Practical Multiscaling.

Intro -- Practical Multiscaling -- Copyright -- Contents -- Preface -- Acknowledgments -- 1 Introduction to Multiscale Methods -- 1.1 The Rationale for Multiscale Computations -- 1.2 The Hype and the Reality -- 1.3 Examples and Qualification of Multiscale Methods -- 1.4 Nomenclature and Definitions -- 1.5 Notation -- 1.5.1 Index and Matrix Notation -- 1.5.2 Multiple Spatial Scale Coordinates -- 1.5.3 Domains and Boundaries -- 1.5.4 Spatial and Temporal Derivatives -- 1.5.5 Special Symbols -- References -- 2 Upscaling/Downscaling of Continua -- 2.1 Introduction -- 2.2 Homogenizaton of Linear Heterogeneous Media -- 2.2.1 Two-Scale Formulation -- 2.2.2 Two-Scale Formulation - Variational Form -- 2.2.3 Hill-Mandel Macrohomogeneity Condition and Hill-Reuss-Voigt Bounds -- 2.2.4 Numerical Implementation -- 2.2.5 Boundary Layers -- 2.2.6 Convergence Estimates -- 2.3 Upscaling Based on Enhanced Kinematics -- 2.3.1 Multiscale Finite Element Method -- 2.3.2 Variational Multiscale Method -- 2.3.3 Multiscale Enrichment Based on Partition of Unity -- 2.4 Homogenization of Nonlinear Heterogeneous Media -- 2.4.1 Asymptotic Expansion for Nonlinear Problems -- 2.4.2 Formulation of the Coarse-Scale Problem -- 2.4.3 Formulation of the Unit Cell Problem -- 2.4.4 Example Problems -- 2.5 Higher Order Homogenization -- 2.5.1 Introduction -- 2.5.2 Formulation -- 2.6 Multiple-Scale Homogenization -- 2.7 Going Beyond Upscaling - Homogenization-Based Multigrid -- 2.7.1 Relaxation -- 2.7.2 Coarse-grid Correction -- 2.7.3 Two-grid Convergence for a Model Problem in a Periodic Heterogeneous Medium -- 2.7.4 Upscaling-Based Prolongation and Restriction Operators -- 2.7.5 Homogenization-based Multigrid and Multigrid Acceleration -- 2.7.6 Nonlinear Multigrid -- 2.7.7 Multigrid for Indefinite Systems -- Problems -- References -- 3 Upscaling/Downscaling of Atomistic/Continuum Media.

3.1 Introduction -- 3.2 Governing Equations -- 3.2.1 Molecular Dynamics Equation of Motion -- 3.2.2 Multiple Spatial and Temporal Scales and Rescaling of the MD Equations -- 3.3 Generalized Mathematical Homogenization -- 3.3.1 Multiple-Scale Asymptotic Analysis -- 3.3.2 The Dynamic Atomistic Unit Cell Problem -- 3.3.3 The Coarse-Scale Equations of Motion -- 3.3.4 Continuum Description of Equation of Motion -- 3.3.5 The Thermal Equation -- 3.3.6 Extension to Multi-Body Potentials -- 3.4 Finite Element Implementation and Numerical Verification -- 3.4.1 Weak Forms and Semidiscretization of Coarse-Scale Equations -- 3.4.2 The Fine-Scale (Atomistic) Problem -- 3.5 Statistical Ensemble -- 3.6 Verification -- 3.7 Going Beyond Upscaling -- 3.7.1 Spatial Multilevel Method Versus Space-Time Multilevel Method -- 3.7.2 The WR Scheme -- 3.7.3 Space-Time FAS -- Problems -- References -- 4 Reduced Order Homogenization -- 4.1 Introduction -- 4.2 Reduced Order Homogenization for Two-Scale Problems -- 4.2.1 Governing Equations -- 4.2.2 Residual-Free Fields and Model Reduction -- 4.2.3 Reduced Order System of Equations -- 4.2.4 One-Dimensional Model Problem -- 4.2.5 Computational Aspects -- 4.3 Lower Order Approximation of Eigenstrains -- 4.3.1 The Pitfalls of a Piecewise Constant One-Partition-Per-Phase Model -- 4.3.2 Impotent Eigenstrain -- 4.3.3 Hybrid Impotent-Incompatible Eigenstrain Mode Estimators -- 4.3.4 Chaboche Modification -- 4.3.5 Analytical Relations for Various Approximations of Eigenstrain Influence Functions -- 4.3.6 Eigenstrain Upwinding -- 4.3.7 Enhancing Constitutive Laws of Phases -- 4.3.8 Validation of the Hybrid Impotent-Incompatible Reduced Order Model with Eigenstrain Upwinding and Enhanced Constitutive Model of Phases -- 4.4 Extension to Nonlocal Heterogeneous Media -- 4.4.1 Staggered Nonlocal Model for Homogeneous Materials.

