Sparse basis pursuit for compliance minimization in the vanishing volume ratio limit

Abstract

We consider a non‐smooth convex variational problem appearing as a formal limit of compliance minimization in the vanishing volume ratio limit. The problem has a classical basis pursuit form, and several successful algorithms have been utilized to solve problems of this class in other application contexts. We discuss the well‐posedness and regularity of solutions to these problems, possible solution algorithms, and their discretizations as relevant in this mechanical engineering context. We then test the algorithms on a few benchmark problems with available analytical solutions. We find that whereas many algorithms are successful in estimating the optimal objective value to the problem to a high accuracy, the same cannot be said about finding the optimal solutions themselves. In particular, in some examples the algorithms struggle to properly identify the areas where the solutions should vanish entirely. We also discuss an example where the found optimal solutions are not sparse even though sparse(r) solutions exist.

Alternating optimization of design and stress for stress-constrained topology optimization

“…Eckstein and Bertsekas (1992) pointed out that ADMM is an application of the Douglas-Rachford splitting algorithm (Douglas and Rachford 1956), which is in turn an instance of the proximal point algorithm. Recently, Evgrafov and Sigmund (2020) proposed to split stress variables into positively and negatively semi-definite parts, and applied this splitting for compliance minimization in the vanishing volume ratio limit. In this paper we adopt ADMM to solve stressconstrained problems with a large number of variables and constraints, and compare its performance with conventional formulations and solvers.…”