A Finite Element Formulation for the Determination of Unknown Boundary Conditions for Three-Dimensional Steady Thermoelastic Problems

Abstract

A three-dimensional finite element method (FEM) formulation for the prediction of unknown boundary conditions in linear steady thermoelastic continuum problems is presented. The present FEM formulation is capable of determining displacements, surface stresses, temperatures, and heat fluxes on the boundaries where such quantities are unknown or inaccessible, provided such quantities are sufficiently over-specified on other boundaries. The method can also handle multiple material domains and multiply connected domains with ease. A regularized form of the method is also presented. The regularization is necessary for solving problems where the over-specified boundary data contain errors. Several regularization approaches are shown. The inverse FEM method described is an extension of a method previously developed by the leading authors for two-dimensional steady thermoelastic inverse problems and three-dimensional thermal inverse problems. The method is demonstrated for several three-dimensional test cases involving simple geometries although it is applicable to arbitrary three-dimensional configurations. Several different solution techniques for sparse rectangular systems are briefly discussed.