A Hankel Transform Approach to Inverse Quasi-Static Steady-State Thermal Stresses in a Thick Circular Plate

Abstract

In this paper, we solve the realistic problem of inverse quasi-static steady-state thermal stresses in a thick circular plate, which is subjected to arbitrary interior temperature and determine the unknown temperature and thermal stresses on the upper surface of the thick circular plate, where the fixed circular edge and the lower surface of the circular plate are thermally insulated using Hankel transform. To achieve our objective, first we construct a new stable algorithm for numerical evaluation of Hankel transform of order ν> - 1. The integrand rf(r) J ν (pr) consists of a slowly varying component rf(r) and a rapidly oscillating component J ν (pr). Most of the algorithms proposed in last few decades approximate the slowly varying component rf(r). In the present paper, we take a different approach and replace the rapidly oscillating component J ν (pr) in the integrand by its hat functions approximation. This approach avoids the complexity of evaluating integrals involving Bessel functions. This leads to a very simple, efficient and stable algorithm for numerical evaluation of Hankel transforms. We further give error and stability analysis and corroborate our theoretical findings by various numerical experiments.