Patch-Based Matrix-Free Time-Domain Method and PML Truncation in Unstructured 3-D Meshes

Abstract

In this article, we propose a new time-domain method for solving Maxwell’s equations: patch-based matrix-free time-domain (P-MFTD) method, in unstructured 3-D meshes. This method is free of matrix solutions regardless of the element shape used for discretization. Unlike its volume-based counterpart, both the curl of electric fields and that of magnetic fields are discretized on 2-D patches, whose orientation can be arbitrary. The method avoids the use of 3-D vector basis functions, thus significantly reducing the number of unknowns required to solve the same problem. Furthermore, a time-marching scheme is developed to ensure the stability of transient simulations, even in the presence of an unsymmetrical numerical system. The accuracy and stability of the proposed time-marching method are theoretically analyzed and found to be guaranteed. Additionally, we extend the formulation to include the truncation of the solution domain by a patch-based perfectly matched layer (PML) in unstructured meshes. The proposed method is validated with analytical solutions, numerical experiments, and measured data. Comparisons with the time-domain finite element (FE) methods and the volume-based MFTD method have demonstrated the superior performance of the proposed method in accuracy, efficiency, and stability in unstructured 3-D meshes.