Finding unknown surface temperatures and heat fluxes in steady heat conduction

We have developed a new direct (non-iterative) methodology for determining unknown temperatures and heat fluxes on surfaces of arbitrarily shaped solid objects where the thermal boundary conditions cannot be measured or evaluated otherwise. The method belongs to a general class of algorithms for the solution of inverse boundary value problems with the objective of determining the unknown boundary conditions. A requirement for this technique to work is that both temperatures and heat fluxes must be available and specified together (creating an over-specified problem) on at least a part of the object's surface. Our inverse algorithm utilizes the boundary element method (BEM). The algorithm computes the temperature field within the entire object and simultaneously calculates temperatures and heat fluxes on surfaces where thermal boundary values are unavailable. Both a two-dimensional and a three-dimensional steady-state BEM program have been developed and were tested on several simple geometries where the boundary conditions and the analytic solutions were known everywhere. Results were in excellent agreement with the analytic values. The algorithm is highly flexible in treating complex geometries, mixed thermal boundary conditions and temperature-dependent material properties. The accuracy and reliability of this technique deteriorate when the known surface conditions are only slightly over-specified and far from the inaccessible surfaces.<>