Integral Equation Solver Research Articles

An FFT-Accelerated Integral-Equation Solver for Analyzing Scattering in Rectangular Cavities

An efficient integral-equation method is presented for the fast analysis of scattering from arbitrarily shaped 3-D structures in a rectangular cavity. The proposed method employs: 1) the frequency-domain surface-volume electric field integral equation to model scattering from conductors and dielectrics; 2) the rectangular-cavity Green functions to account for the cavity walls; 3) the Ewald method and 3-D spatial interpolation to accelerate the Green function computations; and 4) the adaptive integral method (AIM) to reduce the computational complexity of the iterative method-of-moments solution procedure. The structure of interest is first meshed with arbitrary triangular/tetrahedral elements. The mesh is then enclosed with an auxiliary regular grid. Next, a four-step algorithm is executed: interpolation (mesh-to-grid), propagation (grid-to-grid), interpolation (grid-to-mesh), and correction (mesh-to-mesh). The computationally dominant propagation step of the AIM is accelerated by decomposing the Green functions into eight components that are in convolution or correlation form in the three Cartesian directions. This results in eight types of propagation matrices; these are in nested Toeplitz or Hankel form and are efficiently multiplied with the necessary vectors using 3-D fast Fourier transforms. Numerical results validate the method, quantify its costs, and demonstrate its utility.

Time-Domain Integral Equation Solver for Planar Circuits Over Layered Media Using Finite Difference Generated Green's Functions

A simple, highly accurate, and efficient 2D finite-difference technique for computing the direct convolution of time-domain Green's functions (TDGFs) of layered media with temporal interpolators is presented. The proposed TDGFs are incorporated into a recently developed marching-on-in-time (MOT)-based time-domain integral equation (TDIE) solver based on noncausal high-order B-Spline temporal interpolators without requiring the prediction of future unknowns. Numerical results demonstrating the applicability of the technique to the analysis of various microwave structures are presented.

Renewal theory-based life-cycle analysis of deteriorating engineering systems

Engineering systems typically deteriorate due to regular use and exposure to harsh environment. Under such circumstances the owner of a system must take important decisions such as whether to repair, replace or abandon the system. Such decisions can affect the safety of, and the benefits to the users and the owner. Life-cycle analysis (LCA) provides a rational basis for such decision making process. In particular, LCA can provide helpful information on the performance of a system over its entire life-cycle, like its time-dependent reliability, the costs associated with its operation, and other quantities related to the service life of the system.This paper proposes a novel probabilistic formulation for LCA of deteriorating systems named Renewal Theory-based Life-cycle Analysis (RTLCA). The formulation includes equations to obtain important life-cycle variables such as the expected time lost in repairs, the reliability of the system and the cost of operation and failure. The proposed RTLCA formulation is based on renewal theory and proposes analytical solutions for the desired LCA variables using numerically solvable integral equations. As an illustration, the proposed RTLCA formulation is implemented to analyze the life-cycle of an example reinforced concrete (RC) bridge located in a seismic region. This analysis accounts for the accumulated seismic damage in the bridge columns caused by the earthquakes occurring during bridge’s life-cycle. The analysis results provide valuable insight into the importance of seismic damage in a bridge’s life-cycle performance and the strategies to operate a system in an optimal manner.

An [formula omitted] direct solver for integral equations on the plane

An efficient direct solver for volume integral equations with O(N) complexity for a broad range of problems is presented. The solver relies on hierarchical compression of the discretized integral operator, and exploits that off-diagonal blocks of certain dense matrices have numerically low rank. Technically, the solver is inspired by previously developed direct solvers for integral equations based on “recursive skeletonization” and “Hierarchically Semi-Separable” (HSS) matrices, but it improves on the asymptotic complexity of existing solvers by incorporating an additional level of compression. The resulting solver has optimal O(N) complexity for all stages of the computation, as demonstrated by both theoretical analysis and numerical examples. The computational examples further display good practical performance in terms of both speed and memory usage. In particular, it is demonstrated that even problems involving 107 unknowns can be solved to precision 10−10 using a simple Matlab implementation of the algorithm executed on a single core.

ADAMANT: A surface and volume integral-equation solver for the analysis and design of helicon plasma sources

We present a full-wave numerical tool, dubbed ADAMANT (Advanced coDe for Anisotropic Media and ANTennas), devised for the analysis and design of radiofrequency antennas which drive the discharge in helicon plasma sources. ADAMANT relies on a set of coupled surface and volume integral equations in which the unknowns are the surface electric current density on the antenna conductors and the volume polarization current within the plasma. The latter can be inhomogeneous and anisotropic whereas the antenna can have arbitrary shape. The set of integral equations is solved numerically through the Method of Moments with sub-sectional surface and volume vector basis functions. This approach allows the accurate evaluation of the current distribution on the antenna and in the plasma as well as the antenna input impedance, a parameter crucial for the design of the feeding and matching network. We report several numerical examples which serve to validate ADAMANT against other well-established numerical approaches as well as experimental data. The numerical accuracy of the computed solution versus the number of basis functions in the plasma is also assessed. Finally, we employ ADAMANT to characterize the antenna of a real-life helicon plasma source.

