A fast multiple-scale polynomial solution for the inverse Cauchy problem of elasticity in an arbitrary plane domain

The polynomial expansion method together with the collocation technique is a cheap yet simple method to solve the Navier equations of elasticity, which is easily arranged to satisfy the governing equations and boundary conditions pointwise. In this paper we propose a novel numerical algorithm for the solution of an overspecified/underspecified Cauchy problem of linear elasticity in an arbitrary plane domain, by using the multiple-scale Pascal polynomial expansion method (MSPEM), of which the scales are determined a priori by the collocation points, according to the idea of equilibrated matrix. In the numerical tests of a direct problem as well as the overspecified/underspecified Cauchy problems, the MSPEM is very accurate and stable against large relative noise up to 20%20% for the unknown displacements recovery problem, and up to 100%100% absolute noise for the recovery of unknown loading force. The present method is convergent very fast for most cases within 100 iteration steps.