Advances in Discretization Methods (eBook)

Preface 6
Contents 8
Part I Enriched Methods for Flow and Mechanics in Heterogeneous Media 10
A Mixed Finite Element Method for Modeling the Fluid Exchange Between Microcirculation and Tissue Interstitium 11
1 Introduction 11
2 Model Set Up 13
3 Variational Formulation 15
4 Numerical Approximation 19
4.1 Algebraic Formulation 21
5 Numerical Experiments 24
5.1 Coupled 3D-1D Problem on a Single Branch 25
5.1.1 Numerical Results 26
5.2 Coupled 3D-1D Problem on a Y-Shaped Bifurcation 29
5.2.1 Numerical Results 29
6 Conclusions 31
References 32
On a PDE-Constrained Optimization Approach for Flow Simulations in Fractured Media 34
1 Introduction 34
2 Problem Formulation 35
3 A PDE-Constrained Optimization Approach 37
3.1 Discretization Strategies 40
4 Numerical Results 40
4.1 Flow in Complex DFNs 41
4.2 Unsteady Flow in DFNs 48
References 51
A Review of the XFEM-Based Approximation of Flow in Fractured Porous Media 53
1 Introduction 53
2 Governing Equations 56
2.1 Equi-Dimensional Models 56
2.1.1 Dual Formulation 57
2.1.2 Primal Formulation 58
2.2 Hybrid-Dimensional Models 59
2.2.1 Dual Formulation 60
2.2.2 Primal Formulation 62
2.3 Branching and Intersections 63
2.3.1 Assuming Pressure Continuity 65
2.3.2 Robin Boundary Conditions 65
2.3.3 Dual Formulation for X-Shaped Intersections 66
2.4 Boundary Conditions 67
2.4.1 Dirichlet Conditions for Fractured Porous Media 67
2.4.2 Boundary and Coupling Conditions for Fracture Tips 69
3 Numerical Discretization by Means of XFEM 69
3.1 Modification and Addition of Basis Functions 70
3.2 Primal Formulation with XFEM 72
3.3 Dual Mixed Formulation 73
3.4 Fracture Grids and Approximation of the Coupling Terms 74
3.5 Basis Function Enrichment Around Fracture Tips 76
4 Solvers 77
4.1 Conditioning 77
4.2 Iterative Approaches 78
5 Conclusions 79
References 79
Part II Enhanced Finite Element Formulations for Fracture and Interface Problems 83
Modeling of Fracture in Polycrystalline Materials 84
1 Introduction 84
2 Problem Statement 85
2.1 Balance of Linear Momentum for a Fractured Body 86
2.2 Constitutive Equations for Finite Deformations Crystal Plasticity 87
2.3 Continuum Damage Mechanics Coupled to Crystal Plasticity 89
2.4 Non-Local Extension of the GTN Model 91
2.5 Weak Form of the Resulting Boundary Value Problem 92
3 Discretisation of the Boundary Value Problem with the XFEM 92
4 Quasistatic Crack Propagation with the XFEM 95
4.1 Crack Propagation Criterion 95
4.2 Reconstruction of the Fracture Surface 96
4.2.1 Level Set Update by Solving a Hamilton-Jacobi Equation 96
4.2.2 Global Crack Tracking Algorithm 98
4.2.3 Accuracy of the Two Methods 99
4.3 Crack Propagation Algorithm 100
5 Numerical Examples 101
5.1 Verification of the Non-Local Damage Formulation 102
5.2 Crack Propagation in Crystals 104
6 Conclusions 105
References 105
eXtended Hybridizable Discontinuous Galerkin (X-HDG) for Void and Bimaterial Problems 108
1 Introduction 108
2 X-HDG Formulation for Void Problems 111
2.1 Local Problem for Standard Elements 113
2.2 Local Problem for a Cut Element 114
2.3 Global Problem 117
3 X-HDG Formulation for Bimaterial Problems 118
3.1 Local Problem at a Bimaterial Element 119
4 Numerical Tests 120
4.1 X-HDG with Voids 121
4.2 Bimaterial X-HDG 123
5 Conclusions 125
References 126
Crack Lip Contact Modeling Based on Lagrangian Multipliers with X-FEM 128
1 Introduction 128
2 Formulation of the Continuous Problem 129
3 Discretization 131
3.1 Discretization of the Displacement Field 132
3.2 Discretization of the Lagrangian Multipliers 133
4 Stress Intensity Factors 135
5 Numerical Experiments 137
5.1 Cracked Block Under Uniform Pressure 137
