Vineet Rawat

One way domain decomposition method with second order transmission conditions for solving electromagnetic wave problems

Non-overlapping domain decomposition (DD) methods with complex first order Robin-type transmission conditions (TCs) provide an efficient iterative solution for Maxwells equation. Unfortunately, the first order TCs do not effectively account for some eigenmodes of the system matrix, which limits the scalability of the methods. In this work, we examine two TCs with a second order transverse derivative to improve the methods performance. A detailed convergence analysis of the two TCs is presented. We then investigate the use of the two second order TCs in non-conformal and non-overlapping one way DD methods. Numerical results illustrate the effectiveness of the proposed methods on some model problems and on several problems of practical interest.

A Domain Decomposition Method for Electromagnetic Radiation and Scattering Analysis of Multi-Target Problems

A domain decomposition method is presented for analyzing electromagnetic problems involving multiple separable scatterers. The method first decomposes the original problem into several disjoint subregions. In each subregion, the domain decomposition method is further applied by decomposing the region into smaller, possibly repeated, subdomains. The domain decomposition method is general enough for arbitrary geometries, but is also capable of exploiting repetitions. This renders geometrically complicated and electrically large subregion problems tractable. The subregions communicate through the near-field Greens function. To overcome the vast computational costs required in exchanging information between electrically large subregions, the adaptive cross approximation algorithm is adopted to expedite the process. The method is applied to study radiation characteristics of a reflector antenna system and an antenna array in the presence of a frequency selective surface to demonstrate the utility of the present approach.

Treatment of cement variables in the domain decomposition method for Maxwell’s equations

In this contribution we focus our attention on the cement variables of the DDM and consider their theoretical and practical treatment. We first determine the correct functional space of the continuous variables and then examine two possible representations in the VFEM. One choice is shown to provide superior solver convergence though its use results in sub-optimal error convergence. We then propose and validate a remedy to this problem by special treatment of non-planar interfaces. Finally, we demonstrate the scalability of the resulting method.

Hybrid domain decomposition method and boundary element method for the solution of large array problems

The non-overlapping domain decomposition method (DDM) has emerged as a powerful and attractive technique for numerically-rigorous solution of Maxwells equations due to its inherent parallelism and its beauty as an efficient and effective preconditioner. DDM is based on a divide-and-conquer philosophy. Instead of tackling a large and complex problem directly as a whole, the original problem is partitioned into smaller, possibly repetitive, and easier to solve sub-domains. Some suitable boundary conditions called transmission conditions are prescribed at the interfaces between adjacent sub-domains to enforce the continuity of electromagnetic fields. However in the existing approaches, the radiation condition is approximated by the first order absorbing boundary condition (ABC), producing the unwanted spurious reflection from the truncation boundary. In order to minimize such unphysical reflection, the truncation boundary must be placed sufficiently far away from the object, resulting a large number of sub-domains. In this paper, the unbounded exterior space will be treated as an additional domain. This domain is formulated by a boundary element method (BEM) which incorporates the radiation condition through its Greens function.

2 nd order ABC for inhomogeneous medium in vector finite element method

In this work, we extend ABC into second order in consideration of inhomogeneous medium. The detailed procedure to obtain the second order ABC is described in this paper. Transparent conditions are first established by treating the material properties as functions of position, and the second order ABC is derived from the transparent condition.

Domain decomposition methods with second order transmission conditions for solving multiscale electromagnetic wave problems

This paper presents a second order TE transmission condition (TC) which provides converging mechanism for TE evanescent modes. The application of the second order TE TC greatly improves the robustness of the DDM when small elements are encountered within the domain interfaces.

PML in discontinuous galerkin time domain methods

Discontinuous Galerkin methods are an attractive approach for the solution of time dependent problems. Discontinuous Galerkin methods were recently applied to the solutions of Maxwells equations in the time domain[1]. They support elements of various types, non-matching grids and varying polynomial orders in each element. Moreover, they result in block diagonal mass matrices with block size equal to the number of degrees of freedom per element and therefore can lead to an fully explicit time marching scheme. Furthermore, continuity at element interfaces is weakly enforced with the addition of proper penalty terms on the variational formulation. This paper describes an interior penalty discontinuous Galerkin method for solving the two first order Maxwells equations in the time domain. Firstly, the use of central fluxes along with a leap-frog time scheme leads to an energy conserving method. The proposed method is explicit and conditionally stable. Secondly, a local time-stepping strategy is applied to increase efficiency and reduce the computational time.

A hybrid finite element boundary element method for periodic structures on non-periodic meshes using interior penalty method

Recent interests in analyzing/designing large finite arrays, frequency selective surfaces (FSS), and metamaterials speak volume of the need for a robust and efficient numerical method for arbitrary and inhomogeneous periodic structures in 3D. The development in general numerical methods for analyzing arbitrary 3-dimensional periodic structures is nothing new. Many existing literature has thoroughly addressing this issue, including the hybridization of finite and boundary element method utilizes the Ewald transform to quickly compute the needed periodic Greenpsilas function. However, most of the work on modeling periodic structures relies on the availability of periodic meshes. Although such a constraint may not seem much a burden in many problem geometries, however, the relief of such a constraint still contributes greatly to the flexibility of applying the computer codes to periodic structures. This attribute has been the focus of a few recent publications on using non-periodic meshes for analyzing periodic structures.

A domain decomposition preconditioner for time-harmonic electromagnetics

In this article we introduce a new domain decomposition (DD) based iterative method for the solution of the time-harmonic vector wave equation. The method derives its efficiency by decomposing large problems into several smaller, more manageable pieces. It differs from previous non-overlapping DD methods in two significant respects: (1) it does not introduce any auxiliary variables, and (2) the method is applied to a decomposition of the computational mesh. After briefly summarizing the formulation of the method we provide numerical results which verify the accuracy of the method. We then examine its convergence behavior with respect to varying mesh size, partition number, problem size, and frequency. Finally, we apply the method to two challenging problems.