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Preface; Contents; Part I Enriched Methods for Flow and Mechanics in Heterogeneous Media; A Mixed Finite Element Method for Modeling the Fluid Exchange Between Microcirculation and Tissue Interstitium; 1 Introduction; 2 Model Set Up; 3 Variational Formulation; 4 Numerical Approximation; 4.1 Algebraic Formulation; 5 Numerical Experiments; 5.1 Coupled 3D-1D Problem on a Single Branch; 5.1.1 Numerical Results; 5.2 Coupled 3D-1D Problem on a Y-Shaped Bifurcation; 5.2.1 Numerical Results; 6 Conclusions; References; On a PDE-Constrained Optimization Approach for Flow Simulations in Fractured Media
1 Introduction2 Problem Formulation; 3 A PDE-Constrained Optimization Approach; 3.1 Discretization Strategies; 4 Numerical Results; 4.1 Flow in Complex DFNs; 4.2 Unsteady Flow in DFNs; References; A Review of the XFEM-Based Approximation of Flow in Fractured Porous Media; 1 Introduction; 2 Governing Equations; 2.1 Equi-Dimensional Models; 2.1.1 Dual Formulation; 2.1.2 Primal Formulation; 2.2 Hybrid-Dimensional Models; 2.2.1 Dual Formulation; 2.2.2 Primal Formulation; 2.3 Branching and Intersections; 2.3.1 Assuming Pressure Continuity; 2.3.2 Robin Boundary Conditions
2.3.3 Dual Formulation for X-Shaped Intersections2.4 Boundary Conditions; 2.4.1 Dirichlet Conditions for Fractured Porous Media; 2.4.2 Boundary and Coupling Conditions for Fracture Tips; 3 Numerical Discretization by Means of XFEM; 3.1 Modification and Addition of Basis Functions; 3.2 Primal Formulation with XFEM; 3.3 Dual Mixed Formulation; 3.4 Fracture Grids and Approximation of the Coupling Terms; 3.5 Basis Function Enrichment Around Fracture Tips; 4 Solvers; 4.1 Conditioning; 4.2 Iterative Approaches; 5 Conclusions; References
Part II Enhanced Finite Element Formulations for Fracture and Interface ProblemsModeling of Fracture in Polycrystalline Materials; 1 Introduction; 2 Problem Statement; 2.1 Balance of Linear Momentum for a Fractured Body; 2.2 Constitutive Equations for Finite Deformations Crystal Plasticity; 2.3 Continuum Damage Mechanics Coupled to Crystal Plasticity; 2.4 Non-Local Extension of the GTN Model; 2.5 Weak Form of the Resulting Boundary Value Problem; 3 Discretisation of the Boundary Value Problem with the XFEM; 4 Quasistatic Crack Propagation with the XFEM; 4.1 Crack Propagation Criterion
4.2 Reconstruction of the Fracture Surface4.2.1 Level Set Update by Solving a Hamilton-Jacobi Equation; 4.2.2 Global Crack Tracking Algorithm; 4.2.3 Accuracy of the Two Methods; 4.3 Crack Propagation Algorithm; 5 Numerical Examples; 5.1 Verification of the Non-Local Damage Formulation; 5.2 Crack Propagation in Crystals; 6 Conclusions; References; eXtended Hybridizable Discontinuous Galerkin (X-HDG) for Void and Bimaterial Problems; 1 Introduction; 2 X-HDG Formulation for Void Problems; 2.1 Local Problem for Standard Elements; 2.2 Local Problem for a Cut Element; 2.3 Global Problem
1 Introduction2 Problem Formulation; 3 A PDE-Constrained Optimization Approach; 3.1 Discretization Strategies; 4 Numerical Results; 4.1 Flow in Complex DFNs; 4.2 Unsteady Flow in DFNs; References; A Review of the XFEM-Based Approximation of Flow in Fractured Porous Media; 1 Introduction; 2 Governing Equations; 2.1 Equi-Dimensional Models; 2.1.1 Dual Formulation; 2.1.2 Primal Formulation; 2.2 Hybrid-Dimensional Models; 2.2.1 Dual Formulation; 2.2.2 Primal Formulation; 2.3 Branching and Intersections; 2.3.1 Assuming Pressure Continuity; 2.3.2 Robin Boundary Conditions
2.3.3 Dual Formulation for X-Shaped Intersections2.4 Boundary Conditions; 2.4.1 Dirichlet Conditions for Fractured Porous Media; 2.4.2 Boundary and Coupling Conditions for Fracture Tips; 3 Numerical Discretization by Means of XFEM; 3.1 Modification and Addition of Basis Functions; 3.2 Primal Formulation with XFEM; 3.3 Dual Mixed Formulation; 3.4 Fracture Grids and Approximation of the Coupling Terms; 3.5 Basis Function Enrichment Around Fracture Tips; 4 Solvers; 4.1 Conditioning; 4.2 Iterative Approaches; 5 Conclusions; References
Part II Enhanced Finite Element Formulations for Fracture and Interface ProblemsModeling of Fracture in Polycrystalline Materials; 1 Introduction; 2 Problem Statement; 2.1 Balance of Linear Momentum for a Fractured Body; 2.2 Constitutive Equations for Finite Deformations Crystal Plasticity; 2.3 Continuum Damage Mechanics Coupled to Crystal Plasticity; 2.4 Non-Local Extension of the GTN Model; 2.5 Weak Form of the Resulting Boundary Value Problem; 3 Discretisation of the Boundary Value Problem with the XFEM; 4 Quasistatic Crack Propagation with the XFEM; 4.1 Crack Propagation Criterion
4.2 Reconstruction of the Fracture Surface4.2.1 Level Set Update by Solving a Hamilton-Jacobi Equation; 4.2.2 Global Crack Tracking Algorithm; 4.2.3 Accuracy of the Two Methods; 4.3 Crack Propagation Algorithm; 5 Numerical Examples; 5.1 Verification of the Non-Local Damage Formulation; 5.2 Crack Propagation in Crystals; 6 Conclusions; References; eXtended Hybridizable Discontinuous Galerkin (X-HDG) for Void and Bimaterial Problems; 1 Introduction; 2 X-HDG Formulation for Void Problems; 2.1 Local Problem for Standard Elements; 2.2 Local Problem for a Cut Element; 2.3 Global Problem