Fast Integral Equation Research Articles

Fast High-Order Integral Equation Methods for Solving Boundary Value Problems of Two Dimensional Heat Equation in Complex Geometry

Efficient high-order integral equation methods have been developed for solving boundary value problems of the heat equation in complex geometry in two dimensions. First, the classical heat potential theory is applied to convert such problems to Volterra integral equations of the second kind via the heat layer potentials, where the unknowns are only on the space–time boundary. However, the heat layer potentials contain convolution integrals in both space and time whose direct evaluation requires $$O(N_S^2N_T^2)$$ work and $$O(N_SN_T)$$ storage, where $$N_S$$ is the total number of discretization points on the spatial boundary and $$N_T$$ is the total number of time steps. In order to evaluate the heat layer potentials accurately and efficiently, they are split into two parts—the local part containing the temporal integration from $$t-\delta $$ to t and the history part containing the temporal integration from 0 to $$t-\delta $$ . The local part can be dealt with efficiently using conventional fast multipole type algorithms. For problems with complex stationary geometry, efficient separated sum-of-exponentials approximations are constructed for the heat kernel and used for the evaluation of the history part. Here all local and history kernels are compressed only once. The resulting algorithm is very efficient with quasilinear complexity in both space and time for both interior and exterior problems. For problems with complex moving geometry, the spectral Fourier approximation is applied for the heat kernel and nonuniform FFT is used to speed up the evaluation of the history part of heat layer potentials. The performance of both algorithms is demonstrated with several numerical examples.

Improved convergence of fast integral equation solvers for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

In recent years, several fast solvers for the solution of the Lippmann–Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by penetrable inhomogeneous obstacles, have been proposed. While many of these fast methodologies exhibit rapid convergence for smoothly varying scattering configurations, the rate for most of them reduce to either linear or quadratic when material properties are allowed to jump across the interface. A notable exception to this is a recently introduced Nyström scheme (Anand et al., 2016 [22]) that utilizes a specialized quadrature in the boundary region for a high-order treatment of the material interface. In this text, we present a solution framework that relies on the specialized boundary integrator to enhance the convergence rate of other fast, low order methodologies without adding to their computational complexity of O(Nlog⁡N) for an N-point discretization. In particular, to demonstrate the efficacy of the proposed framework, we explain its implementation to enhance the order to convergence of two schemes, one introduced by Duan and Rokhlin (2009) [13] that is based on a pre-corrected trapezoidal rule while the other by Bruno and Hyde (2004) [12] which relies on a suitable decomposition of the Green's function via Addition theorem. In addition to a detailed description of these methodologies, we also present a comparative performance study of the improved versions of these two and the Nyström solver in Anand et al. (2016) [22] through a wide range of numerical experiments.

A Fast IE-FFT Algorithm to Analyze Electrically Large Planar Microstrip Antenna Arrays

A fast integral-equation fast Fourier transform (IE-FFT) algorithm is developed, analyzed, and applied to the electromagnetic solution of radiation problems. The solution begins by the method-of-moments solution of a mixed potential integral equation on electrically large antenna arrays. Similar to other grid-based algorithms, the IE-FFT uses a Cartesian grid to significantly reduce memory requirement and speed up CPU time for both matrix assembly and matrix-vector multiplication when used with an iterative solver. The IE-FFT algorithm employs two discretizations, one for unknown current on an unstructured triangular mesh and the other on a uniform Cartesian grid for interpolating the layered medium's Green's functions. The uniform interpolation of the Green's functions allows us a global FFT for well-separated far-interaction terms. However, the near-interaction terms should be effectively corrected. A simple and efficient Lagrangian interpolation of the Green's functions is proposed. For a 2.5-D planar layered structure problem, the complexity of an IE-FFT is found to approximately $O(N)$ and $O({N\,{\rm{log}}\,N})$ for memory and CPU time, respectively.

