Boundary Element Method for Acoustic Sensitivity Analysis of Vibrating Structures

Abstract: In the boundary element method, treatment of all the possible singular integrals is very important for the correctness and accuracy of the solutions. Generally, the directional derivative of a weakly singular integral is computed by an integral in the sense of Cauchy principle value if the directional derivative of the weakly singular integral kernel is strongly singular or in the sense of Hadamard finite part integral if the the directional derivative of the weakly singular integral kernel is hypersingular. We will try to discover how the strongly singular and hypersingular integrals are generated and propose a method to avoid the appearance of such kind of strongly singular and hypersingular integrals. Here, using some examples, we will show that with an ’exact derivation’ the directional derivative of a weakly singular integral will still be a weakly singular integral or a normal integral. That is none strongly or hypersingular integrals are generated in such a process. And therefore, Cauchy principle value and Hadamard finite part integral are not indispensable. The result will help us to form a new approach to overcome the so-called strongly and hypersingular integrals occurring in the boundary element method.

A coupled system of finite elements and boundary elements is presented for the modelling of sound propagation in a nonuniform mean flow. Finite elements are used to discretize the acoustic pressure in a bounded computational domain including the nonuniform flow field. The radiation of acoustic waves in the exterior uniform flow field is modelled by a boundary integral representation formula. The interface condition on the shared boundary yields an integral equation which is solved using boundary elements. The coupling of finite elements and boundary elements is described in detail for sound propagation in a stationary fluid where the acoustic pressure is governed by the Helmholtz equation. The coupled computational system is applied to estimate scattering and refractive effects on the acoustic pressure waves through a boundary layer surrounding an aircraft’s fuselage. The coupled mathematical formulation for three dimensional sound radiation around an engine in subsonic flow is discussed.

A study on the applications of the acoustic design sensitivity analysis of vibrating bodies

A numerical study of 2D electrothermal flow using boundary element method

The vibration response and far‐field sound radiation of a semisubmerged, finite cylindrical shell with low‐frequency excitation are studied. The solution to this problem can be divided into two steps. The first step is to apply the wave propagation approach to determine the vibration response of the cylindrical shell. In the cylindrical coordinate system, the Flügge shell equations and Laplace equation are used to describe the cylindrical shell and surrounding fluid so that the vibration responses of the shell can be addressed analytically. The fluid free surface effect is taken into account by applying the sine series to force the velocity potential on the free surface to be zero. Furthermore, compared with the FEM (the finite element method), the present method is not only reliable but also effective. In the second step, the far‐field sound radiation is solved by the Fourier transform technique and the stationary phase method in accordance with the vibration responses of the shell from the previous step. The boundary element method is applied to validate the reliability of the acoustical radiation calculation. The circumferential directivity of far‐field sound pressure is discussed, and it is found that the maximum value of the sound pressure always appears directly under the structure when the driving frequencies are relatively low. Besides, in consideration of simplicity and less computation effort, the present method can be used for the rapid prediction of the vibration and far‐field sound pressure of a semisubmerged cylindrical shell with low‐frequency excitation.

An analytical model was developed for an MRI gradient coil duct with a micro‐perforated panel (MPP) acoustic absorber placed inside the duct. The absorption coefficients of a series of MPP absorbers with varying design parameters including diameter of the holes, perforation rate and thickness of the air gap were calculated. Boundary Element Method (BEM) models were then developed for the coil duct with MPP absorbers and porous materials with varying design parameters and material properties. The acoustic power, sound pressure levels and sound field inside the duct were calculated using the BEM. The results show that the principal advantage of an MPP acoustic absorber is its effectiveness in absorbing relatively low frequency noise. By using the proper combination of parameters, a MPP acoustic absorber can be designed to be a wide bandabsorber or a narrow band absorber at any specific central frequency. © 2010 Wiley Periodicals, Inc. Concepts Magn Reson Part B (Magn Reson Engineering) 37B: 103–115, 2010

The indirect boundary element method for the Helmholtz equation in three dimensions is of great interest and practical value for many problems in acoustics as it is capable of treating infinitely thin plates and allows coupling of interior and exterior scattering problems. In the present paper we provide a new approach for treatment of boundary integrals, including hypersingular, singular, and nearly singular integrals via analytical expressions for generic boundary conditions on the both sides of the surface. The fast multipole accelerated boundary element solver in Gumerov and Duraiswami (2009) is extended to incorporate the indirect formulation. The new formulation is compared with the analytical solution of scattering off a disk. Previous authors have not provided such comparisons for an extended range of frequencies. The performance of the method and its scalability are investigated. It is demonstrated that problems with hundreds of thousands of boundary elements can be solved efficiently on a personal computer using the present method.

