A Finite Element Formulation for the Determination of Unknown Boundary Conditions for 3-D Steady Thermoelastic Problems

A 3-D 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 2-D steady thermoelastic inverse problems and 3-D thermal inverse problems. The method is demonstrated for several 3-D test cases involving simple geometries although it is applicable to arbitrary 3-D configurations. Several different solution techniques for sparse rectangular systems are briefly discussed. NOMENCLATURE α Coefficient of thermal expansion {δ} Displacement vector 2 strain Γ Boundary surface λ Lame’s constant Λ Damping parameter ν Poisson’s ratio σ Normal stress σ̄ Standard deviation τ Shear stress Θ Temperature ∆Θ difference between local and reference temperature [D] Damping matrix E Elastic modulus of elasticity G Shear modulus k Fourier coefficient of heat conduction Q Heat source q Heat flux R Uniform random number between 0 and 1 n̂ Unit normal vector u, v, w Deformations in the x, y, z directions X, Y, Z Body force in x, y, z directions x, y, z Cartesian body axes INTRODUCTION It is often difficult or even impossible to place temperature probes, heat flux probes, or strain gauges on certain parts of a surface of a solid body. This can be due to its small size, geometric inaccessibility, or a exposure to a hostile environment. With an appropriate inverse method these unknown boundary values can be determined from additional information provided at the boundaries where the values can be measured directly. In the case of steady thermal and elastic problems, the objective of the inverse problem is to determine displacements, surface stresses, heat fluxes, and temperatures on boundaries where they are unknown. The problem of inverse determination of unknown boundary conditions in two-dimensional steady heat conduction has been solved by a variety of methods [1, 2, 3, 4, 5]. Similarly, a separate inverse boundary condition determination problem in linear elastostatics has been solved by different methods [6]. The inverse boundary condition determination problem for steady thermoelasticity was also solved for several two-dimensional problems [4]. A 3-D finite element formulation is presented here that allows one to solve this inverse problem in a direct manner by overspecifying boundary conditions on boundaries where that information is available. Our objective is to develop and demonstrate an approach for the prediction of thermal boundary conditions on parts of a three-dimensional solid body surface by using FEM. It should be pointed out that the method for the solution of inverse problems to be discussed in this paper is different from the approach based on boundary element method that has been used separately in linear heat conduction [3] and linear elasticity [6]. For inverse problems, the unknown boundary conditions on parts of the boundary can be determined by overspecifying the boundary conditions (enforcing both Dirichlet and Neumann type boundary conditions) on at least some of the remaining portions of the boundary, and providing either Dirichlet or Neumann type boundary conditions on the rest of the boundary. It is possible, after a series of algebraic manipulations, to transform the original system of equations into a system which enforces the overspecified boundary conditions and includes the unknown boundary conditions as a part of the unknown solution vector. This formulation is an adaptation of a method used by Martin and Dulikravich [7] for the inverse detection of boundary conditions in steady heat conduction. Specifically, this work represents an extension of the conceptual work presented by the authors [4, 8] by extending the original formulation from two dimensions into three dimensions. FEM FORMULATION FOR THERMOELASTICITY The Navier equations for linear static deformations u, v, w in three-dimensional Cartesian x, y, z coordinates are (λ + G) ( ∂ 2u ∂x2 + ∂ 2v ∂x∂y + ∂ 2w ∂x∂z ) + G∇u + X = 0 (1) (λ + G) ( ∂ 2u ∂x∂y + ∂ 2v ∂y2 + ∂ 2w ∂y∂z ) + G∇2v + Y = 0 (2) (λ + G) ( ∂ 2u ∂x∂z + ∂ 2v ∂y∂z + ∂ 2w ∂z2 ) + G∇w + Z = 0 (3) where, λ = Eν (1 + ν)(1− 2ν) , G = E 2(1 + ν) Here, X, Y, Z are body forces per unit volume due to stresses from thermal expansion. X = −(3λ + 2G) ∂x (4) Y = −(3λ + 2G) ∂y (5) Z = −(3λ + 2G) ∂z (6) This system of differential equations (1)-(3) can be written in the following matrix form [L] ([C][L]{δ} − [C]{ε0})− {fb} = 0 (7) where the differential operator matrix, [L], is defined as [L] =   ∂ ∂x 0 0 0 ∂ ∂y 0 0 0 ∂ ∂z ∂ ∂y ∂ ∂x 0 ∂ ∂z 0 ∂ ∂x 0 ∂ ∂z ∂ ∂y   (8) and the elastic modulus matrix, [C], is defined as [C] = λ ν   1− ν ν ν 0 0 0 ν 1− ν ν 0 0 0 ν ν 1− ν 0 0 0 0 0 0 1−2ν 2 0 0 0 0 0 0 1−2ν 2 0 0 0 0 0 0 1−2ν 2   (9) Casting the system of equations (7) in integral form using the weighted residual method [9, 10] yields ∫ Ω [V ][L] ([C][L]{δ} − [C]{ε0}) dΩ− ∫ Ω [V ]{fb} dΩ = 0 (10) where the matrix, [V ], is the weight matrix which is a collection of test functions. [V ] = [ v1 0 0 0 v2 0 0 0 v3 ] (11) One should now integrate (10) by parts to get the weak form of (7) ∫ Ω ([L][V ] ) [C][L]{δ} dΩ− ∫ Ω ([L][V ] ) [C]{ε0} dΩ − ∫