Integral Equation Algorithm Research Articles

Fast integral equation algorithms for the solution of electromagnetic wave propagation over general terrains

In this paper a fast numerical algorithm to solve an integral equation model for wave propagation along a perfectly conducting two-dimensional terrain is suggested. It is applied to different actual terrain profiles and the results indicate very good agreement with published work. In addition, the proposed algorithm has achieved considerable saving in processing time. The formulation is extended to solve the propagation over lossy dielectric surfaces. A combined field integral equation (CFIE) for wave propagation over dielectric terrain is solved efficiently by utilizing the method of moments with complex basis functions. The numerical results for different cases of dielectric surfaces are compared with the results of perfectly conducting surface evaluated by the IE conventional algorithm.

The boundary integral method for the D.C. geoelectric problem in the 3-layered earth with a prismoid inhomogeneity in the second layer

Abstract The paper presents algorithm and numerical results for the boundary integral equations (BIE) method of the forward D.C. geoelectric problem for the three-layered earth which contains the prismoidal body with sloped faces in the second layer. This situation occurs in the sedimentary basins. Although the numerical calculations are more complicated in comparison with faces orthogonal to the x, y, z axes, the generalization to the sloped faces enables treatment of the anomalous fields for the bodies of more general shapes as rectangular prisms. The graphs with numerical results present isoline maps of the perturbing potential as well as the resistivity profiles when the source field is due to the pair of D.C. electrodes at the surface of the earth. Also presented apparent resistivity curves for the Schlumberger array AMNB sounding.

Fast Electromagnetic Integral-Equation Solvers on Graphics Processing Units

A survey of electromagnetic integral-equation solvers, implemented on graphics processing units (GPUs), is presented. Several key points for efficient GPU implementations of integral-equation solvers are outlined. Three spatial-interpolation-based algorithms, including the Nonuniform-Grid Interpolation Method (NGIM), the box Adaptive-Integral Method (B-AIM), and the fast periodic interpolation method (FPIM), are described to show the basic principles for optimizing GPU-accelerated fast integral-equation algorithms. It is shown that proper implementations of these algorithms lead to very high computational performance, with GPU-CPU speed-ups in the range of 100-300. Critical points for these accomplishments are (i) a proper arrangement of the data structure, (ii) an “on-the-fly” approach, trading excessive memory usage with increased arithmetic operations and data uniformity, and (iii) efficient utilization of the types of GPU memory. The presented methods and their GPU implementations are geared towards creating efficient electromagnetic integral-equation solvers. They can also find a wide range of applications in a number of other areas of computational physics.

Mechanical Quadrature Methods and Extrapolation Algorithms for Boundary Integral Equations with Linear Boundary Conditions in Elasticity

By potential theory, elastic problems with linear boundary conditions are converted into boundary integral equations (BIEs) with logarithmic and Cauchy singularity. In this paper, a mechanical quadrature method (MQMs) is presented to deal with the logarithmic and the Cauchy singularity simultaneously for solving the boundary integral equations. The convergence and stability are proved based on Anselone’s collective compact and asymptotical compact theory. Furthermore, an asymptotic expansion with odd powers of errors is presented, which possesses high accuracy order O(h3). Using h3−Richardson extrapolation algorithms (EAs), the accuracy order of the approximation can be greatly improved to O(h5), and an a posteriori error estimate can be obtained for constructing a self-adaptive algorithm. The efficiency of the algorithm is illustrated by examples.

An efficient two step Laplace decomposition algorithm for singular Volterra integral equations

The aim of this article is to introduce a new reliable algorithm, namely two-step Laplace decomposition algorithm (TSLDA). This new algorithm provides us with a convenient way to find exact solution with less computation as compared with standard Laplace decomposition algorithm (LDA). The proposed algorithm is use to solve Abel's second kind integral equations efficiently. Key words: Two step Laplace decomposition algorithm, Laplace decomposition algorithm, Abel's second kind integral equations.

Splitting extrapolation algorithm for first kind boundary integral equations with singularities by mechanical quadrature methods

The accuracy of numerical solutions near singular points is crucial for numerical methods. In this paper we develop an efficient mechanical quadrature method (MQM) with high accuracy. The following advantages of MQM show that it is very promising and beneficial for practical applications: (1) the $ O(h_{\rm {max}}^{3})$ convergence rate; (2) the $O(h_{\rm {max}}^{5})$ convergence rate after splitting extrapolation; (3) Cond = $O(h_{\rm {min}}^{-1})$ ; (4) the explicit discrete matrix entries. In this paper, the above theoretical results are briefly addressed and then verified by numerical experiments. The solutions of MQM are more accurate than those of other methods. Note that for the discontinuous model in Li et al. (Eng Anal Bound Elem 29:59---75, 2005), the highly accurate solutions of MQM may even compete with those of the collocation Trefftz method.

