A gradient formulation of the Helmholtz integral equation for acoustic radiation and scattering

1993 ◽  
Vol 93 (4) ◽  
pp. 1700-1709 ◽  
Author(s):  
David T. I. Francis
Keyword(s):  
Integral Equation ◽  
Acoustic Radiation ◽  
Helmholtz Integral

CAT'S EYE RADIATION WITH BOUNDARY ELEMENTS: COMPARATIVE STUDY ON TREATMENT OF IRREGULAR FREQUENCIES

2005 ◽  
Vol 13 (01) ◽  
pp. 21-45 ◽  
Author(s):  
STEFFEN MARBURG ◽  
SIA AMINI
Keyword(s):  
Integral Equation ◽  
Large Range ◽  
Acoustic Radiation ◽  
Collocation Point ◽  
Normal Derivative ◽  
Boundary Integral ◽  
High Frequencies ◽  
Modified Method ◽  
Geometric Singularity ◽  
Helmholtz Integral

This paper reviews a number of techniques developed to overcome the well-known nonuniqueness problem in boundary integral formulations of acoustic radiation. Although the problem has received much attention, comparative studies are hardly known in this field. Furthermore, the problem has often been studied using an unsuitable example, namely a simple radiating sphere. In this case, often the addition of one collocation point in the centre of the sphere suffices to remove the nonuniqueness problem for a large range of wavenumbers. In contrast to the radiating sphere, the radiating cat's eye structure is considered in this paper. Solution of the discretized ordinary Kirchhoff–Helmholtz integral equation, also known as the Surface Helmholtz Equation, reveals a large number of so-called irregular frequencies, i.e. frequencies where the BEM fails. The paper compares the performance of different methods in alleviating this failure. The CHIEF method and its variation due to Rosen et al. are found to encounter difficulties at high frequencies. A much better performance is obtained by combining the Kirchhoff–Helmholtz integral equation with its normal derivative. In particular the method of Burton and Miller and a modification of it which avoids evaluating the hypersingular operator at nonsmooth points are tested. Both methods seem to provide reliable solutions. The modified method encounters minor failures in the frequency response function at a geometric singularity, although performing surprisingly well in many cases. More tests need to be carried out to assess fully the effectiveness of this method which allows easy use of continuous quadratic elements. However, it is the Burton and Miller formulation which appears to be the most reliable for acoustic radiation analysis. The use of CHIEF and its variations cannot be recommended.

Computation of Acoustic Shape Design Sensitivity Using a Boundary Element Method

1992 ◽  
Vol 114 (1) ◽  
pp. 127-132 ◽  
Author(s):  
D. C. Smith ◽  
R. J. Bernhard
Keyword(s):  
Integral Equation ◽  
Boundary Element ◽  
Acoustic Radiation ◽  
Design Sensitivity ◽  
Shape Design ◽  
Acoustic Response ◽  
Design Variables ◽  
Prediction Scheme ◽  
Shape Design Sensitivity ◽  
Helmholtz Integral

Numerical acoustic radiation prediction schemes have developed to the point where they can be used to reliably predict the acoustic response of a vibrating structure. However, the objective of many of the applications of these techniques is the design of a best structural/acoustic configuration for a given application. Thus, the acoustic radiation prediction scheme must be incorporated into a design methodology. Traditionally, the most difficult design variables to incorporate into numerical prediction schemes have been those which define the geometry of the configuration, normally referred to as shape variables. In this investigation, a technique for computing the design sensitivity of the radiated pressure solution to shape variables is formulated and verified. The method uses a boundary element implementation of the Helmholtz integral equation. The potential problems at the characteristic nonuniqueness frequencies of the Helmholtz integral equation are also addressed. The technique is verified for several pulsating sphere examples where analytical solutions are available.

