A fast solver framework for acoustic hybrid integral equations

A novel surface integral equation (IE) formulation for magneto-quasi-static analysis of current flow in two-dimensional conductors situated in lossy layered substrates is proposed. Traditional approach is based on the volume IE formulated with respect to the unknown current distribution in the conductors cross-section. In this work, we transform the volume IE to a surface IE via representation of the volumetric current density as a superposition of the waves emanating from the conductors boundaries. The resulting surface IE features a single unknown surface current density unlike the traditional surface IEs which require both electric and magnetic surface current densities. Also, the resultant single-source surface IE formulation involves only the electric field Greens functions instead of both electric and magnetic field Greens functions as required by the traditional surface IE formulations. The latter property makes the proposed IE formulation particularly suitable for analysis of current flow in conductors embedded in lossy layered substrates.

A newly developed iterative method, the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method is applied to simulate electromagnetic scattering from large inhomogeneous objects embedded in a planarly layered medium. In this fast solver, the weak-form formulation is applied to obtain a less singular discretization of the volume electric field integral equation. Several techniques are utilized to speed up the dyadic function evaluation. To accelerate the operation of the dyadic function on an induced current (i.e., the Green's operation), the function is split into convolutional and correlational components so that FFT can be applied. The CPU time and memory cost of this BCGS-FFT method is O(NlogN) and O(N), respectively, where N is the number of unknowns, significantly more efficient than the method of moments (MoM). As a result, this method is capable of solving large-scale electromagnetic scattering problems in a planarly layered background. A large-scale scattering problem in a layered medium with more than three million unknowns has been solved on a Sun Ultra 60 workstation with 1.2 GBytes memory.

Recent years have seen an increasing interest in the development of fast direct integral equation solvers. These do not rely on the convergence of iterative procedures for obtaining the solution. Instead, they compute a compressed factorized form of the impedance matrix resulting from the discretization of an underlying integral equation. The compressed form can then be applied to multiple right-hand sides, at a relatively low additional cost. The most common class of direct integral equation solvers exploits the rank-deficiency of off-diagonal blocks of the impedance matrix, in order to express them in a compressed manner. Such rank deficiency is inherent to problems of small size compared to the wavelength, as well as to problems of reduced dimensionality, e.g., elongated and quasi-planar problems.

In this paper, a fast electromagnetic integral equation solution methodology is proposed for the frequency-domain modeling of lossy, interconnect structures. The proposed method utilizes a two-layer model for thick conductors where the unknown current density inside the rectangular wire strips is approximated in terms of equivalent surface currents placed at the top and bottom sides of the strip which, for the purposes of this paper, are assumed to be substantially larger than the side strips. A matrix impedance relationship, which depends on the conductor thickness and its material properties, is established between the two surface current densities to account for the skin effect behavior of the field within the conductor. The selected assignment of the unknown current densities on the planes associated with the top and bottom sides of the metallization, combined with the planarity of multilayered interconnect structures, makes possible the application of the conjugate gradient fast Fourier transform (FFT) algorithm for the computationally efficient prediction of the electromagnetic response of multiport interconnect structures. Applications of the resulting fast integral equation solver to the electromagnetic modeling of interconnect circuits with thick lossy conductors from near dc to multigigahertz frequencies are used to demonstrate the validity of the method and quantify its computational efficiency.

Review of large scale computing in electromagnetics with fast integral equation solvers

Fast hybrid integral equation solver for acoustic scattering

We present fast methods for solving Laplace’s and the biharmonic equations on irregular regions with smooth boundaries. The methods used for solving both equations make use of fast Poisson solvers on a rectangular region in which the irregular region is embedded. They also both use an integral equation formulation of the problem where the integral equations are Fredholm integral equations of the second kind. The main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region. Fast solvers are then used to compute the extended solution. Aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth.

