Use of Wave Boundary Elements for Acoustic Computations

2003 ◽  
Vol 11 (02) ◽  
pp. 305-321 ◽  
Author(s):  
Emmanuel Perrey-Debain ◽  
Jon Trevelyan ◽  
Peter Bettess
Keyword(s):  
Boundary Element ◽  
Degrees Of Freedom ◽  
Plane Waves ◽  
Current Method ◽  
Boundary Element Methods ◽  
Element Formulation ◽  
Scattering Problems ◽  
Direct Collocation ◽  
New Type ◽  
New Formulation

Discrete methods of numerical analysis have been used successfully for decades for the solution of problems involving wave diffraction, etc. However, these methods, including the finite element and boundary element methods, can require a prohibitively large number of elements as the wavelength becomes progressively shorter. In this work, a new type of interpolation for the acoustic field is described in which the usual conventional shape functions are modified by the inclusion of a set of plane waves propagating in multiple directions. Including such a plane wave basis in a boundary element formulation has been found in the current work to be highly successful. Results are shown for a variety of classical scattering problems, and also for scattering from nonconvex obstacles. Notable results include a conclusion that, using this new formulation, only approximately 2.5 degrees of freedom per wavelength are required. Compared with the 8 to 10 degrees of freedom normally required for conventional boundary (and finite) elements, this shows the marked improvement in storage requirement. Moreover, the new formulation is shown to be extremely accurate. It is estimated that for 2D Helmholtz problems, and for a given computational resource, the frequency range allowed by this method is extended by a factor of three over conventional direct collocation Boundary Element Method. Recent successful developments of the current method for plane elastodynamics problems are also briefly outlined.

scholarly journals Nodal and Divergence-Conforming Boundary-Element Methods Applied to Electromagnetic Scattering Problems

2004 ◽  
Vol 40 (2) ◽  
pp. 1053-1056
Author(s):  
M.M. Afonso ◽  
J.A. Vasconcelos ◽  
R.C. Mesquita ◽  
C. Vollaire ◽  
L. Nicolas
Keyword(s):  
Boundary Element ◽  
Electromagnetic Scattering ◽  
Boundary Element Methods ◽  
Scattering Problems

scholarly journals Boundary element methods for the wave equation based on hierarchical matrices and adaptive cross approximation

2021 ◽  
Author(s):  
Daniel Seibel
Keyword(s):  
Wave Equation ◽  
Boundary Element ◽  
Domain Boundary ◽  
Boundary Element Methods ◽  
Scattering Problems ◽  
Computational Costs ◽  
Adaptive Cross Approximation ◽  
Transient Problems ◽  
Matrix Compression ◽  
Convolution Quadrature Method

AbstractTime-domain Boundary Element Methods (BEM) have been successfully used in acoustics, optics and elastodynamics to solve transient problems numerically. However, the storage requirements are immense, since the fully populated system matrices have to be computed for a large number of time steps or frequencies. In this article, we propose a new approximation scheme for the Convolution Quadrature Method powered BEM, which we apply to scattering problems governed by the wave equation. We use $${\mathscr {H}}^2$$ H 2 -matrix compression in the spatial domain and employ an adaptive cross approximation algorithm in the frequency domain. In this way, the storage and computational costs are reduced significantly, while the accuracy of the method is preserved.

scholarly journals Resolution of Born Scattering in Curve Geometries: Aspect-Limited Observations and Excitations

Electronics ◽  
2021 ◽  
Vol 10 (24) ◽  
pp. 3089
Author(s):  
Ehsan Akbari Sekehravani ◽  
Giovanni Leone ◽  
Rocco Pierri
Keyword(s):  
Degrees Of Freedom ◽  
Plane Waves ◽  
Single Frequency ◽  
Limited Data ◽  
Scattering Problems ◽  
Approximation Accuracy ◽  
Imaging Results ◽  
Exact Evaluation ◽  
Spread Function ◽  
Scattered Fields

In inverse scattering problems, the most accurate possible imaging results require plane waves impinging from all directions and scattered fields observed in all observation directions around the object. Since this full information is infrequently available in actual applications, this paper is concerned with the mathematical analysis and numerical simulations to estimate the achievable resolution in object reconstruction from the knowledge of the scattered far-field when limited data are available at a single frequency. The investigation focuses on evaluating the Number of Degrees of Freedom (NDF) and the Point Spread Function (PSF), which accounts for reconstructing a point-like unknown and depends on the NDF. The discussion concerns objects belonging to curve geometries, in this case, circumference and square scatterers. In addition, since the exact evaluation of the PSF can only be accomplished numerically, an approximated closed-form evaluation is introduced and compared with the exact one. The approximation accuracy of the PSF is verified by numerical results, at least within its main lobe region, which is the most critical as far as the resolution discussion is concerned. The main result of the analysis is the space variance of the PSF for the considered geometries, showing that the resolution is different over the investigation domain. Finally, two numerical applications of the PSF concept are shown, and their relevance in the presence of noisy data is outlined.

AN INNOVATIVE BOUNDARY ELEMENT FORMULATION FOR BUILDING SLABS

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Ahmed U. Abdelhady ◽  
Youssef F. Rashed
Keyword(s):  
Boundary Element ◽  
Stiffness Matrix ◽  
Degrees Of Freedom ◽  
Research Work ◽  
Element Formulation ◽  
Supporting Cells ◽  
Stiffness Analysis ◽  
Stiffness Matrices ◽  
Limited Applicability ◽  
Generalized Displacements

Slab supported by beams (i.e. beam-slab floor) is a common practice in the construction of buildings. Modeling this slab type using the boundary element method (BEM) is an essential step to provide seamless frameworks for the analysis of buildings with complex geometries. However, one of the most challenging difficulties that have been facing research efforts in this area is its theoretical setting with limited applicability to practical building slabs. This limitation is addressed in this research work and a practical BEM-based formulation for the slab-beam floor is presented. The presented formulation discretizes the connection area between the slab, and beams and columns into cells (supporting cells). The centroids of these cells are used to carry out an additional collocation scheme which is required to solve the resulting system of equations. To generate the slab stiffness matrix, the generalized displacements that correspond to the supporting cells’ degrees of freedom are set to unity (one at a time). By assembling the generated slab stiffness matrix with beams and columns stiffness matrices using the stiffness analysis method, the overall stiffness matrix is obtained. Hooke’s law is then applied to calculate the generalized displacements and straining actions are obtained in the post-processing phase. The developed formulation is applied to an example to validate its results by comparing it with the analytical solution.

A New Boundary Element Method Formulation for Linear Elasticity

1986 ◽  
Vol 53 (1) ◽  
pp. 69-76 ◽  
Author(s):  
N. Ghosh ◽  
H. Rajiyah ◽  
S. Ghosh ◽  
S. Mukherjee
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Linear Elasticity ◽  
The Other ◽  
Element Formulation ◽  
Standard Formulation ◽  
Field Point ◽  
Elasticity Problems ◽  
Method Formulation ◽  
New Formulation

A new boundary element formulation for linear elasticity problems is presented in this paper. The standard formulation for planar problems uses two kernels — one of which is logarithmic singular and the other is 1/r singular, where r is the distance between a source and a field point. The new formulation avoids the use of the strongly singular kernel so that both kernels are now only logarithmic singular. The new formulation has several potential advantages over the standard one, the most significant of which is that it delivers stresses accurately at internal points which are extremely close to the boundary of a body. Numerical results for sample problems, from each of the formulations, are presented and compared here.

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