Convergence of the direct limit analysis design method for discrete topology optimization

Abstract

The present paper addresses the robustness and convergence behavior of the direct limit analysis (LA) based methodology developed for the topology design of continuum structures subject to prescribed statically and plastically admissible loads. The design methodology, based on a direct method formulation of the static LA problem, has recently been proposed for continuous topology optimization and its merits were highlighted. One of its remarkable features is the outstanding similarity of the topology design mathematical problem with its underlying direct static form of the LA problem. Subsequently, it has been extended to solve two dimensional discrete, i.e. black-and-white, topology design problems by modifying the objective function into a square root form in a way to penalize the intermediate densities and solving a sequence of conic quadratic programming problems of identical scale and algebraic structure as the continuous design problem, leaving a number of issues to be investigated, pertaining to convergence. In the present work different families of penalization function forms are proposed and assessed as alternatives to the original square root function. The performance is evaluated in terms of robustness, accuracy in the sense of closeness of the final design to a 0–1 topology and efficiency or number of approximate problems required for convergence. Convergence of the discrete topology design is shown to be improved using higher order power functions as well as trigonometric and exponential type penalty functions. The performance of the design method in solving example problems using the various penalization schemes is compared and the factors that affect it are analyzed.