Numerical shape optimization among convex sets

Beniamin Bogosel (1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

This article proposes a new discrete framework for approximating solutions to shape optimization problems under convexity constraints. The numerical method, based on the support function or the gauge function, is guaranteed to generate discrete convex shapes and is easily implementable using standard optimization software. The framework can handle various objective functions ranging from geometric quantities to functionals depending on partial differential equations. Width or diameter constraints are handled using the support function. Functionals depending on a convex body and its polar body can be handled using a unified framework.

Mots clés: 52A27; Shape optimization; convex shapes; numerical simulations; support function; gauge function MSC Classifications: 49Q10

Domaines: Optimisation et contrôle [math.OC]

Beniamin Bogosel. Numerical shape optimization among convex sets. Applied Mathematics and Optimization, 2023, 87 (1), pp.1. ⟨10.1007/s00245-022-09920-w⟩. ⟨hal-03607776⟩