In marine and ocean engineering area, some underwater cylindrical shell structures or vehicles may approach a free surface (water surface) or a rigid wall (seabed). The existence of the free surface or rigid wall makes the surrounding fluid around the structure an acoustic half-space, causing that the vibroacoustic characteristics of the shell may have much difference from those in an infinite fluid field. In this study, the far-field acoustic radiation of a finite cylindrical shell with an arbitrary, time-harmonic surface velocity distribution positioned within an acoustic semi-space is studied. The Fourier transform method is utilized to express acoustical pressure in the wavenumber domain, and the acoustic boundary effect is considered through the use of the Graf’s addition theorem and the image approach. Then, the relationship between acoustical pressure and the normal velocity on the structure surface is established in the wavenumber domain. Finally, the far-field acoustical pressure could be expressed by the stationary phase method approximately. The accuracy of the present approach is verified by contrast with the boundary element method. It could be observed that there are many similarities between the directivity of the far-field acoustical pressure of the cylindrical shell and the acoustical dipole due to the interference of sound waves. For the case of a rigid wall boundary, a similar interference phenomenon also exists. The present method has the advantages of simpleness and fewer calculations; it can be utilized to rapidly predict the far-field acoustical pressure in an acoustic half-space. Besides, after discussing the multiple scattering effects between the acoustic boundary and the shell, a simpler physical model is successfully put forward to resolve the far-field acoustical pressure of an inclined shell within an acoustic semi-space, assuming that the distance between the shell and the acoustical boundary is larger than a certain value (five-time radiuses in the present study).

SummaryThis paper presents a numerical approach for the design sensitivity analysis of modal frequencies of elastic structures submerged in an unbounded heavy fluid domain. Because the feedback of sound pressure onto the fluid‐loaded structures has to be taken into account in this case, a fully coupled scheme which combines the structural finite element method (FEM) and the acoustic boundary element method (BEM) is employed in the numerical simulation. The resulting nonlinear eigenvalue problem created by the coupled finite element and boundary element (FE‐BE) method is converted into a generalized eigenvalue problem (GEP) of reduced dimension by using a contour integral method. The design sensitivity formulation for non‐repeated eigenvalues is derived based on an adjoint method which uses both the left and right eigenvectors, and a small GEP is formed to calculate the gradients of repeated eigenvalues. The Burton‐Miller formulation is employed to shift the fictitious eigenfrequencies and their derivatives, and the parameter setting which is beneficial to the filtering of fictitious eigenfrequencies is investigated. Numerical examples are given to verify the accuracy and effectiveness of the developed method.

Integration of singular integrals for the Galerkin-type boundary element method in 3D elasticity

A State Space Boundary Element Method with Analytical Formulas for Nearly Singular Integrals

This paper presents a method for the analysis of two-dimensional crack problems using the boundary element method. In the present method, a domain to be analyzed is divided into two regions : the near-tip region, which is circular in shape and centered at the crack tip, and the outer region. The displacements and stresses in the near-tip region are represented by the truncated infinite series of the eigenfunction with unknown coefficients. The stress intensity factor is represented by one of the coefficients. The outer region is formulated by the boundary element method. The system of linear algebraic equations with respect to nodal displacements, nodal tractions and the coefficients of the series are constructed by discretized boundary integration in the outer region and the continuity conditions. To test the accuracy of the present method the center slant cracked rectangular plate under a uniform tension has been analyzed. The isoparametric linear elements are used. The calculated dimensionless stress intensity factors are in good accuracy.

The computation of the acoustic scattering from objects in the ocean is useful to analyze the characteristics of the targets, which can be numerically modelled by acoustic boundary element method (BEM). The BEM has a defect of fictitious frequency problem, Burton-Miller method is studied to overcome the singular problems. In this paper, the Hadamard finite-part integral method based on collocation method is studied to overcome the singularity. It is found that, the near singularity is also quite significant and deteriorates largely the accuracy of BEM. A method based on polar transformation is proposed in this paper to overcome such a defect. Once the problems of singularity and near-singularity are solved, the conventional BEM model is established to serve as a foundation for fast BEM.It is well known that conventional BEM yields a full matrix, which is of O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) memory complexity. Furthermore, the time complexity could be up to O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ), if Gaussian elimination method is adopted to solve the linear equation of BEM. Many fast methods are proposed to solve the problems, such as the GMRES iterative solver, fast multipole method and H-matrix (Hierarchical Matrices) method. H-matrix method is applied to overcome the faults of conventional BEM, by using the low-rank approximations to compress blocks of the matrix, which is a pure algebraic method. The ACA (Adaptive Cross Approximation) method is proposed to compute the outer-product form of low-rank matrices, instead of SVD (Singular Value Decomposition), which consumes high CPU time and is of no usability. ACA-BEM is used to predict the acoustic scattering from objects in the ocean, especially the large-scale problems. The directivities of a rigid sphere and a rigid ellipsoid are computed by ACA BEM, the time consumed is very short. For a problem of more than 20,000 DOFs, it can be solved within 2 minutes. Therefore, ACA BEM can be used to solve large-scale acoustic scattering problems very efficiently.

