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Numerical analysis of a constrained strain energy minimization problem

Tilman Aleman, Arnold Reusken

TL;DR

The paper develops a rigorous saddle-point framework for the constrained strain-energy minimization used to assign tangential velocity on an implicitly evolving surface represented by a level-set φ. It establishes well-posedness and optimal-order finite element error estimates for both the continuous and discrete problems, carefully handling the kernel of rigid motions that can impede uniqueness. Through 2D and 3D numerical experiments, including deforming ellipses and rigid motions, it demonstrates convergence, kernel effects, and the computation of near-isometric particle trajectories on evolving surfaces. The results show that, in generic configurations, the kernel is trivial, enabling accurate recovery of near-rigid motions from level-set data and robust trajectory computation, while domain geometry and level-set critical points influence regularity and convergence when present.

Abstract

We consider a setting in which an evolving surface is implicitly characterized as the zero level of a level set function. Such an implicit surface does not encode any information about the path of a single point on the evolving surface. In the literature different approaches for determining a velocity that induces corresponding paths of points on the surface have been proposed. One of these is based on minimization of the strain energy functional. This then leads to a constrained minimization problem, which has a corresponding equivalent formulation as a saddle point problem. The main topic of this paper is a detailed analysis of this saddle point problem and of a finite element discretization of this problem. We derive well-posedness results for the continuous and discrete problems and optimal error estimates for a finite element discretization that uses standard $H^1$-conforming finite element spaces.

Numerical analysis of a constrained strain energy minimization problem

TL;DR

The paper develops a rigorous saddle-point framework for the constrained strain-energy minimization used to assign tangential velocity on an implicitly evolving surface represented by a level-set φ. It establishes well-posedness and optimal-order finite element error estimates for both the continuous and discrete problems, carefully handling the kernel of rigid motions that can impede uniqueness. Through 2D and 3D numerical experiments, including deforming ellipses and rigid motions, it demonstrates convergence, kernel effects, and the computation of near-isometric particle trajectories on evolving surfaces. The results show that, in generic configurations, the kernel is trivial, enabling accurate recovery of near-rigid motions from level-set data and robust trajectory computation, while domain geometry and level-set critical points influence regularity and convergence when present.

Abstract

We consider a setting in which an evolving surface is implicitly characterized as the zero level of a level set function. Such an implicit surface does not encode any information about the path of a single point on the evolving surface. In the literature different approaches for determining a velocity that induces corresponding paths of points on the surface have been proposed. One of these is based on minimization of the strain energy functional. This then leads to a constrained minimization problem, which has a corresponding equivalent formulation as a saddle point problem. The main topic of this paper is a detailed analysis of this saddle point problem and of a finite element discretization of this problem. We derive well-posedness results for the continuous and discrete problems and optimal error estimates for a finite element discretization that uses standard -conforming finite element spaces.

Paper Structure

This paper contains 22 sections, 13 theorems, 86 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Motivation: An inverse level set problem
  3. A constrained strain energy minimization problem
  4. The kernel space
  5. The inverse level set problem
  6. General case
  7. Examples
  8. Reformulation as saddle point problem
  9. Discretization of the saddle point problem
  10. Discretization error analysis
  11. Numerical experiments
  12. Artificial example of Lemma \ref{['lemspecial']}
  13. Deforming ellipse
  14. Regular domain without critical points
  15. Domain with reentrant corners without critical points
  16. ...and 7 more sections

Key Result

Lemma 3.1

\newlabellemma1 The minimization problem problem1 has a unique solution.

Figures (9)

  • Figure 6.1: Errors $\Vert \mathbf u-\mathbf u_h\Vert_{1}$ for the synthetic example
  • Figure 6.2: Plots of the level sets
  • Figure 6.3: $H^1$ errors for a regular domain without critical points of $\phi$
  • Figure 6.4: $H^1$ errors for a domain with reentrant corners without critical points of $\phi$
  • Figure 6.5: Pointwise errors in the velocity and Lagrange multiplier
  • ...and 4 more figures

Theorems & Definitions (33)

  • Remark 2.1
  • Remark 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Remark 3.2
  • Lemma 3.4
  • ...and 23 more