Virtual boundary element method in conjunction with conjugate gradient algorithm for three-dimensional inverse heat conduction problems

In this article, the virtual boundary element method (VBEM) in conjunction with conjugate gradient algorithm (CGA) is employed to treat three-dimensional inverse problems of steady-state heat conduction. On the one hand, the VBEM may face numerical instability if a virtual boundary is improperly selected. The numerical accuracy is very sensitive to the choice of the virtual boundary. The condition number of the system matrix is high for the larger distance between the physical boundary and the fictitious boundary. On the contrary, it is difficult to remove the source singularity. On the other hand, the VBEM will encounter ill-conditioned problem when this method is used to analyze inverse problems. This study combines the VBEM and the CGA to model three-dimensional heat conduction inverse problem. The introduction of the CGA effectively overcomes the above shortcomings, and makes the location of the virtual boundary more free. Furthermore, the CGA, as a regularization method, successfully solves the ill-conditioned equation of three-dimensional heat conduction inverse problem. Numerical examples demonstrate the validity and accuracy of the proposed method.