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.

Inverse Problems in Aerodynamics, Heat Transfer, Elasticity and Materials Design

“…A similar procedure is also applied to the elasticity system of equations [40][41][42][43] where thermal stresses are accounted for in the vector {F}. These linear systems will remain sparse (since FEM creates sparse matrix problems), but will be non-symmetric and possibly rectangular depending on the ratio of the actual number of known to unknown boundary conditions.…”

“…FEM formulation with Galerkin's method [40][41][42][43] after assembling all element equations results in two linear algebraic systems that could be expressed (in case of no heat sources) as (23) Here, [K c ] is the stiffness matrix for the thermal problem and [K] is the stiffness matrix for the elasticity problem. Similarly, {u} is the vector of unknown Kirchoff's temperature functions and {δ} is the vector of the unknown deformations (displacements).…”

“…For example, in three-dimensional heat conduction while using a tetrahedral (three-dimensional) finite element, one could specify both the temperature, us, and the heat flux, q s , at node 1, flux only at nodes 2 and 3, and assume the boundary conditions at node 4 as unknown. The original system of linear algebraic equations is modified by adding a row and a column corresponding to the additional equation for the over-specified flux at node 1 and the additional unknown due to the unknown boundary flux at node 4 [40][41][42][43].…”

“…26 in a least squares sense minimizes the following error function. is used only for equations involving the unknown boundary values, then the results will be smooth and the error explicitly introduced by the regularization will be small [40][41][42].…”

The method of fundamental solutions for an inverse boundary value problem in static thermo-elasticity

“…In this study, this inverse problem of coupled thermo-elasticity in the static regime is solved numerically, apparently, for the first time. For related Cauchy inverse boundary condition numerical reconstructions in static thermo-elasticity the reader is referred to [3][4][5].…”