Shape optimum design of porous structure for minimizing maximum thermal stress

Abstract

In this study, we propose a solution to a shape optimization problem for the strength design of porous structures under thermal loading. The homogenization method is used to bridge the macrostructure and the porous microstructures, in which the elastic and thermal expansion tensor are homogenized. The local thermal stress in the porous structure is minimized. By replacing the maximum thermal stress with a Kreisselmeier-Steinhauser function, the difficulty of non-differentiability of maximum stress is avoided. This problem is formulated as a distributed parameter optimization problem subject to an area constraint including the all microstructures. The shape gradient function for this design problem is derived using Lagrange multiplier method, the material derivative methods, which are independently defined for the sub-regions in the macrostructure and the adjoint variable method. The H1 gradient method is used to determine the optimal shape of the porous unit cell while reducing the objective function and maintaining smooth design boundaries. Effectiveness of the proposed method for minimizing the thermal stress of porous structures is confirmed by the numerical examples.

Keywords

• multi-scale optimization
• homogenization method
• shape optimization
• h1 gradient method
• periodic microstructure
• min-max problem
• kreisselmeier-steinhauser function
• stress concentration