Second-order Transmission Condition Research Articles

Efficient 3-D Finite Element Modeling of Periodic XBAR Resonators

An efficient technique is presented for 3D finite element modeling of large-scale periodic XBAR resonators in the time-harmonic domain. In this technique, a domain decomposition scheme is employed to decompose the computational domain into many small subdomains whose FE subsystems can be factorized with a direct sparse solver at a low cost. Transmission conditions are enforced to interconnect adjacent subdomains, and a global interface system is formulated and solved iteratively. To accelerate the convergence, a second-order transmission condition (SOTC) is designed to make the subdomain interfaces transparent for propagating and evanescent waves. An effective forward-backward preconditioner is constructed that, when combined with the SOTC, significantly reduce the number of iterations at no additional cost. Numerical results are given to demonstrate the accuracy, efficiency, and capability of the proposed algorithm.

A Flexible FEM-BEM-DDM for EM Scattering by Multiscale Anisotropic Objects

In this article, a nonconformal domain decomposition method (DDM) based on the hybrid finite element method (FEM) and boundary element method (BEM) is proposed to solve arbitrary 3-D anisotropic (including bianisotropic and single anisotropic) multiscale composite objects. The object is divided into interior and exterior regions. The interior region is solved by the bianisotropic finite element domain decomposition method (FEM-DDM), and the exterior region is solved by the discontinuous Galerkin (DG) method based on the combined field integral equation (CFIE). In order to decouple the electric and magnetic fields of the bianisotropic object, a new finite element–boundary element–domain decomposition method (FEM-BEM-DDM) formulation is derived. To ensure the continuity of electric current and electric field between anisotropic FEM subdomains, a new auxiliary current is introduced into the interface of different subdomains of FEM and the interface of FEM subdomains with BEM subdomains. The former uses second-order transmission conditions (SOTCs) to ensure continuity, whereas the latter uses Robin transmission conditions (RTCs) to ensure continuity. The BEM dense matrix–vector multiplication is accelerated by a multilevel fast multipole algorithm (MLFMA). The proposed FEM-BEM-DDM allows for nonconformal discretization between any touching subdomains and provides an effective preconditioner. In numerical examples, the accuracy and effectiveness of the method are verified.

Hybrid FEM-DDM and BEM-BoR for the Analysis of Multiscale Composite Structures

We propose a novel scheme to accelerate the computation of the finite-element method-boundary element method (FEM-BEM) for 3-D electromagnetic scattering. It builds on the FEM-domain decomposition method (FEM-DDM) and the degenerated boundary element method (BEM) developed for the body of revolution (BoR), which is, thus, referred to as BEM-BoR. When coupled with FEM in the framework of domain decomposition, the infinite computational domain is truncated by a closed rotationally symmetric surface enclosing the 3-D scatterer, and BEM-BoR is then applied on this surface as a radiation boundary condition (RBC). The finite-domain interior to this surface is further decomposed into several subdomains, which are connected by the second-order transmission conditions (SOTCs) imposed on nonconformal meshes. We refer to the resultant solver as a hybrid FEM-DDM and BEM-BoR (FEM-BEM-BoR). Since BEM-BoR can be divided into several 2-D problems instead of the original 3-D one, the proposed FEM-BEM-BoR is more efficient than the original FEM-BEM. Numerical examples are presented, and the comparisons of the simulation results with FEM-BEM confirm the accuracy and efficiency improvement of FEM-BEM-BoR. In addition, its practical applicability is demonstrated by modeling large-scale objects.

Twofold Domain Decomposition Method for the Analysis of Multiscale Composite Structures

A nonconformal twofold domain decomposition method (TDDM) based on the hybrid finite element method-boundary element method (FEM-BEM) is proposed for analyzing 3-D multiscale composite structures. The proposed TDDM starts by partitioning the composite object into a closed exterior boundary domain and an interior volume domain. The interior and exterior boundary value problems are coupled to each other through the Robin transmission conditions (RTCs). Both domains are then independently decomposed into subregions to facilitate computation. Specifically, FEM-DDM with the second order transmission conditions (SOTCs) is employed for the interior domain, and BEM-discontinuous Galerkin (BEM-DG) based on the combined field integral equation (CFIE) is applied for the exterior boundary domain. The proposed TDDM allows for nonconformal discretization between any touching subdomains. Without the introduction of a stabilization term that relies on a line integral over intersection of nonmatching meshes and relevant terms involving surface-line integrals, the proposed TDDM provides an effective domain decomposition (DD) preconditioner for the global system. Numerical examples are presented, and the comparisons of the simulation results with FEM-BEM confirm the validity and accuracy of TDDM. Moreover, its ability to model practical large-scale and multiscale targets is also demonstrated.