Meshless Galerkin algorithms for boundary integral equations with moving least square approximations

In this paper, we first give error estimates for the moving least square (MLS) approximation in the H k norm in two dimensions when nodes and weight functions satisfy certain conditions. This two-dimensional error results can be applied to the surface of a three-dimensional domain. Then combining boundary integral equations (BIEs) and the MLS approximation, a meshless Galerkin algorithm, the Galerkin boundary node method (GBNM), is presented. The optimal asymptotic error estimates of the GBNM for three-dimensional BIEs are derived. Finally, taking the Dirichlet problem of Laplace equation as an example, we set up a framework for error estimates of the GBNM for boundary value problems in three dimensions.

Time-domain augmented EFIE and its marching-on-in-degree solution

A time-domain augmented electric field integral equation (TDAEFIE) and its marching-on-in-degree (MOD) solution are presented for analysis of transient electromagentic responses from three-dimensional closed conducting bodies of arbitrary shape.By enforcing a condition on the normal component of the electric flux density, the TDAEFIE eliminates the potential internal resonance problem of the time-domain electric field integral equation (TDEFIE) algorithm. With the use of weighted Laguerre polynomials as entire-domain temporal basis functions, the MOD solution overcomes the late-time instability problem that often occurs in the marching-on-in-time (MOT) approach. Compared with the MOD solution of the time-domain combined field integral equation (TDCFIE), the MOD solution of the TDAEFIE is more efficient because it takes less computational time for calculating the matrix elements and the matrix-vector multiplications related to the excitation at the right-hand side of the matrix equation. Numerical results are presented to illustrate the good performance of the TDAEFIE algorithm. © 2011 Wiley Periodicals, Inc. Microwave Opt Technol Lett 53:1439–1444, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26015

Near-surface resistivity contrast mapping with a capacitive sensor array and an inductive source

Electromagnetic survey methodology is adapted to use the electric component to directly detect buried resistors and map resistivity contrasts in the near surface. System implementations do not require ground contact because they use capacitive electric-field sensors and an inductive source and may be operated at walking pace. This study outlined theoretical basis, computational modeling, and verification for the methodology. The systems are designed to operate at low enough frequency that any responses are at the resistive limit; as such, the electric fields they measure are insensitive to horizontal layering and absolute conductivity. A surface integral equation algorithm is used to model regular discrete objects in a half-space. Anomalies are controlled by geometry and lateral resistivity contrast rather than by absolute resistivity values. A prototype electrode array system called CARIS 1 reliably detects resistive objects submerged in a saltwater tank, and the measured responses are consistent with numerical modeling. These results provided the basis for further development of the CARIS II system with flexible geometry that is adaptable to detect resistive or conductive targets in any background environment. CARIS is designed to be useful where conventional electromagnetics, ground-penetrating radar, and conventional resistivity face difficulties or fail, and it has easily detected a range of buried targets in the near surface. However, it is quite sensitive to disturbed ground and surface undulation and inhomogeneity.

Iteration methods and algorithms for dielectric-resonator integral equations of the field

Quasi-eigenmodes of open cylindrical and rectangular dielectric resonators (DRs) are determined by the method of iterative solution of the volume integral and integro-differential equations with corresponding functionals. New forms of equations and iteration algorithms for the nonlinear input of the desired complex parameter are proposed. Frequencies and Q-factors of the H01δ and H011 modes of a cylindrical DR and the H mode of a rectangular DR for the uniform and nonuniform cases are obtained numerically. The influence of a thin semiconductor layer located at the ends of the DR and irradiated by high-power laser pulses on the frequencies and Q-factors of the DR modes is examined. It is shown that an up to ten or more percent tuning of resonant frequencies can be reached by transformation of a low conducting state to a high conducting state.

A fully discrete Galerkin method for high frequency exterior acoustic scattering in three dimensions

Standard Galerkin discretization techniques (with locally- or globally-supported basis functions) for boundary integral equations are inefficient for high frequency three dimensional exterior scattering simulations because they require a fixed number of unknowns per wavelength in each dimension, leading to large CPU time and memory requirements to set up the dense Galerkin matrix, with each entry requiring evaluation of multi-dimensional highly oscillatory integrals. In this work, using globally-supported basis functions, we describe an efficient fully discrete Galerkin surface integral equation algorithm for simulating high frequency acoustic scattering by three dimensional convex obstacles that includes a powerful integration scheme for evaluation of four dimensional Galerkin integrals with high-order accuracy. Such high-order order accuracy for various practically relevant frequencies ( k ∈ [1, 100,000]) substantially improves on approximations based on standard asymptotic techniques. We demonstrate the efficiency of our algorithm for spherical and non-spherical convex scattering for several wavenumbers 1 ⩽ k ⩽ 100,000 for low to high order prescribed tolerance. Our fully discrete algorithm requires only mild growth in the number of unknowns and CPU time as the frequency increases.

