301 research outputs found
Wideband Fast Kernel-Independent Modeling of Large Multiscale Structures Via Nested Equivalent Source Approximation
We propose a wideband fast kernel-independent modeling of large multiscale structures; we employ a nested equivalent source approximation (NESA) to compress the dense system matrix. The NESA was introduced by these authors for low and moderate frequency problems (smaller than a few wavelengths); here, we introduce a high-frequency NESA algorithm, and propose a hybrid version with extreme wideband properties. The equivalent sources of the wideband NESA (WNESA) are obtained by an inverse-source process, enforcing equivalence of radiated fields on suitably defined testing surfaces. In the low-frequency region, the NESA is used unmodified, with a complexity of mathcal{O}(N) . In the high-frequency region, in order to obtain a fixed rank matrix compression, we hierarchically divide the far coupling space into pyramids with angles related to the peer coupling group size, and the NESA testing surfaces are defined as the boundaries of the pyramids. This results in a directional nested low-rank (fixed rank) approximation with mathcal{O}(Nlog {N}) computational complexity that is kernel independent; overall, the approach yields wideband fast solver for the modeling of large structures that inherits the efficiency and accuracy of low-frequency NESA for multiscale problems. Numerical results and discussions demonstrate the validity of the proposed work
Theory and implementation of -matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels
In this work, we study the accuracy and efficiency of hierarchical matrix
(-matrix) based fast methods for solving dense linear systems
arising from the discretization of the 3D elastodynamic Green's tensors. It is
well known in the literature that standard -matrix based methods,
although very efficient tools for asymptotically smooth kernels, are not
optimal for oscillatory kernels. -matrix and directional
approaches have been proposed to overcome this problem. However the
implementation of such methods is much more involved than the standard
-matrix representation. The central questions we address are
twofold. (i) What is the frequency-range in which the -matrix
format is an efficient representation for 3D elastodynamic problems? (ii) What
can be expected of such an approach to model problems in mechanical
engineering? We show that even though the method is not optimal (in the sense
that more involved representations can lead to faster algorithms) an efficient
solver can be easily developed. The capabilities of the method are illustrated
on numerical examples using the Boundary Element Method
Complexity Analysis of a Fast Directional Matrix-Vector Multiplication
We consider a fast, data-sparse directional method to realize matrix-vector
products related to point evaluations of the Helmholtz kernel. The method is
based on a hierarchical partitioning of the point sets and the matrix. The
considered directional multi-level approximation of the Helmholtz kernel can be
applied even on high-frequency levels efficiently. We provide a detailed
analysis of the almost linear asymptotic complexity of the presented method.
Our numerical experiments are in good agreement with the provided theory.Comment: 20 pages, 2 figures, 1 tabl
Computationally efficient boundary element methods for high-frequency Helmholtz problems in unbounded domains
This chapter presents the application of the boundary element method to high-frequency Helmholtz problems in unbounded domains. Based on a standard combined integral equation approach for sound-hard scattering problems we discuss the discretization, preconditioning and fast evaluation of the involved operators. As engineering problem, the propagation of high-intensity focused ultrasound fields into the human rib cage will be considered. Throughout this chapter we present code snippets using the open-source Python boundary element software BEM++ to demonstrate the implementation
An accurate boundary value problem solver applied to scattering from cylinders with corners
In this paper we consider the classic problems of scattering of waves from
perfectly conducting cylinders with piecewise smooth boundaries. The scattering
problems are formulated as integral equations and solved using a Nystr\"om
scheme where the corners of the cylinders are efficiently handled by a method
referred to as Recursively Compressed Inverse Preconditioning (RCIP). This
method has been very successful in treating static problems in non-smooth
domains and the present paper shows that it works equally well for the
Helmholtz equation. In the numerical examples we specialize to scattering of E-
and H-waves from a cylinder with one corner. Even at a size kd=1000, where k is
the wavenumber and d the diameter, the scheme produces at least 13 digits of
accuracy in the electric and magnetic fields everywhere outside the cylinder.Comment: 19 pages, 3 figure
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