Multilevel compressed block decomposition-based finite-element domain decomposition method for the fast analysis of finite periodic structures

An efficient finite-element domain decomposition method based on the multilevel compressed block decomposition (MLCBD) algorithm is presented. The dual-primal finite element tearing and interconnecting (FETI-DP) method is first introduced to generate a nonoverlapping domain decomposition method with good convergence properties. A second-order transmission condition is adopted to further accelerate the convergence rate of the FETI-DP. With the MLCBD algorithm, the computational complexity for the direct solution of FETI-DP subdomain equations can be reduced to be almost logarithmic linear. The accuracy of the MLCBD algorithm is highly adjustable according to practical requirements. Exploiting the repetitiveness of finite periodic structures, the proposed method can significantly reduce the computational costs. Numerical examples demonstrate the effectiveness of the proposed method for the electromagnetic simulation of finite periodic structures.

A Hybrid Conformal/Nonconformal Domain Decomposition Method for Multi-Region Electromagnetic Modeling

This paper presents a unified framework for solving large-scale multi-region electromagnetic problems using a hybrid conformal/nonconformal domain decomposition method. In this method, an original electrically large problem is first divided into several regions based on its geometry. Each region is then meshed and further decomposed into smaller subdomains. As such, the decomposed problem is mesh-conformal and geometry-conformal within each region but could be mesh-nonconformal and geometry-nonconformal between two neighboring regions. To solve such a problem, a hybrid method is proposed that employs the finite-element tearing and interconnecting (FETI) method to deal with mesh-nonconformal and/or geometry-nonconformal interfaces, where a second-order transmission condition and a crosspoint correction technique are applied to improve the iterative convergence of the interface system and ensure a correct interconnection across subdomain interfaces. For mesh-conformal and geometry-conformal interfaces inside each region, the hybrid method employs the dual-primal finite element tearing and interconnecting (FETI-DP) method to construct an effective coarse grid correction for the interface problem. A unified global system of equations is finally formulated for the interface unknowns from both nonconformal and conformal interfaces. The proposed method is validated with simulation of wave propagation and radiation by broadband antennas and phased-array antennas covered with a radome and a near-field focal lens.

Efficient Modeling of Large-Scale Electromagnetic Well-Logging Problems Using an Improved Nonconformal FEM-DDM

The numerical modeling of the advanced electromagnetic well-logging problems is very challenging because it requires tremendous workload of grid meshing and computation. For example, global meshing and simulation have to be done repeatedly for every logging position, because the model includes complex sensor arrays moving continuously along the borehole. In this paper, an efficient nonconformal finite element domain decomposition method is developed to solve these problems efficiently. First, the well-logging model is divided into two nonconformal subdomains so that the sensing arrays inside the borehole are separated from the beds outside. Since the mesh size between different subdomains can be inconsistent, well-logging modeling can significantly reduce the unknowns and computational cost. Second, a second-order transmission condition is implemented successfully in a quasi-symmetrical form via a novel treatment of Gaussian integration on the nonconformal interface. Finally, a hierarchical hexahedral basis function is introduced to further reduce the unknowns and extend the applicability of the method to multiscale problems. Numerical examples show that this method greatly reduces the memory cost and speeds up the computation compared with the traditional finite element method, especially when the response needs to be simulated repeatedly while the logging tools keep moving along the borehole.

Fast and accurate finite element analysis of large-scale three-dimensional photonic devices with a robust domain decomposition method

A fast and accurate full-wave technique based on the dual-primal finite element tearing and interconnecting method and the second-order transmission condition is presented for large-scale three-dimensional photonic device simulations. The technique decomposes a general three-dimensional electromagnetic problem into smaller subdomain problems so that parallel computing can be performed on distributed-memory computer clusters to reduce the simulation time significantly. With the electric fields computed everywhere, photonic device parameters such as transmission and reflection coefficients are extracted. Several photonic devices, with simulation volumes up to 1.9×10(4) (λ/n(avg))3 and modeled with over one hundred million unknowns, are simulated to demonstrate the application, efficiency, and capability of this technique. The simulations show good agreement with experimental results and in a special case with a simplified two-dimensional simulation.

FEM-DDM WITH AN EFFICIENT SECOND-ORDER TRANSMISSION CONDITION IN BOTH HIGH-FREQUENCY AND LOW-FREQUENCY APPLICATIONS

In this paper, a novel second-order transmission condition is developed in the framework of non-conformal flnite element domain decomposition method to meet the challenges brought by complex and large-scale electromagnetic modeling. First, it is implemented e-ciently on the non-conformal interface via a Gauss integral scheme. Then, the eigenvalue analysis of the DDM system show a more clustered eigenvalue distribution of this transmission condition compared with several existing transmission conditions. After that, it is applied to large-scale complex problems such as S-type waveguides in the high frequency band and dielectric well-logging applications in the low frequency band. The flnal numerical results demonstrate that this transmission condition has high e-ciency and huge capability for modeling large-scale problems with multi-resolution in any frequency band.