An isogeometric Burton-Miller method for the transmission loss optimization with application to mufflers with internal extended tubes

9R1. Natural Boundary Integral Method and Its Applications. - De-hao Yu (Inst of Comput Math and Sci/Eng Comput, Chinese Acad of Sci, Beijing, ROC). Kluwer Acad Publ, Dordrecht, Netherlands. Distributed in USA by Kluwer Acad Publ, Norwell MA. 2002. 539 pp. ISBN 1-4020-0457-5. $155.00.Reviewed by D Givoli (Dept of Aerospace Eng, Technion-Israel, Haifa, 32000, Israel).Kang Feng (1923-1993) was a well-known Chinese mathematician, and until his death was the President of the Chinese Society of Computational Mathematics. De-hao Yu, the author of this book, was his PhD student. This is an English version of a previous monograph by Yu that appeared in Chinese ten years ago. The book summarizes Yu’s research work based on the mathematical foundation laid by Feng. Most of Feng’s and Yu’s publications have appeared in Chinese and have not been generally accessible to western readers. This monograph, published by Kluwer, is especially welcome since it allows general access to the interesting work of both Feng and Yu. The book focuses on the concept of Natural Boundary Reduction (NBR) in the context of two-dimensional (2D) elliptic problems, and in particular problems governed by Laplace’s equation, the biharmonic equation, plane elasticity equations, and Stokes’ equations. The idea is to replace the differential equation in the given 2D domain Ω by an integral equation on its boundary Γ, and to solve this integral equation numerically on Γ via a variational formulation. This sounds like the basic idea of the Boundary Element Method (BEM); however, in NBR the reduction of the differential equation in Ω to an integral equation on Γ is performed differently, using the so-called natural integral operator, which is also known as the Dirichlet-to-Neumann (DtN) map. The book is written beautifully. It is very clear, interesting, and not dry despite being mathematically rigorous. Although the author is a mathematician, he wrote this book in a way that makes it accessible to mathematically-oriented graduate students and researchers in computational and applied mechanics and engineering. The basic ideas are described clearly and in a well-organized way. In addition, one can learn quite a lot from this book about areas as diverse as singular integrals, harmonic functions, and domain decomposition. The summary on singular integrals in Section 1.4, for example, is superb. The book, with 540 pages, is divided into seven chapters. Chapter 1 introduces the main ideas of NBR, and is followed by Chapters 2–5, which apply the method to the four differential equations mentioned above. Chapter 6 discusses the coupling of NBR and Finite Elements. This is similar to the DtN method, which has been devised independently in the west primarily for wave problems (which are not discussed at all in this book). The last chapter shows how to solve unbounded domain problems using NBR combined with iterative Domain Decomposition. The book also includes a preface (telling about Prof Feng), a list of more than 200 references (about half of which are Chinese publications), and an index. The NBR method is described on the back cover as a competitor to the standard BEM. Such a comparison is not totally appropriate. The BEM is a general method that can handle problems in irregular geometries. It makes use of the full-space Green’s function associated with the operator involved, which does not depend on the given geometry. On the other hand, NBR makes use of the natural integral operator, which depends on the specific geometry under consideration. Finding this operator on Γ (analytically) is equivalent to solving the problem in Ω analytically. Thus, in its simplest form, the NBR method can be applied in practice only in cases where the analytic solution is already known! Of course, there is no point in using NBR in such a manner. However, this does not mean that the idea underlying NBR is not useful. It is definitely useful when the domain of the problem under consideration involves a region where the problem is analytically-solvable and a region where it is not. Such problems are discussed in the last two chapters, where the practical usefulness of the method is demonstrated. In addition, understanding the various ways in which a problem can be represented (via partial differential equations, as a variational problem, by an integral equation, etc) is always beneficial and may lead to interesting insights regarding the properties of solutions and to new computational methods. Natural Boundary Integral Method and Its Applications is neither a course textbook nor a state-of-the-art research book on a wide scientific area, but it is a satisfying self-contained summary of a very interesting piece of work that has been hidden from the western reader so far. This monograph is highly recommended as an enjoyable and eye-opening reading for the mathematically-oriented researcher and practitioner of applied mechanics.