ℋ 2 ‐matrix‐based fast volume integral equation solver for electrodynamic analysis

An ℋ2-matrix-based mathematical framework is introduced and further developed to reduce the computational complexity of the volume integral equation (VIE)-based analysis of electrodynamic problems. Numerical experiments have demonstrated a significant reduction in CPU time and memory consumption. A linear scaling with respect to matrix size in both CPU time and memory is achieved for small and medium-sized electrodynamic problems with an interpolation-based ℋ2-representation of the VIE operator in conjunction with a rank function for accuracy control. The proposed solver is applicable to arbitrarily shaped three-dimensional structures immersed in inhomogeneous materials.

Fast integral equation solver for Maxwell’s equations in layered media with FMM for Bessel functions

The paper presents a new fast integral equation solver for Maxwell’s equations in 3-D layered media. First, the spectral domain dyadic Green’s function is derived, and the 0-th and the 1-st order Hankel transforms or Sommerfeld-type integrals are used to recover all components of the dyadic Green’s function in real space. The Hankel transforms are performed with the adaptive generalized Gaussian quadrature points and window functions to minimize the computational cost. Subsequently, a fast integral equation solver with O(Nz2NxNy log(NxNy)) in layered media is developed by rewriting the layered media integral operator in terms of Hankel transforms and using the new fast multipole method for the n-th order Bessel function in 2-D. Computational cost and parallel efficiency of the new algorithm are presented.

Electromagnetic Analysis for Inhomogeneous Interconnect and Packaging Structures Based on Volume-Surface Integral Equations

Electromagnetic analysis for interconnect and packaging structures usually relies on the solutions of surface integral equations (SIEs) in integral equation solvers. Though the SIEs are necessary for the conductors in the structures, one has to assume a homogeneity of material for each layer of a substrate if SIEs are used for the substrate. When the inhomogeneity of materials in the substrate has to be taken into account, then volume integral equations (VIEs) are indispensable. In this paper, we consider the inhomogeneous materials of substrate and replace the SIEs with the VIEs to form volume-surface integral equations (VSIEs) for the entire structures. Also, the use of VIEs could alleviate low-frequency effects, remove the need of selecting a basis function for magnetic current, and facilitate geometric discretization in some scenarios. The VSIEs are solved with the method of moments by using the Rao-Wilton-Glisson basis function to represent the surface current on the conductors and Schaubert-Wilton-Glisson basis function to expand the volume current inside the substrate. To avoid the inconvenience of charge density in the traditional implementation of the VIEs, we suggest that the dyadic Green's function be kept in its original form without moving the gradient operator onto the basis and testing functions. Numerical examples are presented to demonstrate the effectiveness of the approach.

Efficient Generation of Aggregative Basis Functions in the Marching-on-in-Degree Time-Domain Integral Equation Solver

The aggregative basis function method for a marching-on-in-degree time-domain integral equation solver can significantly reduce the matrix size and storage and make the reduced system easily manageable and solvable. In the conventional generation of an aggregative basis function, a repeated solution of the matrix equation is required with various excitations, and hence, an efficient approach to generate the aggregative basis function is proposed in this article. The singular value decomposition is performed on multiple right-hand sides, which are linearly dependent or rank deficient, and after the singular value decomposition truncation, fewer matrix equation solutions are necessary with the retained excitation to generate the aggregative basis function. The size-reduced system is then constructed in the marching-on-in-degree time-domain integral equation solver with the aggregative basis function method to analyze transient electromagnetic scattering of conducting objects. Several numerical examples are included to demonstrate the performance of the proposed method.

High-Order Calderón Preconditioned Time Domain Integral Equation Solvers

Two high-order accurate Calderon preconditioned time domain electric field integral equation (TDEFIE) solvers are presented. In contrast to existing Calderon preconditioned time domain solvers, the proposed preconditioner allows for high-order surface representations and current expansions by using a novel set of fully-localized high-order div- and quasi curl-conforming (DQCC) basis functions. Numerical results demonstrate that the linear systems of equations obtained using the proposed basis functions converge rapidly, regardless of the mesh density and of the order of the current expansion.

Solvability of Integral Equations with Endogenous Delays

This paper performs a systematic study of some nonlinear integral equations with unknown lower limits of integration, which arise in economics, operations research, population biology and environmental sciences. A general investigation technique is developed for systems of such equations and several theorems about the existence and uniqueness of a solution are proven. The links to other functional equations with state-dependent delays are discussed. Applied interpretation of the obtained results is provided.

