Fast hybrid integral equation solver for acoustic scattering

Reliable modeling of the scattering by acoustically large and geometrically complex objects can be achieved by means of subdomain-dependent problem formulation and a numerically rigorous solution. While the objects’ inhomogeneity has driven the development of differential equation formulations and solvers, integral equation formulations, where the object’s background is modeled via a Green’s function, are advantageous for unbounded domains. In the hybrid integral equations approach (Usner et al., 2006), the interaction of separate subdomains with external fields is described by pertinent integral equations. Their Galerkin discretization leads to a dense blocked stiffness matrix. The development of compressed representations of the matrix, which are necessary for the treatment of large systems, becomes non-trivial due to the multitude of integral equation kernels and the different geometrical and physical characteristic of the subdomains. As part of the development of a fast hybrid integral equation solver framework, we consider the case of objects composed of large inhomogeneous volumes, modeled as incompressible fluids, and of simplified solids, modeled via surface integral equations. A hybrid integral equation formulation is derived and solved numerically. The iterative solution is accelerated by employing the butterfly-compressed hierarchical representation of the stiffness matrix, recently used for acoustic volume integral equations (Kaplan, 2022).

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.

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.

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.

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.

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

We describe a fast integral equation-based method for full-wave extraction in layered media. The method uses a combination of the fast integral equation solver IES/sup 3/, layered Green's functions, and a formulation that gives a well-conditioned linear system even in the electrically-small regime (i.e., when circuit structures are a fraction of the wavelength of light). The overall approach gives O(NlogN) complexity, where N is the number of panels in a discretization of the conductor surfaces. We apply our method to the simulation of an integrated inductor on a lossy CMOS substrate and compare the results to measurement.

A fast convergent integral equation solution to the scattering problem of a transverse electric/transverse magnetic (TE/TM) plane wave by a one-dimensional periodic array of thin metal strips on a dielectric substrate is described. The formulation of the integral equation is similar to that derived by Montgomery for a two-dimensional periodic array of thin conductors on a dielectric substrate. However, the basis functions which satisfy the appropriate edge conditions are incorporated here for the unknown current expansion on the strips. Following the standard Galerkin's procedure, one may readily determine the induced currents on the strips and thus the reflected and transmitted fields. Sample numerical results are given and good agreement with previously published data is obtained. It is found that the convergence rate of this method is improved by an order of magnitude. Also it is shown that the dielectric substrate has a strong effect on the scattering from the large spacing strip grating.

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

This paper discusses a time domain fast boundary integral equation method for three dimensional elastodynamics and its parallelisation for a large shared memory parallel computer. Some details of the PWTD (Plane Wave Time Domain) approach and an extension of the theory to the anisotropic case are presented. We then examine the parallelisation strategies of the code using OpenMP and MPI-OpenMP hybridisation. In the case of MPI-OpenMP hybrid parallelisation, a numerical example with more than one million spatial DOF is shown. It is concluded that the method is promising, its parallelisation with OpenMP is effective and that larger problem can be analysed with MPI-OpenMP hybrid parallelisation.

This paper presents a fast boundary integral equation method for approximating the conformal mapping from bounded and unbounded multiply connected regions of connectivity $m+1$ onto the canonical region obtained by removing $m$ rectilinear slits from a strip. The method is based on a combination of a uniquely solvable boundary integral equation with generalized Neumann kernel and the fast multipole method. The presented method requires $O((m+1)n\ln n)$ operations, where $n$ is the number of nodes in the discretization of each boundary component. The numerical results of some test calculations illustrate that our method has the ability to handle regions with complex geometry and very high connectivity. Besides the above canonical region, the presented fast method also can be used to compute the numerical conformal mapping from multiply connected regions onto Koebe's 39 canonical slit regions.

This contribution outlines a reformulation of the so-called thin stratified medium fast multipole approach e.g. given in [1] for the characterization of Planar-3D structures in multilayered environments. This reformulation is mainly based on another fast solver which makes use of translation operatores in Cartesian wavenumbers [2][3]. For this reformulation, most of the theoretical basics and proceedings of the framework in [2][3] can be applied for the new implementations. Furthermore this new formulation leads to significant simplifications during the computation of the group interactions.

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.

This paper presents a new fast matrix-vector multiplication scheme for the solution of the electric field integral equation. Similarly to other fast methods, our approach reduces the matrix-vector multiplication cost from O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) to O(N log N). Differently from other fast solvers, however, the effectiveness of EFIE preconditioning techniques such as quasi-Helmholtz decompositions or Calderon approaches is maintained by our method even for very high matrix compression rates. This is thanks to the fact that, in the scheme we are proposing, the contribution from the scalar potential when applied to or tested with solenoidal functions is always zero independent of the compression error. In addition, the new method will take advantage of the redundancies of the EFIE matrix in the low-frequency/dense discretization regime, and it will further decrease both the memory storage and the multiplication cost with respect to currently available fast solvers. Numerical results will show the effectiveness of our approach and its impact on the solution of realistic problems.

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.