A Hybrid Spectral-Element / Finite-Element Time-Domain Method for Multiscale Electromagnetic Simulations

Abstract

<p>In this study we propose a fast hybrid spectral-element time-domain (SETD) / finite-element time-domain (FETD) method for transient analysis of multiscale electromagnetic problems, where electrically fine structures with details much smaller than a typical wavelength and electrically coarse structures comparable to or larger than a typical wavelength coexist.</p><p>Simulations of multiscale electromagnetic problems, such as electromagnetic interference (EMI), electromagnetic compatibility (EMC), and electronic packaging, can be very challenging for conventional numerical methods. In terms of spatial discretization, conventional methods use a single mesh for the whole structure, thus a high discretization density required to capture the geometric characteristics of electrically fine structures will inevitably lead to a large number of wasted unknowns in the electrically coarse parts. This issue will become especially severe for orthogonal grids used by the popular finite-difference time-domain (FDTD) method. In terms of temporal integration, dense meshes in electrically fine domains will make the time step size extremely small for numerical methods with explicit time-stepping schemes. Implicit schemes can surpass stability criterion limited by the Courant-Friedrichs-Levy (CFL) condition. However, due to the large system matrices generated by conventional methods, it is almost impossible to employ implicit schemes to the whole structure for time-stepping.</p><p>To address these challenges, we propose an efficient hybrid SETD/FETD method for transient electromagnetic simulations by taking advantages of the strengths of these two methods while avoiding their weaknesses in multiscale problems. More specifically, a multiscale structure is divided into several subdomains based on the electrical size of each part, and a hybrid spectral-element / finite-element scheme is proposed for spatial discretization. The hexahedron-based spectral elements with higher interpolation degrees are efficient in modeling electrically coarse structures, and the tetrahedron-based finite elements with lower interpolation degrees are flexible in discretizing electrically fine structures with complex shapes. A non-spurious finite element method (FEM) as well as a non-spurious spectral element method (SEM) is proposed to make the hybrid SEM/FEM discretization work. For time integration we employ hybrid implicit / explicit (IMEX) time-stepping schemes, where explicit schemes are used for electrically coarse subdomains discretized by coarse spectral element meshes, and implicit schemes are used to overcome the CFL limit for electrically fine subdomains discretized by dense finite element meshes. Numerical examples show that the proposed hybrid SETD/FETD method is free of spurious modes, is flexible in discretizing sophisticated structure, and is more efficient than conventional methods for multiscale electromagnetic simulations.</p>