The purpose of this book is to present the current state of the art of the Virtual Element Method (VEM) by collecting contributions from many of the most active researchers in this field and covering a broad range of topics: from the mathematical foundation to real life computational applications.
The book is naturally divided into three parts. The first part of the book presents recent advances in theoretical and computational aspects of VEMs, discussing the generality of the meshes suitable to the VEM, the implementation of the VEM for linear and nonlinear PDEs, and the construction of discrete hessian complexes. The second part of the volume discusses Virtual Element discretization of paradigmatic linear and non-linear partial differential problems from computational mechanics, fluid dynamics, and wave propagation phenomena. Finally, the third part contains challenging applications such as the modeling of materials with fractures, magneto-hydrodynamics phenomena and contact solid mechanics.
The book is intended for graduate students and researchers in mathematics and engineering fields, interested in learning novel numerical techniques for the solution of partial differential equations. It may as well serve as useful reference material for numerical analysts practitioners of the field.
Author(s): Paola F. Antonietti, Lourenço Beirão da Veiga, Gianmarco Manzini
Series: SEMA SIMAI Springer Series, 31
Publisher: Springer
Year: 2022
Language: English
Pages: 620
City: Cham
Preface
Contents
Editors and Contributors
About the Editors
Contributors
1 VEM and the Mesh
1.1 Introduction
1.2 Model Problem
1.3 State of the Art
1.3.1 Geometrical Assumptions
1.3.2 Convergence Results in the VEM Literature
1.4 Violating the Geometrical Assumptions
1.4.1 Datasets Definition
1.4.2 VEM Performance over the Datasets
1.5 Mesh Quality Metrics
1.5.1 Polygon Quality Metrics
1.5.2 Performance Indicators
1.5.3 Results
1.6 Mesh Quality Indicators
1.6.1 Definition
1.6.2 Results
1.7 PEMesh Benchmarking Tool
References
2 On the Implementation of Virtual Element Method for Nonlinear Problems over Polygonal Meshes
2.1 Introduction
2.1.1 Structure of the Chapter
2.1.2 Basic Notation
2.2 Governing Equations
2.3 Virtual Element Framework
2.4 Computation of the Projection Operators and Discrete Bilinear Forms
2.5 Fully Discrete Scheme
2.6 Implementation
2.7 Numerical Examples
2.8 Conclusion
References
3 Discrete Hessian Complexes in Three Dimensions
3.1 Introduction
3.2 Matrix and Vector Operations
3.2.1 Matrix-Vector Products
3.2.2 Differentiation
3.2.3 Matrix Decompositions
3.2.4 Projections to a Plane
3.3 Two Hilbert Complexes for Tensors
3.3.1 Hessian Complexes
3.3.2 divdiv Complexes
3.4 Polynomial Complexes for Tensors
3.4.1 De Rham and Koszul Polynomial Complexes
3.4.2 Hessian Polynomial Complexes
3.4.3 Divdiv Polynomial Complexes
3.5 A Conforming Virtual Element Hessian Complex
3.5.1 ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper H left parenthesis d i v right parenthesis) /StPNE pdfmark [/StBMC pdfmarkH(div)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark-Conforming Element for Trace-Free Tensors
3.5.2 H2-Conforming Virtual Element
3.5.3 Trace Complexes
