Yaniv Brick

Multilevel nonuniform grid algorithm for acceleration of integral equation-based solvers for acoustic scattering

A fast algorithm for the evaluation of acoustic fields produced by given source distributions is developed with the aim of accelerating iterative boundary element method (BEM) solvers. The algorithm is based on field smoothing by phase and amplitude compensation, which allows for sampling of the fields radiated by finite-size source distributions over coarse nonuniform (spherical) grids (NGs). Subsequently, the fields at the desired target points can be obtained by an interpolation and phase and amplitude restoration. Combining this approach with the divide-and-conquer strategy, the total field is computed via a hierarchical decomposition of the source domain. In this computational scheme, the phase and amplitude compensated fields produced by neighboring subdomains are gradually aggregated through a multilevel process involving interpolation between increasingly dense NGs and the scatterer surface. This multilevel NG algorithm is used to reduce the computational cost of applying the field evaluation operator and its adjoint, as required in each iteration of the conjugate gradient solver based on the BEM-discretized integral representation of scattering problems. Accuracy and computational efficiency of the NG algorithm are demonstrated on representative examples of elongated, quasi-planar, and full 3-D scatterers.

Fast direct solution of 3-D scattering problems via nonuniform grid-based matrix compression

A fast non-iterative algorithm for the solution of large 3-D acoustic scattering problems is presented. The proposed approach can be used in conjunction with the conventional boundary element discretization of the integral equations of acoustic scattering. The algorithm involves domain decomposition and uses the nonuniform grid (NG) approach for the initial compression of the interactions between each subdomain and the rest of the scatterer. These interactions, represented by the off-diagonal blocks of the boundary element method matrix, are then further compressed while constructing sets of interacting and local basis and testing functions. The compressed matrix is obtained by eliminating the local degrees of freedom through the Schurs complement-based technique procedure applied to the diagonal blocks. In the solution process, the interacting unknowns are first determined by solving the compressed system equations. Subsequently, the local degrees of freedom are determined for each subdomain. The proposed technique effectively reduces the oversampling typically needed when using low-order discretization techniques and provides significant computational savings.

Fast Green's Function Evaluation for Sources and Observers Near Smooth Convex Bodies

An efficient procedure for the evaluation of the Greens function for a source and multiple observation points near a convex impedance boundary cylinder is presented. The evaluation is performed using non-uniform grids spread along the cylinders perimeter and regionally tailored for capturing the Greens functions behavior in the line-of-sight and shadow regions. On these grids, the Greens function is assumed to be accurately pre-computed using analytical or numerical techniques. The procedure is demonstrated for the circular case associated with the generalized equivalence integral equation.

A fast direct integral-equation solver for hydraulic fracture diagnosis

An ℋ-matrix based fast direct solver is presented to accelerate the solution of volume integral equations pertinent to the analysis of hydraulic fracture resistivity measurements. The solvers performance for this problem type is compared to that of an FFT-accelerated iterative solver.

3D Multilevel Non-Uniform Grid Algorithm for Fast Field Evaluation

The analysis of large scattering problems is often performed using iterative method-of-moments (MoM) solvers. The main computational bottleneck of such iterative solvers stems from the need to perform at each iteration at least one matrix-vector product, which is equivalent to field evaluation for a given source distribution. The O(N2) complexity and memory needs of direct matrix-vector multiplication (N being the number of unknowns), underscores the need for fast field evaluation techniques. The multilevel fast multipole algorithm (MLFMA) achieves an O(NlogN) complexity for dynamic frequency domain problems. Also, the recently introduced non-uniform grid (NG) algorithm promises to achieve a similar asymptotic complexity and storage requirements while allowing analysis time-domain problems as well as virtually seamless transition from dynamic to quasi-static regime. In comparison with the MLFMA, the implementation of the NG is straightforward. This paper presents a 3D dynamic scalar field evaluation performed at O(NlogN) operations using a multilevel NG (MLNG) algorithm. The scalar case allows us to explore the main features of the MLNG while somewhat simplifying the formulation

A fast and stable solver for acoustic scattering problems based on the nonuniform grid approach.

A fast and stable boundary element method (BEM) algorithm for solving external problems of acoustic scattering by impenetrable bodies is developed. The method employs the Burton-Miller integral equation, which provides stable convergence of iterative solvers, and a generalized multilevel nonuniform grid (MLNG) algorithm for fast evaluation of field integrals. The MLNG approach is used here for the removal of computational bottlenecks involved with repeated matrix-vector multiplications as well as for the low-order basis function regularization of the hyper-singular integral kernel. The method is used for calculating the fields scattered by large acoustic scatterers, including nonconvex bodies with piece-wise smooth surfaces. As a result, the algorithm is capable of accurately incorporating high-frequency effects such as creeping waves and multiple-edges diffractions. In all cases, stable convergence of the method is observed. High accuracy of the method is demonstrated by comparison with the traditional BEM solution. The computational complexity of the method in terms of both the computation time and storage is estimated in practical computations and shown to be close to the asymptotic O(N log N) dependence.

A Schur-complement method for integral-equation analysis of antennas near anatomical human models

The Schur complement method is applied to a multiregion integral equation formulation for analyzing antennas near anatomical human models to improve the conditioning of the system. The multiregion problem is split into a primary (external) domain and a secondary (internal) domain. The antenna unknowns are transferred as an effective load to the equivalent surface unknowns via the Schur complement. The proposed methods implementation and costs are detailed.

All multiscale problems are hard, some are harder: A nomenclature for classifying multiscale electromagnetic problems

Different types of multiscale problems encountered in classical electromagnetics are discussed. A nomenclature is proposed that is useful for identifying the difficulty of a problem and the suitability of a computational method for solving it.

Fast computation of modified Green's function for generalized source integral equation solvers

A technique for the fast computation of modified Greens function arising from the generalized source integral equation (GSIE) formulation is proposed. The method is based on an efficient sampling and tabulation of the modified Greens function, which comprises two components: a direct free-space component and an additional contribution by electric and magnetic sources designed to “shield” the source from certain observation regions. The efficient sampling of the sum of both contributions is performed by using non uniform grids designed in accordance with the wave-field phenomenology unique to the source-“shield” configuration. The proposed methods performance is demonstrated for representative examples.

Iterative physical optics (IPO) with integral evaluation of self-shadowing

A novel iterative physical optics (IPO) algorithm is proposed, for the analysis of scattering from large complex geometries involving multiple reflections and complex self-shadowing effects. The algorithm involves two types of nested iterations: reflection (“bounce”) iterations and self-shadowing iterations. At each bounce iteration, the physical optics sources induced on the surface of the scatterer produce a correction to the incident field, which in turn creates a correction to the physical optics sources. Each correction is evaluated to account for self-shadowing effects by a nested iterative evaluation of shadow-radiation integrals. The nested iterative formulation is naturally accelerable by using fast field evaluation methods, e.g., the multilevel non-uniform grid algorithm. The procedures applicability to complex geometries is demonstrated numerically by comparison to numerically exact results and to standard PO results.