Multiscale Methods in Computational Mechanics - Progress and Accomplishments

Table of Contents

Preface

List of Authors

PART 1 Computational Fluid Dynamics

Residual-Based Variational Multiscale Theory of LES Turbulence Modeling

1 Variational Multiscale Formulation of the Incompressible Navier–Stokes Equations

1.1 Incompressible Navier–Stokes Equations

Global Space-Time Variational Formulation

Sliced Space-Time Variational Formulation

1.2 Scale Separation

1.3 Perturbation Series

2 Turbulent Channel Flow

3 Conclusions

Acknowledgements

References

A Posteriori Error Estimation for Computational Fluid Dynamics: The Variational Multiscale Approach

1 Introduction

2 The Variational Multiscale Approach to Error Estimation

2.1 The Abstract Problem

2.2 The Variational Multiscale Error Estimation Paradigm

3 The Smooth Paradigm for Error Estimation

3.1 Intrinsic Error Time Scales

Estimates in the L2 Norm

Example: One-Dimensional Advection-Diffusion

3.2 Error Upper Bounds

3.3 Relation to the Flow Time Scale Parameter

3.4 Extensions

4 Multidimensional Model

4.1 A Model for the Error Distribution

Element Interior Error

Element Boundary Error

4.2 Norms Based on the L8 Norm of the Residual

4.3 Summary of the Model

5 Multidimensional Error Scales for the Bilinear Quad

5.1 Hyperbolic Limit

5.2 Elliptic Limit

6 Numerical Example: L-shaped Domain Problem

7 Adaptivity

8 Conclusions

References

Advances in Variational Multiscale Methods for Turbulent Flows

1 Introduction

2 Residual-Based Variational Multiscale Method with Dynamic Subgrid Scales

3 The Algebraic Variational Multiscale-Multigrid Method

4 Using NURBS in Residual-Based Variational Multiscale Methods

5 Towards a Residual-Based Variational Multiscale Method for Turbulent Fluid-Structure Interaction

6 Conclusion

Acknowledgement

References

Variational Germano Approach for Multiscale Formulations

1 Introduction

2 General Discrete Germano Identity

2.1 Numerical Method as a Discrete Projector

2.2 Inverse Implication: Projector Implies Numerical Method

2.3 Multilevel Commutativity

2.4 Discrete Germano Identity

2.5 Partitioned Approach

3 Discrete Germano Approach for Stabilized Methods

3.1 Computation of Coarse Stabilization Parameter: Dissipation Method

Non-Homogenous Boundary Conditions

3.2 Computation of Coarse Stabilization Parameter: Least-Squares Method

Basis Independent Least-Squares Method

Basis Independent Least-Squares Method for a Spatial Varying Stability Parameter

3.3 Computation of Fine Stabilization Parameter

4 1D Convection-Diffusion

4.1 Stabilized Formulation

4.2 Stabilization Parameter Structure

4.3 Dissipation Method for Homogeneous Boundary Conditions

4.4 Dissipation Method for Non-Homogeneous Boundary Conditions

Reconstruction of the Lagrange Multipliers on Coarse Mesh

Injection of the Lagrange Multipliers from Fine Mesh

4.5 Naive Least-Squares Method

4.6 Basis-Independent Least-Squares Method

5 Numerical Results

5.1 Homogenous Boundary Conditions

5.2 Non-Homogenous Boundary Conditions

5.3 Basis Dependence of the Least-Squares Method

5.4 Computational Cost

6 Conclusion

Acknowledgments

References

Dissipative Structure and Long Term Behavior of a Finite Element Approximation of Incompressible Flows with Numerical Subgrid Scale Modeling

1 Introduction

2 Formulation

2.1 Continuous problem

2.2 Subgrid Scale Decomposition

2.3 Simplifying Assumptions

2.4 Final Formulation

3 Dissipative Structure and Backscatter

3.1 Local Kinetic Energy Balance Equations

3.2 Global Kinetic Energy Balance Equations

3.3 Backscatter

3.4 Flow over a Surface Mounted Obstacle

4 Long Term Stability

5 Long Term Simulations

5.1 Flow over a Plate

5.2 Flow around a Telescope

6 Conclusions

Acknowledgments

References

Large-Eddy Simulation of Multiscale Particle Dynamics at High Volume Concentration in Turbulent Channel Flow

1 Introduction

2 Mathematical Formulation

2.1 The Gas Phase

2.2 The Solids Phase

2.3 Subgrid-Modeling

2.4 The Numerical Method

3 Results

3.1 Turbulence Modulation

3.2 Effects of Collisions

3.3 Coherent Particle Structures

4 Concluding Remarks

Acknowledgments

References

PART 2 Materials with Microstructure

An Incremental Strategy for Modeling Laminate Microstructures in Finite Plasticity – Energy Reduction, Laminate Orientation and Cyclic Behavior

1 Introduction

2 Non-Convex Potentials and Relaxation

3 First-Order Laminate Microstructures

4 Incremental Numerical Scheme

5 Results

5.1 Evolution of the Internal Variables and Laminate Orientation

5.2 Energy Reduction

5.3 Cyclic Behavior

6 Discussion and Conclusions

References

The Micromorphic versus Phase Field Approach to Gradient Plasticity and Damage with Application to Cracking in Metal Single Crystals

