Checkerboard free topology optimization for compliance minimization applying the finite-volume theory

Abstract

The purpose of this research is to demonstrate that the numerical issue associated with the checkerboard patterns can be entirely controlled by applying the finite-volume theory. Usually, in the gradient-based topology optimization algorithms, it is common to occur some problems associated with numerical instabilities, such as checkerboard pattern, mesh dependence, and local minima. The occurrence of checkerboard subdomains is directly related to the assumptions of the finite-element method, as the satisfaction of equilibrium equations and continuity conditions through the element nodes. Differently, the finite-volume theory satisfies the equilibrium equations at the subvolume level, and the continuity conditions are established through the subvolumes adjacent interfaces, as expected from the Continuum Mechanics point of view. Thus, a topology optimization approach based on the standard (or zeroth order) finite-volume theory for linear elasticity is proposed, resulting in a numerically efficient computational modeling, able to obtain checkerboard free topologies in the absence of filtering techniques. A sensitivity filtering technique is employed to solve issues associated with mesh dependence and length scale in the finite-volume approach, providing optimized topologies with desired manufacturing features.