Finite Elements with Nonreflecting Boundary Conditions Formulated by the Helmholtz Integral Equation

1999 ◽  
Vol 121 (2) ◽  
pp. 214-220
Author(s):  
Shu-Wei Wu
Keyword(s):  
Integral Equation ◽  
Finite Elements ◽  
Surrounding Medium ◽  
Constraint Equation ◽  
Thin Body ◽  
Source Surface ◽  
Element Formulation ◽  
Artificial Boundary ◽  
Acoustic Domain ◽  
Helmholtz Integral

In the proposed approach, an acoustic domain is split into two parts by an arbitrary artificial boundary. The surrounding medium around the vibrating surface is discretized with finite elements up to the artificial boundary. The constraint equation specified on the artificial boundary is formulated with the Helmholtz integral equation straightforwardly, in which the source surface coincides with the vibrating surface discretized with boundary elements. To ensure the uniqueness of the numerical solution, the composite Helmholtz integral equation proposed by Burton and Miller was adopted. Due to the avoidance of singularity problems inherent in the boundary element formulation, this method is very efficient and easy to implement in an isoparametric element environment. It should be noted that the present method also can be applied to thin-body problems by using quarter-point elements.

scholarly journals Numerical Analysis of Structure Acoustic Radiation Based on Wave Superposition Method

2011 ◽  
Vol 2-3 ◽  
pp. 733-738
Author(s):  
Sheng Yao Gao ◽  
De Shi Wang ◽  
Yi Qun Du
Keyword(s):  
Integral Equation ◽  
Acoustic Field ◽  
Acoustic Radiation ◽  
Boundary Surface ◽  
Impact Factors ◽  
Power Calculation ◽  
Superposition Method ◽  
Sound Power ◽  
Wave Superposition ◽  
Equivalent Sources

To overcome the non-uniqueness of solution at eigenfrequencies in the boundary integral equation method for structural acoustic radiation, wave superposition method is introduced to study the acoustics characteristics including acoustic field reconstruction and sound power calculation. The numerical method is implemented by using the acoustic field from a series of virtual sources which are collocated near the boundary surface to replace the acoustic field of the radiator, namely the principle of equivalent. How to collocate these equivalent sources is not indicated definitely. Once wave superposition method is applied to sound power calculation, it is necessary to evaluate its accuracy and impact factors. In the paper, the basic principle of wave superposition method is described, and then the integral equation is discretized. Also, the impact factors including element numbers, frequency limitation, and distance between virtual source and integral surface are analyzed in the process of calculate the acoustic radiation from the simply supported thin plate under concentrated force. The extensive measures of acoustic field at the thin plate are compared with results obtain using different numerical methods. The results show that: (a) The agreement between the results from the above numerical methods is excellent. The wave superposition method requires fewer elements and hence is faster. But the extensive numerical modeling suggests that as long as the volume velocity matching yields more than adequate accuracy. (b) The equivalent sources should be collocated inside the radiator. And the accuracy of a given Gauss integration formula will decrease as the source approaches the boundary surface. (c) The numerical method is applicable to the acoustic radiation of structure with complicated shape. (d) The method described in this paper can be used to perform effectively sound power calculation, and its application range can be extended on the basis of these conclusions.

scholarly journals The Burton and Miller Method: Unlocking Another Mystery of Its Coupling Parameter

2016 ◽  
Vol 24 (01) ◽  
pp. 1550016 ◽  
Author(s):  
Steffen Marburg
Keyword(s):  
Integral Equation ◽  
Literature Review ◽  
Coupling Parameter ◽  
Solution Process ◽  
Normal Derivative ◽  
The Neumann Problem ◽  
Spurious Modes ◽  
Specific Formulation ◽  
Wrong Sign ◽  
Helmholtz Integral

The phenomenon of irregular frequencies or spurious modes when solving the Kirchhoff–Helmholtz integral equation has been extensively studied over the last six or seven decades. A class of common methods to overcome this phenomenon uses the linear combination of the Kirchhoff–Helmholtz integral equation and its normal derivative. When solving the Neumann problem, this method is usually referred to as the Burton and Miller method. This method uses a coupling parameter which, theoretically, should be complex with nonvanishing imaginary part. In practice, it is usually chosen proportional or even equal to [Formula: see text]. A literature review of papers about the Burton and Miller method and its implementations revealed that, in some cases, it is better to use [Formula: see text] as coupling parameter. The better choice depends on the specific formulation, in particular, on the harmonic time dependence and on the fundamental solution or Green’s function, respectively. Surprisingly, an unexpectedly large number of studies is based on the wrong choice of the sign in the coupling parameter. Herein, it is described which sign of the coupling parameter should be used for different configurations. Furthermore, it will be shown that the wrong sign does not just make the solution process inefficient but can lead to completely wrong results in some cases.

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