Multiscale Modeling and Analysis for Materials Simulation

The following sections are included:

• CONTENTS

• FOREWORD

• PREFACE

Dislocations are line defects and the primary carriers of plastic deformation in crystalline materials. In this article, we give an introduction to the dislocation theory and review the recent advances in modeling and simulation of dislocation dynamics in two or nearly two dimensions, including the Peierls-Nabarro model and its generalizations for the structure and dynamics of dislocations in a single slip plane, dislocation dynamics in thin films, and dislocation models for grain boundaries. We call these problems 2+ε dimensional problems because the physical extent of one of the 3 dimensions is much smaller than the other two. This reduction of dimension plays an important role both on the nature of the structures formed and/or on how the evolution of these structures is simulated. This reduction of dimension also leads to the formation of special dislocation microstructures, whose energetics and dynamics play important roles in the plastic deformation behavior and to several other related physical properties. Dislocation dynamics simulation provides an excellent tool for such investigations.

The lecture note introduces some basic theory and simulation methods for molecular dynamics models. Principles of statistical mechanics, which connect the trajectories of atoms to macroscopic quantities, will be presented along with the computational methods. In addition to equilibrium simulations, we will present several recent models for non-equilibrium simulations.

We present in this note a unified approach on how to design simple, efficient and energy stable time discretization schemes for the Allen-Cahn or Cahn-Hilliard Navier-Stokes system which models two-phase incompressible flows with matching or non-matching density. Special emphasis is placed on designing schemes which only require solving linear systems at each time step while satisfy discrete energy laws that mimic the continuous energy laws. We construct the time discretization schemes in weak formulations so that they can be used with any consistent Galerkin type spacial discretization schemes such as finite element methods and spectral/spectral-element methods.

We review some recent developments in computational stochastic homogenization. The results have been obtained in a series of works [6, 7, 8, 9, 16, 17, 19, 20, 25, 26, 27, 71]. They all aim at designing numerical approaches that both are practically relevant and keep the computational workload limited. For consistency, we begin with a pedagogic introduction on general, and next more specifically stochastic, homogenization.