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Mixed finite elements for Kirchhoff-Love plate bending

Thomas Führer, Norbert Heuer

TL;DR

The paper develops a conforming mixed finite element method for Kirchhoff–Love plate bending by constructing low-order, $\mathbb{H}(\operatorname{div\mathbf{div}};\Omega)$-conforming spaces on triangles and parallelograms with explicit DOFs. It proves quasi-optimal convergence for Dirichlet/polygonal plates and provides an a posteriori estimator derived from a Helmholtz-type decomposition, with reliability and efficiency analyses. Numerical tests on convex and non-convex domains confirm expected rates for uniform meshes and demonstrate the robustness of adaptive refinement guided by the estimator, including a postprocessing step that yields high-order accuracy for smooth solutions. The framework handles corner forces and general boundary conditions, and the explicit basis/interpolation constructions facilitate practical, implementable conforming discretizations for bending moments.

Abstract

We present a mixed finite element method with triangular and parallelogram meshes for the Kirchhoff-Love plate bending model. Critical ingredient is the construction of low-dimensional local spaces and appropriate degrees of freedom that provide conformity in terms of a sufficiently large tensor space and that allow for any kind of physically relevant Dirichlet and Neumann boundary conditions. For Dirichlet boundary conditions and polygonal plates, we prove quasi-optimal convergence of the mixed scheme. An a posteriori error estimator is derived for the special case of the biharmonic problem. Numerical results for regular and singular examples illustrate our findings. They confirm expected convergence rates and exemplify the performance of an adaptive algorithm steered by our error estimator.

Mixed finite elements for Kirchhoff-Love plate bending

TL;DR

The paper develops a conforming mixed finite element method for Kirchhoff–Love plate bending by constructing low-order, -conforming spaces on triangles and parallelograms with explicit DOFs. It proves quasi-optimal convergence for Dirichlet/polygonal plates and provides an a posteriori estimator derived from a Helmholtz-type decomposition, with reliability and efficiency analyses. Numerical tests on convex and non-convex domains confirm expected rates for uniform meshes and demonstrate the robustness of adaptive refinement guided by the estimator, including a postprocessing step that yields high-order accuracy for smooth solutions. The framework handles corner forces and general boundary conditions, and the explicit basis/interpolation constructions facilitate practical, implementable conforming discretizations for bending moments.

Abstract

We present a mixed finite element method with triangular and parallelogram meshes for the Kirchhoff-Love plate bending model. Critical ingredient is the construction of low-dimensional local spaces and appropriate degrees of freedom that provide conformity in terms of a sufficiently large tensor space and that allow for any kind of physically relevant Dirichlet and Neumann boundary conditions. For Dirichlet boundary conditions and polygonal plates, we prove quasi-optimal convergence of the mixed scheme. An a posteriori error estimator is derived for the special case of the biharmonic problem. Numerical results for regular and singular examples illustrate our findings. They confirm expected convergence rates and exemplify the performance of an adaptive algorithm steered by our error estimator.
Paper Structure (22 sections, 26 theorems, 167 equations, 6 figures)

This paper contains 22 sections, 26 theorems, 167 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preparations
  3. Sobolev spaces and trace operators
  4. Polynomial spaces and basis functions
  5. Piola--Kirchhoff transformation
  6. $\mathbb{H}(\operatorname{div\mathbf{div}};\Omega)$ elements
  7. Local finite element for triangles
  8. Local finite element for parallelograms
  9. Global space and canonical interpolation
  10. Mixed finite element method
  11. Accuracy enhancement by postprocessing
  12. A posteriori error estimation
  13. Some tools
  14. Proof of reliability in Theorem \ref{['thm:aposteriori']}
  15. Proof of efficiency in Theorem \ref{['thm:aposteriori']}
  16. ...and 7 more sections

Key Result

Proposition 1

Let $\boldsymbol{M}\in \mathbb{H}(\operatorname{div\mathbf{div}};\mathcal{T},\mathcal{E})$. Then, $\boldsymbol{M}\in\mathbb{H}(\operatorname{div\mathbf{div}};\Omega)$ if and only if for all $E\in\mathcal{E}_\Omega$, $z\in \mathcal{V}_\Omega$ where $K,K'\in\mathcal{T}$, $K\neq K'$ with $\overline K\cap \overline{K'} = \overline E$.

Figures (6)

  • Figure 1: Errors and estimator for the smooth solution from Section \ref{['sec:ex:smooth']}; uniformly refined meshes with triangles (left) and squares (right). The black dotted line indicates $\mathcal{O}((\#\mathcal{T})^{-1})$.
  • Figure 2: Domain and initial meshes for the problem from Section \ref{['sec:ex:sing']}.
  • Figure 3: Errors for the singular problem from Section \ref{['sec:ex:sing']} on sequences of uniformly refined triangular and parallelogram meshes. The two black dotted lines indicate $\mathcal{O}((\#\mathcal{T})^{-1})$ and $\mathcal{O}((\#\mathcal{T})^{-0.337})$.
  • Figure 4: Errors for the singular problem from Section \ref{['sec:ex:sing']} on sequences of uniformly and adaptively refined triangular meshes. The two black dotted lines indicate $\mathcal{O}((\#\mathcal{T})^{-1})$ and $\mathcal{O}((\#\mathcal{T})^{-0.337})$.
  • Figure 5: Meshes $\mathcal{T}_1,\ldots,\mathcal{T}_6$ generated by the adaptive loop for the problem from Section \ref{['sec:ex:sing']}. Axes are not scaled uniformly for presentation purposes.
  • ...and 1 more figures

Theorems & Definitions (50)

  • Proposition 1
  • Lemma 2
  • Lemma 3
  • Proposition 4
  • proof
  • Theorem 5
  • proof
  • Remark 6
  • Proposition 7
  • proof
  • ...and 40 more