Inverse Determination of Thermal Conductivity of Functionally Graded Materials with Polynomial Approximation

The dual reciprocity boundary element method (DRBEM) combined with the conjugate gradient method (CGM) is proposed to estimate the spatially varying thermal conductivity of functionally graded materials for the transient heat conduction problem. The DRBEM is employed to solve the direct problem, retaining the advantage of boundary-only discretization. The unknown spatially varying thermal conductivity is expressed as a polynomial function related to the position coordinates, which greatly reduces the number of inversion parameters. The CGM is utilized to solve for the unknown polynomial coefficients, thereby determining the thermal conductivity. During the CGM iteration process, the complex variable derivation method is used to calculate the sensitivity coefficients accurately. The accuracy of DRBEM is validated by comparing it with the finite element method. Moreover, numerical results show that accurate results can be obtained by polynomial approximation. As the polynomial degree increases, the inversion results become more accurate, but it also leads to an increased number of iterations and longer computation time. Several examples validate that the proposed method is efficient and accurate by investigating the influences of initial guess, measurement error, and the number and location of measurement points.