Adaptive Cross Approximation Algorithm Research Articles

A study on partial pivot ACA boundary element method for elasticity problems

The boundary element method (BEM) employing the partial pivot adaptive cross approximation (PACA) algorithm has been observed to experience convergence failures and reduced solution accuracy when solving elasticity problems, especially at large scales. To address this issue, this paper proposes an improved algorithm for both 2D and 3D elasticity problems.Our investigations revealed that when the normal of an element is parallel to the coordinate axes and the element is in the same plane as the source point, zero entries with regular distributions will appear in the coefficient matrix. This results in the premature satisfaction of the convergence criterion of the PACA algorithm, leading to the omission of some rows and columns of the coefficient matrix. To address this issue, this paper proposes improvements to the PACA algorithm through the Coordinate Rotation Method (CRM) and the Component Traversal Method (CTM), and validates the effectiveness of these two improved methods.Furthermore, we investigate reasons behind the decrease in accuracy when employing the CTM to solve large-scale problems. We attribute this to the oscillatory nature of the estimated relative error and propose the modified PACA (M-PACA) algorithm. M-PACA not only maintains the advantages of the PACA algorithm in reducing storage space and accelerating coefficient matrix generation but also addresses its limitations, ensuring more accurate and reliable solutions for elasticity problems.

Dual-Layer Compressive Sensing Scheme Incorporating Adaptive Cross Approximation Algorithm for Solving Monostatic Electromagnetic Scattering Problems

Dual-Layer Compressive Sensing Scheme Incorporating Adaptive Cross Approximation Algorithm for Solving Monostatic Electromagnetic Scattering Problems

Modeling of Eddy Current NDT Simulations by Kriging Surrogate Model

ABSTRACT In this article, the Kriging surrogate model is proposed to accelerate the model-assisted probability of detection (MAPoD) analysis for eddy current nondestructive testing (NDT). The Kriging surrogate model is the key for enabling the efficient MAPoD analysis, considering huge number of uncertainties propagate in the eddy current NDT system, by replacing the time-consuming physical model. To generate the model responses of NDT system, a precise physical model based on the adaptive cross approximation algorithm accelerated boundary element method (BEM) is applied. In the numerical case, the MAPoD study of eddy current for detecting surface flaws in the conducting plate is analyzed. By comparing the PoD metrics achieved by the pure physical model and by the Kriging surrogate model, the accuracy and efficiency of the proposed surrogate model are demonstrated.

Characteristic Modes Analysis of Aperiodic Arrays Using Entire Domain Basis Function Method With Adaptive Cross Approximation Algorithm

The characteristic modes of the aperiodic arrays are analyzed by combining the entire domain basis function method with the adaptive cross approximation (ACA) algorithm. The array cells are identical. The entire domain basis functions defined on the array cells are obtained by analyzing the characteristic modes of the reference cell firstly. Then, the generalized eigenvalue equation whose size is smaller than the conventional one can be obtained. After solving the equation, the characteristic modes of the whole array can be analyzed. Calculation of the matrices is very time-consuming. To speed up the computation, the ACA algorithm is applied. Several numerical examples are performed. The accuracy and computational efficiency of the proposed technique are validated by simulation results.

A Hybrid Method of Adaptive Cross Approximation Algorithm and Chebyshev Approximation Technique for Fast Broadband BCS Prediction Applicable to Passive Radar Detection

A hybrid method combining the adaptive cross approximation method (ACA) and the Chebyshev approximation technique (CAT) is presented for fast wideband BCS prediction of arbitrary-shaped 3D targets based on non-cooperative radiation sources. The incident and scattering angles can be computed by using their longitudes, latitudes and altitudes according to the relative positions of the satellite, the target and the passive bistatic radar. The ACA technique can be employed to reduce the memory requirement and computation time by compressing the low-rank matrix blocks. By exploiting the CAT into ACA, it is only required to calculate the currents at several Chebyshev–Gauss frequency sampling points instead of direct point-by-point simulations. Moreover, a wider frequency band can be obtained by using the Maehly approximation. Three numerical examples are presented to validate the accuracy and efficiency of the hybrid ACA-CAT method.