The ultimate intrinsic signal‐to‐noise ratio of loop‐ and dipole‐like current patterns in a realistic human head model

The ultimate intrinsic signal-to-noise ratio (UISNR) represents an upper bound for the achievable SNR of any receive coil. To reach this threshold a complete basis set of equivalent surface currents is required. This study systematically investigated to what extent either loop- or dipole-like current patterns are able to reach the UISNR threshold in a realistic human head model between 1.5 T and 11.7 T. Based on this analysis, we derived guidelines for coil designers to choose the best array element at a given field strength. Moreover, we present ideal current patterns yielding the UISNR in a realistic body model. We distributed generic current patterns on a cylindrical and helmet-shaped surface around a realistic human head model. We excited electromagnetic fields in the human head by using eigenfunctions of the spherical and cylindrical Helmholtz operator. The electromagnetic field problem was solved by a fast volume integral equation solver. At 7 T and above, adding curl-free current patterns to divergence-free current patterns substantially increased the SNR in the human head (locally >20%). This was true for the helmet-shaped and the cylindrical surface. On the cylindrical surface, dipole-like current patterns had high SNR performance in central regions at ultra-high field strength. The UISNR increased superlinearly with B0 in most parts of the cerebrum but only sublinearly in the periphery of the human head. The combination of loop and dipole elements could enhance the SNR performance in the human head at ultra-high field strength.

A fast boundary integral equation method for point location problem

A numerical method based on the boundary integral equation is proposed for the point location problem. For a bounded domain, the integral value is close to 1.0 if a point is inside the domain, and is close to 0.0 when the point is outside the domain. For convenience of integration, the boundary of the domain can be discretized into boundary integral cells. The idea of isogeometric analysis can be easily coupled with the proposed method, i.e., using the parametric functions in geometric modeling to create the integral cells, which results in a mesh-free procedure for which the geometry can be exactly produced at all stages. Thus, the method can be applied to arbitrary shapes and easily embedded in computer-aided design (CAD) packages. The method is time-consuming if implemented directly; a fast multipole method is coupled with the proposed method to accelerate the integral procedure. Some examples of 2D and 3D cases are tested to show the accuracy and efficiency of the proposed method.

An Efficient Integral-Based Method for Three-Dimensional MR-EPT and the Calculation of the RF-Coil-Induced ${B_z}$ Field.

Magnetic resonance electrical property tomography (MR-EPT) has significant potential for the estimation of the electrical properties (EPs) of tissue, which are essential for the calculation of specific absorption rates (SAR), a critical safety factor requiring monitoring and controlling in applications of ultrahigh field magnetic resonance imaging. In this paper, a novel, efficient method based on integral equations is proposed for the calculation of the EPs and the RF-coil-induced field. An inverse problem framework is first constructed to include the forward problem operator, while the EPs are reconstructed by using a nonlinear conjugate gradient method. The RF-coil-induced component is then calculated based on the achieved EPs and the forward operator. The proposed MR-EPT algorithm improves upon and differs from the existing methods in three aspects. First, a three-dimensional algorithm with improved efficiency is proposed. The higher efficiency arises from using a fast integral equation solver as well as an approximation of initial solution. Second, in addition to the EP values, the proposed method calculates the RF-coil-induced component, which is usually neglected in the existing MR-EPT algorithms. Here, we show that considering this field can significantly improve the accuracy of the SAR calculation. Finally, in contrast to differential approaches, the proposed method is more robust against noisy measurement of the transmit magnetic fields, because of the nature of the integral equations. The proposed method is verified through a full-wave simulation and an anatomically accurate numerical brain model, demonstrating its accuracy and efficiency.