A Scalable Nonoverlapping and Nonconformal Domain Decomposition Method for Solving Time-Harmonic Maxwell Equations in $\mathbb{R}^3$

We present a novel nonoverlapping and nonconformal domain decomposition method (DDM) for solving the time-harmonic Maxwell equations in $\mathbb{R}^3$. There are three major technical ingredients in the proposed nonconformal DDM: (a) a true second order transmission condition (SOTC) to enforce fields continuities across domain interfaces; (b) a corner edge penalty term to account for corner edges between neighboring subdomains; and (c) a global plane wave deflation technique to further improve the convergence of DDM for electrically large problems. It has been shown previously that a SOTC, which involves two second order transverse derivatives, facilitates convergence in the conformal domain decomposition method for both propagating and evanescent electromagnetic waves across domain interfaces. However, the discontinuous nature of the cement variables across the corner edges between neighboring subdomains remains troublesome. To mitigate the technical difficulty encountered and to enforce the needed divergence-free condition, we introduced a corner edge penalty term into the interior penalty formulation for the nonconformal DDM. The introduction of the corner edge penalty term successfully restored the superior performance of the SOTC. Finally, through an analysis of the DDM with the SOTC, we show that there still exists a weakly convergent region where the convergence in the DDM can still be unbearably slow for electrically large problems. Furthermore, it is found that the weakly convergent region is centered at the cutoff modes, or electromagnetic waves propagate in parallel to the domain interfaces. Subsequently, a global plane wave deflation technique is utilized to derive an effective global-coarse-grid preconditioner to promote fast convergence of the cutoff or near cutoff modes in the vicinity of domain interfaces. Finally, the strength of the proposed method is illustrated by means of three numerical examples.

Signal Integrity Analysis of High-Speed Interconnects by Using Nonconformal Domain Decomposition Method

3-D interconnect techniques in multilayer very large scale integration design, such as stacked layers and various chip-stacking systems, have paved the way for the improvement of integrated circuit density and operation speed. However, these techniques often accompany with impedance discontinuities that induce signal integrity (SI)/power integrity and electromagnetic interference effects. Consequently, SI analysis serves as one of the key guidelines of chip-package-board design in deep submicrometer technology. However, full-wave numerical analysis of 3-D high-speed and high-density interconnects is also considered to be a great challenge. In this paper, a nonconformal domain decomposition method with second-order transmission condition for SI analysis of high-speed interconnects on a multiscale multilayer printed circuit board is presented. The accuracy and robustness of the proposed method are first witnessed by modeling a six-layer two-trace differential pair. Then we study a 14-layer 16-trace interconnect model to demonstrate the efficiency of the method. Eye diagrams of the two models are also presented in the time-domain SI analysis.

Non-Conformal Domain Decomposition Method With Mixed True Second Order Transmission Condition for Solving Large Finite Antenna Arrays

A non-overlapping and non-conformal domain decomposition method (DDM) is presented for modeling large finite antenna arrays. Two major benefits of the proposed DDM: (a) A mixed true second order transmission condition (SOTC) with corner edge penalty terms is developed to facilitate fast convergence in the DDM iterations. Numerical experiments indicate that the convergence of DDM with the aforementioned SOTC is insensitive to the sizes of arrays; and, (b) The non-conformal property of the proposed DDM permits the use of completely independent discretization for each of the sub-domains. We demonstrate the performance of the proposed approach through several large-scale problems.

Non-conformal domain decomposition method with second-order transmission conditions for time-harmonic electromagnetics

Non-overlapping domain decomposition (DD) methods provide efficient algorithms for solving time-harmonic Maxwell equations. It has been shown that the convergence of DD algorithms can be improved significantly by using high order transmission conditions. In this paper, we extend a newly developed second-order transmission condition (SOTC), which involves two second-order transverse derivatives, to facilitate fast convergence in the non-conformal DD algorithms. However, the non-conformal nature of the DD methods introduces an additional technical difficulty, which results in poor convergence in many real-life applications. To mitigate the difficulty, a corner-edge penalty method is proposed and implemented in conjunction with the SOTC to obtain truly robust solver performance. Numerical results verify the analysis and demonstrate the effectiveness of the proposed methods on a few model problems. Finally, drastically improved convergence, compared to the conventional Robin transmission condition, was observed for an electrically large problem of practical interest.

A fast-domain decomposition method for the solution of electromagnetic scattering by large objects

A nonoverlapping domain decomposition method is proposed for the finite element solution of the scattering problem by electrically large, inhomogeneous, infinite cylinders of arbitrary cross section. To minimize the size of the total computational domain, a second-order-absorbing boundary condition (ABC) is applied upon an outer boundary of arbitrary shape which may be conformal to the surface of the scatterer. This domain is then partitioned into concentric subdomains circumscribing the object. A second-order transmission condition, derived from the ABC, is prescribed upon the interfaces between two adjacent subdomains. This particular configuration is responsible for the fast convergence of the domain decomposition iterative algorithm, which is parallelizable. Numerical results obtained with a nonparallelized computer code are presented, which emphasize the superiority of this technique in terms of memory storage requirements and computing times over the standard finite element approach, as well as over the rigorous hybrid finite element-integral equation formulation.