Method of automorphic functions for an inverse problem of antiplane elasticity

TL;DR

This paper introduces a novel method using automorphic functions and conformal mappings to solve a nonlinear inverse problem in antiplane elasticity, specifically determining inclusion profiles in multiply connected domains under shear stress.

Contribution

It develops a new approach employing automorphic functions and Riemann-Hilbert problems to explicitly construct conformal maps for complex elasticity problems.

Findings

Series-form representation of conformal maps for Schottky groups

Numerical solutions for two and three inclusion cases

Effective method for inverse elasticity problems in multiply connected domains

Abstract

A nonlinear inverse problem of antiplane elasticity for a multiply connected domain is examined. It is required to determine the profile of $n$ uniformly stressed inclusions when the surrounding infinite body is subjected to antiplane uniform shear at infinity. A method of conformal mappings of circular multiply connected domains is employed. The conformal map is recovered by solving consequently two Riemann-Hilbert problems for piecewise analytic symmetric automorphic functions. For domains associated with the first class Schottky groups a series-form representation of a ($3n-4$) parametric family of conformal maps solving the problem is discovered. Numerical results for two and three uniformly stressed inclusions are reported and discussed.