Characterization of Near-Field Coupling Effects From Complicated Three-Dimensional Structures in Rectangular Cavities Using Fast Integral Equation Method

This paper characterizes the frequency domain near-field coupling effects from complicated three-dimensional transmission lines, monopole/dipole radiators, and printed circuit boards (PCBs) located in a metallic rectangular cavity with an efficient iterative fast Fourier transform-accelerated integral-equation method. The simulation method combines the rectangular-cavity Green's functions with the frequency-domain electric field integral equation, which is further accelerated by the extension of adaptive integral method for rectangular cavities. By using this method, the near-field coupling effects and internal field distributions are captured and compared with independent references for various structures, including transmission lines, monopole/dipole radiators, and PCBs in rectangular cavities. Numerical results show that strong couplings do appear at multiple resonant frequencies of the structures. In addition, this method can be further employed to mitigate and suppress near-field coupling effects in the optimization of electromagnetic compatibility/interference designs.

Imaging of locally rough surfaces from intensity-only far-field or near-field data

This paper is concerned with a nonlinear imaging problem, which aims to reconstruct a locally perturbed, perfectly reflecting, infinite plane from intensity-only (or phaseless) far-field or near-field data. A recursive Newton iteration algorithm in frequencies is developed to reconstruct the locally rough surface from multi-frequency intensity-only far-field or near-field data, where the fast integral equation solver developed by Zhang and Zhang (2013 SIAM J. Appl. Math. 73 1811–29) is used to solve the direct scattering problem in each iteration. For the case with far-field data, a main feature of our work is that the incident field is taken as a superposition of two plane waves with different directions rather than one plane wave, so the location and shape of the local perturbation of the infinite plane can be reconstructed simultaneously from intensity-only far-field data with multiple wave numbers. This is different from previous work on inverse scattering from phaseless far-field data, where only the shape reconstruction was considered due to the translation invariance property of the phaseless far-field pattern corresponding to one plane wave as the incident field. Finally, numerical examples are presented to demonstrate that our reconstruction algorithm is stable and accurate even for the case of multiple-scale profiles.

An HSS Matrix-Inspired Butterfly-Based Direct Solver for Analyzing Scattering From Two-Dimensional Objects

A butterfly-based fast direct integral equation solver for analyzing high-frequency scattering from two-dimensional objects is presented. The solver leverages a randomized butterfly scheme to compress blocks corresponding to near- and far-field interactions in the discretized forward and inverse electric field integral operators. The observed memory requirements and computational cost of the proposed solver scale as O(Nlog^2N) and O(N^1.5 logN), respectively. The solver is applied to the analysis of scattering from electrically large objects spanning over ten thousand of wavelengths and modeled in terms of five million unknowns.

Induced Magnetic Field Computation for Ships Based on Fast Multipole Method and Integral Equation Method

The nonsymmetric and dense coefficient matrix of integral equation method results in low computation efficiency and large memory cost in induced magnetic field computation of large scale ships. In order to improve the computation efficiency, a new method called FMMIEM is developed based on fast multipole method and integral equation method. The basic formulas of the FMMIEM were deduced. The numerical implementations of the multipole expansion, multipole to local translation and local expansion were described in detail. The FMMIEM not only has the virtue of the vector integral equation method which needs only meshing source area and is easy to deal with the infinite domain problems, such as the induced magnetic field modeling, but also has the excellent performance of the fast multipole method in conjunction with iterative solver which can accelerate the matrix vector product. The analytical numerical experiment of the sphere magnetized in the uniform applied magnetic field and the magnetostatic field modeling of the main cabin of the simple submarine mockup were designed and the results showed that the FMMIEM is with the high calculation accuracy and reduces costing time obviously. The proposed method is very helpful in engineering.