This comprehensive resource conveniently combines advanced topics of electromagnetic theory, a high level of mathematical detail, and the well-established ubiquitous Method of Moments applied to the solution of practical wave-scattering and antenna problems formulated with surface, volume, and hybrid integral equations

We propose a fast, patient-specific workflow for on-line specific absorption rate (SAR) supervision. An individualized electromagnetic model is created while the subject is on the table, followed by rapid SAR estimates for that individual. Our goal is an improved correspondence between the patient and model, reducing reliance on general anatomical body models. A 3D fat-water 3T acquisition (~2minutes) is automatically segmented using a computer vision algorithm (~1minute) into what we found to be the most important electromagnetic tissue classes: air, bone, fat, and soft tissues. We then compute the individual's EM field exposure and global and local SAR matrices using a fast electromagnetic integral equation solver. We assess the approach in 10 volunteers and compare to the SAR seen in a standard generic body model (Duke). The on-the-table workflow averaged 7'44″. Simulation of the simplified Duke models confirmed that only air, bone, fat, and soft tissue classes are needed to estimate global and local SAR with an error of 6.7% and 2.7%, respectively, compared to the full model. In contrast, our volunteers showed a 16.0% and 20.3% population variability in global and local SAR, respectively, which was mostly underestimated by the Duke model. Timely construction and deployment of a patient-specific model is computationally feasible. The benefit of resolving the population heterogeneity compared favorably to the modest modeling error incurred. This suggests that individualized SAR estimates can improve electromagnetic safety in MRI and possibly reduce conservative safety margins that account for patient-model mismatch, especially in non-standard patients.

New parallel stochastic global integral and local differential equation modeling and inversion

On singular integral equations and fundamental solutions of poroelasticity

This paper presents an overview of several integral equation (IE) formulations where the use of Legendre basis functions is beneficial. Specifically, we introduce the basis functions and deal with formulations for surface IE, volume IE, volume-surface IE, and layered media, respectively. In some cases, the high complexity of a problem arises due its large size in terms of wavelengths which can be handled with the MoM-PO approach described. Finally, we present a numerical example solved with two different formulations.

SUMMARYThis text introduces the following: (1) new regularized combined field integral equations (CFIE‐R) for frequency‐domain sound‐hard scattering problems; and (2) fast, high‐order algorithms for the numerical solution of the CFIE‐R and related integral equations. Similar to the classical combined field integral equation (CFIE), the CFIE‐R are uniquely‐solvable integral equations based on the use of single and double layer potentials. Unlike the CFIE, however, the CFIE‐R utilize a composition of the double‐layer potential with a regularizing operator that gives rise to highly favorable spectral properties—thus making it possible to produce accurate solutions by means of iterative solvers in small numbers of iterations. The CFIE‐R‐based fast high‐order integral algorithms introduced in this text enable highly accurate solution of challenging sound‐hard scattering problems, including hundred‐wavelength cases, in single‐processor runs on present‐day desktop computers. A variety of numerical results demonstrate the qualities of the numerical solvers as well as the advantages that arise from the new integral equation formulation. Copyright © 2012 John Wiley & Sons, Ltd.

Fast solvers of integral equations of the second kind: wavelet methods

Many different problems in mathematics, physics, engineering can be expressed in the form of integral equations.Among these are diffraction problems, population growth, heat transfer, particle transport problems, electrical engineering, elasticity, control, elastic waves, diffusion problems, quantum mechanics, heat radiation, electrostatics and contact problems.Therefore, the solutions which are obtained by the mathematical methods play an important role in these fields.The most two basic types of integral equations are called Fredholm (FIEs) and Volterra (VIEs).In many instances, the ordinary and partial differential equations can be converted into Fredhom and Volterra integral equations that are solved more effectively.We aim through this research to present an improved Adomian decomposition method based on modified Bernstein polynomials (ADM-MBP) to solve nonlinear integral equations of the second kind.We introduced efficient method, constructed on modified Bernstein polynomials.The formulation is developed to solve nonlinear Fredholm and Volterra integral equations of second kind.This method is tested for some examples from nonlinear integral equations.Maple software was used to obtain the solutions of these examples.The results demonstrate reliability of the proposed method.Generally, the proposed method is very convenient to apply to find the solutions of Fredholm and Volterra integral equations of second kind.