Phase Field Model for Multi-Material Shape Optimization of Inextensible Rods

¹ Abteilung für Angewandte Mathematik and liv MatS Cluster of Excellence Albert-Ludwigs-Universität Freiburg Hermann-Herder-Strasse 10, 79104 Freiburg i. Br., Germany
² Centre de Recerca Matemàtica Edifici C, Campus Bellaterra, 08193 Bellaterra, Spain
³ Abteilung für Angewandte Mathematik Albert-Ludwigs-Universität Freiburg Hermann-Herder-Strasse 10, 79104 Freiburg i. Br., Germany

^* Corresponding author: amaione@crm.cat

Received: 4 August 2023
Accepted: 24 April 2024

Abstract

We derive a model for the optimization of the bending and torsional rigidities of nonhomogeneous elastic rods. This is achieved by studying a sharp interface shape optimization problem with perimeter penalization, that treats both rigidities as objectives. We then formulate a phase field approximation of the optimization problem and show the convergence to the aforementioned sharp interface model via Γ-convergence. In the final part of this work we numerically approximate minimizers of the phase field problem by using a steepest descent approach and relate the resulting optimal shapes to the development of the morphology of plant stems.