Multilevel Adaptive Cross Approximation Research Articles

Nested kernel degeneration-based boundary element method solver for rapid computation of eddy current signals

In this article, the nested kernel degeneration (NKD) based integral equation fast solver is proposed to compress the dense system matrix generated by boundary element method (BEM) for calculating eddy current signals in 3D nondestructive evaluation (NDE) problems. The near and diagonal block interactions are computed and stored by full matrices. However, for the far block interactions, not like in the single level kernel degeneration-based method that all the interaction matrices are compressed by the interpolations, the nested approximation framework makes the expression of the higher-level matrices by the ones at leaf levels and the transfer matrices between neighboring levels. Different integral kernels are degenerated and nested to ensure the performance of NKD based method. The robustness and efficiency of the proposed solver are validated and verified by making numerical predictions and comparing them with the ones achieved by analytical method, numerical methods and experiments. Our numerical experiments show that the NKD solver is more efficient than the butterfly solver (multilevel adaptive cross approximation algorithm (MLACA)) for electrically small eddy current NDE problem. With the NKD based method, O(N) computational complexity is achieved for multiscale low frequency eddy current simulations without sacrificing accuracy, where N is the number of unknowns. So far, to our best knowledge, the proposed method with linear complexity is the most efficient integral equation-based BEM forward solver to predict the eddy current signals from cracks and flaws.

Analysis of electromagnetic non-destructive evaluation modelling using Stratton-Chu formulation-based fast algorithm.

The eddy current non-destructive evaluation (NDE) modelling using Stratton-Chu formulation-based fast algorithm is analysed. Stratton-Chu formulations, which have no low frequency breakdown issue, are selected for modelling electromagnetic NDE problems with low frequency and high conductivity approximations. As the main contribution of this article, the robustness and efficiency of the approximations, which result in big savings in both memory and CPU time, are validated and analysed using examples from practical EC testing. The boundary element method (BEM) is used to discretize the integral equations into a linear system of equations: the first order Rao-Wilton-Glisson (RWG) vector basis functions with the flat triangle meshes of the object and pulse basis functions are selected to expand the equivalent surface currents and the normal component of magnetic fields, respectively. Then the multilevel adaptive cross approximation (MLACA) algorithm is applied to accelerate the iterative solution process. The performance and efficiency of adaptively applying a multi-stage (level) algorithm based on the criteria concluded for the operators are shown. This article is part of the theme issue 'Advanced electromagnetic non-destructive evaluation and smart monitoring'.

Efficient Solution of Electromagnetic Scattering Problems Using Multilevel Adaptive Cross Approximation and LU Factorization

The multilevel adaptive cross approximation (MLACA), previously described in the literature, extends the single-level ACA with a recursive multilevel algorithm that significantly improves compression of off-diagonal matrix blocks resulting from electromagnetic integral equations (IE) discretized via the method of moments (MoM). In this article, the MLACA approach is extended and applied to a direct solution of the MoM matrix system via LU factorization. It will be shown through numerical experiments that the off-diagonal LU blocks are also compressible using MLACA, yielding a compression rate superior to the single-level ACA and a memory complexity of O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4/3</sup> log N). In addition, the MLACA LU block updates are performed in rank-reduced form, yielding a very efficient software implementation via a Level 3 BLAS optimized for the CPU or GPU.

An Efficient MLACA-SVD Solver for Superconducting Integrated Circuit Analysis

Inductance extraction for superconducting integrated circuits requires the accurate solution of structural current distributions. FastHenry is a well known, magnetoquasistatic solver suitable for this task. It is based upon the partial element equivalent circuit, integral equation method, with the structure discretized into hexahedral filaments. It employs the multilevel fast multipole algorithm (MLFMA) for compressed storage of the mutual inductance matrix, which accounts for most of the required memory. This MLFMA implementation is especially memory efficient, given certain approximations and algorithmic parameter choices. However, errors are introduced into the matrix representation. Here, a multilevel adaptive cross approximation solver with singular value decomposition recompression (MLACA-SVD) is presented as an alternative to FastHenry's existing MLFMA solver. MLACA-SVD compresses off-diagonal matrix blocks to a specified error tolerance, based on evaluating selected entries. Quadrature recipes are presented for guaranteed accuracy of matrix entry evaluation. Numerical results for examples of practical interest show that the MLACA-SVD memory scaling versus b (number of filaments) is practically identical to that of FastHenry's MLFMA, and is close to O(blog b). The MLACA-SVD requires less memory for the same solution accuracy, and furthermore offers complete control over matrix approximation errors. For the examples considered, it is found to be a more efficient solver. It is well suited to parallelization.

