1,921 research outputs found
Fast Multipole Method For 3-D Helmholtz Equation In Layered Media
In this paper, a fast multipole method (FMM) is proposed to compute
long-range interactions of wave sources embedded in 3-D layered media. The
layered media Green's function for the Helmholtz equation, which satisfies the
transmission conditions at material interfaces, is decomposed into a free space
component and four types of reaction field components arising from wave
reflections and transmissions through the layered media. The proposed algorithm
is a combination of the classic FMM for the free space component and FMMs
specifically designed for the four types reaction components, made possible by
new multipole expansions (MEs) and local expansions (LEs) as well as the
multipole-to-local translation (M2L) operators for the reaction field
components. { Moreover, equivalent polarization source can be defined for each
reaction component based on the convergence analysis of its ME. The FMMs for
the reaction components, implemented with the target particles and equivalent
polarization sources, are found to be much more efficient than the classic FMM
for the free space component due to the fact that the equivalent polarization
sources and the target particles are always separated by a material interface.}
As a result, the FMM algorithm developed for layered media has a similar
computational cost as that for the free space. Numerical results validate the
fast convergence of the MEs and the complexity of the FMM for
interactions of low-frequency wave sources in 3-D layered media
Taylor expansion based fast Multipole Methods for 3-D Helmholtz equations in Layered Media
In this paper, we develop fast multipole methods for 3D Helmholtz kernel in
layered media. Two algorithms based on different forms of Taylor expansion of
layered media Green's function are developed. A key component of the first
algorithm is an efficient algorithm based on discrete complex image
approximation and recurrence formula for the calculation of the layered media
Green's function and its derivatives, which are given in terms of Sommerfeld
integrals. The second algorithm uses symmetric derivatives in the Taylor
expansion to reduce the size of precomputed tables for the derivatives of
layered media Green's function. Numerical tests in layered media have validated
the accuracy and O(N) complexity of the proposed algorithms
Nested domain decomposition with polarized traces for the 2D Helmholtz equation
We present a solver for the 2D high-frequency Helmholtz equation in
heterogeneous, constant density, acoustic media, with online parallel
complexity that scales empirically as , where is
the number of volume unknowns, and is the number of processors, as long as
. This sublinear scaling is achieved by domain
decomposition, not distributed linear algebra, and improves on the scaling reported earlier in [L. Zepeda-N\'u\~nez and L.
Demanet, J. Comput. Phys., 308 (2016), pp. 347-388 ]. The solver relies on a
two-level nested domain decomposition: a layered partition on the outer level,
and a further decomposition of each layer in cells at the inner level. The
Helmholtz equation is reduced to a surface integral equation (SIE) posed at the
interfaces between layers, efficiently solved via a nested version of the
polarized traces preconditioner [L. Zepeda-N\'u\~nez and L. Demanet, J. Comput.
Phys., 308 (2016), pp. 347-388.]. The favorable complexity is achieved via an
efficient application of the integral operators involved in the SIE.Comment: 34 pages, 9 figure
A fast solver for multi-particle scattering in a layered medium
In this paper, we consider acoustic or electromagnetic scattering in two
dimensions from an infinite three-layer medium with thousands of
wavelength-size dielectric particles embedded in the middle layer. Such
geometries are typical of microstructured composite materials, and the
evaluation of the scattered field requires a suitable fast solver for either a
single configuration or for a sequence of configurations as part of a design or
optimization process. We have developed an algorithm for problems of this type
by combining the Sommerfeld integral representation, high order integral
equation discretization, the fast multipole method and classical multiple
scattering theory. The efficiency of the solver is illustrated with several
numerical experiments
Fast alternating bi-directional preconditioner for the 2D high-frequency Lippmann-Schwinger equation
This paper presents a fast iterative solver for Lippmann-Schwinger equation
for high-frequency waves scattered by a smooth medium with a compactly
supported inhomogeneity. The solver is based on the sparsifying preconditioner
and a domain decomposition approach similar to the method of polarized traces.
