Partition of Unity Methods: The Extended Finite Element Method (Wiley Series in Computational Mechanics)

Partition of Unity Methods
Contents
List of Contributors
Preface
Acknowledgments
1 Introduction
	1.1 The Finite Element Method
	1.2 Suitability of the Finite Element Method
	1.3 Some Limitations of the FEM
	1.4 The Idea of Enrichment
	1.5 Conclusions
	References
2 A Step-by-Step Introduction to Enrichment
	2.1 History of Enrichment for Singularities and Localized Gradients
		2.1.1 Enrichment by the ``Method of Supplementary Singular Functions''
		2.1.2 Finite Element with a Singularity
		2.1.3 Partition of Unity Enrichment
		2.1.4 Mesh Overlay Methods
		2.1.5 Enrichment for Strong Discontinuities
	2.2 Weak Discontinuities for One-dimensional Problems
		2.2.1 Conventional Finite Element Solution
		2.2.2 eXtended Finite Element Solution
		2.2.3 eXtended Finite Element Solution with Nodal Subtraction/Shifting
		2.2.4 Solution
	2.3 Strong Discontinuities for One-dimensional Problem
	2.4 Conclusions
	References
3 Partition of Unity Revisited
	3.1 Completeness, Consistency, and Reproducing Conditions
	3.2 Partition of Unity
	3.3 Enrichment
		3.3.1 Description of Geometry of Enrichment Features
		3.3.2 Choice of Enrichment Functions
		3.3.3 Imposition of boundary conditions
		3.3.4 Numerical Integration of theWeak Form
	3.4 Numerical Examples
		3.4.1 One-Dimensional Multiple Interface
		3.4.2 Two-Dimensional Circular Inhomogeneity
		3.4.3 Infinite Plate with a Center Crack Under Tension
	3.5 Conclusions
	References
4 Advanced Topics
	4.1 Size of the Enrichment Zone
	4.2 Numerical Integration
		4.2.1 Polar Integration
		4.2.2 Equivalent Polynomial Integration
		4.2.3 Conformal Mapping
		4.2.4 Strain Smoothing in XFEM
	4.3 Blending Elements and Corrections
		4.3.1 Blending Between Different Partitions of Unity
		4.3.2 Interpolation Error in Blending Elements
		4.3.3 Addressing Blending Phenomena
	4.4 Preconditioning Techniques
		4.4.1 The First Preconditioner Proposed for the XFEM
		4.4.2 A domain Decomposition Preconditioner for the XFEM
	References
5 Applications
	5.1 Linear Elastic Fracture in Two Dimensions with XFEM
		5.1.1 Inclined Crack in Tension
		5.1.2 Example of a Crack Inclusion Interaction Problem
		5.1.3 Effect of the Distance Between the Crack and the Inclusion
	5.2 Numerical Enrichment for Anisotropic Linear Elastic Fracture Mechanics
	5.3 Creep and Crack Growth in Polycrystals
	5.4 Fatigue Crack Growth Simulations
	5.5 Rectangular Plate with an Inclined Crack Subjected to Thermo-Mechanical Loading
	References
6 Recovery-Based Error Estimation and Bounding in XFEM
	6.1 Introduction
	6.2 Error Estimation in the Energy Norm. The ZZ Error Estimator
		6.2.1 The SPR Technique
		6.2.2 The MLS Approach
	6.3 Recovery-based Error Estimation in XFEM
		6.3.1 The SPR-CX Technique
		6.3.2 The XMLS Technique
		6.3.3 The MLS-CX Technique
		6.3.4 On the Roles of Enhanced Recovery and Admissibility
	6.4 Recovery Techniques in Error Bounding. Practical Error Bounds.
	6.5 Error Estimation in Quantities of Interest
		6.5.1 Recovery-based Estimates for the Error in Quantities of Interest
		6.5.2 The Stress Intensity Factor as QoI: Error Estimation
	References
7 ????-FEM: An Efficient Simulation Tool Using Simple Meshes for Problems in Structure Mechanics and Heat Transfer
