Optimization of the porous material described by the Biot model

TL;DR

This paper develops a shape optimization method for microstructures in porous materials modeled by the Biot equations, using homogenization, sensitivity analysis, and a two-scale approach to improve material properties like stiffness and permeability.

Contribution

It introduces a sensitivity analysis framework for pore shape optimization within the Biot model and proposes a sequential linearization approach for two-scale microstructure optimization.

Findings

Optimized microstructures enhance stiffness while maintaining permeability.

Sensitivity analysis enables effective shape updates for pore geometries.

Numerical results demonstrate improved material performance through the proposed method.

Abstract

The paper is devoted to the shape optimization of microstructures generating porous locally periodic materials saturated by viscous fluids. At the macroscopic level, the porous material is described by the Biot model defined in terms of the effective medium coefficients, involving the drained skeleton elasticity, the Biot stress coupling, the Biot compressibility coefficients, and by the hydraulic permeability of the Darcy flow model. By virtue of the homogenization, these coefficients are computed using characteristic responses of the representative unit cell consisting of an elastic solid skeleton and a viscous pore fluid. For the purpose of optimization, the sensitivity analysis on the continuous level of the problem is derived. We provide sensitivities of objective functions constituted by the Biot model coefficients with respect to the underlying pore shape described by a B-spline box which embeds the whole representative cell. We consider material design problems in the framework of which the layout of a single representative cell is optimized. Then we propose a sequential linearization approach to the two-scale problem in which local microstructures are optimized with respect to macroscopic design criteria. Numerical experiments are reported which include stiffness maximization with constraints allowing for a sufficient permeability, and vice versa. Issues of the design anisotropy, the spline box parametrization are discussed. In order to avoid remeshing a geometric regularization technique based on injectivity constraints is applied.