Numerical shape optimization of the Canham-Helfrich-Evans bending energy

TL;DR

This paper introduces a new numerical method for optimizing the shape of surfaces based on the Canham-Helfrich-Evans bending energy, utilizing a three-field lifting approach to simplify the complex fourth order problem.

Contribution

It presents a novel numerical scheme that reduces a fourth order shape optimization problem to a second order saddle point problem using a three-field lifting procedure.

Findings

The scheme effectively computes shape derivatives for arbitrary polynomial orders.

Numerical simulations demonstrate the scheme's accuracy and applicability.

The method simplifies complex shape optimization problems in membrane modeling.

Abstract

In this paper we propose a novel numerical scheme for the Canham-Helfrich-Evans bending energy based on a three-field lifting procedure of the distributional shape operator to an auxiliary mean curvature field. Together with its energetic conjugate scalar stress field as Lagrange multiplier the resulting fourth order problem is circumvented and reduced to a mixed saddle point problem involving only second order differential operators. Further, we derive its analytical first variation (also called first shape derivative), which is valid for arbitrary polynomial order, and discuss how the arising shape derivatives can be computed automatically in the finite element software NGSolve. We finish the paper with several numerical simulations showing the pertinence of the proposed scheme and method.