Calculus of Variations and Geometric Measure Theory

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

Multi-material model and shape optimization for bending and torsion of inextensible rods

created by maione on 10 Sep 2022
modified on 13 Sep 2022

[BibTeX]

Submitted Paper

Inserted: 10 sep 2022
Last Updated: 13 sep 2022

Year: 2022

ArXiv: 2209.04538 PDF

Abstract:

We derive a model for the optimization of bending and torsional rigidity of non-homogeneous elastic rods, by studying a sharp interface shape optimization problem with perimeter penalization for the rod cross section, that treats the resulting torsional and bending rigidities as objectives. We then formulate a phase field approximation to the optimization problem and show Γ-convergence to the aforementioned sharp interface model. This also implies existence of minimizers for the sharp interface optimization problem. Finally, we numerically find minimizers of the phase field problem using a steepest descent approach and relate the resulting optimal shapes to the development of plant morphology.

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