An ACA-BM-SBM for 2D acoustic sensitivity analysis

<abstract> <p>In this paper, we present a novel computational approach (named ACA-BM-SBM) for the calculation of 2D acoustic sensitivity by combining the Burton-Miller-type singular boundary method (BM-SBM) with the adaptive cross-approximation (ACA) algorithm. The BM-SBM circumvents the source singularities of the fundamental solutions by introducing the origin intensity factors, and it eliminates the fictitious frequency problem in external acoustic fields by introducing the Burton-Miller formula. As a semi-analysis meshless method, the BM-SBM can accurately solve the external acoustic problem governed by the Helmholtz equation. Nevertheless, the computational inefficiency introduced by the dense coefficient matrix renders this method suboptimal, particularly for large-scale simulations. As the number of nodes increases, the computation time and store memory increase dramatically. ACA is a purely algebraic method based on hierarchical matrices which can be used to partition the coefficient matrix step by step. By employing ACA, the BM-SBM can be effectively accelerated, and this results in less computation time, as well as fewer memory requirements. Numerical experiments, including Dirichlet and Neumann boundary conditions, illustrate that the proposed approach is an accurate, efficient and fast numerical method for acoustic sensitivity analysis.</p> </abstract>

Optimization of block size for volume integral equation‐based characteristic basis function method

AbstractThe block size of the characteristic basis function method (CBFM) based on volume integral equation (VIE) is optimized to improve the efficiency on analyzing the scattering from dielectric objects in this study. Several literatures have discussed the optimum block size of surface integral equation (SIE)‐based CBFM, which achieved the minimum CPU time for analyzing conductor objects. However, due to the differences of basis functions, the conclusions derived from SIE‐based CBFM are not applicable for VIE‐based CBFM. This study presents the relationship between the Schaubert–Wilton–Glisson (SWG) basis functions and characteristic basis functions (CBFs), which paves the way for evaluating the computational complexity of VIE‐based CBFM. In addition, the Sherman–Morrison–Woodbury formula‐based algorithm (SMWA) and adaptive cross approximation (ACA) algorithm are combined to accelerate the CBFs generation and reduced matrix construction. Based on the computational complexity of the VIE‐based CBFM combined with SMWA and ACA, the optimal number of blocks for VIE‐based CBFM is derived theoretically. Numerical results of homogeneous and inhomogeneous dielectric objects scattering problems are presented to validate the feasibility and accuracy of the proposed optimization method.

Efficient Characteristic Mode Analysis for PEC Objects Using Sherman–Morrison–Woodbury Formula-Based Algorithm

In characteristic mode analysis (CMA), solving the characteristic modes (CMs) from the generalized eigenvalue problem takes the main part of the workload. In this letter, the algorithm based on the Sherman–Morrison–Woodbury (SMW) formula is embedded into the implicitly restarted Arnoldi method to improve the efficiency of solving the CMs. In this method, the impedance matrix is compressed into a product of block-diagonal matrices by using the SMW formula with adaptive cross approximation algorithm. Benefiting from the efficient compression, calculating the inverse matrix of the impedance matrix and the matrix vector products related to the solution of the generalized eigenvalue problem is accelerated. Consequently, the computational time and memory requirement in CMA are both reduced. Some numerical examples are demonstrated to show the accuracy and efficiency of the proposed method.

PMCHWT Solver Accelerated by Adaptive Cross Approximation for Efficient Computation of Scattering from Metal Nanoparticles.