1 Generalized Continua and Material Microstructure

2 Micromorphic Approach

2.1 Thermomechanics with Additional Degrees of Freedom

2.2 Non-Dissipative Contribution of Generalized Stresses and Micromorphic Model

Micromorphic Model

2.3 Viscous Generalized Stress and Phase Field Model

Phase Field Model

2.4 Elasto-Plastic Decomposition of Generalized Strains

3 Continuum Damage Model for Single Crystals and Its Regularization

3.1 Constitutive Equations

3.2 Microdamage Continuum

4 Finite Element Implementation

4.1 Variational Formulation and Discretization

4.2 Implicit Incremental Formulation

5 Numerical Examples

6 Conclusion

References

Homogenization and Multiscaling of Granular Media for Different Microscopic Constraints

1 Introduction

2 Quasi-Static Homogenization of Granular Aggregates

2.1 Deformation-Driven Homogenization of Microstructures

Definition of Particle Microstructures

Microscopic Boundary Conditions

Microscopic Equilibrium State

Macroscopic Boundary Conditions

2.2 Penalty-Type Implementation of Boundary Constraints

3 Microstructural Modeling of Granular Materials

3.1 Micromechanical Model for Interparticle Contact

3.2 Dynamic Relaxation of the Microstructural Response

4 Numerical Examples and Comparative Study

4.1 Specification of Basic Micromechanical Functions

4.2 Compression-Shear Mode for Cubic Microstructures

5 Multiple Scale Simulation of a Granular Medium

5.1 Two-Scale Simulations Based on DE-FE Coupling

5.2 Simulation of a Biaxial Compression Test of a Soil

Experimental Setup

Coupled FE-DE Two-Scale Model

Results and Discussion

6 Conclusion

Acknowledgement

References

Effective Hydraulic and Mechanical Properties of Heterogeneous Media with Interfaces

1 Introduction

2 Hydraulic Model for a Porous Matrix with Impermeable Inclusionary Phase

2.1 Mori–Tanaka Estimate

2.2 The Variational Approach

2.3 The Self-Consistent Approach

3 Mechanical Model for a Granular Cemented Rock

3.1 General Framework

3.2 The Self-Consistent Homogenization Scheme

4 Concluding Remarks

References

An Extended Finite Element Method for the Analysis of Submicron Heat Transfer Phenomena

1 Introduction

2 Level-Set Description of Material Layout

3 Gray Phonon Model

4 Discretization Methods

4.1 Discrete Ordinate Method

4.2 Extended Finite Element Method

4.3 Lagrange Multiplier Method

5 Numerical Examples

5.1 Verification Example

5.2 Analysis of Nano-Composites

5.3 Design Study

6 Conclusions

Acknowledgments

References

PART 3 Composites, Laminates, and Structures: Optimization

Multiscale Modeling and Simulation of Composite Materials and Structures

1 Introduction

2 Information-Passing Multiscale Approaches in Space

2.1 Direct Mathematical Homogenization for Nonlinear Problems

2.2 Eigendeformation-Based Reduced Order Homogenization

3 Concurrent Multiscale Methods in Space

3.1 Multiscale Enrichment Based on Partition of Unity (MEPU)

3.2 Adaptive Model Selection

3.3 Numerical Example

4 Temporal Multiscale Model for Fatigue Life Prediction

References

Multiscale Modelling of the Failure Behaviour of Fibre-Reinforced Laminates

1 Introduction

2 Review of the Interface Damage Model

3 Mesoscale Simulations of a Centre-Cracked 2/1 GLARE Laminate

3.1 Geometry and Boundary Conditions

3.2 Results for a 2/1 Lay-up with Elastic Aluminium Layers

3.3 Results for a 2/1 Lay-up with Elasto-Plastic Aluminium Layers

4 Microscale Simulations of Single-Fibre Epoxy Systems

4.1 Fibre-Epoxy Interfacial Strength versus Epoxy Strength

5 Microscale Simulations of Multiple-Fibre Epoxy Systems

5.1 Influence of the Fibre Volume Fraction

6 Coupling between Microscale and Mesoscale Crack Modelling

6.1 Fibre-Epoxy Specimen Subjected to Uniaxial Tension

6.2 Influence of Sample Size

6.3 Influence of Imperfections

7 Concluding Remarks

Acknowledgements

References

Improved Multiscale Computational Strategies for Delamination

1 Introduction

2 Application of the Two-Scale Domain Decomposition Strategy to Delamination Analysis

2.1 The Substructured Delamination Problem

2.2 Two-Scale Iterative Resolution of the Substructured Problem

Introduction of the Macroscopic Scale

The Iterative Algorithm

2.3 First Example of a Delamination Analysis

3 Analysis of the Parameters of the Iterative Algorithm

4 The Three-Scale Domain Decomposition Strategy

4.1 Resolution of the Macroproblem through the Balancing Domain Decomposition Method

Partitioning of the Macroproblem

Resolution of the Super-Interface Problem

4.2 Results

5 Efficiency of the Strategy: Study of a Complex Test Case

6 Conclusion

References

Damage Propagation in Composites – Multiscale Modeling and Optimization

1 Introduction

2 Modeling of Discontinuities on Small Scale

2.1 Geometrical Description

2.2 Kinematic Description

2.3 Cohesive Law

2.4 Numerical Examples

Two-Phase Material under Tension [13]

