Virtual Element Methods in Engineering Sciences

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

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. Blaz Hudobivnik studied Civil Engineering at the University of Ljubljana. He was awarded his Doctoral degree in 2016 under the supervision of Prof. Joze 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.