Shape Optimization for Microstructure Design of Porous Materials Described by the Biot model in the Homogenization Framework

Abstract

The paper deals with shape optimization of microstructures generating porous locally periodic materials saturated by viscous fluids. The porous material is described as the Biot continuum derived by the homogenization method. The effective medium properties are given by the drained skeleton elasticity, the Biot stress coupling, the Biot compressibility coefficients, and by the hydraulic permeability of the Darcy flow model. These are computed using characteristic responses - solutions of the state problems defined in the representative unit cell constituted by an elastic skeleton and by a fluid channel generating the porosity. The design of the channel is described by a B-spline box which embeds the whole representative cell. The sensitivity analysis for all the homogenized material coefficients is derived using the domain method of the design velocity approach. The optimality criterion is aimed to maximize stiffness of the drained porous material and allow for a sufficient permeability and vice versa. Issues of the spline box parametrization, the channel shape regularity and FE mesh updates are discussed. The maximization problems are solved using the sparse nonlinear optimizer SNOPT.