A Virtual Boundary Element Method for Three-Dimensional Inverse Heat Conduction Problems in Orthotropic Media

This paper aims to apply a virtual boundary element method (VBEM) to solve the inverse problems of three-dimensional heat conduction in orthotropic media. This method avoids the singular integrations in the conventional boundary element method, and can be treated as a potential approach for solving the inverse problems of the heat conduction owing to the boundary-only discretization and semi-analytical algorithm. When the VBEM is applied to the inverse problems, the numerical instability may occur if a virtual boundary is not properly chosen. The method encounters a highly illconditioned matrix for the larger distance between the physical boundary and the virtual boundary, and otherwise is hard to avoid the singularity of the source point. Thus, it must adopt an appropriate regularization method to deal with the ill-posed systems of inverse problems. In this study, the VBEM and different regularization techniques are combined to model the inverse problem of three-dimensional heat conduction in orthotropic media. The proper regularization techniques not only make the virtual boundary to be allocated freer, but also solve the ill-conditioned equation of the inverse problem. Numerical examples demonstrate that the proposed method is efficient, accurate and numerically stable for solving the inverse problems of three-dimensional heat conduction in orthotropic media.