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Geometrical mixed finite element methods for fourth order obstacle problems in linearised elasticity

Paolo Piersanti, Tianyu Sun

TL;DR

The work tackles fourth-order obstacle problems arising in linearised elasticity by formulating a new mixed Finite Element Method that uses penalty terms and Courant triangles to approximate constrained biharmonic and shallow-shell problems. It demonstrates strong convergence of penalised solutions to the constrained solution and provides explicit discretisation error bounds, notably $\|u_\kappa^h-u_\kappa\|_{H^1_0(\omega)}+\|\vec{\xi}_\kappa^h-\vec{\xi}_\kappa\|_{\vec{H}^1_0(\omega)} \le C h/\kappa$ for the biharmonic case. A key insight is that the curvature of the middle surface governs discretisation choices: nonzero curvature requires a symmetry constraint on the dual variable's gradient and prevents Courant-triangle schemes, whereas a flat surface restores rigidity and allows Courant triangles. Numerical experiments corroborate the theoretical results, including convergence, stability against locking, and the influence of geometry on the shallow-shell models.

Abstract

This paper is devoted to the study of a novel mixed Finite Element Method for approximating the solutions of fourth order variational problems subjected to a constraint. The first problem we consider consists in establishing the convergence of the error of the numerical approximation of the solution of a biharmonic obstacle problem. The contents of this section are meant to generalise the approach originally proposed by Ciarlet \& Raviart, and then complemented by Ciarlet \& Glowinski. The second problem we consider amounts to studying a two-dimensional variational problem for linearly elastic shallow shells subjected to remaining confined in a prescribed half-space. We first study the case where the parametrisation of the middle surface for the linearly elastic shallow shell under consideration has non-zero curvature, and we observe that the numerical approximation of this model via a mixed Finite Element Method based on conforming elements requires the implementation of the additional constraint according to which the gradient matrix of the dual variable has to be symmetric. However, differently from the biharmonic obstacle problem previously studied, we show that the numerical implementation of this result cannot be implemented by solely resorting to Courant triangles. Finally, we show that if the middle surface of the linearly elastic shallow shell under consideration is flat, the symmetry constraint required for formulating the constrained mixed variational problem announced in the second part of the paper is not required, and the solution can thus be approximated by solely resorting to Courant triangles. The theoretical results we derived are complemented by a series of numerical experiments.

Geometrical mixed finite element methods for fourth order obstacle problems in linearised elasticity

TL;DR

The work tackles fourth-order obstacle problems arising in linearised elasticity by formulating a new mixed Finite Element Method that uses penalty terms and Courant triangles to approximate constrained biharmonic and shallow-shell problems. It demonstrates strong convergence of penalised solutions to the constrained solution and provides explicit discretisation error bounds, notably for the biharmonic case. A key insight is that the curvature of the middle surface governs discretisation choices: nonzero curvature requires a symmetry constraint on the dual variable's gradient and prevents Courant-triangle schemes, whereas a flat surface restores rigidity and allows Courant triangles. Numerical experiments corroborate the theoretical results, including convergence, stability against locking, and the influence of geometry on the shallow-shell models.

Abstract

This paper is devoted to the study of a novel mixed Finite Element Method for approximating the solutions of fourth order variational problems subjected to a constraint. The first problem we consider consists in establishing the convergence of the error of the numerical approximation of the solution of a biharmonic obstacle problem. The contents of this section are meant to generalise the approach originally proposed by Ciarlet \& Raviart, and then complemented by Ciarlet \& Glowinski. The second problem we consider amounts to studying a two-dimensional variational problem for linearly elastic shallow shells subjected to remaining confined in a prescribed half-space. We first study the case where the parametrisation of the middle surface for the linearly elastic shallow shell under consideration has non-zero curvature, and we observe that the numerical approximation of this model via a mixed Finite Element Method based on conforming elements requires the implementation of the additional constraint according to which the gradient matrix of the dual variable has to be symmetric. However, differently from the biharmonic obstacle problem previously studied, we show that the numerical implementation of this result cannot be implemented by solely resorting to Courant triangles. Finally, we show that if the middle surface of the linearly elastic shallow shell under consideration is flat, the symmetry constraint required for formulating the constrained mixed variational problem announced in the second part of the paper is not required, and the solution can thus be approximated by solely resorting to Courant triangles. The theoretical results we derived are complemented by a series of numerical experiments.
Paper Structure (6 sections, 13 theorems, 133 equations, 16 figures)

This paper contains 6 sections, 13 theorems, 133 equations, 16 figures.

Table of Contents

  1. Introduction
  2. A prototypical example of fourth order obstacle problem. The biharmonic model
  3. Numerical approximation of an obstacle problem for linearly elastic shallow shells. The general case
  4. Numerical approximation of an obstacle problem for linearly elastic shallow shells. The case of a flat surface
  5. Numerical experiments for the biharmonic obstacle problem
  6. Numerical experiments simulating the displacement of linearly elastic shallow shells subjected to an obstacle

Key Result

Theorem 2.1

Let $u$ be the unique solution of Problem problem1. For each $\kappa>0$, denote by $u_\kappa$ the unique solution of Problem problem2. Then, the sequence $\{u_\kappa\}_{\kappa>0}$ is such that

Figures (16)

  • Figure 1: According to Theorem \ref{['th:biharmonic-strong']}, given $0<h<<1$, the first component of the solution $(u_\kappa,\vec{\xi}_\kappa)$ of Problem \ref{['problem4']} converges with respect to the standard norm of $H^1_0(\omega)$ as $\kappa\to 0^+$. In particular, we verify that $\{u_\kappa\}_{\kappa>0}$ is a Cauchy sequence in $H^1_0(\omega)$.
  • Figure 2: The manufactured solution $u_{\textup{exact}}$ enjoys radial symmetry and is of class $H^2_0(\omega)$. The numerical solution is very close to the manufactured solution.
  • Figure 3: Given $0<q<1$ as in Theorem \ref{['th:3']}, the error $\|u_{h^{q}}^{h}-u_{\textup{exact}}\|_{H^1_0(\omega)}$ converges to zero as $h\to0^+$ with order of convergence $\mathcal{O}(h^{1-q})$.
  • Figure 4: The manufactured solution $u_{\textup{exact}}$ enjoys radial symmetry and is of class $H^2_0(\omega)$. The numerical solution is very close to the manufactured solution.
  • Figure 5: Given $0<q<1$ as in Theorem \ref{['th:3']}, the error $\|u_{h^{q}}^{h}-u_{\textup{exact}}\|_{H^1_0(\omega)}$ converges to zero as $h\to0^+$ with order of convergence $\mathcal{O}(h^{1-q})$.
  • ...and 11 more figures

Theorems & Definitions (26)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 16 more