Numerical approximation of the solution of an obstacle problem modelling the displacement of elliptic membrane shells via the penalty method

TL;DR

This paper develops and proves the convergence of a finite element-based numerical scheme using penalty methods for an obstacle problem modeling elastic membrane shells, validated by numerical simulations.

Contribution

It introduces a convergence analysis for a penalty-based finite element scheme for obstacle problems in elastic shells, including an iterative Brezis-Sibony method.

Findings

Convergence of the penalized finite element scheme to the original problem.

Validation of theoretical results through numerical simulations.

Effective approximation without nonlinear optimization tools.

Abstract

In this paper we establish the convergence of a numerical scheme based, on the Finite Element Method, for a time-independent problem modelling the deformation of a linearly elastic elliptic membrane shell subjected to remaining confined in a half space. Instead of approximating the original variational inequalities governing this obstacle problem, we approximate the penalized version of the problem under consideration. A suitable coupling between the penalty parameter and the mesh size will then lead us to establish the convergence of the solution of the discrete penalized problem to the solution of the original variational inequalities. We also establish the convergence of the Brezis-Sibony scheme for the problem under consideration. Thanks to this iterative method, we can approximate the solution of the discrete penalized problem without having to resort to nonlinear optimization tools. Finally, we present numerical simulations validating our new theoretical results.