Partition of Unity Methods

Stéphane P. A. Bordas is a Professor in Computational Mechanics and earned his PhD from Northwestern University, USA, in 2004. He has published over 200 papers in unfitted simulation of free boundary problems and data driven modelling of complex systems. He has supervised over 30 PhD students and is Editor- in-Chief of Advances in Applied Mechanics. Alexander Menk is employed by Bosch GmbH. He is a PhD graduate from Glasgow University, UK, supervised by Prof. Bordas. His contributions range from automatic numerically determined enrichment to preconditioners for extended finite element methods for fracture. Sundararajan Natarajan has been a Professor of Computational Mechanics since 2014 and earned his PhD from Cardiff University, UK, supervised by Prof. Bordas and Prof. Kerfriden. He has made strong contributions to a number of methods on unfitted methods for free boundary problems, in particular on numerical integration and strain smoothing.

• List of Contributors xiPreface xiiiAcknowledgments xv1 Introduction 11.1 The Finite Element Method 21.2 Suitability of the Finite Element Method 91.3 Some Limitations of the FEM 111.4 The Idea of Enrichment 161.5 Conclusions 192 A Step-by-Step Introduction to Enrichment 232.1 History of Enrichment for Singularities and Localized Gradients 252.2 Weak Discontinuities for One-dimensional Problems 382.3 Strong Discontinuities for One-dimensional Problem 582.4 Conclusions 613 Partition of Unity Revisited 673.1 Completeness, Consistency, and Reproducing Conditions 673.2 Partition of Unity 683.3 Enrichment 693.4 Numerical Examples 863.5 Conclusions 954 Advanced Topics 994.1 Size of the Enrichment Zone 994.2 Numerical Integration 1004.3 Blending Elements and Corrections 1084.4 Preconditioning Techniques 1165 Applications 1255.1 Linear Elastic Fracture in Two Dimensions with XFEM 1255.2 Numerical Enrichment for Anisotropic Linear Elastic Fracture Mechanics 1305.3 Creep and Crack Growth in Polycrystals 1335.4 Fatigue Crack Growth Simulations 1385.5 Rectangular Plate with an Inclined Crack Subjected to Thermo-Mechanical Loading 1406 Recovery-Based Error Estimation and Bounding in XFEM 1456.1 Introduction 1456.2 Error Estimation in the Energy Norm. The ZZ Error Estimator 1476.3 Recovery-based Error Estimation in XFEM 1516.4 Recovery Techniques in Error Bounding. Practical Error Bounds. 1746.5 Error Estimation in Quantities of Interest 1797 Φ-FEM: An Efficient Simulation Tool Using Simple Meshes for Problems in Structure Mechanics and Heat Transfer 1917.1 Introduction 1917.2 Linear Elasticity 1947.3 Linear Elasticity with Multiple Materials 2047.4 Linear Elasticity with Cracks 2087.5 Heat Equation 2127.6 Conclusions and Perspectives 2148 eXtended Boundary Element Method (XBEM) for Fracture Mechanics and Wave Problems 2178.1 Introduction 2178.2 Conventional BEM Formulation 2188.3 Shortcomings of the Conventional Formulations 2268.4 Partition of Unity BEM Formulation 2288.5 XBEM for Accurate Fracture Analysis 2288.6 XBEM for ShortWave Simulation 2358.7 Conditioning and its Control 2438.8 Conclusions 2459 Combined Extended Finite Element and Level Set Method (XFE-LSM) for Free Boundary Problems 2499.1 Motivation 2499.2 The Level Set Method 2509.3 Biofilm Evolution 2569.4 Conclusion 26910 XFEM for 3D Fracture Simulation 27310.1 Introduction 27310.2 Governing Equations 27410.3 XFEM Enrichment Approximation 27510.4 Vector Level Set 28010.5 Computation of Stress Intensity Factor 28210.6 Numerical Simulations 28810.7 Summary 30011 XFEM Modeling of Cracked Elastic-Plastic Solids 30311.1 Introduction 30311.2 Conventional von Mises Plasticity 30311.3 Strain Gradient Plasticity 31211.4 Conclusions 32312 An Introduction to Multiscale analysis with XFEM 32912.1 Introduction 32912.2 Molecular Statics 33012.3 Hierarchical Multiscale Models of Elastic Behavior -- The Cauchy-Born Rule 33612.4 Current Multiscale Analysis -- The Bridging Domain Method 33812.5 The eXtended Bridging Domain Method 340References 344Index 345