Efficient and accurate method for 2D periodic structures based on the physical features of the transient heat conduction

Abstract

An efficient and accurate method is developed to solve the transient heat conduction problem in two-dimensional (2D) periodic structures. For a 2D periodic structure, according to the physical features of the transient heat conduction, the periodic property of the structure, and the physical meaning of the matrix exponential, it is demonstrated that the matrix exponential for a reasonable time step is a sparse matrix containing many identical elements. Next, based on the superposition principle of linear systems and the algebraic structure of the matrix exponential, computation of the response of original 2D periodic structure is transformed into computation of the responses of small-scale models with several unit cells. Finally, the precise integration method (PIM) is used to compute the temperature responses of the small-scale models. The proposed method not only inherits the accuracy and stability of the PIM but also achieves significantly improved computational time and storage requirements. A series of numerical examples demonstrate that the proposed method is more efficient than the Crank-Nicholson method and provides highly precise solutions even with a larger time step.