Pull-out of Fiber in Matrix

3 Multiscale Formulation [11, 12]

3.1 Formulation Using Continuum Damage

Material Model

Concept

Numerical Examples [11]

3.2 Discontinuum Model in Multiscale Formulation

Concept

Numerical Examples [13]

4 Optimal Fiber Layout [18–20]

4.1 Material Model

4.2 Optimization Concept

4.3 Multiphase Material Optimization

4.4 Shape Optimization of Fiber Geometry

4.5 Numerical Examples

Multiphase Material Optimization for a Beam

Multiphase and Shape Optimization for Deep Beam

5 Conclusions

Acknowledgement

References

Computational Multiscale Model for NATM Tunnels: Micromechanics-Supported Hybrid Analyses

1 Introduction

2 Continuum Micromechanics of Microheterogeneous Materials

2.1 Representative Volume Element (Separation of Scales)

2.2 Homogenization of Elasticity

2.3 Homogenization of Strength

3 Micromechanics at the Cement Paste Level

3.1 Micromechanical Representation

3.2 Constitutive Behavior of Clinker, Water, Hydrates, and Air

3.3 Homogenized Elasticity of Cement Paste

3.4 Homogenized Strength of Cement Paste

4 Micromechanics at the Shotcrete Level

5 Experimental Validation of Micromechanics-Based Material Models

5.1 Mixture-Dependent Shotcrete Composition

5.2 Experimental Validation on Shotcrete Level

6 Micromechanics-Based Studies: Influence of Water-Cement and Aggregate-Cement Ratios on Evolutions of Elasticity and Strength of Shotcrete

7 Continuum Micromechanics-Based Safety Assessment of NATM Tunnel Shells

7.1 Water-Cement Ratio-Dependence of Structural Safety

7.2 Aggregate-Cement Ratio-Dependence of Structural Safety

8 Conclusions

Acknowledgements

References

Optimization of Corrugated Paperboard under Local and Global Buckling Constraints

1 Introduction

2 Problem Description and Methods Used

2.1 Unit Cell Approach

2.2 Formulation of the Optimization Problem

2.3 Homogenization Process Using Unit Cell Models

2.4 Unit Cell Approach for Local Buckling Analysis

2.5 Numerical Meso Structural Optimization Approach

3 Results of Optimization

4 Fold Formation in the Post-Buckling Regime

5 Conclusions

References

Framework for Multi-Level Optimization of Complex Systems

1 Introduction

2 A Unifying Multi-Level Notation

3 Decomposition

3.1 Physical Coupling

3.2 Problem Matrix

4 Coordination

5 Software Framework

6 Supersonic Business Jet Optimization

6.1 Multi-Level Optimization Problem

Hierarchic Decomposition

Non-Hierarchic Decomposition

Coordination

6.2 Numerical Results

Results

7 Conclusions

Acknowledgments

References

PART 4 Coupled Problems and Porous Media

Multiscale/Multiphysics Model for Concrete

1 Introduction

2 General Mathematical Model

3 Effective Stress Principle

4 Application of the Model to Concrete Structures at Elevated Temperature

4.1 Simulation of a Concrete Column under Fire with Fast Cooling

5 Application of the Model to Concrete Structures Subject to Leaching Process

5.1 Modelling Kinetics of Calcium Leaching Process

5.2 Numerical Simulation of the Non-Isothermal Leaching Process in a Concrete Wall

6 Conclusions

References

Swelling Phenomena in Electro-Chemically Active Hydrated Porous Media

1 Introduction

2 TPM Fundamentals

2.1 Immiscible Components and Volume Fractions

2.2 Miscible Components and Molar Concentrations

2.3 Constituent Balance Relations

3 Swelling Media as Biphasic, Four-Component Aggregates

3.1 Restrictions Obtained from the Entropy Inequality

3.2 The Fluid Components

3.3 Ion Diffusion and Fluid Flow

3.4 The Electrical Potential

3.5 The Solid Skeleton

4 Weak Forms and Basic Numerical Setting

5 Numerical Examples

5.1 Free Swelling Hydrogel

5.2 Electro-Active Polymers

5.3 Borehole Instability in Active Soil

5.4 Swelling of an Intervertebral Disc

6 Conclusion

References

Propagating Cracks in Saturated Ionized Porous Media

1 Introduction

2 Bulk Material

3 Discontinuity in the Solid Part

3.1 Cohesive Zone

3.2 Nonlocal Stress

4 Shearing Mode

4.1 Discontinuity in the Fluid Part

4.2 Numerical Example

5 Tensile Mode

5.1 Discontinuity in the Fluid Part

5.2 Numerical Example

6 Concluding Remarks

List of Symbols

Acknowledgement

References

Author Index

Subject Index