A fast integral equation method for solid particles in viscous flow using quadrature by expansion

Boundary integral methods are advantageous when simulating viscous flow around rigid particles, due to the reduction in number of unknowns and straightforward handling of the geometry. In this work we present a fast and accurate framework for simulating spheroids in periodic Stokes flow, which is based on the completed double layer boundary integral formulation. The framework implements a new method known as quadrature by expansion (QBX), which uses surrogate local expansions of the layer potential to evaluate it to very high accuracy both on and off the particle surfaces. This quadrature method is accelerated through a newly developed precomputation scheme. The long range interactions are computed using the spectral Ewald (SE) fast summation method, which after integration with QBX allows the resulting system to be solved in Mlog⁡M time, where M is the number of particles. This framework is suitable for simulations of large particle systems, and can be used for studying e.g. porous media models.

Fast Coiflet magnetic field integral equation for scattering from open rough surfaces

The magnetic field integral equation (MFIE) is solved by Coiflets for rough surface scattering. The vanishing moments of Coiflets provide one-point quadrature, which slashes the matrix filling effort from O ( N 2 ) to O ( N ), and consequently reduces the complexity scaling in between O ( N ) and O ( N log N ) for problems with unknowns up to 10 6 . The bi-conjugate solver converges very fast owing to the well-posedness of the MFIE. The resulting impedance matrix is further sparsified by the matrix-formed standard fast wavelet transform. By properly selecting multiresolution levels of the total transformation matrix, the solution precision can be enhanced while matrix sparsity and memory consumption have not been noticeably sacrificed. Numerical results are compared with the Rao-Wilton-Glisson multilevel fast multipole algorithm (RWG-MLFMA) based commercial software FEKO, and good agreement is observed.

An efficient and high‐order frequency‐domain approach for transient acoustic–structure interactions in three dimensions

This paper is concerned with numerical solution of the transient acoustic–structure interaction problems in three dimensions. An efficient and higher-order method is proposed with a combination of the exponential window technique and a fast and accurate boundary integral equation solver in the frequency-domain. The exponential window applied to the acoustic–structure system yields an artificial damping to the system, which eliminates the wrap-around errors brought by the discrete Fourier transform. The frequency-domain boundary integral equation approach relies on accurate evaluations of relevant singular integrals and fast computation of nonsingular integrals via the method of equivalent source representations and the fast Fourier transform. Numerical studies are presented to demonstrate the accuracy and efficiency of the method. Copyright © 2016 John Wiley & Sons, Ltd.

The Mixed-Order BCGS-FFT Method for the Scattering of Three-Dimensional Inhomogeneous Anisotropic Magnetodielectric Objects

This paper presents the first FFT-based fast volume integral equation solver for inhomogeneous anisotropic magnetodielectric objects. The volume integral equations are formulated by employing mixed-order basis functions that expand the flux densities and vector potentials in the coupled field integral equations in terms of different sets of basis functions with different orders. Volumetric roof-top basis functions are used for flux densities whereas second-order curl conforming basis functions are used for vector potentials. A fast volume integral equations solver namely the BCGS-FFT method is then applied to accelerate the solution of this mixed-order weak-form formulation. Examples show that the mixed-order BCGS-FFT method has high accuracy compared to both analytical and numerical solutions. Examples also show the mixed-order BCGS-FFT method has high computational efficiency compared to commercial software.

Approach for a fast computation of induced magnetic field created by naval vessels

Integral equation method is widely used in magnetic field modeling of ships, but its nonsymmetric and dense co- efficient matrix results in low computation efficiency and large memory, which limits its application in large scale problems. An approach for a fast computation of induced magnetic field created by ships was presented based on multiple layers fast multipole method and vector integral equation method. The numerical implementation of the proposed approach was described in detail. It is shown by an analytical sphere example and a simplified mock-up experiment that the proposed approach can reduce computation time and memory cost in induced magnetic field modeling of ferromagnetic objects.

Stochastic Estimation of the Frobenius Norm in the ACA Convergence Criterion

The adaptive cross approximation (ACA) algorithm has been used in many fast Integral Equation solvers for electromagnetic Radiation and Scattering problems. It efficiently computes a low rank approximation to the interaction matrix between mutually distant parts of a scattering object. The ACA is an iterative algorithm that needs an accurate and efficient convergence criterion. The evaluation of this criterion may consume a considerable part of the computational resources. This communication presents an efficient new way to evaluate the convergence criterion, using a stochastic approach.