The iterative solver has two levels, the outer level in which a sparsifying
preconditioner for the Lippmann-Schwinger equation is constructed, and the
inner level, in which the resulting sparsified system is solved fast using an
iterative solver preconditioned with a bi-directional matrix-free variant of
the method of polarized traces. The complexity of the construction and
application of the preconditioner is and
respectively, where is the number of degrees of
freedom. Numerical experiments in 2D indicate that the number of iterations in
both levels depends weakly on the frequency resulting in method with an overall
complexity.Comment: small modification
Fast multipole method for 3-D Laplace equation in layered media
In this paper, a fast multipole method (FMM) is proposed for 3-D Laplace
equation in layered media. The potential due to charges embedded in layered
media is decomposed into a free space component and four types of reaction
field components, and the latter can be associated with the potential of a
polarization source defined for each type. New multipole expansions (MEs) and
local expansions (LEs), as well as the multipole to local (M2L) translation
operators are derived for the reaction components, based on which the FMMs for
reaction components are then proposed. The resulting FMM for charge
interactions in layered media is a combination of using the classic FMM for the
free space components and the new FMMs for the reaction field components. With
the help of a recurrence formula for the run-time computation of the
Sommerfeld-type integrals used in M2L translation operators, pre-computations
of a large number of tables are avoided. The new FMMs for the reaction
components are found to be much faster than the classic FMM for the free space
components due to the separation of equivalent polarization charges and the
associated target charges by a material interface. As a result, the FMM for
potential in layered media costs almost the same as the classic FMM in the free
space case. Numerical results validate the fast convergence of the MEs for the
reaction components, and the O(N) complexity of the FMM with a given truncation
number p for charge interactions in 3-D layered media.Comment: arXiv admin note: text overlap with arXiv:1902.0513
Spectrally-accurate numerical method for acoustic scattering from doubly-periodic 3D multilayered media
A periodizing scheme and the method of fundamental solutions are used to
solve acoustic wave scattering from doubly-periodic three-dimensional
multilayered media. A scattered wave in a unit cell is represented by the sum
of the near and distant contribution. The near contribution uses the free-space
Green's function and its eight immediate neighbors. The contribution from the
distant sources is expressed using proxy source points over a sphere
surrounding the unit cell and its neighbors. The Rayleigh-Bloch radiation
condition is applied to the top and bottom layers. Extra unknowns produced by
the periodizing scheme in the linear system are eliminated using a Schur
complement. The proposed numerical method avoids using singular quadratures and
the quasi-periodic Green's function or complicated lattice sum techniques.
Therefore, the proposed scheme is robust at all scattering parameters including
Wood anomalies. The algorithm is also applicable to electromagnetic problems by
using the dyadic Green's function. Numerical examples with 10-digit accuracy
are provided. Finally, reflection and transmission spectra are computed over a
wide range of incident angles for device characterization
Sweeping preconditioners for the iterative solution of quasiperiodic Helmholtz transmission problems in layered media
We present a sweeping preconditioner for quasi-optimal Domain Decomposition
Methods (DDM) applied to Helmholtz transmission problems in periodic layered
media. Quasi-optimal DD (QO DD) for Helmholtz equations rely on transmission
operators that are approximations of Dirichlet-to-Neumann (DtN) operators.
Employing shape perturbation series, we construct approximations of DtN
operators corresponding to periodic domains, which we then use as transmission
operators in a non-overlapping DD framework. The Robin-to-Robin (RtR) operators
that are the building blocks of DDM are expressed via robust boundary integral
equation formulations. We use Nystr\"om discretizations of quasi-periodic
boundary integral operators to construct high-order approximations of RtR.
Based on the premise that the quasi-optimal transmission operators should act
like perfect transparent boundary conditions, we construct an approximate LU
factorization of the tridiagonal QO DD matrix associated with periodic layered
media, which is then used as a double sweep preconditioner. We present a
variety of numerical results that showcase the effectiveness of the sweeping
preconditioners applied to QO DD for the iterative solution of Helmholtz
transmission problems in periodic layered media
Adapting free-space fast multipole method for layered media Green's function: algorithm and analysis
In this paper, we present a numerical algorithm for the accurate and
efficient computation of the convolution of the frequency domain layered media
Green's function with a given density function. Instead of compressing the
convolution matrix directly as in the classical fast multipole method, fast
direct solvers, and fast H-matrix algorithms, the new algorithm considers a
translated form of the original matrix so that many existing building blocks
from the highly optimized free-space fast multipole method can be easily
adapted to the Sommerfeld integral representations of the layered media Green's
function. An asymptotic analysis is performed on the Sommerfeld integrals for
large orders to provide an estimate of the decay rate in the new "multipole"
and "local" expansions. In order to avoid the highly oscillatory integrand in
the original Sommerfeld integral representations when the source and target are
close to each other, or when they are both close to the interface in the
scattered field, mathematically equivalent alternative direction integral
representations are introduced. The convergence of the multipole and local
expansions and formulas and quadrature rules for the original and alternative
direction integral representations are numerically validated.Comment: 27 page
High-performance modeling acoustic and elastic waves using the Parallel Dichotomy Algorithm
A high-performance parallel algorithm is proposed for modeling the
propagation of acoustic and elastic waves in inhomogeneous media. An initial
boundary-value problem is replaced by a series of boundary-value problems for a
constant elliptic operator and different right-hand sides via the integral
Laguerre transform. It is proposed to solve difference equations by the
conjugate gradient method for acoustic equations and by the GMRES method
for modeling elastic waves. A preconditioning operator was the Laplace operator
that is inverted using the variable separation method. The novelty of the
proposed algorithm is using the Dichotomy Algorithm (Terekhov, 2010), which was
designed for solving a series of tridiagonal systems of linear equations, in
the context of the preconditioning operator inversion. Via considering
analytical solutions, it is shown that modeling wave processes for long
instants of time requires high-resolution meshes. The proposed parallel
fine-mesh algorithm enabled to solve real application seismic problems in
acceptable time and with high accuracy. By solving model problems, it is
demonstrated that the considered parallel algorithm possesses high performance
and efficiency over a wide range of the number of processors (from 2 to 8192).Comment: The formula (2) has been correcte
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