Multiple spatial and temporal scales method for numerical simulation of non-Fourier heat conduction problems: Multidimensional case

A multiple spatial and temporal scales method is developed to numerically simulate the phenomenon of non-Fourier heat conduction for periodic heterogeneous materials in multi-dimensions by high-order asymptotic homogenization theory. Amplified spatial and reduced temporal scales are introduced respectively to better account for the fluctuations of the temperature field due to material heterogeneity and non-local effect of the homogenized solution. In the previous work by Zhang et al. [25], a one-dimensional case has been addressed, and the aim of the present manuscript is to extend one-dimensional solution to multidimensional case. A multidimensional high-order non-local model of non-Fourier heat conduction is derived. The relationships of homogenized heat conduction coefficients for different orders are determined and a nested finite element solution procedure is outlined for the homogenized coefficients. The validity and effectiveness of the model is demonstrated by illustrating the two-dimensional numerical examples.