Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Michael Neunteufel, Joachim Schöberl, Kevin Sturm

TL;DR

This work presents a novel three-field numerical scheme for minimizing the Canham-Helfrich-Evans bending energy on closed membranes by lifting the distributional curvature to an auxiliary mean curvature field, thereby avoiding fourth-order derivatives and yielding a reduced two-field, second-order formulation. A carefully derived shape-derivative framework enables gradient-based surface optimization with area and volume constraints, implemented in NGSolve with automatic shape differentiation via ALE. The method is validated through curvature computations and equilibrium-shape benchmarks, including spontaneous curvature, and demonstrates good convergence and qualitative agreement with established phase diagrams. The approach provides a robust, flexible tool for simulating membrane morphologies under complex geometric constraints with potential applicability to vesicles and red-blood cells in biophysical contexts.

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.

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

TL;DR

This work presents a novel three-field numerical scheme for minimizing the Canham-Helfrich-Evans bending energy on closed membranes by lifting the distributional curvature to an auxiliary mean curvature field, thereby avoiding fourth-order derivatives and yielding a reduced two-field, second-order formulation. A carefully derived shape-derivative framework enables gradient-based surface optimization with area and volume constraints, implemented in NGSolve with automatic shape differentiation via ALE. The method is validated through curvature computations and equilibrium-shape benchmarks, including spontaneous curvature, and demonstrates good convergence and qualitative agreement with established phase diagrams. The approach provides a robust, flexible tool for simulating membrane morphologies under complex geometric constraints with potential applicability to vesicles and red-blood cells in biophysical contexts.

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.

Paper Structure

This paper contains 24 sections, 8 theorems, 89 equations, 18 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and Problem statement
  3. Curvature computation
  4. Discrete shape operator
  5. Variational formulation
  6. Reduction for mean curvature
  7. Perturbed problem
  8. Shape derivatives
  9. Shape derivative of normal/tangential vectors and constraints
  10. Shape derivative of shape operator
  11. Shape derivative, state, and adjoint state problem
  12. Stabilization of element-areas
  13. Solving algorithm
  14. Basic algorithm
  15. Improved surface preservation
  16. ...and 9 more sections

Key Result

Lemma 3.1

There holds for all sufficiently small $|t|$:

Figures (18)

  • Figure 2.1: Normal, tangential, and co-normal (element-normal) vectors $\boldsymbol{\nu}$, $\boldsymbol{\tau}$, and $\boldsymbol{\mu}$ on two neighboured elements.
  • Figure 3.1: Jump of normal vector over two affine elements.
  • Figure 3.2: Construction of approximation of discrete jump. (a) The polygon curve with jump angle $\alpha$. (b) The construction of the circles. (c) The final approximated smooth curve.
  • Figure 3.3: Construction of approximation of discrete jump in 3D. (a) The affine curve with junction. (b) The approximated smooth surface.
  • Figure 3.4: Angle computation in two-dimensional setting. (a) Unperturbed surface. (b) Perturbed averaged normal vector on perturbed surface. (c) Unperturbed averaged normal vector on perturbed surface.
  • ...and 13 more figures

Theorems & Definitions (16)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 6 more