365 B I-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 History and recent developments of virtual elements . . . . . . . . . . . . . 21.2 Introductory examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.1 Virtual element formulation of a truss using a linear ansatz . 71.2.2 Quadratic ansatz for a one-dimensional virtual truss element 111.3 Contents of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Continuum mechanics background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.2 Balance laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 Constitutive Eqations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.1 Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.2 Finite elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.3 Elasto-plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.1 Potential and weak form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.2 Incompressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3.3 Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.3.4 Heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 VEM Ansatz functions and projection for solids . . . . . . . . . . . . . . . . . . . 373.1 Two-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.1.1 General ansatz space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1.2 Computation of the projection . . . . . . . . . . . . . . . . . . . . . . . . . 433.1.3 Equivalent projector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.1.4 Projection for a linear ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.1.5 Computation of the projection using symbolic software . . . . 523.1.6 Projection for a quadratic ansatz . . . . . . . . . . . . . . . . . . . . . . . . 543.1.7 Serendipity virtual element for a quadratic ansatz . . . . . . . . . 59viiviii Contents3.1.8 Computation of the second order projection usingautomatic di erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.1.9 Higher order ansatz for virtual elements . . . . . . . . . . . . . . . . . 653.2 Three-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.2.1 General ansatz space in three dimensions . . . . . . . . . . . . . . . . 673.2.2 Computation of the projection in three dimensions . . . . . . . . 693.2.3 Projection for linear ansatz in three dimensions . . . . . . . . . . . 704 VEM Ansatz functions and projection for the Poisson equation . . . . . . 774.1 Two-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.1.1 Computation of the projection . . . . . . . . . . . . . . . . . . . . . . . . . 784.1.2 Projection for a linear ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.1.3 Projection for a quadratic ansatz . . . . . . . . . . . . . . . . . . . . . . . . 804.2 Three-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825 Construction of the virtual element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.1 Consistency Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.1.1 Weak form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.1.2 Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.2 Stabilization techniques for virtual elements . . . . . . . . . . . . . . . . . . . . 905.2.1 Stabilization by a discrete bi-linear form . . . . . . . . . . . . . . . . . 905.2.2 Energy stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.3 Assembly to the global equation system . . . . . . . . . . . . . . . . . . . . . . . . 955.4 Numerical example for the Poisson equation . . . . . . . . . . . . . . . . . . . . 966 Virtual elements for elasticity problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.1 Linear elastic response of two-dimensional solids . . . . . . . . . . . . . . . . 1046.1.1 Consistency term using Voigt notation. . . . . . . . . . . . . . . . . . . 1056.1.2 Consistency term using tensor notation . . . . . . . . . . . . . . . . . . 1086.1.3 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.1.4 Definition and labeling of di erent mesh types . . . . . . . . . . . . 1166.1.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.2 Finite Strain: compressible elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.2.1 Consistency term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.2.2 Stability term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.2.3 Nonlinear virtual elements for three-dimensional problemsin elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.2.4 General solution for nonlinear equations . . . . . . . . . . . . . . . . . 1306.2.5 Numerical examples, compressible case . . . . . . . . . . . . . . . . . 1326.3 Incompressible elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416.3.1 Linear virtual element with constant pressure . . . . . . . . . . . . . 1436.3.2 Quadratic serendipity virtual element with linear pressure . . 1446.3.3 Nearly incompressible behaviour . . . . . . . . . . . . . . . . . . . . . . . 1506.3.4 Numerical examples, incompressible case . . . . . . . . . . . . . . . . 1516.4 Anisotropic elastic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158Contents ix6.4.1 Numerical examples, anisotropic case . . . . . . . . . . . . . . . . . . . 1617 Virtual elements for problems in dynamics . . . . . . . . . . . . . . . . . . . . . . . . 1677.1 Continuum formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1677.2 Mass matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1697.3 Solution algorithms for small strains . . . . . . . . . . . . . . . . . . . . . . . . . . . 1737.3.1 Matrix formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1737.3.2 Numerical integration in time, time stepping schemes . . . . . . 1747.4 Solution algorithms for finite strains . . . . . . . . . . . . . . . . . . . . . . . . . . . 1777.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1807.5.1 Transversal Beam Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 1817.5.2 Cook's membrane problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1837.5.3 3D Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1858 Virtual element formulation for finite plasticity . . . . . . . . . . . . . . . . . . . . 1898.1 Formulation of the virtual element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1898.1.1 Consistency part due to projection . . . . . . . . . . . . . . . . . . . . . . 1898.1.2 Algorithmic treatment of finite strain elasto-plasticity. . . . . . 1908.1.3 Energy stabilization of the virtual element for finite plasticity1928.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1938.2.1 Necking of cylindrical bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1938.2.2 Taylor Anvil Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1979 Virtual elements for thermo-mechanical problems . . . . . . . . . . . . . . . . . 2039.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2039.