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On a calculation method of the thickness via partial differential equations

Atsushi Nakayasu, Takayuki Yamada

TL;DR

The paper analyzes a PDE-based method for computing geometric thickness by solving the elliptic system $-a\Delta\mathbf{s}+(1-\chi_\Omega)\mathbf{s}=-\nabla\chi_\Omega$ on a domain $D$ containing the shape $\Omega$, with thickness defined as $T^a = \frac{2}{\sqrt{a}\,\mathrm{div}\mathbf{s}}$. It proves convergence of the PDE-based thickness to the geometric thickness $\bar{T}_\Omega$ as $a\to0$ and provides quantitative rates, including explicit expressions for annuli via modified Bessel functions. The results cover interval, band, and annulus geometries, giving exact solutions or sharp bounds (often with $O(\sqrt{a})$ convergence and exponentially small corrections) and rigorous $L^2$ error estimates in general domains. These findings establish a solid mathematical foundation for the PDE-based thickness, enhancing its reliability for shape analysis and topology optimization.

Abstract

This paper presents a mathematical analysis of an elliptic partial differential equation (PDE) designed to compute the geometric thickness of a given shape. The PDE-based formulation provides a direct and systematic approach to evaluate thickness through the elliptic equation, whose solution yields a vector field from which the thickness is extracted as the divergence. While the convergence of this PDE-based thickness to the geometric thickness had been rigorously justified only for simple geometries such as intervals and straight bands, its validity for more general shapes remained open. In this work, we extend the analysis to annular domains, where curvature effects are nontrivial. We prove that the PDE-based thickness converges to the geometric thickness as the diffusion parameter tends to zero by estimating the difference between two notions of thickness with the square root of the diffusion parameter. Explicit expressions involving modified Bessel functions are obtained for annuli, together with sharp inequalities for their ratios. These results provide a rigorous mathematical foundation for the PDE-based thickness and demonstrate its potential as a reliable tool in shape analysis and topology optimization.

On a calculation method of the thickness via partial differential equations

TL;DR

The paper analyzes a PDE-based method for computing geometric thickness by solving the elliptic system on a domain containing the shape , with thickness defined as . It proves convergence of the PDE-based thickness to the geometric thickness as and provides quantitative rates, including explicit expressions for annuli via modified Bessel functions. The results cover interval, band, and annulus geometries, giving exact solutions or sharp bounds (often with convergence and exponentially small corrections) and rigorous error estimates in general domains. These findings establish a solid mathematical foundation for the PDE-based thickness, enhancing its reliability for shape analysis and topology optimization.

Abstract

This paper presents a mathematical analysis of an elliptic partial differential equation (PDE) designed to compute the geometric thickness of a given shape. The PDE-based formulation provides a direct and systematic approach to evaluate thickness through the elliptic equation, whose solution yields a vector field from which the thickness is extracted as the divergence. While the convergence of this PDE-based thickness to the geometric thickness had been rigorously justified only for simple geometries such as intervals and straight bands, its validity for more general shapes remained open. In this work, we extend the analysis to annular domains, where curvature effects are nontrivial. We prove that the PDE-based thickness converges to the geometric thickness as the diffusion parameter tends to zero by estimating the difference between two notions of thickness with the square root of the diffusion parameter. Explicit expressions involving modified Bessel functions are obtained for annuli, together with sharp inequalities for their ratios. These results provide a rigorous mathematical foundation for the PDE-based thickness and demonstrate its potential as a reliable tool in shape analysis and topology optimization.

Paper Structure

This paper contains 14 sections, 13 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Interval shape
  4. The statements of the results
  5. Proof of Theorem \ref{['t_int_whole']}
  6. Proof of Theorem \ref{['t_int_gen']}
  7. Band shape
  8. The statements of the results
  9. Proof of Theorem \ref{['t_band_whole']}
  10. Proof of Theorem \ref{['t_band_gen']}
  11. Annulus shape
  12. The statements of the results
  13. Proof of Theorem \ref{['t_anu_whole']}
  14. Proof of Theorem \ref{['t_anu_gen']}

Key Result

Lemma 2.4

Let $\vb*{d} \in H^1(D)^N$ be a weak solution of the homogeneous equation e_hswf not necessarily satisfying the boundary conditions. Then, we have where $\operatorname*{ess\ sup}$ is the essential supremum and the right-hand side is understood as the trace sense.

Theorems & Definitions (22)

  • Example 2.1: Interval shape
  • Example 2.2: Band shape
  • Example 2.3: Annulus shape
  • Lemma 2.4: Maximum principle for homogeneous equations
  • proof
  • Lemma 2.5: Interior $H^1$ estimate for homogeneous equations
  • proof
  • Theorem 3.1: Interval shape in the whole line
  • Theorem 3.2: Interval shape in general interval
  • Theorem 4.1: Band shape in the whole space
  • ...and 12 more