Advanced quadrature methods and splitting extrapolation algorithms for first kind boundary integral equations of Laplace's equation with discontinuity solutions

The Laplace equation can be transformed to the first kind boundary integral equations (BIEs), and the corner and discontinuity singularity of Laplace's equation can be studied by the open arcs of the BIEs. For the first kind BIEs, there exist the Galerkin method (GM) and the collocation method (CM); but they suffer in low convergence. The advanced (i.e., the mechanical) quadrature methods (AQMs) and the splitting extrapolation methods (SEMs) originated in [16,17] are proposed in Huang et al. [15] for first kind BIEs with open arcs, to achieve O( h 3) or even O( h 5) convergence, and the excellent stability with Cond. = O( h −1), where h is the uniform meshspacing, accompanied with the strict analysis. Moreover, the algorithms of AQMs and the SEMs are simple without any integration computation. Hence the AQMs and the SEMs are superior to the existing methods, such as GM and CM. A challenging discontinuity model of Laplace's equation is proposed in Li et al. [20], and the collocation Trefftz method (CTM) is used to give highly accurate solutions. For the AQMs and the SEMs, the strict theoretical analysis is given in [15], and the discontinuity model [20] is dealt with in this paper, to also achieve highly accurate solutions. The numerical solutions in this paper display that the AQMs and the SEMs are significant not only to the first kind BIEs with the open arc singularity, but also to Laplace's equation with highly strong singularity such as the discontinuity of solutions. Moreover, the link of the first kind BIEs with open arcs and the Laplace equation with discontinuity solutions are explored clearly, to display the significance of the proposed algorithms; this paper strengthens the first kind BIEs and its engineering applications.

TRANSIENT TIME-DEPENDENT ELECTRIC FIELD OF DIELECTRIC BODIES USING THE VOLTERRA INTEGRAL EQUATION IN THREE DIMENSIONS

An accurate and ∞exible three-dimensional Volterra Time Domain Integral Equation (TDIE) algorithm is presented and implemented here to model the time-dependent electromagnetic fleld of arbitrarily shaped dielectric bodies. This development is motivated by the need for a modern high-resolution numerical tool that is capable of providing a full and comprehensive investigation of devices containing a diverse range of feature sizes or boundaries, in all three space dimensions plus time. Stability, accuracy and convergence of the algorithm are discussed and verifled by means of canonical working examples.

Computing Electromagnetic Wave Resistivity MWD Tool's Response Using Accelerated Iteration Algorithm for Integral Equations

AbstractThe 2D integral equations are computed by an integrated iteration algorithm of modified successive approximation method (MSAM) and Aitken acceleration technique. The longitudinally stratified virgin formation is taken as the background and the computational region is restricted within the borehole and invasion zone in the algorithm, thus the algorithm is qualified for the virtues of small number of unknowns, fast converging speed and high accuracy. The response of the electromagnetic wave resistivity MWD tool in cylindrically symmetrical 2D formation is simulated by the algorithm. The results have shown that the amplitude attenuation and phase shift are affected differently by the borehole, invasion and surrounding shale, and that their radial depth of investigation and vertical resolution are also different. The bed boundary can be accurately located by the crossover point of the compensated amplitude attenuation and phase shift resistivities.

Three dimensional electromagnetic scattering T-matrix computations

The infinite T-matrix method is a powerful tool for electromagnetic scattering simulations, particularly when one is interested in changes in orientation of the scatterer with respect to the incident wave or changes of configuration of multiple scatterers and random particles, because it avoids the need to solve the fully reconfigured systems. The truncated T-matrix (for each scatterer in an ensemble) is often computed using the null-field method. The main disadvantage of the null-field based T-matrix computation is its numerical instability for particles that deviate from a sphere. For large and/or highly non-spherical particles, the null-field method based truncated T-matrix computations can become slowly convergent or even divergent. In this work, we describe an electromagnetic scattering surface integral formulation for T-matrix computations that avoids the numerical instability. The new method is based on a recently developed high-order surface integral equation algorithm for far field computations using basis functions that are tangential on a chosen non-spherical obstacle. The main focus of this work is on the mathematical details required to apply the high-order algorithm to compute a truncated T-matrix that describes the scattering properties of a chosen perfect conductor in a homogeneous medium. We numerically demonstrate the stability and convergence of the T-matrix computations for various perfect conductors using plane wave incident radiation at several low to medium frequencies and simulation of the associated radar cross of the obstacles.

