Practical Multiscaling

Jacob Fish, Columbia University, USAJacob Fish is the Robert A. W. and Christine S. Carleton Professor in Civil Engineering at Columbia University. He is the Founder and Editor-in-Chief of the International Journal of Multiscale Computational Engineering and serves as an Associate Editor of the International Journal for Numerical Methods in Engineering. He is also on the editorial board of the Computer Methods in Applied Mechanics and Engineering, the International Journal of Computational Methods and the International Journal of Computational Engineering Science.He has has written over 160 journal articles, book chapters and has authored two previous books. Dr. Fish specializes in Multiscale Science and Engineering with applications to aerospace, automotive industry, civil engineering, biological and material sciences.

• Preface xiAcknowledgments xv1 Introduction to Multiscale Methods 11.1 The Rationale for Multiscale Computations 11.2 The Hype and the Reality 21.3 Examples and Qualification of Multiscale Methods 31.4 Nomenclature and definitions 51.5 Notation 61.5.1 Index and matrix notation 61.5.2 M ultiple Spatial Scale Coordinates 81.5.3 Domains and boundaries 91.5.4 Spatial and Temporal Derivatives 91.5.5 Special symbols 10References 112 Upscaling/Downscaling of Continua 132.1 Introduction 132.2 Homogenizaton of Linear Heterogeneous Media 162.2.1 Two-Scale Formulation 162.2.2 Two-Scale Formulation – Variational Form 232.2.3 Hill–Mandel Macrohomogeneity Condition and Hill–Reuss–Voigt Bounds 252.2.4 N umerical Implementation 272.2.5 B oundary Layers 382.2.6 Convergence Estimates 412.3 Upscaling Based on Enhanced Kinematics 472.3.1 M ultiscale Finite Element Method 482.3.2 Variational Multiscale Method 482.3.3 M ultiscale Enrichment Based on Partition of Unity 492.4 Homogenization of Nonlinear Heterogeneous Media 502.4.1 Asymptotic Expansion for Nonlinear Problems 502.4.2 Formulation of the Coarse-Scale Problem 542.4.3 Formulation of the Unit Cell Problem 582.4.4 Example Problems 612.5 Higher Order Homogenization 642.5.1 Introduction 642.5.2 Formulation 652.6 Multiple-Scale Homogenization 692.7 Going Beyond Upscaling – Homogenization-Based Multigrid 712.7.1 Relaxation 732.7.2 Coarse-grid Correction 772.7.3 Two-grid Convergence for a Model Problem in a Periodic Heterogeneous Medium 792.7.4 Upscaling-Based Prolongation and Restriction Operators 812.7.5 Homogenization-based Multigrid and Multigrid Acceleration 832.7.6 N onlinear Multigrid 842.7.7 M ultigrid for Indefinite Systems 86Problems 87References 913 Upscaling/Downscaling of Atomistic/Continuum Media 953.1 Introduction 953.2 Governing Equations 963.2.1 M olecular Dynamics Equation of Motion 963.2.2 M ultiple Spatial and Temporal Scales and Rescaling of the MD Equations 983.3 Generalized Mathematical Homogenization 1003.3.1 M ultiple-Scale Asymptotic Analysis 1003.3.2 The Dynamic Atomistic Unit Cell Problem 1023.3.3 The Coarse-Scale Equations of Motion 1033.3.4 Continuum Description of Equation of Motion 1063.3.5 The Thermal Equation 1073.3.6 Extension to Multi-Body Potentials 1123.4 Finite Element Implementation and Numerical Verification 1133.4.1 Weak Forms and Semidiscretization of Coarse-Scale Equations 1133.4.2 The Fine-Scale (Atomistic) Problem 1153.5 Statistical Ensemble 1183.6 Verification 1203.7 Going Beyond Upscaling 1263.7.1 Spatial Multilevel Method Versus Space–Time Multilevel Method 