Semi-analytical solution for steady state heat conduction in a heterogeneous half space with embedded cuboidal inhomogeneity

A three dimensional model for solving steady state heat conduction in a semi-infinite domain containing an elementary cuboidal inhomogeneity is established in this paper. A set of analytical formulas for steady state heat conduction in a full space with an embedded cuboidal inclusion is derived. The temperature field of a half space distributed with arbitrarily shaped inhomogeneities is then obtained via the application of numerical discretization and the method of images. Benchmark comparisons involving a cuboidal/ellipsoidal inhomogeneity and double-inhomogeneity particle with the results produced by the finite element method (FEM) are conducted. Good agreements between the results of the two methods demonstrate the effectiveness and capability of the proposed model. Further, a case of heterogeneous material containing arrayed inhomogeneities is studied to explore the disturbance and interaction of the temperature field due to distributed inhomogeneities.