The numerical manifold method for 2D transient heat conduction problems in functionally graded materials

Abstract

Benefiting from the use of two cover systems, that is, the mathematical cover and the physical cover, the numerical manifold method (NMM) is capable of solving both continuous and discontinuous problems in the same platform. Presently, the NMM is further developed to tackle two-dimensional transient heat conduction problems in the functionally graded materials (FGMs). Firstly, the governing equation, the associated boundary conditions and the initial condition are presented. Then, the fundamentals of the NMM are briefly reviewed. Following, the NMM discrete formulations are derived based on the Galerkin-form weighted residual method and then solved with the backward difference scheme. Finally, for verification, three numerical examples with increasing complexity are tested on uniform mathematical covers composed of square mathematical elements, and our results well demonstrate the advantages of the proposed method in discretization and accuracy; besides, the effects of material gradient on the thermal behavior of FGMs are also examined.