3.5.4 ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper H left parenthesis c u r l right parenthesis) /StPNE pdfmark [/StBMC pdfmarkH(curl)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark-Conforming Element for Symmetric Tensors
3.5.5 Discrete Conforming Hessian Complex
3.5.6 Discrete Poincaré Inequality
3.6 Discretization for the Linearized Einstein-Bianchi System
3.6.1 Linearized Einstein-Bianchi System
3.6.2 Conforming Discretization
References
4 Some Virtual Element Methods for Infinitesimal ElasticityProblems
4.1 Introduction
4.2 Elasticity Formulation with Infinitesimal Strain
4.2.1 Primal Form
4.2.2 Mixed Form
4.3 Virtual Element Methods for Elasticity
4.3.1 Primal Methods Based on Virtual Work Principle
4.3.1.1 The Local Space
4.3.1.2 The Local Bilinear Form
4.3.1.3 The Local Loading Term
4.3.1.4 The Discrete Scheme
4.3.1.5 Mixed Methods Based on Hellinger Reissner Principle: 2D Case
4.3.1.6 The Local Spaces
4.3.1.7 The Local Bilinear Forms
4.3.1.8 The Local Loading Term
4.3.1.9 The Discrete Scheme
4.3.2 Mixed Methods Based on Hellinger Reissner Principle: 3D Case
4.3.2.1 The Local Spaces
4.3.2.2 The Local Forms
4.3.2.3 The Local Loading Term
4.3.2.4 The Discrete Scheme
4.4 Numerical Results
4.4.1 2D Numerical Tests
4.4.1.1 Primal Formulation
4.4.1.2 Hellinger-Reissner Mixed Formulation
4.4.2 3D Numerical Results
4.5 Conclusions
References
5 An Introduction to Second Order Divergence-Free VEM for Fluidodynamics
5.1 Introduction
5.2 The Navier-Stokes Equation
5.3 Notations and Preliminaries
5.4 Virtual Element Spaces in 2D
5.4.1 Virtual Elements for Stokes
5.4.2 Enhanced Virtual Elements for Navier-Stokes
5.5 Virtual Elements on Curved Polygons
5.6 Virtual Element Spaces in 3D
5.6.1 Face Spaces
5.6.2 Virtual Elements for Stokes
5.6.3 Enhanced Virtual Elements for Navier-Stokes
5.7 Virtual Element Problem
5.7.1 Global Spaces
5.7.2 Discrete Forms
5.7.3 Divergence-Free Velocity Solution
5.8 Convergence Results and Exploring the Divergence-FreeProperty
5.8.1 Convergence Results
5.8.2 Reduced Virtual Elements
5.8.3 Stokes Complex and curl Formulation
5.8.4 Stability in the Darcy Limit and Brinkman Equation
5.9 Numerical Tests
5.10 Conclusions
References
6 A Virtual Marriage à la Mode: Some Recent Results on the Coupling of VEM and BEM
6.1 Introduction
6.2 The Coupling Procedures
6.2.1 BIEM for Laplace and Helmholtz
6.2.2 The Costabel & Han Coupling
6.2.3 The Modified Costabel & Han Coupling
6.2.4 Solvability Analysis
6.3 The Costabel & Han VEM/BEM Schemes in 2D
6.3.1 Preliminaries
6.3.2 The Costabel & Han VEM/BEM Schemefor Poisson
6.3.2.1 The Discrete Setting
6.3.2.2 Solvability and a Priori Error Analyses
6.3.3 The Costabel & Han VEM/BEM Schemefor Helmholtz
6.3.3.1 The Discrete Setting
6.3.3.2 Solvability and a Priori Error Analyses
6.4 The Modified Costabel & Han VEM/BEM Schemes in 3D
6.4.1 Preliminaries
6.4.2 The Discrete Setting
6.4.3 Solvability and a Priori Error Analyses
6.5 Numerical Results
6.5.1 Convergence Tests for the Poisson Model
6.5.2 Convergence Tests for the Helmholtz Model
References
7 Virtual Element Approximation of Eigenvalue Problems
7.1 Introduction
7.2 Abstract Setting
7.2.1 Model Problem
7.3 Virtual Element Approximation of the Laplace Eigenvalue Problem
7.3.1 Virtual Element Method
7.3.2 The VEM Discretization of the LaplaceEigenproblem
