Inverse Determination of Boundary Conditions and Sources in Steady Heat Conduction With Heat Generation

A Boundary Element Method (BEM) implementation for the solution of inverse or ill-posed two-dimensional Poisson problems of steady heat conduction with heat sources and sinks is proposed. The procedure is noniterative and cost effective, involving only a simple modification to any existing BEM algorithm. Thermal boundary conditions can be prescribed on only part of the boundary of the solid object while the heat sources can be partially or entirely unknown. Overspecified boundary conditions or internal temperature measurements are required in order to compensate for the unknown conditions. The weighted residual statement, inherent in the BEM formulation, replaces the more common iterative least-squares (L2) approach, which is typically used in this type of ill-posed problem. An ill-conditioned matrix results from the BEM formulation, which must be properly inverted to obtain the solution to the ill-posed steady heat conduction problem. A singular value decomposition (SVD) matrix solver was found to be more effective than Tikhonov regularization for inverting the matrix. Accurate results have been obtained for several steady two-dimensional heat conduction problems with arbitrary distributions of heat sources where the analytic solutions were available.