An iteration-free finite element domain decomposition method (DDM) is proposed for the simulation of 3-D electromagnetic problems. The finite element tearing and interconnecting (FETI) with Robin-type transmission condition is introduced to construct a domain decomposition framework. Based on hierarchical ( $\mathcal {H}$ -) matrix algorithm, the numerical Green’s function of each subdomain can be represented by a data-sparse matrix form. Subsequently, the original 3-D problem is reduced to a 2-D problem on the skeleton. Derived from the finite element discretization of pseudo-differential operator, the skeleton matrix can also be efficiently approximated with an $\mathcal {H}$ -matrix by assembling all the numerical Green’s functions of subdomains. Based on $\mathcal {H}$ -LU factorization algorithm, the computational complexity and memory requirement for the direct solution of the skeleton equation can be reduced to be logarithmic linear. The resulting direct FETI (D-FETI) method completely depends on direct method without any iterative process. Hence, it does not suffer from the issue of convergence rate, and avoids redundant computations for multiple right-hand side problems. Numerical examples demonstrate the effectiveness of the proposed method.

An Iteration-Free Domain Decomposition Method for the Fast Finite Element Analysis of Electromagnetic Problems

An efficient direct solution scheme is presented to solve the finite element-boundary integral (FEBI) system for the analysis of open-region electromagnetic problems in half-space. Data-sparse representation based on hierarichal ( $\mathcal {H}$ -) matrix is employed to construct the direct solution framework. The FEBI system matrix is first represented to be an $\mathcal {H}$ -matrix form, in which the FE submatrices are accurately represented by $\mathcal {H}$ -matrices, while the $\mathcal {H}$ -matrix approximations of the BI submatrices are constructed by the adaptive cross approximation algorithm combined with singular value decomposition recompression (ACA-SVD). Based on the data-sparse formatted algorithm, the lower and upper (LU) triangular factors of the FEBI system matrix can be computed and stores as $\mathcal {H}$ -matrix forms with low computational costs. The resulting method, refered to as the direct FEBI method (D-FEBI), is free of convergence problem and well-suited for multiple right-hand side (RHS) computations. The performance of the D-FEBI is examined by two examples of monostatic electromagnetic scattering above a lossy half-space.

Direct Solution of Finite Element-Boundary Integral System for Electromagnetic Analysis in Half-Space

In this paper, a review of the low-rank factorization method is presented, with emphasis on their application to multiscale problems. Low-rank matrix factorization methods exploit the rankdeficient nature of coupling impedance matrix blocks between two separated groups. They are widely used, because they are purely algebraic and kernel free. To improve the computation precision and efficiency of low-rank based methods, the improved sampling technologies of adaptive cross approximation (ACA), post compression methods, and the nested low-rank factorizations are introduced. ${\mathrm {O}}(N)$ and ${\mathrm {O}}(N \log N)$ computation complexity of the nested equivalence source approximation can be achieved in low and high frequency regime, which is parallel to the multilevel fast multipole algorithm, N is the number of unknowns. Efficient direct solution and high efficiency preconditioning techniques can be achieved with the low-rank factorization matrices. The trade-off between computation efficiency and time are discussed with respect to the number of levels for low-rank factorizations.

/pdf/low-rank-matrix-factorization-method-for-multiscale-3qvw40kyxr.pdf

Low-Rank Matrix Factorization Method for Multiscale Simulations: A Review

Numerical simulation plays an important role for the prediction of optical trapping based on plasmonic nano-optical tweezers. However, complicated structures and drastic local field enhancement of plasmonic effects bring great challenges to traditional numerical methods. In this article, an accurate and efficient numerical simulation method based on a dual-primal finite element tearing and interconnecting (FETI-DP) and Maxwell stress tensor is proposed, to calculate the optical force and potential for trapping nanoparticles. A low-rank sparsification approach is introduced to further improve the FETI-DP simulation performance. The proposed method can decompose a large-scale and complex problem into small-scale and simple problems by using non-overlapping domain division and flexible mesh discretization, which exhibits high efficiency and parallelizability. Numerical results show the effectiveness of the proposed method for the prediction and analysis of optical trapping at nanoscale.

/pdf/efficient-prediction-and-analysis-of-optical-trapping-at-23v8heva1v.pdf

Efficient Prediction and Analysis of Optical Trapping at Nanoscale via Finite Element Tearing and Interconnecting Method

Plasma generated by hypersonic aerocraft is one of the research subjects in computational electromagnetics. When the aerocraft flies at an ultra-high speed in the near space, plasma sheath will be generated in the surroundings and it will trigger varying changes in aerocraft radar electromagnetic scattering properties. Plasma sheath brings about tremendous impacts on the communication between the aerocraft and the outside world, and even ends with "blackout effects" to interrupt communication in severe cases. The interplay between the plasma sheath and the electromagnetic wave involves multiple disciplines covering aerodynamics, plasma sheath physics, and electromagnetic field theory. Therefore, it is challengeable to build a rigorous numerical modeling of the plasma sheath. In this paper, the latest technologies in the electromagnetic simulation of hypersonic aerocraft with plasma sheath are reviewed. Applied cases review and illustrate extensive approaches from approximate calculations to accurate calculations, including analytical methods, differential equation methods, integral equation methods and high-frequency approximation methods. The paper assesses the RCS of electromagnetic wave in analyzing the electromagnetic scattering problem in plasma. With the considerations the reality of electromagnetic simulation and future challenges, the paper talks about the efficiency, advantages and disadvantages of these approaches in simulating hypersonic aerocraft targets.

Numerical modeling of EM scattering from plasma sheath: A review

A fast low-frequency surface integral equation solver based on hierarchical matrix algorithm is proposed. First, the augmented electric field integral equation (A-EFIE) formulation is introduced to eliminate the low-frequency breakdown of traditional EFIE. To deal with large-scale problems, the lowfrequency multilevel fast multipole algorithm (LF-MLFMA) is employed to construct a hierarchical (H-) matrix representation of the A-EFIE system matrix. Moreover, a recompression method is developed to further compress the H-matrix generated by LF-MLFMA. The H-matrix-based triangular factorization algorithm can be performed with almost linear computational complexity and memory requirement, which produces a fast direct solver for multiple right-hand-side (RHS) problems, and a good preconditioner to accelerate the convergence rate of an iterative solver. Numerical examples demonstrate the effectiveness of the proposed method for the analysis of various low-frequency problems.

/pdf/fast-low-frequency-surface-integral-equation-solver-based-on-bsb6ixwj4t.pdf

Fast Low-Frequency Surface Integral Equation Solver Based on Hierarchical Matrix Algorithm

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Fast Low-Frequency Surface Integral Equation Solver Based on Hierarchical Matrix Algorithm