Determination of Temperatures and Heat Fluxes on Surfaces and Interfaces of Multi-Domain Three-Dimensional Electronic Components

A finite element method (FEM) formulation for the prediction of unknown steady boundary conditions in heat conduction for multi-domain three-dimensional solid objects is presented. The FEM formulation is capable of determining temperatures and heat fluxes on the boundaries where such quantities are unknown, provided such quantities are sufficiently over-specified on other boundaries. An inverse finite element program has been previously developed and successfully tested on 3-D simple geometries. The finite element code uses an efficient sparse matrix storage scheme that allows treatment of realistic three-dimensional problems on personal computer. The finite element formulation also allows for very straight-forward treatment of geometries composed of many different materials. The inverse FEM formulation was applied to the prediction of die junction temperature distribution in a simple ball grid array (BGA) electronic package. Examples are presented with simulated measurement that include random measurement errors. Regularization was applied to control numerical error when large measurement errors were added to the over-specified boundary conditions.