1273.7.2 The WR Scheme 1293.7.3 Space–Time FAS 130Problems 131References 1334 Reduced Order Homogenization 1374.1 Introduction 1374.2 Reduced Order Homogenization for Two-Scale Problems 1394.2.1 Governing Equations 1394.2.2 Residual-Free Fields and Model Reduction 1414.2.3 Reduced Order System of Equations 1484.2.4 One-Dimensional Model Problem 1504.2.5 Computational Aspects 1544.3 Lower Order Approximation of Eigenstrains 1564.3.1 The Pitfalls of a Piecewise Constant One-Partition-Per-Phase Model 1574.3.2 Impotent Eigenstrain 1594.3.3 Hybrid Impotent-Incompatible Eigenstrain Mode Estimators 1634.3.4 Chaboche Modification 1644.3.5 Analytical Relations for Various Approximations of Eigenstrain Influence Functions 1654.3.6 Eigenstrain Upwinding 1724.3.7 Enhancing Constitutive Laws of Phases 1754.3.8 Validation of the Hybrid Impotent-Incompatible Reduced Order Model with Eigenstrain Upwinding and Enhanced Constitutive Model of Phases 1804.4 Extension to Nonlocal Heterogeneous Media 1844.4.1 Staggered Nonlocal Model for Homogeneous Materials 1864.4.2 Staggered Nonlocal Multiscale Model 1884.4.3 Validation of the Nonlocal Model 1894.4.4 Rescaling Constitutive Equations 1934.5 Extension to Dispersive Heterogeneous Media 1974.5.1 Dispersive Coarse-Scale Problem 1994.5.2 The Quasi-Dynamic Unit Cell Problem 2014.5.3 Linear Model Problem 2044.5.4 N onlinear Model Problem 2054.5.5 Implicit and Explicit Formulations 2084.6 Extension to Multiple Spatial Scales 2094.6.1 Residual-Free Governing Equations at Multiple Scales 2104.6.2 M ultiple-Scale Reduced Order Model 2114.7 Extension to Large Deformations 2144.8 Extension to Multiple Temporal Scales with Application to Fatigue 2194.8.1 Temporal Homogenization 2204.8.2 M ultiple Temporal and Spatial Scales 2244.8.3 Fatigue Constitutive Equation 2254.8.4 Verfication of the Multiscale Fatigue Model 2264.9 Extension to Multiphysics Problems 2274.9.1 Reduced Order Coupled Vector-Scalar Field Model at Multiple Scales 2284.9.2 Environmental Degradation of PMC 2324.9.3 Validation of the Multiphysics Model 2354.10 Multiscale Characterization 2394.10.1 Formulation of the Inverse Problem 2394.10.2 Characterization of Model Parameters in ROH 241Problems 241References 2435 Scale-separation-free Upscaling/Downscaling of Continua 2495.1 Introduction 2495.2 Computational Continua (C2) 2515.2.1 N onlocal Quadrature 2515.2.2 Coarse-Scale Problem 2545.2.3 Computational Unit Cell Problem 2575.2.4 One-dimensional model problem 2605.3 Reduced Order Computational Continua (RC2) 2655.3.1 Residual-Free Computational Unit Cell Problem 2665.3.2 The Coarse-Scale Weak Form 2745.3.3 Coarse-Scale Consistent Tangent Stiffness Matrix 2755.4 Nonlocal Quadrature in Multidimensions 2785.4.1 Tetrahedral Elements 2785.4.2 Triangular Elements 2875.4.3 Quadrilateral and Hexahedral Elements 2925.5 Model Verification 2975.5.1 The Beam Problem 300Problems 302References 3036 Multiscale Design Software 3056.1 Introduction 3056.2 Microanalysis with MDS-Lite 3086.2.1 Familiarity with the GUI 3096.2.2 Labeling Data Files 3126.2.3 The First Walkthrough MDS-Micro Example 3126.2.4 The Second Walkthrough MDS-Micro Example 3186.2.5 Parametric Library of Unit Cell Models 3316.3 Macroanalysis with MDS-Lite 3406.3.1 First Walkthrough MDS-Macro Example 3416.3.2 Second Walkthrough MDS-Macro Example 3626.3.3 Third Walkthrough Example 3736.3.4 Fourth Walkthrough Example 379Problems 391References 393Index 395