Analysis of a structural-aerodynamic fully-coupled formulation for aeroelastic response of rotorcraft

This paper deals with a computational aeroelastic tool aimed at the analysis of the response of rotary wings in arbitrary steady motion. It has been developed by coupling a nonlinear beam model for blades structural dynamics with a potential-flow boundary integral equation solver for the prediction of unsteady aerodynamic loads around three-dimensional, lifting bodies. The Galerkin method is used for the spatial integration of the resulting differential aeroelastic system, whereas the periodic blade response is determined by a harmonic balance approach. This aeroelastic model yields a unified approach for aeroelastic response and blade pressure prediction, that may conveniently be used for aeroacoustic purposes. It is able to examine configurations where blade–vortex interactions occur. Numerical results show the capability of the aeroelastic tool to evaluate blade response and vibratory hub loads for a helicopter main rotor in level and descent flight conditions, and examine the efficiency and robustness of the different numerical solution algorithms that may be applied in the developed aeroelastic solver. Comparisons among aeroelastic predictions based on different aerodynamic models are also presented.

Integral Equation Method for the First and Second Problems of the Stokes System

Using the integral equation method we study solutions of boundary value problems for the Stokes system in Sobolev space H1(G) in a bounded Lipschitz domain G with connected boundary. A solution of the second problem with the boundary condition \(\partial {\bf u}/\partial {\bf n} -p{\bf n}={\bf g}\) is studied both by the indirect and the direct boundary integral equation method. It is shown that we can obtain a solution of the corresponding integral equation using the successive approximation method. Nevertheless, the integral equation is not uniquely solvable. To overcome this problem we modify this integral equation. We obtain a uniquely solvable integral equation on the boundary of the domain. If the second problem for the Stokes system is solvable then the solution of the modified integral equation is a solution of the original integral equation. Moreover, the modified integral equation has a form f + Sf = g, where S is a contractive operator. So, the modified integral equation can be solved by the successive approximation. Then we study the first problem for the Stokes system by the direct integral equation method. We obtain an integral equation with an unknown \({\bf g}=\partial {\bf u}/\partial {\bf n} -p{\bf n}\). But this integral equation is not uniquely solvable. We construct another uniquely solvable integral equation such that the solution of the new eqution is a solution of the original integral equation provided the first problem has a solution. Moreover, the new integral equation has a form \({\bf g}+\tilde S{\bf g}={\bf f}\), where \(\tilde S\) is a contractive operator, and we can solve it by the successive approximation.

A fast solver for the Hilbert-type singular integral equations based on the direct Fourier spectral method

In this paper, we develop a fast and direct Fourier spectral method for solving the Hilbert-type singular integral equation. This method leads to a fully discrete linear system, whose coefficient matrix is expressed as the sum of a sparse matrix and a quasi-circulant matrix. We show that it requires a nearly linear computational cost to obtain and then solve the fully discrete linear system. We also prove that the proposed algorithm preserves the optimal convergent order. One numerical experiment is presented to demonstrate its approximate accuracy and computational efficiency, verifying the theoretical estimates.

Fast ${\cal H}$-Matrix-Based Direct Integral Equation Solver With Reduced Computational Cost for Large-Scale Interconnect Extraction

In this paper, we propose a fast H-matrix-based direct solution with a significantly reduced computational cost for an integral-equation-based capacitance extraction of large-scale 3-D interconnects in multiple dielectrics. We reduce the computational cost of an H-matrix-based computation by simultaneously optimizing the H-matrix partition to minimize the number of matrix blocks and minimizing the rank of each matrix block based on a prescribed accuracy. With the proposed cost-reduction method, we develop a fast LU-based direct solver. This solver possesses a complexity of kCspO (NlogN) in storage, a complexity of k2Csp2O(Nlog2N) in LU factorization, and a complexity of kCspO(NlogN) in LU solution, where k is the maximal rank, Csp is a constant dependent on matrix partition, and the constant kCsp is minimized based on accuracy by the proposed cost-reduction method. The proposed solver successfully factorizes dense matrices that involve millions of unknowns in fast CPU time and modest memory consumption, and with the prescribed accuracy satisfied. As an algebraic method, the underlying fast technique is kernel independent.

