A 3-D finite element formulation for the determination of unknown boundary conditions in heat conduction

Abstract

This chapter presents a 3D finite element method (FEM) formulation for the detection of unknown steady boundary conditions in heat conduction. The present FEM formulation is capable of determining temperatures and heat fluxes on the boundaries where such quantities are unknown or inaccessible, provided such quantities are sufficiently over-specified on other boundaries. A regularized form of the method is also presented. Regularization is necessary for solving problems where the over-specified boundary data contains errors. Details of the discretization and regularization as well as sample results for a 3D problem are presented. A formulation for the inverse determination of unknown steady boundary conditions in heat conduction for 3D problems has been developed using the FEM. The formulation has been tested numerically using an annular geometry with a known analytic solution. The formulation can predict the temperatures on the unknown boundary with high accuracy in the annular domain without the need for regularization. However, regularization is required to compute a good solution when measurement errors in the over-specified boundary conditions are added. Two different regularization methods have been applied. Both allow a stable QR factorization to be computed, but neither result in highly accurate temperature predictions on the unknown boundaries for large values of measurement errors. However, both regularization methods prevent amplification of the measurement errors. Further research is needed to develop better regularization methods so that the present formulation can be made more robust with respect to measurement errors used with more complex geometries.