Numerical approximation of the solution of Koiter's model for an elliptic membrane shell subjected to an obstacle via the penalty method

TL;DR

The paper addresses the numerical approximation of Koiter's obstacle problem for a linearly elastic elliptic membrane shell confined to a half-space, formulating a fourth-order variational inequality and solving it through a penalty–mixed approach. It develops an intrinsic Blouza–Le Dret framework to handle vector-valued constraints, proves regularity enhancements and strong convergence of penalised solutions to the obstacle solution, and designs a finite element method that converges to the penalised problem. The main contributions include existence/uniqueness results, a quantified penalty error bound, and a rigorous FEM convergence analysis, complemented by numerical experiments that validate the theory and illustrate the method’s behavior under varying mesh, penalty, and load parameters. This work advances reliable numerical treatment of high-order obstacle problems in vector fields, with potential impact on simulations of constrained shell deformations in engineering. $

Abstract

This paper is devoted to the analysis of a numerical scheme based on the Finite Element Method for approximating the solution of Koiter's model for a linearly elastic elliptic membrane shell subjected to remaining confined in a prescribed half-space. First, we show that the solution of the obstacle problem under consideration is uniquely determined and satisfies a set of variational inequalities which are governed by a fourth order elliptic operator, and which are posed over a non-empty, closed, and convex subset of a suitable space. Second, we show that the solution of the obstacle problem under consideration can be approximated by means of the penalty method. Third, we show that the solution of the corresponding penalised problem is more regular up to the boundary. Fourth, we write down the mixed variational formulation corresponding to the penalised problem under consideration, and we show that the solution of the mixed variational formulation is more regular up to the boundary as well. In view of this result concerning the augmentation of the regularity of the solution of the mixed penalised problem, we are able to approximate the solution of the one such problem by means of a Finite Element scheme. Finally, we present numerical experiments corroborating the validity of the mathematical results we obtained.

Figures (12)

• Figure 1: Given $0<h<<1$, the first component of the solution $(\tilde{\bm{\zeta}}^{\varepsilon,h}_\kappa,\tilde{\bm{\varphi}}^{\varepsilon,h}_\kappa)$ of Problem \ref{['problem3']} converges with respect to the standard norm of $\bm{H}^1_0(\omega)$ as $\kappa\to0^+$.
• Figure 2: Given $0<q<2/3$ as in Theorem \ref{['th:conv']}, the error $\|\tilde{\bm{\zeta}}^{\varepsilon,h_1}_{h_1^{q}}-\tilde{\bm{\zeta}}^{\varepsilon,h_2}_{h_2^{q}}\|_{\bm{H}^1_0(\omega)}$ converges to zero as $h_1, h_2\to0^+$. As $q$ increases, the number of iterations needed to meet the stopping criterion of the Cauchy sequence increases.
• Figure 3: Cross sections of a deformed membrane shell subjected not to cross a given planar obstacle. Given $0<h<<1$ and $0<q<2/3$ we observe that as the applied body force magnitude increases the contact area increases.
• Figure :
• Figure :

Theorems & Definitions (24)

• Theorem 3.1: Theorem 2.6-4 in Ciarlet2000
• Theorem 4.1
• Theorem 4.2
• Lemma 5.1
• proof
• Theorem 5.1
• proof
• Lemma 6.1
• Lemma 6.2
• proof