7.3.3 Convergence Analysis
7.3.4 Numerical Results
7.4 Extension to Nonconforming and hp Version of VEM
7.4.1 Nonconforming VEM
7.4.2 hp Version of VEM
7.5 The Choice of the Stabilization Parameters
7.5.1 A Simplified Setting
7.5.2 The Role of the VEM Stabilization Parameters
7.6 Applications
7.6.1 The Mixed Laplace Eigenvalue Problem
7.6.2 The Steklov Eigenvalue Problem
7.6.3 An Acoustic Vibration Problem
7.6.4 Eigenvalue Problems Related to Plate Models
7.6.5 Eigenvalue Problems Related to Linear ElasticityModels
References
8 Virtual Element Methods for a Stream-Function Formulation of the Oseen Equations
8.1 Introduction
8.2 Model Problem
8.3 Virtual Element Methods
8.3.1 Virtual Spaces and Polynomial Projections Operator
8.3.2 Construction of the Local and Global Discrete Forms
8.3.3 Discrete Formulation
8.4 Error Analysis
8.4.1 Preliminary Results
8.4.2 A Priori Error Estimates
8.5 Recovering the Velocity, Vorticity and Pressure Fields
8.5.1 Computing the Velocity Field
8.5.2 Computing the Fluid Vorticity
8.5.3 Computing the Fluid Pressure
8.6 Numerical Results
8.6.1 Test 1: Smooth Solution
8.6.2 Test 2: Solution with Boundary Layer
8.6.3 Test 3: Solution with Non Homogeneous Dirichlet Boundary Conditions
References
9 The Nonconforming Trefftz Virtual Element Method: General Setting, Applications, and Dispersion Analysis for the Helmholtz Equation
9.1 Introduction
9.2 Polygonal Meshes and Broken Sobolev Spaces
9.3 The Nonconforming Trefftz Virtual Element Method for the Laplace Problem
9.4 General Structure of Nonconforming Trefftz Virtual Element Methods
9.5 The Nonconforming Trefftz Virtual Element Method for the Helmholtz Problem
9.6 Stability and Dispersion Analysis for the Nonconforming Trefftz VEM for the Helmholtz Equation
9.6.1 Abstract Dispersion Analysis
9.6.2 Minimal Generating Subspaces
9.6.3 Numerical Results
9.6.3.1 Dependence of Dispersion and Dissipation on the Bloch Wave Angle
9.6.3.2 Exponential Convergence of the Dispersion Error Against the Effective Degree q
9.6.3.3 Algebraic Convergence of the Dispersion Error Against the Wave Number k
References
10 The Conforming Virtual Element Method for Polyharmonic and Elastodynamics Problems: A Review
10.1 Introduction
10.1.1 Paradigmatic Examples
10.1.1.1 Cahn-Hilliard Equation
10.1.1.2 Anisotropic Cahn-Hilliard Equation
10.1.1.3 A High Order Phase Field Model for Brittle Fracture
10.1.2 Notation and Technicalities
10.1.3 Mesh Assumptions
10.2 The Virtual Element Method for the Polyharmonic Problem
10.2.1 The Continuous Problem
10.2.2 The Conforming Virtual Element Approximation
10.2.2.1 Virtual Element Spaces
10.2.2.2 Modified Lowest Order Virtual Element Spaces
10.2.2.3 Discrete Bilinear Form
10.2.2.4 Discrete Load Term
10.2.2.5 VEM Spaces with Arbitrary Degree of Continuity
10.2.2.6 Convergence Results
10.3 The Virtual Element Method for the Cahn-Hilliard Problem
10.3.1 The Continuous Problem
10.3.2 The Conforming Virtual Element Approximation
10.3.2.1 A C1 Virtual Element Space
10.3.2.2 Virtual Element Bilinear Forms
10.3.2.3 The Discrete Problem
10.3.3 Numerical Results
10.4 The Virtual Element Method for the Elastodynamics Problem
10.4.1 The Continuous Problem
10.4.2 The Conforming Virtual Element Approximation