An accelerated algorithm that can efficiently calculate the light scattering of a single metal nanoparticle was proposed. According to the equivalent principle, the method of moment (MoM) transforms the Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) integral equations into linear algebraic equations, which are solved by the flexible generalized minimal residual solver (FGMRES). Each element of near field MoM impedance matrix was described by Rao–Wilton–Glisson (RWG) basis functions and calculated by double surface integrals. Due to the low-rank property, the adaptive cross approximation (ACA) algorithm based on the octree data structure was applied to compress the MoM impedance matrix of far field action leading to the significant reduction of solution time and memory. Numerical results demonstrated that the proposed method is both accurate and efficient. Compared with the traditional MoM, the ACA algorithm can significantly reduce the impedance matrix filling time and accelerate the scattering field’s computation from actual metal nanoparticles using PMCHWT integral equations.

Partially pivoted ACA based acceleration of the energetic BEM for time-domain acoustic and elastic waves exterior problems

We consider acoustic and elastic wave propagation problems in 2D unbounded domains, reformulated in terms of space-time Boundary Integral Equations (BIEs). For their numerical solution, we employ a weak formulation related to the energy of the system and we discretize the weak problems by a Galerkin-type Boundary Element Method (BEM): this approach, called Energetic BEM, has revealed accurate and stable even on large time intervals of analysis. In particular, it results that, when standard Lagrangian basis functions are considered, the BEM matrices have Toeplitz lower triangular block structure, where blocks for growing time become fully populated; hence the overall memory cost of the energetic BEM is O(M2N), M and N being the number of the space degrees of freedom and the total number of performed time steps, respectively. This drawback prevents the application of such method to large scale problems. As a possible remedy, we propose a fast technique based on the Adaptive Cross Approximation (ACA). The core of this procedure is the approximation of sufficiently large time blocks of the energetic BEM matrix through the partially pivoted ACA algorithm, which allows to compute only few of the original entries. This leads to reduced assembly time, which for the energetic BEM is generally relevant, coupled with reduced memory storage requirements. Additionally, the consequent acceleration of the matrix/vector multiplication together with a marching on time procedure, leads to remarkable reduction of the computational solution time. The effectiveness of the proposed method is theoretically proved and several numerical results are presented and discussed.

Fast analysis of narrow-band transient electromagnetic scattering using the ACA enhanced envelope-tracking marching-on-in-time solver

In this paper, an efficient adaptive cross approximation (ACA) algorithm enhanced envelope-tracking marching-on-in-time (ET-MOT) solver is proposed to analyze the narrow-band transient electromagnetic scattering from three-dimensional perfect electrically conducting (PEC) objects. In the ET-MOT method, the time step size is inversely proportional to the bandwidth of the incident signal. However, the size of spatial discretization is still inversely proportional to the maximum frequency component of the incident wave. Therefore, the transmission distance of electromagnetic wave in one-time step is usually much larger than the size of spatial discretization. This results in the immediate interaction matrix Z0 being a dense matrix or even a full rank matrix for narrow-band problems. The ACA algorithm is used to accelerate the matrix-vector multiplication during the iteration solution at each time marching step. A series of numerical examples show the accuracy and efficiency of the proposed method.

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.

Boundary element methods for the wave equation based on hierarchical matrices and adaptive cross approximation

Time-domain Boundary Element Methods (BEM) have been successfully used in acoustics, optics and elastodynamics to solve transient problems numerically. However, the storage requirements are immense, since the fully populated system matrices have to be computed for a large number of time steps or frequencies. In this article, we propose a new approximation scheme for the Convolution Quadrature Method powered BEM, which we apply to scattering problems governed by the wave equation. We use {mathscr {H}}^2-matrix compression in the spatial domain and employ an adaptive cross approximation algorithm in the frequency domain. In this way, the storage and computational costs are reduced significantly, while the accuracy of the method is preserved.

