Nonlinear transient heat transfer analysis of functionally graded material sandwich slabs by incremental differential quadrature element method

As a first attempt, the incremental differential quadrature element method (IDQEM) was adopted to perform the one-dimensional nonlinear transient heat transfer analysis of functionally graded material (FGM) sandwich slabs. The thermophysical properties of the slab were considered to be position- and temperature-dependent. To implement the IDQEM, the sandwich slab was divided into three spatial sub-domains along the layer interfaces, and the entire heating process was also divided into several temporal sub-domains. For each temporal sub-domain, the governing equations as well as the initial condition, interfacial condition, and boundary condition were discretized by the differential quadrature technique. Because the obtained discrete equations were built in different regions of grid points, a modification of the equations was proposed which were then expressed in the matrix forms so that they can be built in the same regions. Using the Kronecker product, the simultaneous matrix equations were transformed into a set of nonlinear algebraic equations, which were then solved by the Newton-Raphson iteration method to obtain the temperature profile for each temporal sub-domain. Because the initial condition of each temporal sub-domain was defined by the temperature results at the end of the previous sub-domain, the temperature profile of the slab during the entire heating process can be obtained by repeating the calculation procedure from the first temporal sub-domain to the last one. Numerical examples were carried out to verify the fast convergence of the present method. The correctness of the present method was verified through comparison with the analytical and numerical results reported in previous works. The effects of temperature-dependent thermophysical properties, volume fraction index, and thermal boundary on the temperature profile of the slab were discussed.