Efficient Solution Techniques for Multiscale Structural Optimization in Materials Science

We consider the modeling and simulation of multiscale phenomena which arise in finding the optimal shape design of microcellular composite materials with heterogeneous microstructures. The paper focuses on the solution of the resulting partial differential equation (PDE) constrained structural optimization problem and development of efficient multiscale numerical algorithms which are challenging tools in reducing the computational complexity. The modeling strategy is applied in materials science for microstructural ceramic materials of multiple constituents. Our multiscale method is based on the efficient combination of both macroscopic and microscopic models. The homogenization technique based on the concept of strong separation of scales and the asymptotic expansion of the unknown displacements is applied to extract the macroscopic information from the microscale model. In the framework of all-at-once approach we find a proper combination of the iterative procedure for the nonlinear problem arising from the first order necessary optimality conditions, also known as Karush-Kuhn-Tucker (KKT) conditions, and efficient large-scale solvers for the stress-strain constitutive equation. We use the path-following predictor-corrector schemes by means of Newton's method and fast multigrid (MG) solution techniques. The performance of two preconditioners, incomplete Cholesky (IC) and algebraic multigrid (AMG), for the resulting homogenized state equation is studied. The comparative analysis for both preconditioners in terms of number of iterations and computing times is presented and discussed. Our interests focus on the parallel implementation of the preconditioning techniques and the use of BoomerAMG as a part of the free software library Hypre developed at the Center for Applied Scientific Computing (CASC), Lawrence Livermore National Laboratory (LLNL).