A recursive method for computing singular solutions in corners with homogeneous Dirichlet-Robin boundary condition with power-law coefficient variation

TL;DR

This work develops a recursive framework to compute singular corner solutions of the Laplace equation under mixed Dirichlet-Robin boundary conditions with a power-law Robin coefficient $K(r)\propto r^{\alpha}$. It introduces two recursive schemes, based on Dirichlet-Neumann and Dirichlet-Dirichlet base problems, that build each singular eigensolution as a main term plus a (finite or infinite) shadow-term series featuring a power-logarithmic structure. The authors derive convergence criteria tied to the exponent $\alpha$ and the corner angle $\omega$, provide a closed-form solution for the critical case $\alpha=-1$, analyze energy behavior, and illustrate the approach with examples including a bridged-crack model in antiplane elasticity. The methods are amenable to implementation in computer algebra systems and applicable to heat transfer, acoustics, electrostatics, elasticity, and other problems with Robin interfaces, offering a practical tool for handling corner singularities in numerical simulations.

Abstract

This study introduces a recursive method for computing asymptotic solutions of the Laplace equation in corner domains with the homogeneous Dirichlet boundary condition on one side and the Robin boundary condition with a power-law coefficient variation with exponent $α\in \mathbb{R}$ on the other side (D-R corner problem). An asymptotic solution of this D-R corner problem is given as the sum of a main term, the solution of either a homogeneous Dirichlet-Neumann (D-N) or Dirichlet-Dirichlet (D-D) corner problem, and a finite or infinite series of the associated higher-order shadow terms by using harmonic basis functions with power-logarithmic terms. To determine this series of shadow terms, it is shown that the recursive procedures based on recursive non-homogeneous D-N or D-D corner problems are always convergent for $α> -1$ or $α< -1$, respectively. For the critical case $α=-1$, the closed form expression of the asymptotic solution is given. Asymptotic solutions for several relevant D-R corner problems are derived and analysed. Two of these examples are applied to the problem of bridged cracks in antiplane Mode III in linear elastic fracture mechanics. The results presented can be applied to many other physical and engineering applications, such as heat transfer with the thermal resistance condition, acoustics and electrostatics with the impedance condition, and elasticity and structural analysis with the Winkler spring boundary condition.

Figures (5)

• Figure 1: Scheme of a corner elastic problem with spring distribution of varying stiffness with $\alpha=-1$, where $D = \Gamma_{1}$ and $R = \Gamma_{2}$ denote the Dirichlet and Robin boundaries, respectively.
• Figure 28: 3D plots of the eigensolution (a) $u_1$ and its derivatives (b) $u_{1,r}$ and (c) $r^{-1} u_{1,\theta}$ associated to $\lambda_{1}$, for $\omega=\pi/2$, $\alpha = -1$, and $\gamma = 1/2$.
• Figure 29: Graph of intersection between $\tan(\lambda\omega )$ and $-\lambda/\gamma$, for several values of $\gamma$.
• Figure 30: Mode III bridged crack problem reduces to a D-R problem for half-plane: $\omega = \pi$ and $K(r) = \kappa r^\alpha$.
• Figure 31: Overview of stress singularities in bridged cracks.

Theorems & Definitions (21)

• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Remark 3.5
• Remark 3.6
• Remark 3.7
• Remark 1
• Proposition 1
• proof