A novel Gaussian interpolation formula‐based IE‐FFT algorithm for solving EM scattering problems

AbstractThe fast Fourier transform‐based integral equation algorithm (IE‐FFT) is a recently developed fast algorithm to solve electrically large problems. The key of the IE‐FFT algorithm is to choose a proper interpolation method, since it not only determines the algorithm's accuracy but also influences the computational complexity and the memory requirement directly. In this article, a novel Gaussian interpolation formula is proposed to optimize the IE‐FFT algorithm for solving EM scattering problems. Compared with the Lagrange interpolation, it can bring with higher accuracy and less memory costs. Furthermore, this interpolation scheme is also well suited for the higher‐order hierarchical basis functions. © 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 2233–2236, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24523

Efficient high-order evaluation of scattering by periodic surfaces: deep gratings, high frequencies, and glancing incidences

We present a superalgebraically convergent integral equation algorithm for evaluation of TE and TM electromagnetic scattering by smooth perfectly conducting periodic surfaces z=f(x). For grating-diffraction problems in the resonance regime (heights and periods up to a few wavelengths) the proposed algorithm produces solutions with full double-precision accuracy in single-processor computing times of the order of a few seconds. The algorithm can also produce, in reasonable computing times, highly accurate solutions for very challenging problems, such as (a) a problem of diffraction by a grating for which the peak-to-trough distance equals 40 times its period that, in turn, equals 20 times the wavelength; and (b) a high-frequency problem with very small incidence, up to 0.01 degrees from glancing. The algorithm is based on the concurrent use of Floquet and Chebyshev expansions together with certain integration weights that are computed accurately by means of an asymptotic expansion as the number of integration points tends to infinity.

Simulation studies promote technological development of radiofrequency phased array hyperthermia

A treatment planning program package for radiofrequency hyperthermia has been developed. It consists of software modules for processing three-dimensional computerized tomography (CT) data sets, manual segmentation, generation of tetrahedral grids, numerical calculation and optimisation of three-dimensional Ε field distributions using a volume surface integral equation algorithm as well as temperature distributions using an adaptive multilevel finite-elements code, and graphical tools for simultaneous representation of CT data and simulation results. Heat treatments are limited by hot spots in healthy tissues caused by Ε field maxima at electrical interfaces (bone/muscle). In order to reduce or avoid hot spots suitable objective functions are derived from power deposition patterns and temperature distributions, and are utilised to optimise antenna parameters (phases, amplitudes). The simulation and optimisation tools have been applied to estimate the improvements that could be reached by upgrades of the clinically used SIGMA-60 applicator (consisting of a single ring of four antenna pairs). The investigated upgrades are increased number of antennas and channels (triple-ring of 3 × 8 antennas and variation of antenna inclination. Significant improvement of index temperatures (1–2°C) is achieved by upgrading the single ring to a triple ring with free phase selection for every antenna or antenna pair. Antenna amplitudes and inclinations proved as less important parameters.

An integral equation algorithm for 3-D simulation of pore pressure in a porous elastic medium with heterogeneities

We present an algorithm for simulation of pore pressure due to a 3-D surface water reservoir, on a porous elastic half-space with heterogeneity of different permeability/hydraulic diffusivity. The perturbation in pore pressure due to the anomalous permeability of the heterogeneity is calculated by introducing a secondary source term in the porous elastic diffusion equation. The resulting equation is solved using integral equation method. Green's functions of porous elastic half-space are used in this algorithm. As an illustrative example, the case of Koyna–Warna region, a well-known site of reservoir triggered seismicity in India, is simulated. Pore pressure due to reservoir loading is calculated in the fault zones and the surrounding medium. This algorithm will have a wide application as it can be applied to various cases, such as fluid injection, increased stream discharge, porous elastic relaxation, etc., where the diffusion of pore pressure in the porous elastic medium with heterogeneities is significant.

Modular Fast Direct Electromagnetic Analysis Using Local-Global Solution Modes

Local-global solution (LOGOS) modes provide a computationally efficient framework for developing fast, direct solution methods for electromagnetic simulations. LOGOS modes provide a single basis for which both electromagnetic operators and their inverses are sparse with a controllable accuracy. For fixed-frequency modeling with h-refinement, efficient application of this observation is shown to provide efficient factorization algorithms for both integral equation and finite element methods. In addition to providing fast, direct solutions, the LOGOS framework is also shown to provide a new strategy for modular electromagnetic analysis. The possibility of developing efficient direct solution strategies at high frequencies using the LOGOS framework is also discussed.