A finite element method for prediction of unknown boundary conditions in two-dimensional steady-state heat conduction problems

This paper presents a finite element method in predicting unknown boundary conditions of homogeneous and two-layered materials subjected to steady-state heat conduction. Firstly, a finite element formulation is introduced to solve a steady-state boundary inverse heat conduction problem of homogeneous material. Effects of bias error of temperature specified on interior nodes and locations of such nodes on accuracy of predicted temperature distribution are examined. Then, modified cubic spline is specified on material boundary to stabilize predicted temperature distribution. Cubic spline functions using different numbers of control points are used in examining their effects on accuracy of predicted temperature distribution and computing time when specifying no bias temperatures. Finally, the formulation with cubic spline function specification is employed in predicting unknown boundary conditions of two-layered materials with thermal conductivity ratio of 0.1, 1, and 10. Concept of coincident nodes is applied in handling physical condition characterized by thermal contact resistance and heat source strength at layer interface. Effect of bias error of temperatures specified on nodes within thicker layer is examined under three interface conditions. Cubic spline function with five control points can predict temperature distributions accurately for all interface conditions when specifying no bias temperatures. RMS errors vary linearly with bias errors for interface conditions with no heat source but are drastically affected by bias error when heat source exists at the interface.