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Boundary elements for clamped Kirchhoff--Love plates

Thomas Führer, Gregor Gantner, Norbert Heuer

TL;DR

The paper addresses boundary-element discretization for the Dirichlet biharmonic problem underlying clamped Kirchhoff--Love plates on piecewise smooth boundaries. It develops a direct boundary element method based on the representation formula, using the single-layer operator $oldsymbol{V}$ and double-layer operator $oldsymbol{K}_ u$ within a trace-duality framework, and introduces a Lagrange multiplier to handle potential non-invertibility of $oldsymbol{V}$. The contributions include explicit representations of boundary integral operators, high-order hp-discretization spaces for both Dirichlet data in $H^{3/2,1/2}(oldsymbol{ extGamma})$ and Neumann traces in $H^{-3/2,-1/2}(oldsymbol{ extGamma})$, and rigorous approximation and convergence results showing quasi-optimality with optimal order under mild regularity. Numerical experiments on smooth and non-smooth domains confirm the predicted convergence rates, including treatment of singular integrals via specialized quadrature. The work delivers a boundary-element-only solver for plate bending problems with non-smooth geometries, enabling high-order accuracy and efficient computation in engineering applications, with potential impact on simulations of thin-plate structures and related elasticity problems. $ oldsymbol{V}$, $ oldsymbol{K}_ u$, boundary traces, and representation formulas are central to the method, providing a rigorous, computable boundary-only framework for biharmonic-type problems.$

Abstract

We present a Galerkin boundary element method for clamped Kirchhoff--Love plates with piecewise smooth boundary. It is a direct method based on the representation formula and requires the inversion of the single-layer operator and an application of the double-layer operator to the Dirichlet data. We present trace approximation spaces of arbitrary order, required for both the Dirichlet data and the unknown Neumann trace. Our boundary element method is quasi-optimal with respect to the natural trace norm and achieves optimal convergence order under minimal regularity assumptions. We provide explicit representations of both boundary integral operators and discuss the implementation of the appearing integrals. Numerical experiments for smooth and non-smooth domains confirm predicted convergence rates.

Boundary elements for clamped Kirchhoff--Love plates

TL;DR

The paper addresses boundary-element discretization for the Dirichlet biharmonic problem underlying clamped Kirchhoff--Love plates on piecewise smooth boundaries. It develops a direct boundary element method based on the representation formula, using the single-layer operator and double-layer operator within a trace-duality framework, and introduces a Lagrange multiplier to handle potential non-invertibility of . The contributions include explicit representations of boundary integral operators, high-order hp-discretization spaces for both Dirichlet data in and Neumann traces in , and rigorous approximation and convergence results showing quasi-optimality with optimal order under mild regularity. Numerical experiments on smooth and non-smooth domains confirm the predicted convergence rates, including treatment of singular integrals via specialized quadrature. The work delivers a boundary-element-only solver for plate bending problems with non-smooth geometries, enabling high-order accuracy and efficient computation in engineering applications, with potential impact on simulations of thin-plate structures and related elasticity problems. , , boundary traces, and representation formulas are central to the method, providing a rigorous, computable boundary-only framework for biharmonic-type problems.$

Abstract

We present a Galerkin boundary element method for clamped Kirchhoff--Love plates with piecewise smooth boundary. It is a direct method based on the representation formula and requires the inversion of the single-layer operator and an application of the double-layer operator to the Dirichlet data. We present trace approximation spaces of arbitrary order, required for both the Dirichlet data and the unknown Neumann trace. Our boundary element method is quasi-optimal with respect to the natural trace norm and achieves optimal convergence order under minimal regularity assumptions. We provide explicit representations of both boundary integral operators and discuss the implementation of the appearing integrals. Numerical experiments for smooth and non-smooth domains confirm predicted convergence rates.
Paper Structure (19 sections, 18 theorems, 116 equations, 3 figures)

This paper contains 19 sections, 18 theorems, 116 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation
  4. Model problem
  5. Spaces, traces, and jumps
  6. Standard spaces
  7. Traces and induced spaces
  8. Jumps
  9. Layer potentials and boundary integral operators
  10. Boundary element method
  11. Variational formulation
  12. Approximation spaces
  13. Approximation estimates in $H^{3/2,1/2}(\Gamma)$
  14. Approximation estimates in $H^{-3/2,-1/2}(\Gamma)$
  15. Discretization and convergence
  16. ...and 4 more sections

Key Result

Lemma 1

Any $v,w\in H^2(\Omega)$ with $\Delta^2v,\Delta^2w\in L_2(\Omega)$ satisfy

Figures (3)

  • Figure 1: Initial mesh (top) and convergence plot (bottom) for the circle.
  • Figure 2: Initial mesh (top) and convergence plot (bottom) for the square.
  • Figure 3: Initial mesh (top) and convergence plot (bottom) for the pacman.

Theorems & Definitions (33)

  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • proof
  • Remark 4
  • Lemma 5
  • Lemma 6: Schmidt_01_BIO
  • Lemma 7: Schmidt_01_BIO
  • Theorem 8
  • ...and 23 more