4.4.2 Staggered Nonlocal Multiscale Model -- 4.4.3 Validation of the Nonlocal Model -- 4.4.4 Rescaling Constitutive Equations -- 4.5 Extension to Dispersive Heterogeneous Media -- 4.5.1 Dispersive Coarse-scale Problem -- 4.5.2 The Quasi-Dynamic Unit Cell Problem -- 4.5.3 Linear Model Problem -- 4.5.4 Nonlinear Model Problem -- 4.5.5 Implicit and Explicit Formulations -- 4.6 Extension to Multiple Spatial Scales -- 4.6.1 Residual-Free Governing Equations at Multiple Scales -- 4.6.2 Multiple-Scale Reduced Order Model -- 4.7 Extension to Large Deformations -- 4.8 Extension to Multiple Temporal Scales with Application to Fatigue -- 4.8.1 Temporal Homogenization -- 4.8.2 Multiple Temporal and Spatial Scales -- 4.8.3 Fatigue Constitutive Equation -- 4.8.4 Verfication of the Multiscale Fatigue Model -- 4.9 Extension to Multiphysics Problems -- 4.9.1 Reduced Order Coupled Vector-Scalar Field Model at Multiple Scales -- 4.9.2 Environmental Degradation of PMC -- 4.9.3 Validation of the Multiphysics Model -- 4.10 Multiscale Characterization -- 4.10.1 Formulation of the Inverse Problem -- 4.10.2 Characterization of Model Parameters in ROH -- Problems -- References -- 5 Scale-separation-free Upscaling/Downscaling of Continua -- 5.1 Introduction -- 5.2 Computational Continua (C2) -- 5.2.1 Nonlocal Quadrature -- 5.2.2 Coarse-Scale Problem -- 5.2.3 Computational Unit Cell Problem -- 5.2.4 One-Dimensional Model Problem -- 5.3 Reduced Order Computational Continua (RC2) -- 5.3.1 Residual-Free Computational Unit Cell Problem -- 5.3.2 The Coarse-Scale Weak Form -- 5.3.3 Coarse-Scale Consistent Tangent Stiffness Matrix -- 5.4 Nonlocal Quadrature in Multidimensions -- 5.4.1 Tetrahedral Elements -- 5.4.2 Triangular Elements -- 5.4.3 Quadrilateral and Hexahedral Elements -- 5.5 Model Verification -- 5.5.1 The Beam Problem -- Problems -- References.

6 Multiscale Design Software -- 6.1 Introduction -- 6.2 Microanalysis with MDS-Lite -- 6.2.1 Familiarity with the GUI -- 6.2.2 Labeling Data Files -- 6.2.3 The First Walkthrough MDS-Micro Example -- 6.2.4 The Second Walkthrough MDS-Micro Example -- 6.2.5 Parametric Library of Unit Cell Models -- 6.3 Macroanalysis with MDS-Lite -- 6.3.1 First Walkthrough MDS-Macro Example -- 6.3.2 Second Walkthrough MDS-Macro Example -- 6.3.3 Third Walkthrough Example -- 6.3.4 Fourth Walkthrough Example -- Problems -- References -- Index.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.