Adaptive Cross Approximation 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.

A Low-Rank Matrix Approach to Compute Polynomial Approximations of Smooth Two-Dimensional Functions

Polynomial approximation of smooth functions is becoming increasingly important in fields like numerical analysis and scientific computing. These approximations are vital in models that rely on spectral methods. To reduce the memory costs for large dimensional problems, various methods to provide data-sparse representations have been proposed, including methods based on singular value decomposition, adaptive cross approximation, and matrices with hierarchical low-rank structures, to mention a few. This work presents implementation details on the polynomial approximation of univariate smooth functions through the Polynomial1 class, and of bivariate smooth functions by low-rank matrix representation via the Polynomial2 class. These approaches are explained within Tau Toolbox, a mathematical software library for solving integro-differential problems by the spectral Tau method.

Use of the Adaptive Cross Approximation for the Efficient Computation of the Reduced Matrix with the Characteristic Basis Function Method

A technique for the reduction in the CPU-time in the analysis of electromagnetic problems using the Characteristic Basis Function Method (CBFM) is presented here, allowing for analysis of electrically large cases where an iterative solution process cannot be avoided. This technique is based on the use of the Adaptive Cross Approximation (ACA) for the fast computation of the coupling matrix between CBFs belonging to adjacent blocks, as well as the Multilevel Fast Multipole Method (MLFMM) for the computation of matrix−vector products in the solution of the full system. This combination allows for a noticeable reduction in the computational resources during the analysis of electrically large and complex scenarios while maintaining a very good degree of accuracy. A number of test cases serve to validate the presented approach in terms of accuracy, memory and CPU-time compared with conventional techniques.

A Novel Parallel Direct Solver Using Adaptive Cross Approximation for Analysis of Electromagnetic Radiation Problems With Complex Structures

A Novel Parallel Direct Solver Using Adaptive Cross Approximation for Analysis of Electromagnetic Radiation Problems With Complex Structures

Adaptive [formula omitted]-matrix computations in linear elasticity

This article deals with the efficient numerical treatment of the Lamé equations. The equations of linear elasticity are considered as boundary integral equations and solved in the setting of the boundary element method (BEM). Using BEM, one is faced with the solution of a system of equations with a fully populated system matrix, which is in general very costly. In order to overcome this difficulty, adaptive and approximate algorithms based on hierarchical matrices and the adaptive cross approximation are proposed. These new methods rely on error estimators and refinement techniques known from adaptivity but are not used here to improve the mesh. We apply these new techniques to both, the efficient solution of Lamé equations and to the multiplication with given data.

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

Fast Construction of Measurement Matrix and Sensing Matrix with Dual Adaptive Cross Approximation

Fast Construction of Measurement Matrix and Sensing Matrix with Dual Adaptive Cross Approximation

Adaptive Cross Approximation Accelerates Compressive Sensing-based Method of Moments for Solving Electromagnetic Scattering Problems

Adaptive Cross Approximation Accelerates Compressive Sensing-based Method of Moments for Solving Electromagnetic Scattering Problems

On the Adaptive Cross Approximation for the Magnetic Field Integral Equation

On the Adaptive Cross Approximation for the Magnetic Field Integral Equation

Reflective conditions for radiative transfer in integral form with H-matrices

In a recent article the authors showed that the radiative Transfer equations with multiple frequencies and scattering can be formulated as a nonlinear integral system. In the present article, the formulation is extended to handle reflective boundary conditions. The fixed point method to solve the system is shown to be monotone. The discretization is done with a P1 Finite Element Method. The convolution integrals are precomputed at every vertex of the mesh and stored in compressed hierarchical matrices, using Partially Pivoted Adaptive Cross-Approximation. Then the fixed point iterations involve only matrix vector products. The method is O(NN3ln⁡N), with respect to the number of vertices, when everything is smooth. A numerical implementation is proposed and tested on two examples. As there are some analogies with ray tracing the programming is complex.