Investigation on accelerating FFT‐based methods for the EFIE on graphics processors

SUMMARYThe use of graphic processor units (GPUs) has been recently proposed in computational electromagnetics to accelerate the solution of the electric field integral equation. In these methods, the linear systems obtained by using boundary elements are considered, and then an accelerated solution for a specific excitation is obtained. The existing studies are mostly focused on speeding up the filling time or the LU decomposition of that matrix. This limits the application to simple simulation scenarios if a fast method is not employed. In this paper, we propose a GPU acceleration for FFT‐based integral equation solvers. We will investigate the operations involved in the solver, and we will motivate the use of GPUs. Results of numerical tests will be reported firstly on a perfect electric conductor sphere with different radii; then a realistic aircraft will be considered. We found that using GPUs for FFT‐based methods allows achieving a reasonable speed‐up. Copyright © 2012 John Wiley & Sons, Ltd.

Ray tracing with multi-radiation transmitters

Abstract. A restriction in using electromagnetic ray tracing for field prediction is given by the far-field condition: the results are only valid in the far-field region of the radiator. In this paper, it will be shown how ray tracing for accurate field computation can also be applied in the near-field regions of transmitters. The reduction of required large distances between transmitter and receiver is achieved by subdividing the transmitter in smaller subtransmitters. Even for complex transmitters, e.g. antennas with objects in close proximity such as metallic carrier platforms, subtransmitter models can be very efficiently generated by using the Multilevel Fast Multipole Method (MLFMM). This well-known integral equation solving technique makes very large problems in computational electromagnetics manageable. The subtransmitters can be directly generated based on this algorithm. A simulation example will show the improved modeling accuracy and options for simplification and refinement will also be discussed.

Selected theoretical and numerical aspects of fast volume and surface integral equation solvers for simulation of elasto-acoustic waves in complex inhomogeneous media

Comparative analysis is considered of two fast FFT-matrix-compression based elasto-acoustic integral equation solvers, employing volumetric and surface formulations, and designed to analyze sound propagation inside a human head; in particular to examine mechanisms of energy transfer to the inner ear through airborne as well as non-airborn path-ways, and to assess effectiveness of noise-protection devices. Verification tests involving the fast surface and volume integral equation solvers are carried out comparing their predictions with those following from an analytical solution of field distribution in an elasto-acoustic layered sphere.Resultsiare presented of representative numerical simulations of acoustic energy transfer to the cochlea for a human head model containing a detailed geometry representation of the outer, middle, and inner ear. The geometry model consists of: (1) the outer surface of the skin surrounding the skull and containing (2) the outer ear represented by its exterior surface, the surface of the auditory canal, and the tympanic membrane modeled as a finite-thickness surface, (3) the middle ear consisting of the system of ossicles and supporting structures, (4) the skull described by its external surfaces and including (5) a set of surfaces representing the inner ear (boundaries of the cochlea, the vestibule, and the semi-circular canals). *This work is supported by a grant from US Air Force Office of Scientific Research.

Fast Integral Equation Solution by Multilevel Green's Function Interpolation Combined With Multilevel Fast Multipole Method

A fast wideband integral equation (IE) solver combining the multilevel interpolatory fast Fourier transform accelerated approach (MLIPFFT) with the multilevel fast multipole method (MLFMM) is discussed. On electrically fine levels within an oct-tree multilevel structure, coupling computations are performed by MLIPFFT. This method is based on a 3D Lagrange factorization of the pertinent Green's functions with a smooth approximation error in space and it does not suffer a low frequency breakdown as known from MLFMM. For high frequency integral equation problems, MLIPFFT has decreased computational efficiency as the Nyquist theorem requires increasing numbers of samples in 3 dimensions. Due to a transition from the interpolation point based MLIPFFT source/receive formulation towards an appropriate <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -space representation at a certain level within the oct-tree, the high frequency efficient MLFMM can be employed for coarse levels. The hybrid algorithm is hence well suited for fast wideband integral equation solutions. Both, mixed-potential and direct-field formulations are considered. Furthermore, a method for MLIPFFT extrapolation error reduction based on fine level interpolation domain spreading is introduced. In several numerical examples, the performance of the proposed algorithm is demonstrated.

Stability Properties of the Time Domain Electric Field Integral Equation Using a Separable Approximation for the Convolution With the Retarded Potential

The state of art of time domain integral equation (TDIE) solvers has grown by leaps and bounds over the past decade. During this time, advances have been made in (i) the development of accelerators that can be retrofitted with these solvers and (ii) understanding the stability properties of the electric field integral equation. As is well known, time domain electric field integral equation solvers have been notoriously difficult to stabilize. Research into methods for understanding and prescribing remedies have been on the uptick. The most recent of these efforts are (i) Lubich quadrature and (ii) exact integration. In this paper, we re-examine the solution to this equation using (i) the undifferentiated form of the TD-EFIE and (ii) a separable approximation to the spatio-temporal convolution. The proposed scheme can be constructed such that the spatial integrand over the source and observer domains is smooth and integrable. As several numerical results will demonstrate, the proposed scheme yields stable results for long simulation times and a variety of targets, both of which have proven extremely challenging in the past.