Robust fast direct integral equation solver for quasi-periodic scattering problems with a large number of layers.

We present a new boundary integral formulation for time-harmonic wave diffraction from two-dimensional structures with many layers of arbitrary periodic shape, such as multilayer dielectric gratings in TM polarization. Our scheme is robust at all scattering parameters, unlike the conventional quasi-periodic Green's function method which fails whenever any of the layers approaches a Wood anomaly. We achieve this by a decomposition into near- and far-field contributions. The former uses the free-space Green's function in a second-kind integral equation on one period of the material interfaces and their immediate left and right neighbors; the latter uses proxy point sources and small least-squares solves (Schur complements) to represent the remaining contribution from distant copies. By using high-order discretization on interfaces (including those with corners), the number of unknowns per layer is kept small. We achieve overall linear complexity in the number of layers, by direct solution of the resulting block tridiagonal system. For device characterization we present an efficient method to sweep over multiple incident angles, and show a 25× speedup over solving each angle independently. We solve the scattering from a 1000-layer structure with 3 × 105 unknowns to 9-digit accuracy in 2.5 minutes on a desktop workstation.

Fast Integral Equation Solution of Scattering of Multiscale Objects by Domain Decomposition Method with Mixed Basis Functions

Nonconformal nonoverlapping domain decomposition method (DDM) with mixed basis functions is presented to realize fast integral equation solution of electromagnetic scattering of multiscale objects. The original multiscale objects are decomposed into several closed subdomains. The higher order hierarchical vector basis functions are used in the electrically large smooth subdomains to significantly reduce the number of unknowns, while traditional Rao-Wilton-Glisson basis functions are used for subdomains with tiny structures. A well-posed matrix is successfully derived by the present DDM. Besides, the nonconformal property of DDM allows flexible mesh generation for complicated objects. Numerical results are presented to validate the proposed method and illustrate its advantages.

Fast Contour Integral Equation Method for Wideband Power Integrity Analysis

The contour integral equation method has been used for system-level power integrity analysis of electronic packages. But, wideband simulation of complex package structures necessitates a fast contour integral equation (CIE) method. For this purpose, we propose both a priori and a posteriori approaches to speed up the CIE method for fast wideband simulation of power ground planes. First, the speedup is achieved by means of a priori approach, i.e., fast matrix filling using Hankel function interpolation. Second, a posteriori approach using the vector-fitting-based adaptive frequency sampling is combined with the CIE method for fast wideband analysis. We specifically propose an automatic band split algorithm to confine the order of the vector fitting model and ensure fast model convergence. Numerical and experimental results are presented to validate the accuracy and demonstrate the efficiency of the proposed acceleration techniques.

Installed Performance Modeling of Complex Antenna Array Mounted on Extremely Large-Scale Platform Using Fast MoM-PO Hybrid Framework

A fast method of moments-physical optics (MoM- PO) hybrid framework is proposed to efficiently analyze complex antenna array mounted on large-scale platforms with multi-scale features. In this fast MoM-PO hybrid framework, the MoM, enhanced by the adaptive integral method (AIM), is employed to effectively simulate the antenna array, and the PO is used to efficiently describe the perturbation from the large platform on the antenna array. An iterative process is implemented to capture the interaction between the antenna array and the platform for achieving stable and converged solution. The proposed hybrid framework provides very good flexibility to cater for any fast integral equation solvers for speeding up the MoM solution process in modeling large and complex antenna arrays. Numerical simulations clearly demonstrate that the proposed fast MoM-PO technique is capable of handling complex onboard antenna arrays installed on electrically large platforms with much less computational costs as compared with the conventional MoM-PO method and multilevel fast multipole algorithm (MLFMA).