A Weighted Average Adaptive Cross Approximation

A Weighted Average Adaptive Cross Approximation

Realizations of the generalized adaptive cross approximation in an acoustic time domain boundary element method

AbstractIn acoustics, the boundary element method (BEM) is much more common compared to elasticity. This is driven by the applications, which are in acoustics very often radiation problems causing trouble in finite element or finite volume methods. Nevertheless, for a time domain calculation, still an efficient BE formulations is lacking. We consider the time domain BEM for the homogeneous wave equation with vanishing initial conditions and given Dirichlet boundary conditions. The generalized convolution quadrature method (gCQ) developed by Lopez‐Fernandez and Sauter is used for the temporal discretisation. The spatial discretisation is done classically. Essentially, the gCQ requires to establish boundary element matrices of the corresponding elliptic problem in Laplace domain at several complex frequencies. Consequently, an array of system matrices is obtained. This array of system matrices can be interpreted as a three‐dimensional array of data, which should be approximated by a data‐sparse representation. The adaptive cross approximation (ACA) can be generalized to handle these three‐dimensional data arrays. Adaptively, the rank of the three‐dimensional data array is increased until a prescribed accuracy is obtained. On a pure algebraic level, it is decided whether a low‐rank approximation of the three‐dimensional data array is close enough to the original matrix. Hierarchical matrices are used in the two spatial dimensions and the third dimension are the complex frequencies. Hence, the algorithm makes not only a data sparse approximation in the two spatial dimensions but detects adaptively how much frequencies are necessary for which matrix block.

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.

Modelling of 3D periodic cathodic protection problems in reinforced concrete structures with accelerated boundary element method

The steel used for concrete reinforcement exhibits either active or passive electrochemical behavior. As long as the concrete environment remains highly alkaline, the steel remains passive. However, carbonation and/or chloride contamination of concrete will cause steel to become active and corrode. In the latter case, a common strategy is the use of cathodic protection (CP) systems to lower the potential of the steel at a protective level. The numerical modelling of cathodically protected concrete buildings and infrastructures usually requires large-scale models due to their size and complex geometry. An efficient modelling approach is to consider any possible periodicity in these problems by applying periodic boundary conditions to a representative volume element of the structure. The boundary element method (BEM) is extensively used for solving CP problems. A BEM formulation is proposed for solving 3D periodic CP problems in this work. The computation cost is reduced by applying acceleration techniques based on adaptive cross approximation (ACA). As an engineering application, a sacrificial anode CP system of a reinforced concrete column is designed, illustrating the importance of detailed modelling in the design of such systems. To this end, the macrocell corrosion problem is initially solved, and a parametric study with respect to concrete conductivity is carried out.

Wideband Sherman–Morrison–Woodbury Formula-Based Algorithm for Electromagnetic Scattering Problems

In this paper, a wideband Sherman-Morrison-Woodbury formula-based algorithm (WSMWA) is proposed to efficiently compute the wideband and wide-angle electromagnetic scattering problems. In the proposed algorithm, the standard adaptive cross approximation (ACA) decomposition is only performed at the highest frequency of the frequency band of interest to find the dominant basis functions for each far-block pair. Then, at any frequency within the entire frequency band, the approximate compression of the impedance matrix can be efficiently constructed by using the impedance interpolation method with the dominant basis functions selected at the highest frequency. As a result, the WSMWA avoids performing the standard ACA repeatedly and saves a lot of computational time in comparison with the conventional Sherman-Morrison-Woodbury formula-based algorithm (SMWA) for wideband and wide-angle applications. Numerical results for electromagnetic scattering are given to demonstrate the efficiency and accuracy of the proposed algorithm.

ACA-accelerated hybrid boundary node method applied to multi-domain steady heat conduction problems

This paper presents a fast implementation of the hybrid boundary node method (HBNM) to solve the multi-domain steady heat conduction problems using the Adaptive Cross Approximation (ACA). It is well known that the matrices arising from the discretization of boundary integral equation are fully populated and require special compression techniques for the efficient treatment. The ACA using only few of the original entries for the approximation of the whole matrix is a purely algebraic algorithm which is kernel-independent and easy to be applied directly for various problems. Accelerated by the ACA, several multi-domain steady heat conduction problems have been solved by the HBNM. The results demonstrate the potentials of the ACA-HBNM for solving multi-domain steady heat conduction problems.