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2049.2.1 Energetic and dissipative response functions . . . . . . . . . . . . . . 2059.2.2 Global constitutive equations . . . . . . . . . . . . . . . . . . . . . . . . . . 2089.2.3 Weak form and pseudo-potential energy function . . . . . . . . . . 2089.3 Virtual element discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2099.4 Representative numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2139.4.1 Forming of a steel bolt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21410 Virtual elements for fracture processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 21710.1 Brittle crack-propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21810.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21810.1.2 Equations of brittle crack propagation . . . . . . . . . . . . . . . . . . . 22010.1.3 Modeling crack propagation with virtual elements . . . . . . . . . 22110.1.4 Computation of stress intensity factors . . . . . . . . . . . . . . . . . . 22110.1.5 Propagation criteria: Maximum circumferential stresscriterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22310.1.6 Stress intensity factor analysis using virtual elements . . . . . . 22310.1.7 Cutting Technique and crack update algorithm . . . . . . . . . . . . 22710.1.8 Crack propagation simulations based on the cutting technique23110.2 Phase field methods for brittle fracture using virtual elements . . . . . . 23510.2.1 Governing equations for elasticity . . . . . . . . . . . . . . . . . . . . . . 235x Contents10.2.2 Regularization of a sharp crack topology . . . . . . . . . . . . . . . . 23510.2.3 Variational formulation to brittle fracture . . . . . . . . . . . . . . . . 23810.2.4 Formulation of the virtual element method . . . . . . . . . . . . . . . 24110.2.5 VEM for phase field brittle fracture simulations . . . . . . . . . . . 24410.3 Phase field methods for ductile fracture using virtual elements . . . . . 24610.3.1 Governing equations for phase field ductile fracture . . . . . . . 24810.3.2 Formulation of the virtual element method . . . . . . . . . . . . . . . 25110.3.3 VEM for phase field ductile fracture simulations . . . . . . . . . . 25110.4 An adaptive scheme to follow crack paths combining phase fieldand cutting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25710.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25710.4.2 Modeling crack propagation using VEM . . . . . . . . . . . . . . . . 25810.4.3 A discontinuous crack propagation using phase field . . . . . . 25810.4.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26210.5 Fracturing analysis using damage mechanics . . . . . . . . . . . . . . . . . . . . 26810.5.1 Governing equations for isotropic damage model . . . . . . . . . . 26810.5.2 Virtual element formulation for damage . . . . . . . . . . . . . . . . . 27110.5.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27411 Virtual element formulation for contact. . . . . . . . . . . . . . . . . . . . . . . . . . . 28111.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28211.2 Theoretical background for contact of solids . . . . . . . . . . . . . . . . . . . . 28311.2.1 Local contact kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28311.2.2 Constitutive relations for contact . . . . . . . . . . . . . . . . . . . . . . . 28611.2.3 Potential form for solids in contact . . . . . . . . . . . . . . . . . . . . . . 28811.3 Contact discretization using VEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29011.3.1 Node insertion algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29011.3.2 Inserted node and gap in the two-dimensional case . . . . . . . . 29211.3.3 Discretization of the contact interface in 2d . . . . . . . . . . . . . . 29411.3.4 Penalty formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29811.3.5 Augmented Lagrangian multiplier formulation . . . . . . . . . . . . 30111.4 Stabilization of VEM in case of contact . . . . . . . . . . . . . . . . . . . . . . . . 30311.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30511.5.1 Patch test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30611.5.2 Hertzian contact problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30611.5.3 Contacting Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31011.5.4 Hertz contact for large deformations . . . . . . . . . . . . . . . . . . . . 31111.5.5 Ironing problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31211.5.6 Wall mounting of a bolt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31512 Virtual elements for homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319Contents xi13 Virtual elements for beams and plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32113.1 Virtual element formulations for Euler-Bernoulli beams . . . . . . . . . . 32213.1.1 Fourth order ansatz for a one-dimensional virtual beamelement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32613.2 Virtual element formulations for Kirchho -Love plates . . . . . . . . . . . 33113.2.1 Mathematical model of the plate and constitutive relations . . 33113.3 Formulation of the virtual element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33513.3.1 General notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33513.3.2 Ansatz and projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33613.3.3 Ansatz function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33813.3.4 Plate element with constant curvature . . . . . . . . . . . . . . . . . . . 33913.3.5 Plate element with linear curvature . . . . . . . . . . . . . . . . . . . . . 34213.3.6 Residual and sti ness matrix of the virtual plate element . . . 34513.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34713.4.1 Notation used in the examples . . . . . . . . . . . . . . . . . . . . . . . . . . 34713.4.2 Clamped plate under uniform load . . . . . . . . . . . . . . . . . . . . . . 34813.4.3 Rhombic plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35213.4.4 Rectangular orthotropic plate . . . . . . . . . . . . . . . . . . . . . . . . . . 35413.4.5 Plate with anisotropic material . . . . . . . . . . . . . . . . . . . . . . . . . 35513.5 1 -continuous virtual elements for FEM codes . . . . . . . . . . . . . . . . . . 35713.5.1 Clamped plate under uniform load . . . . . . . . . . . . . . . . . . . . . . 35813.5.2 Clamped plate under point load . . . . . . . . . . . . . . . . . . . . . . . . 35913.5.3 L-shaped plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360A Formulae in virtual element formulations . . . . . . . . . . . . . . . . . . . . . . . . . 363A.1 Integration over polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363A.2 Computation of volume by surface integrals . . . . . . . . . . . . . . . . . . . . 365B I-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391