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

TL;DR

This paper develops a phase field model for optimizing the bending and torsional rigidities of inextensible elastic rods, with applications to plant stem morphology, using a rigorous convergence analysis and numerical methods.

Contribution

It introduces a novel phase field approximation for multi-material shape optimization of elastic rods, with proven convergence to the sharp interface model.

Findings

Convergence of phase field approximation to the sharp interface model.

Numerical minimizers relate to plant stem morphology.

Effective optimization of bending and torsional rigidities.

Abstract

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 $\Gamma$-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.