Low-rank compression techniques in integral methods for eddy currents problems

Volume integral methods for the solution of eddy current problems are very appealing in practice since only the conducting regions need to be meshed. However, they require the assembly and storage of a dense stiffness matrix. With the objective of cutting down assembly time and memory occupation, low-rank approximation techniques like the Adaptive Cross Approximation (ACA) have been considered a major breakthrough. Recently, the VINCO framework has been introduced to reduce significantly memory occupation and computational time thanks to a novel factorization of the dense stiffness matrix. The aim of this paper is introducing a new matrix compression technique enabled by the VINCO framework. We compare the performance of VINCO framework approaches with state-of-the-art alternatives in terms of memory occupation, computational time and accuracy by solving benchmark eddy current problems at increasing mesh sizes; the comparisons are carried out using both direct and iterative solvers. The results clearly indicate that the so-called VINCO-FAIME approach which exploits the Fast Multipole Method (FMM) has the best performance.

A Hybrid CM-BEM Formulation for Solving Large-Scale 3D Eddy-Current Problems Based on ℋ-Matrices and Randomized Singular Value Decomposition for BEM Matrix Compression

We present a novel a,v-q hybrid method for solving large-scale time-harmonic eddy-current problems. This method combines a hybrid unsymmetric formulation based on the cell method and the boundary element method with a hierarchical matrix-compression technique based on randomized singular value decomposition. The main advantage is that the memory requirements are strongly reduced compared to the corresponding hybrid method without matrix compression while retaining the same robust solution strategy consisting of a simple construction of the preconditioner. In addition, the matrix-compression accuracy and efficiency are enhanced compared to traditional compression methods, such as adaptive cross approximation. The numerical results show that the proposed hybrid approach can also be effectively used to analyze large-scale eddy-current problems of engineering interest.

Numerical simulation of fluid flow with a fast Boundary‐Domain Integral Method

AbstractFor the simulation of fluid flow the velocity‐vorticity formulation of the Navier‐Stokes equations can be used. Approximating the first order time derivative in these equations by finite differences, essentially, the Yukawa equation for the velocity has to be solved with an inhomogeneity governed by the vorticity. Applying the Boundary Element Method to such problems is not straightforward. However, the Boundary‐Domain Integral Method (BDIM) can be used, which requires to solve beside the boundary integrals as well a domain integral due to the inhomogeneity. Classical BEM formulations have a quadratic complexity with respect to the unknowns on the boundary and, for the BDIM additionally, for the unknowns in the domain. This prevents the BDIM from being applied to real‐world problems.To reduce the quadratic complexity in BEM, approximation methods like the fast multipole method, ℋ‐matrices in combination with the adaptive cross approximation or the ℋ2 ‐matrix technique can be used. All these methods reduce the complexity to an almost linear order. Here, the ℋ2 ‐matrix technique is applied to reduce storage and computing time for the right hand side, i.e., for the matrix of the domain integrals. Based on interpolatory kernel expansions and an heuristic criterium for too small matrix entries an almost linear complexity can be achieved. The numerical studies focus on a lid‐driven cavity. The complexity reduction is demonstrated.

A Recompressed Nested Cross Approximation for Electrically Large Bodies

A recompressed nested cross approximation (rNCA) based closely on the recent fast nested cross approximation (fNCA) algorithm is formulated in this article. The proposed method builds on previous work in which the fNCA was formulated in a purely algebraic and kernel-independent fashion, using a top-down recursive application of the adaptive cross-approximation (ACA). Our proposed method employs ACA recompression to avoid the need to compute low-rank approximations of excessively large far-field matrices, and thus mitigates the effects of high-frequency rank growth on run-time scaling for electrically large models. The low run-time and memory cost allows for efficient parallel computation of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}^{2}$ </tex-math></inline-formula> -matrices for systems of excessive electrical sizes. Radar cross sections (RCSs) are evaluated for electrically large instances of a perfectly conducting sphere and the NASA Almond. We observe near-linear scaling of memory cost and construction time.