2024/06/21 by Livia Betz, Betz, Livia · 1 citation
Engineering · #35Q93 #49N99 #49Q10 #Advanced Numerical Analysis Techniques #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #FOS: Mathematics #Optimization and Control (math.OC) #Topology Optimization in Engineering
paper · pdf · doi:10.48550/arxiv.2406.15146
openalex publication_date 2024/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is concerned with a shape optimization problem governed by a non-smooth PDE, i.e., the nonlinearity in the state equation is not necessarily differentiable. We follow the functional variational approach of [40] where the set of admissible shapes is parametrized by a large class of continuous mappings. This methodology allows for both boundary and topological variations. It has the advantage that one can rewrite the shape optimization problem as a control problem in a function space. To overcome the lack of convexity of the set of admissible controls, we provide an essential density property. This permits us to show that each parametrization associated to the optimal shape is the limit of global optima of non-smooth distributed optimal control problems. The admissible set of the approximating minimization problems is a convex subset of a Hilbert space of functions. Moreover, its structure is such that one can derive strong stationary optimality conditions [6]. The present manuscript provides the basis for the investigations from [5], where necessary conditions in form of an optimality system have been recently established.