Nonlinear structural design using multiscale topology optimization. Part I: Static formulation

We present a hierarchical multiscale design framework that couples computational homogenization with topology optimization to design a composite structure's microstructure to optimize its nonlinear elastostatic behavior. To generate a well-posed macroscopic topology optimization problem, we use relaxation which requires homogenization to relate the macroscopic homogenized response to its microstructure. And because closed form expressions for homogenized properties generally do not exist for materials with nonlinear response we rely on computational homogenization to evaluate them. To optimize the homogenized properties of the unit cell we again use topology optimization and to make this unit cell optimization problem well posed we use restriction and thereby obtain a minimum microstructural length scale. The coupled nonlinear analyzes and optimization problems are computationally intensive tasks that we resolve with a scalable parallel framework based on a single-program-multiple-data programming paradigm. Numerical implementation is discussed and examples are provided.