New Galerkin-vector theory and efficient numerical method for analyzing steady-state heat conduction in inhomogeneous bodies subjected to a surface heat flux

Highlights: • We report a new method for solving inhomogeneous heat-conduction problems. • We derived the Galerkin-vector forms of the disturbed temperature and heat flux. • The eigentemperature gradient is tackled with numerical EIM. • We explore the influences of inhomogeneity shape, location, and distributions. -- Abstract: This paper reports a new semi-analytical model, together with core analytical solutions, for solving steady-state heat-conduction problems involving materials of a semi-infinite matrix embedded with arbitrarily distributed inhomogeneities, and a novel fast computing algorithm for model construction. The thermal field is analyzed as the summation of the solution to the homogeneous matrix and the variations caused by the inhomogeneities. The former is obtained through the route of discrete convolution and FFT/Influence coefficients/Green's function, and the surface heat flux via the conjugate gradient method (CGM). The eigentemperature gradient of the latter is tackled with the numerical equivalent inclusion method (EIM) based on the new analytical formulas for the disturbed temperature and heat flux from the Galerkin vectors. The influences of inhomogeneity shape, location, and heat-conduction properties are studied, and the thermal fields affected by multi-inhomogeneities in a layered form and with a regular or a random distribution are investigated.