	7.1 Introduction
	7.2 Linear Elasticity
		7.2.1 Dirichlet Conditions
		7.2.2 Mixed Boundary Conditions
	7.3 Linear Elasticity with Multiple Materials
	7.4 Linear Elasticity with Cracks
	7.5 Heat Equation
	7.6 Conclusions and Perspectives
	References
8 eXtended Boundary Element Method (XBEM) for Fracture Mechanics and Wave Problems
	8.1 Introduction
	8.2 Conventional BEM Formulation
		8.2.1 Elasticity
		8.2.2 Helmholtz Wave Problems
	8.3 Shortcomings of the Conventional Formulations
	8.4 Partition of Unity BEM Formulation
	8.5 XBEM for Accurate Fracture Analysis
		8.5.1 Williams Expansions
		8.5.2 Local XBEM Enrichment at Crack Tips
		8.5.3 Results
		8.5.4 Auxiliary Equations and Direct Evaluation of Stress Intensity Factors
		8.5.5 Fracture in Anisotropic Materials
		8.5.6 Conclusions
	8.6 XBEM for Short Wave Simulation
		8.6.1 Background to the Development of Plane Wave Enrichment
		8.6.2 Plane Wave Enrichment
		8.6.3 Evaluation of Boundary Integrals
		8.6.4 Collocation Strategy and Solution
		8.6.5 Results
		8.6.6 Choice of Basis Functions
		8.6.7 Scattering from Sharp Corners
	8.7 Conditioning and its Control
	8.8 Conclusions
	References
9 Combined Extended Finite Element and Level Set Method (XFE-LSM) for Free Boundary Problems
	9.1 Motivation
	9.2 The Level Set Method
		9.2.1 The Level Set Representation of the Embedded Interface
		9.2.2 The Basic Level Set Evolution Equation
		9.2.3 Velocity Extension
		9.2.4 Level Set Function Update
		9.2.5 Coupling the Level Set Method with the XFEM
	9.3 Biofilm Evolution
		9.3.1 Biofilms
		9.3.2 Biofilm Modeling
		9.3.3 Two-Dimensional Model
		9.3.4 Solution Strategy
		9.3.5 Variational Form
		9.3.6 Enrichment Functions
		9.3.7 Interface Conditions
		9.3.8 Interface Speed Function
		9.3.9 Accuracy and Convergence
		9.3.10 Numerical Results
	9.4 Conclusion
	Acknowledgment
	References
10 XFEM for 3D Fracture Simulation
	10.1 Introduction
	10.2 Governing Equations
	10.3 XFEM Enrichment Approximation
	10.4 Vector Level Set
	10.5 Computation of Stress Intensity Factor
		10.5.1 Brittle Material
		10.5.2 Ductile Material
	10.6 Numerical Simulations
		10.6.1 Computation of Fracture Parameters
		10.6.2 Fatigue Crack Growth in Compact Tension Specimen
	10.7 Summary
	References
11 XFEM Modeling of Cracked Elastic-Plastic Solids
	11.1 Introduction
	11.2 Conventional von Mises Plasticity
		11.2.1 Constitutive Model
		11.2.2 Asymptotic Crack Tip Fields
		11.2.3 XFEM Enrichment
		11.2.4 Numerical Implementation
		11.2.5 Representative Results
	11.3 Strain Gradient Plasticity
		11.3.1 Constitutive Model
		11.3.2 Asymptotic Crack Tip Fields
		11.3.3 XFEM Enrichment
		11.3.4 Numerical Implementation
		11.3.5 Representative Results
	11.4 Conclusions
	References
12 An Introduction to Multiscale analysis with XFEM
	12.1 Introduction
		12.1.1 Types of Multiscale Analysis
	12.2 Molecular Statics
		12.2.1 Atomistic Potentials
		12.2.2 A simple 1D Harmonic Potential Example
		12.2.3 The Lennard-Jones Potential
		12.2.4 The Embedded Atom Method
	12.3 Hierarchical Multiscale Models of Elastic Behavior – The Cauchy-Born Rule
	12.4 Current Multiscale Analysis – The Bridging Domain Method
	12.5 The eXtended Bridging Domain Method
		12.5.1 Simulation of a Crack Using XFEM
	References
Index