Numerical analysis of a constrained strain energy minimization problem

TL;DR

This paper analyzes a constrained strain energy minimization problem for evolving surfaces represented implicitly, focusing on the saddle point formulation and finite element discretization, with well-posedness and error estimates.

Contribution

It provides a detailed analysis of the saddle point formulation and finite element discretization of a constrained strain energy minimization problem on implicit surfaces.

Findings

Established well-posedness for continuous and discrete problems

Derived optimal error estimates for finite element discretization

Validated the approach through theoretical analysis

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.