10.4.2.1 Virtual Element Spaces
10.4.2.2 Discrete Bilinear Forms
10.4.2.3 Discrete Load Term
10.4.2.4 The Discrete Problem
10.4.2.5 Stability and Convergence Analysis for the Semi-Discrete Problem
10.4.3 Numerical Results
References
11 The Virtual Element Method in Nonlinear and Fracture Solid Mechanics
11.1 Introduction
11.2 Position of the Problem
11.3 Basis of the VEM in 2D Solid Mechanics
11.3.1 Kinematics
11.3.2 Stiffness Matrix
11.3.3 Force Vector
11.3.4 The Case k=3
11.4 Nonlinear Inelastic Material Response
11.4.1 Plastic Behavior
11.4.2 Viscoelastic Behavior
11.4.3 Shape Memory Alloy Behavior
11.4.4 Numerical Applications
11.4.4.1 Viscoelastic Cylinder Subjected to Internal Pressure
11.4.4.2 Elastoplastic Plate with Circular Hole
11.4.4.3 Shape Memory Alloy Device
11.5 Homogenization of Long Fiber Composites
11.5.1 Problem Formulation
11.5.2 Computational Homogenization: Smart Use of VEM Meshing Versatility
11.6 Fracture Mechanics
11.6.1 Interface Model
11.6.2 Cracking Process Through VEM Technology
11.6.3 Numerical Applications
11.6.3.1 Non-Symmetric Three-Point Bending Test with Topological Adaptive Mesh Refinement
11.6.3.2 Symmetric Three-Point Bending Test and Comparison with XFEM
11.7 Concluding Remarks
References
12 The Virtual Element Method for the Coupled System of Magneto-Hydrodynamics
12.1 Introduction
12.2 Mathematical Formulation
12.2.1 Weak Formulation
12.3 The Virtual Element Method
12.3.1 Mesh Notation and Regularity Assumptions
12.3.2 The Nodal Space
12.3.2.1 The Polynomial Reconstruction Operators
12.3.3 The Edge Space
12.3.4 The Cell Space
12.3.5 The de Rham Complex
12.3.6 Fluid Flow
12.4 Energy Estimates
12.5 Linearization
12.6 Well-Posedness and Stability of the Linear Solver
12.7 Numerical Experiments
12.7.1 Experimental Study of Convergence
12.7.2 Magnetic Reconnection
12.8 Conclusions
References
13 Virtual Element Methods for Engineering Applications
13.1 Generic Formulation of a Nonlinear Boundary Value Problem
13.2 Formulation of the Virtual Element Method
13.2.1 Ansatz Functions for VEM in Two Dimensions
13.2.2 Ansatz Functions for VEM in Three Dimensions
13.2.3 Residual and Tangent Matrix of the Virtual Elements
13.2.4 Stabilization of the Method
13.3 VEM for Fracturing Solids
13.3.1 Basic Equations of Elastic Solids
13.3.2 Crack Propagation Based on Stress Intensity Factors
13.3.3 Construction of the Crack Path Using SIF
13.3.4 Phase-Field Approach for Brittle Crack Propagation
13.3.5 Numerical Examples
13.3.5.1 Crack Propagation using Phase-Field Approach: Bi-Material Plate
13.3.5.2 Crack Propagation Using Stress Intensity Factors
13.4 VEM for Contact
13.4.1 Governing Equations for Finite Elasticity and Contact
13.4.2 Virtual Element Method for Contact
13.4.2.1 Standard Stabilisation
13.4.2.2 Edge Stabilisation
13.4.2.3 Energy Stabilisation
13.4.2.4 Patch Test and Stabilization Test
13.4.3 Contact for Large Deformations Including Friction
13.4.4 Node Insertion Algorithm
13.4.4.1 Contact Discretization: Frictionless
13.4.4.2 Contact Discretization: Friction
13.4.5 Numerical Examples
13.4.5.1 Hertzian Contact Problem for Small Deformations
13.4.5.2 Large Deformational Contact: Ironing Problem
13.4.5.3 Wall Mounting of a Bolt
13.5 Conclusion
References
