Matrix-free time-domain methods for general electromagnetic analysis

Many engineering challenges demand an efficient computational solution of large-scale problems. If a computational method can be made free of matrix solutions, then it has a potential of solving very large scale problems. Among existing computational electromagnetic methods, the explicit finite-difference time-domain (FDTD) method is free of matrix solutions. However, it requires a structured orthogonal grid for space discretization. In this work, we develop a new time-domain method that naturally requires no matrix solution, regardless of whether the discretization is a structured grid or an unstructured mesh. No dual mesh, interpolation, projection and mass lumping are needed. Furthermore, a time-marching scheme is developed to ensure the stability for simulating an unsymmetrical numerical system, while preserving the matrix-free merit of the proposed method. This time-marching scheme is then made unconditionally stable, and hence allowing for the use of an arbitrarily large time step without sacrificing the matrix-free property. Extensive numerical experiments have been carried out on a variety of two- and three-dimensional unstructured meshes and even mixed-element meshes. Correlations with analytical solutions and the results obtained from the time-domain finite-element method have validated the accuracy, matrix-free property, stability, and generality of the proposed method.^ In addition to an extensive development of the proposed method in arbitrary 2- and 3-D unstructured meshes, we have also made a connection between the proposed new method and the classical FDTD method. We have found that the proposed matrix-free method naturally reduces to the FDTD method in an orthogonal grid. It also results in a new patch-based single-grid formulation of the FDTD algorithm. This new formulation not only makes the implementation of the original FDTD much easier, but also reveals a natural rank-1 decomposition of the curl-curl operator. Such a representation leads to an efficient extraction of unstable eigenmodes from fine cells only, from which a fast explicit and unconditionally stable FDTD method is developed. In addition, to efficiently handle multiscale structures, we develop an accurate FDTD subgridding algorithm suitable for arbitrary subgridding settings with arbitrary contrast ratios between the normal gird and the subgrid. Although the resulting system matrix is unsymmetric, we develop a time marching method to overcome the stability problem without sacrificing the matrix-free merit of the original FDTD. This method is general, which is also applicable to other subgridding algorithms whose underlying numerical systems are unsymmetric. The proposed FDTD subgridding algorithm is then further made unconditionally stable, thus permitting the use of a time step independent of space step.^ Last but not the least, the framework of the proposed method can be flexibly extended to solve partial differential equations in other disciplines, which we have demonstrated for thermal analysis