Finding unknown surface temperatures and heat fluxes in steady heat conduction

We have developed a new direct noniterative 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 steady 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 two-dimensional steady-state inverse boundary value problem-boundary element method (IBVP-BEM) algorithm computes the temperature field within the entire object and simultaneously calculates temperatures and heat fluxes on surfaces where thermal boundary values are unavailable. Our code has been tested on several simple geometries where the boundary conditions and the analytic solutions were known everywhere. 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 boundary conditions are only slightly over-specified and far from the inaccessible boundaries