scholarly journals Fast hybrid numerical-asymptotic boundary element methods for high frequency screen and aperture problems based on least-squares collocation

2020 ◽  
Vol 1 (4) ◽  
Author(s):  
A. Gibbs ◽  
D. P. Hewett ◽  
D. Huybrechs ◽  
E. Parolin
Keyword(s):  
Least Squares ◽  
Boundary Element ◽  
High Frequency ◽  
Singular Integrals ◽  
Weighted Least Squares ◽  
Computational Cost ◽  
Boundary Element Methods ◽  
Two Dimensional ◽  
Numerical Instabilities ◽  
Value Decomposition

Abstract We present a hybrid numerical-asymptotic (HNA) boundary element method (BEM) for high frequency scattering by two-dimensional screens and apertures, whose computational cost to achieve any prescribed accuracy remains bounded with increasing frequency. Our method is a collocation implementation of the high order hp HNA approximation space of Hewett et al. (IMA J Numer Anal 35:1698–1728, 2015), where a Galerkin implementation was studied. An advantage of the current collocation scheme is that the one-dimensional highly oscillatory singular integrals appearing in the BEM matrix entries are significantly easier to evaluate than the two-dimensional integrals appearing in the Galerkin case, which leads to much faster computation times. Here we compute the required integrals at frequency-independent cost using the numerical method of steepest descent, which involves complex contour deformation. The change from Galerkin to collocation is nontrivial because naive collocation implementations based on square linear systems suffer from severe numerical instabilities associated with the numerical redundancy of the HNA basis, which produces highly ill-conditioned BEM matrices. In this paper we show how these instabilities can be removed by oversampling, and solving the resulting overdetermined collocation system in a weighted least-squares sense using a truncated singular value decomposition. On the basis of our numerical experiments, the amount of oversampling required to stabilise the method is modest (around 25% typically suffices), and independent of frequency. As an application of our method we present numerical results for high frequency scattering by prefractal approximations to the middle-third Cantor set.

scholarly journals Multilevel weighted least squares polynomial approximation

2020 ◽  
Vol 54 (2) ◽  
pp. 649-677 ◽  
Author(s):  
Abdul-Lateef Haji-Ali ◽  
Fabio Nobile ◽  
Raúl Tempone ◽  
Sören Wolfers
Keyword(s):  
Least Squares ◽  
Polynomial Approximation ◽  
Weighted Least Squares ◽  
A Priori ◽  
Computational Cost ◽  
Discretization Error ◽  
Logarithmic Factor ◽  
Complexity Bounds ◽  
Single Level ◽  
Random Samples

Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method that utilizes samples computed with different accuracies and is able to match the accuracy of single-level approximations with reduced computational cost. We derive complexity bounds under certain assumptions about polynomial approximability and sample work. Furthermore, we propose an adaptive algorithm for situations where such assumptions cannot be verified a priori. Finally, we provide an efficient algorithm for the sampling from optimal distributions and an analysis of computationally favorable alternative distributions. Numerical experiments underscore the practical applicability of our method.

scholarly journals Determination of Lateral Modulation Apodization Functions Using a Regularized, Weighted Least Squares Estimation

2010 ◽  
Vol 2010 ◽  
pp. 1-7 ◽  
Author(s):  
Chikayoshi Sumi
Keyword(s):  
Least Squares ◽  
Spatial Resolution ◽  
Soft Tissues ◽  
Displacement Vector ◽  
Weighted Least Squares ◽  
Strain Tensor ◽  
Estimation Method ◽  
Modulation Method ◽  
Value Decomposition

Recently, work in this group has focused on the lateral cosine modulation method (LCM) which can be used for next-generation ultrasound (US) echo imaging and tissue displacement vector/strain tensor measurements (blood, soft tissues, etc.). For instance, in US echo imaging, a high lateral spatial resolution as well as a high axial spatial resolution can be obtained, and in tissue displacement vector measurements, accurate measurements of lateral tissue displacements as well as of axial tissue displacements can be realized. For an optimal determination of an apodization function for the LCM method, the regularized, weighted minimum-norm least squares (WMNLSs) estimation method is presented in this study. For designed Gaussian-type point spread functions (PSFs) with lateral modulation as an example, the regularized WMNLS estimation in simulations yields better approximations of the designed PSFs having wider lateral bandwidths than a Fraunhofer approximation and a singular-value decomposition (SVD). The usefulness of the regularized WMNLS estimation for the determination of apodization functions is demonstrated.

Application of the Complex Envelope Vectorization to a Boundary Element Formulation

2008 ◽  
Author(s):  
Oliviero Giannini ◽  
Aldo Sestieri
Keyword(s):  
Finite Elements ◽  
External Field ◽  
Boundary Element ◽  
High Frequency ◽  
Computational Cost ◽  
Field Variable ◽  
Element Formulation ◽  
Complex Envelope ◽  
Reduced Dimensions ◽  
Core Solution

The complex envelope vectorization (CEV) is a recent method that has been successfully applied to structural and internal acoustic problems. Unlike other methods proposed in the last two decades to solve high frequency problems, CEV is not an energy method, although it shares with all the other techniques a variable transformation of the field variable. By such transformation involving a Hilbert transform, CEV allows the representation of a fast oscillating signal through a set of low oscillating signals. Thanks to such transformation it is possible to solve a high frequency dynamic problem at a computational cost that is lower than that required by finite elements. In fact, by using finite elements, a high frequency problem usually implies large matrices. On the contrary the CEV formulation is obtained by solving a set of linear problems of highly reduced dimensions. Although it was proved that CEV is in general a successful procedure, it was shown that it is particularly appropriate when the modes of the system have a negligible role on the solution. Moreover, the numerical advantage of the CEV formulation is much more pronounced when full matrices are used. Thus, for the first time it is applied to a boundary element formulation (BEM). Both external and internal acoustic fields of increasing complexity are considered: the internal and external field generated by a pulsating sphere; the external field of a forced box, where the velocity field is determined by finite elements; a set of 4 plates that form an open cavity. The results are compared with those obtained by a BEM procedure (SYSNOISE), highlighting the good quality of the proposed approach. An estimate of the computational advantage is also provided. Finally it is worthwhile to point out that the reduction of the BE matrices allows for an in-core solution even for large problems.

scholarly journals Energy Flow Boundary Element Method for Vibration Analysis of One and Two Dimension Structures

2008 ◽  
Vol 15 (1) ◽  
pp. 33-50 ◽  
Author(s):  
Ho-Won Lee ◽  
Suk-Yoon Hong ◽  
Do-Hyun Park ◽  
Hyun-Wung Kwon
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
High Frequency ◽  
Energy Flow ◽  
Vibration Analysis ◽  
Thin Plates ◽  
Two Dimensional ◽  
Flow Boundary ◽  
Energy Equations ◽  
Element Method

In this paper, Energy Flow Boundary Element Method (EFBEM) was developed to predict the vibration behavior of one- and two-dimensional structures in the medium-to-high frequency ranges. Free Space Green functions used in the method were obtained from EFA energy equations. Direct and indirect EFBEMs were formulated for both one- and two-dimensional cases, and numerically applied to predict the energy density and intensity distributions of simple Euler-Bernoulli beams, single rectangular thin plates, and L-shaped thin plates vibrating in the medium-to-high frequency ranges. The results from these methods were compared with the EFA solutions to verify the EFBEM.

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