Hybrid Method of ACA and Interpolation-based Algorithms for Analysis of Eddy Current Nondestructive Evaluations

ABSTRACT In this article, the hybridization of adaptive cross approximation (ACA) algorithm and interpolation-based separation of the kernel function is proposed to accelerate solving the matrix equations resulted in the boundary element method (BEM) for 3D arbitrary-shaped eddy current nondestructive evaluation problems. The hybrid method combines the advantages of both the ACA algorithm and the interpolation-based methods, and resolves the shortcoming of pure ACA method, when modeling the planar eddy current nondestructive evaluation problems, that it cannot compress the null entries the BEM generated when the testing and basis patches are co-planar. In the proposed method, the submatrices associated with the null entries are compressed by the interpolation-based method, while the others are compressed by the ACA algorithm. Several benchmarks are shown to demonstrate both the robustness and efficiency of the proposed fast and general solver.

Averaged Adaptive Cross Approximation for Structured Matrices

A modified form of the adaptive cross approximation (ACA) is considered for certain structured matrices for which the original ACA fails to provide controllably accurate matrix representations of low-rank submatrices. The modified ACA algorithm considered here is based on forming an averaged matrix, obtained via left- and right- multiplication of the original matrix block by sparse matrices that have LU factorizations. The performance of the modified ACA algorithm is examined for several sub-matrices for which the original ACA fails to provide controllably accurate representations, and significantly improved error control is observed.

Improving the Accuracy of the Adaptive Cross Approximation With a Convergence Criterion Based on Random Sampling

The accuracy of the adaptive cross approximation (ACA) algorithm, a popular method for the compression of low-rank matrix blocks in method of moment computations, is sometimes seriously compromised by unpredictable errors in the convergence criterion. This article proposes an alternative criterion based on global sampling of the error in the elements of the ACA compressed matrix. The sampling error depends not only on the size of the sample but also on the population distribution of the error, which makes it difficult to control the error independently of the underlying problem. However, as argued and demonstrated in this article, the distribution of the error converges to the same unique probability distribution function for all low-rank matrices. Complementing the sampling criterion with a simple mechanism to detect this convergence, we arrive at a criterion that controls the error irrespective of the underlying problem. As a practical example, the RCS of a moderate size metallic ogive is computed to illustrate the merits of the proposed criterion. The proposed algorithm may also be useful in other methods that approximate low-rank matrices by interpolation of a reduced set of its elements.

Nested Fast Adaptive Cross Approximation Algorithm for Solving Electromagnetic Scattering Problems

In this article, a nested fast adaptive cross approximation (NFACA) algorithm is presented to accelerate the solution of electromagnetic scattering problems. It is based on the fast adaptive cross approximation (FACA) and the nested cross approximation (NCA). In the NFACA, the FACA is applied instead of the ACA or the Chebyshev nodes, to enhance the efficiency. It is shown to have O(N) storage and computational complexity for electrically small problems. Different from the conventional NCA or the fast NCA (FNCA), the pivot selection does not rely on a preliminary selection with Chebyshev nodes or ACA for each group. The NFACA permits the selection of the optimum pivots independently at each level to make the accuracy controllable. Numerical results of several objects are presented to demonstrate the performance of this method. Compared with the FACA and NCA in solving the electromagnetic scattering problems, the NFACA is more efficient both in computational cost and storage. Compared with the FNCA, the NFACA is both faster and more accurate.

Skin effect and proximity effect analysis of stranded conductor based on mixed order MoM with adaptive cross approximation algorithm

In this paper, an efficient adaptive cross approximation (ACA) method is employed for the skin effect and proximity effect analysis. Multi-layer stranded cables with multiple strands and different materials are used for analysis to approximate the effect. The skin and proximity matrices are computed by the ACA low-rank matrix approximation technique. The ACA method can provide better solutions through different iterations. The ACA algorithm is very stable and useful for calculating a low rank matrix approximation, because the adaptive mesh refinement for generating optimal computation. The ACA approach provides the less computational memory and less CPU time. Compare the calculated results with standard work results (such as reference) to verify the effectiveness of the adaptive cross-approximation algorithm.

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'.

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

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 (H-) matrix is employed to construct the direct solution framework. The FEBI system matrix is first represented to be an H-matrix form, in which the FE submatrices are accurately represented by H-matrices, while the 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 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.