Compliance minimization with penalization of stress concentration through simple laminates and correctors

Abstract

Compliance minimization is the most studied problem in two materials optimal design in linear elastic regime. However, some of the penalized solutions present high stress concentration, therefore limiting their usefullnes if the nonlinear regime is also involved in the actual use of the structure, which happens very often in civil engineering applications. In this paper correctors are used to take into account high frequency oscillations induced on the stress field by the microstructure, and then an integral of the squared norm of the stress tensor, multiplied by a penalization parameter, is added to the compliance, and that sum is minimized by performing a partial relaxation. Three numerical examples are provided, including the short cantilever and a benchmark problem in the design of reinforced concrete irregular beams.