Computing multiple solutions of topology optimization problems

Thesis

Topology optimization finds the optimal material distribution of a continuum in a domain, subject to PDE and volume constraints. Density-based models often result in a PDE, volume and inequality constrained, nonconvex, infinite-dimensional optimization problem. These problems can exhibit many local minima. In practice, heuristics are used to aid the search for better minima, but these can fail even in the simplest of cases.

In this thesis we address two core issues related to the nonconvexity of topology optimization problems: the convergence of the discretization and the computation of the solutions. First, we consider the convergence of a finite element discretization of a fluid topology optimization problem. Results available in literature show that there exists a sequence of finite element solutions that weakly(-*) converges to a solution of the infinite-dimensional problem. We improve on these classical results. In particular, by fixing any isolated minimizer, we show that there exists a sequence of finite element solutions that \emph{strongly} converges to that minimizer. Moreover, these results hold for both traditional conforming finite element methods and more sophisticated divergence-free discontinuous Galerkin finite element methods.

We then focus on developing a solver that can systematically compute multiple minimizers of a general density-based topology optimization problem. This leads to the successful computation of 42 distinct solutions of a two-dimensional fluid topology optimization problem. Finally, by developing preconditioners for the linear systems that arise during the optimization process, we are able to apply the solver to three-dimensional fluid topology optimization problems. This culminates in an example where we compute 11 distinct three-dimensional solutions.

Authors: Papadopoulos, IPA

• DOI: 10.5287/ora-8npjngzkb
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: preconditioning; multiple solutions; finite element method; numerical analysis; deflation
• Subjects: Numerical analysis; Constrained optimization; Nonconvex programming; Multigrid methods (Numerical analysis); Differential equations, Partial; Finite element method; Mathematics