Calculus of Variations and Geometric Measure Theory

P. W. Dondl - A. Maione - S. Wolff-Vorbeck

Phase field model for multi-material shape optimization of inextensible rods

created by maione on 10 Sep 2022
modified on 02 May 2024


Accepted Paper

Inserted: 10 sep 2022
Last Updated: 2 may 2024

Journal: ESAIM: Control, Optimisation and Calculus of Variations
Year: 2024
Doi: 10.1051/cocv/2024039

ArXiv: 2209.04538 PDF


We derive a model for the optimization of the bending and torsional rigidities of non-homogeneous 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.

Keywords: shape optimization, numerical simulations, steepest descent, Γ-convergence, bending and torsional rigidity, sharp interface, phase field problems, diffuse interface, optimality conditions, plant morphology