MLACA With Modified Grouping Strategy for Efficient Superconducting Circuit Analysis

Electromagnetic analysis of superconducting integrated circuits is routinely required for inductance extraction. FastHenry is a well-known, applicable magnetoquasistatic analysis tool. It is accelerated with the multilevel fast multipole algorithm (MLFMA). A multilevel adaptive cross approximation solver with singular value decomposition recompression (MLACA-SVD) was recently proposed for FastHenry, as an alternative to the MLFMA. It was found to be more efficient, with regards to accuracy versus required memory. It uses the same octree grouping as FastHenry's MLFMA. Here, two modified grouping strategies are proposed to further improve MLACA-SVD efficiency by compressing interactions between larger groups, while maintaining scaling performance consistent with valid admissibility of interactions. The strategies are denoted “shell merging” and “wall merging.” Results show that the MLACA-SVD solver with merging requires about four times less memory than FastHenry's MLFMA, for similar accuracy. Shell merging is found to be slightly superior.

Efficient numerical analysis of finite FSS with multilayered media by MLACA

In this study, an efficient numerical technique for transmission characteristic analysis of finite planar frequency selective surface (FSS) with multilayered media is presented by using the multilevel adaptive cross approximation (MLACA). The computational scheme is based on the mixed-potential electric field integral equation (MPIE) with multilayered media Green's functions (MLGFs). The three-level modified discrete complex images method (DCIM) is presented for the spatial-domain MLGFs without any quasi-static and surface-wave extraction. For electrically large-size FSS structures, the MLACA method is then employed to alleviate the high computational complexity and memory requirement of method of moments. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method.

Multilevel adaptive cross approximation for efficient modeling of 3D arbitrary shaped eddy current NDE problems

In this article, the multilevel adaptive cross approximation (MLACA) algorithm is presented to accelerate the boundary element method (BEM) for eddy current nondestructive evaluation (NDE) 3D problems involving arbitrary shapes. The Stratton-Chu formula, which does not have the low frequency breakdown issue, has been selected for modeling. The equivalent electric and magnetic surface currents are expanded with Rao-Wilton-Glisson (RWG) vector basis functions while the normal component of the magnetic field is expanded with pulse basis functions. The MLACA compresses the rank deficient matrices with the ACA and the butterfly algorithm. We improve the efficiency of MLACA by truncating the integral kernels after a certain distance and applying the multi-stage (level) algorithm adaptively based on the criteria for different operators to further decrease the memory and CPU time requirements while keeping almost the same accuracy comparing with the traditional MLACA. The proposed method is especially helpful to deal with the large solution domain issue of the BEM for eddy current problems. Numerical predictions are compared with the analytical, the semi-analytical predictions and the experimental results for 3D eddy current NDE problems of practical interest to demonstrate the robustness and efficiency of the proposed method.

Application of multilevel directional adaptive cross approximation technique for electromagnetic problems

In this paper, a novel scheme is presented for forming the matrix equations of multilevel adaptive cross approximation (MLACA) algorithm. The main idea of the proposed technique is to use the directional grouping scheme to subdivide the far-field domain of MLACA algorithm. By using the grouping scheme, the far-field interaction domain can be divided into many cone structures. The matrix between the observation group and far-field group in the cone structure is low-rank, which meets the directional far-field requirement. At the same time, the near-field interaction matrices are formed by the SVD(T) method to further reduce the total memory requirements. With the given techniques, the memory requirement of the novel grouping scheme for the far-field is much less than half of traditional MLACA algorithm. Meanwhile, the memory requirement of the SVD(T) method for the near-field is only about one-third of direct filling.

An Efficient High-Order Marching-on-in-Degree Solver for Conducting and Dielectric Bodies of Revolution

The transient electromagnetic (EM) scattering from bodies of revolution (BoRs) is analyzed by the time-domain integral equation with the Silvester–Lagrange polynomials. The high-order spatial basis functions are used along the generatrix, while the weighted Laguerre polynomials are served as the temporal basis functions. In this way, the late time stability can be guaranteed by using the marching-on-in-degree scheme. In this communication, the conducting targets are simulated by the time-domain combined field integral equation while the dielectric ones are formulated in terms of the magnetic current combined field integral equation. Moreover, the multilevel adaptive cross approximation algorithm is adopted to accelerate the calculations of the well-separated groups. The numerical results demonstrate that the proposed solver is efficient to analyze the transient EM scattering from BoRs.