5.2 Cracked Block Under Non-Uniform Pressure 141
6 Conclusions 145
References 146
Stress Intensity Factors Through Crack Opening Displacements in the XFEM 148
1 Introduction 148
2 XFEM with a Hybrid Explicit-Implicit Crack Description 149
2.1 Hybrid Explicit-Implicit Crack Description 151
2.1.1 Coordinate System (a,b) 151
2.1.2 Coordinate System (r,?) 152
3 Stress Intensity Factors Through Crack Opening Displacements 153
3.1 Approximated State 153
3.1.1 Localisation of the Zero Level-Sets 154
3.1.2 Computation of Approximated CODs 155
3.2 Reference State for Pure Mode I, II and III Cracks 157
3.2.1 Stress Intensity Approach 157
3.2.2 Computation of Expected CODs 158
3.3 Computation of SIFs Using CODs 158
3.4 Location of the Fitting Points 159
3.5 Consideration of the Crack Front 160
4 Numerical Results 162
4.1 Eccentric Three-Point Bending Test 162
4.2 Shear Edge Crack 163
4.3 Loaded Crack in 2D 164
4.4 Penny-Shaped Crack 165
4.5 Crack Propagation in Two Dimensions 167
5 Conclusions 168
References 168
Part III Polygonal and Polyhedral Methods 170
The Virtual Element Method for Underground Flow Simulations in Fractured Media 171
1 Introduction 171
1.1 Discrete Fracture Networks 172
1.2 Further Notation 173
2 Problem Formulation 174
3 The Virtual Element Method for DFN Simulations 175
3.1 The VEM Setting in the DFN Framework 175
3.2 Formulation of the Problem Towards Domain Decomposition 178
3.3 A Globally Conforming Approach 179
3.3.1 Implementation 182
3.4 A Hybrid Mortar Virtual Element Approach 183
3.4.1 Implementation 184
4 Numerical Results 184
5 Conclusions 187
References 188
Adaptive Discontinuous Galerkin Methods on Polytopic Meshes 191
1 Introduction 191
2 Model Problem 193
3 Interior Penalty Discontinuous Galerkin Method 194
4 Error Estimation and Adaptive Mesh Refinement 197
5 Numerical Examples 199
5.1 Example 1: Two-Dimensional Cross Geometry 200
5.2 Example 2: Modelling Trabecular Bone 202
6 Conclusions 208
References 208
Part IV Advances in XFEM/Fictitious Domain Methods 211
Stabilized X-FEM for Heaviside and Nonlinear Enrichments 212
1 Introduction 212
2 Motivation for Ill-Conditioning 213
3 Structure of Ill-Conditioning 216
3.1 Heaviside Function Enrichment 216
3.2 Linear Function Enrichment 218
4 The New Stabilization Technique 219
5 Method Validation 220
5.1 1D Bar with a Cut 220
5.2 1D Bar with Linear Enrichment 221
5.3 Three Dimensional Test 223
6 Conclusions 229
References 229
An Adaptive Fictitious Domain Method for Elliptic Problems 232
1 Introduction 233
2 Fictitious Domains Method 234
3 Adaptive Fictitious Domain Method 235
3.1 Infinite Dimensional Fictitious Domain Algorithm 236
3.2 Adaptive Finite Dimensional Fictitious Domain Method 236
3.2.1 The Module ELLIPTIC 237
3.3 The Module ENRICH 239
3.4 The Module UPDATE 240
3.5 The AFDM Algorithm 240
4 Numerical Results 242
4.1 Implementation Issues 242
4.2 Test Case: L-Shape Domain 244
5 Conclusions 246
References 247
Higher-Order Accurate Integration for Cut Elements with Chen-Babuška Nodes 248
1 Introduction 248
2 Integration in Two Dimensions 250
2.1 Reconstruction of Zero-Isolines in 2D Background Elements 254
2.2 Integration in Cut 2D Background Elements 255
2.3 Numerical Results in Two Dimensions 256
2.3.1 Integration on Zero-Isolines 257
2.3.2 Integration in Cut 2D Elements 261
3 Integration in Three Dimensions 263
3.1 Reconstruction of Zero-Isosurfaces in 3D Background Elements 265
3.2 Integration in Cut 3D Background Elements 265
3.3 Numerical Results in Three Dimensions 267
3.3.1 Integration on Zero-Isosurfaces 268
3.3.2 Integration in Cut 3D Elements 269
4 Conclusions 270
References 270