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 threedimensional 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 measurements that include random measurement errors. Regularization was applied to control numerical error when large measurement errors were added to the over-specified boundary conditions. ∗Visiting Associate Professor. Member ASME. †Research Associate. Member ASME. ‡Professor and Director of MAIDO Institute. Fellow of ASME; Currently Professor and Chair, Mechanical & Materials Eng. Dept., Florida International University, Miami, Florida. NOMENCLATURE Γ Boundary surface Λ Damping parameter Ωe element volume σ̄ Standard deviation Θ Temperature Θamb Ambient fluid temperature [D] Damping matrix h heat convection coefficient kx,ky,kz Fourier coefficients of heat conduction in x,y,z ~n Surface unit normal vector ~q Heat flux qs Normal heat flux R Uniform random number between 0 and 1 S Heat source x,y,z Cartesian body axes