Sparsified Multi-level Adaptive Cross Approximation-Characteristic Basis Function Method for Fast Electromagnetic Scattering Analysis

ABSTRACTWith the increase of block size, the amount of calculation and storage of the recently developed sparsified adaptive cross approximation for compressing the impedance submatrix of well-separated blocks will become as large as O(s2M) and O(sM), respectively, where M is the number of unknowns in each block, and s is the effective rank of the impedance submatrix. To address this issue, an improved sparsified adaptive cross approximation, termed as sparsified multi-level adaptive cross approximation, is presented in this article. The sparsified multi-level adaptive cross approximation uses the multi-level adaptive cross approximation to exhibit a faster and sparser compression of the sparsified adaptive cross approximation inner matrices. As a result, the computational time and storage can be reduced to O(s3) + O(sM) and O(s2) + O(Mlogs), respectively, where log s « s « M. Furthermore, the characteristic basis function method is used to further reduce the storage of the sparsified multi-level adaptive cross approximation by compressing its outer matrices. Numerical results about the electromagnetic scattering are given to verify the advantages of the proposed sparsified multi-level adaptive cross approximation-characteristic basis function method.

Error Bound of the Multilevel Adaptive Cross Approximation (MLACA)

An error bound of the multilevel adaptive cross approximation (MLACA), which is a multilevel version of the adaptive cross approximation-singular value decomposition (ACA-SVD), is rigorously derived. For compressing an off-diagonal submatrix of the method of moments (MoM) impedance matrix with a binary tree, the $L$ -level MLACA includes $L+1$ steps, and each step includes $2^L$ ACA-SVD decompositions. If the relative Frobenius norm error of the ACA-SVD used in the MLACA is smaller than $\varepsilon $ , the rigorous proof in this communication shows that the relative Frobenius norm error of the $L$ -level MLACA is smaller than ${(1 + \varepsilon)^{L + 1}} - 1$ . In practical applications, the error bound of the MLACA can be approximated as $\varepsilon (L + 1)$ , because $\varepsilon $ is always $\ll 1$ . The error upper bound can be used to control the accuracy of the MLACA. To ensure an error of the $L$ -level MLACA smaller than $\varepsilon $ for different $L$ , the ACA-SVD threshold can be set to ${(1 + \varepsilon)^{\tfrac{1}{{L + 1}}}} - 1$ , which approximately equals ${\varepsilon/{(L + 1)}}$ for practical applications.

Parallel Marching-on-in-Degree Solver of Time-Domain Combined Field Integral Equation for Bodies of Revolution Accelerated by MLACA

Time-domain combined field integral equation (TD-CFIE) for bodies of revolution (BORs) is solved by marching-on-in-degree (MOD) method. A multilevel partitioning is adopted to group the spatial basis functions along the longitudinal dimension. The interactions of the adjacent groups are calculated directly in the traditional manner, while the impedance matrices associated with the well-separated groups at each level are computed by the multilevel adaptive cross approximation (MLACA) algorithm. The hybrid MPI and OpenMP parallel programming technique is utilized to further accelerate the solving process on a shared-memory computer system. Two numerical results demonstrate that the proposed method can greatly reduce the memory requirement and CPU time, thus enhance the capability of MOD method.

Augmented EFIE with Adaptive Cross Approximation Algorithm for Analysis of Electromagnetic Problems

The augmented electric field integral equation (A-EFIE) with charge neutrality enforcement provides a stable formulation to conquer low-frequency breakdown characteristic of conventional EFIE. It is augmented with additional charge unknowns through current continuity equation. The A-EFIE combined with the multilevel adaptive cross-approximation (MLACA) algorithm is developed to further reduce the memory requirement and computation time for analyzing electromagnetic problems. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed method.

Multilevel Adaptive Cross Approximation (MLACA)

The Multilevel Adaptive Cross Approximation (MLACA) is proposed as a fast method to accelerate the matrix-vector products in the iterative solution of the linear system that results from the discretization of electromagnetic Integral Equations (IE) with the Method of Moments (MoM). The single level ACA, already described in the literature, is extended with a multilevel recursive algorithm in order to improve the asymptotical complexity of both the computational cost and the memory requirements. The main qualities of ACA are maintained: it is purely algebraic and independent of the integral equation kernel Green's function as long as it produces pseudo-rank-deficient matrix blocks. The algorithm is presented in such a way that it can be easily implemented on top of an existing MoM code with most commonly used Green's functions.

A fast integral equation method for simulating high-field radio frequency coil arrays in magnetic resonance imaging

A fast full-wave numerical approach was developed for simulating high-field multi-channel radio-frequency (RF) receive coil arrays in magnetic resonance imaging. To improve the efficiency, the impedance matrix was compressed by a multilevel adaptive cross approximation method. Furthermore, careful organization of multiple coil simulations was applied so that the impedance matrix associated with biological subjects is constructed and pre-conditioned only once. Numerical examples demonstrate the efficacy of the proposed approach for RF coil simulations.