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

TL;DR

This paper develops a recursive method to compute asymptotic solutions for Laplace equations in corner domains with mixed boundary conditions involving power-law coefficients, applicable to various physical and engineering problems.

Contribution

It introduces a convergent recursive approach for solving D-R corner problems with power-law coefficients, including a closed-form solution for the critical case.

Findings

Recursive procedures are convergent for α > -1 or α < -1.

Closed-form asymptotic solution derived for α = -1.

Applications demonstrated in fracture mechanics and other fields.

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 $\alpha\in \mathbb{R}$ 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 $\alpha > -1$ or $\alpha < -1$, respectively. For the critical case $\alpha=-1$, 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.