Pulsating Sphere Research Articles

An efficient technique for multi-frequency acoustic analysis by boundary element method

A new technique for multi-frequency acoustic analysis by boundary element method is proposed. This new technique is based on the elimination of the frequency-dependent character of the boundary element coefficient matrices. By making use of the algebraic polynomials and taking out the frequency term from integrands, numerical integration involved in setting up the coefficient matrices is only confined to a frequency-independent part and the boundary element global coefficient matrices obtained by this technique are frequency independent. The final global coefficient matrices at any frequency can be simply formed by a summation of the frequency-independent global matrices and the need to compute the numerical integration at each frequency is eliminated. The technique therefore seems to be especially interesting when the bulk of the computational effort is spent on the integration phase of the solution process for a multi-frequency analysis. The test case of sound radiation from a pulsating sphere is given to demonstrate the technique. The numerical results show that this new technique can significantly save the CPU time for a multi-frequency problem.

Solving the hypersingular boundary integral equation in three-dimensional acoustics using a regularization relationship.

Regularization of the hypersingular integral in the normal derivative of the conventional Helmholtz integral equation through a double surface integral method or regularization relationship has been studied. By introducing the new concept of discretized operator matrix, evaluation of the double surface integrals is reduced to calculate the product of two discretized operator matrices. Such a treatment greatly improves the computational efficiency. As the number of frequencies to be computed increases, the computational cost of solving the composite Helmholtz integral equation is comparable to that of solving the conventional Helmholtz integral equation. In this paper, the detailed formulation of the proposed regularization method is presented. The computational efficiency and accuracy of the regularization method are demonstrated for a general class of acoustic radiation and scattering problems. The radiation of a pulsating sphere, an oscillating sphere, and a rigid sphere insonified by a plane acoustic wave are solved using the new method with curvilinear quadrilateral isoparametric elements. It is found that the numerical results rapidly converge to the corresponding analytical solutions as finer meshes are applied.

Hydrodynamic coefficients of a submerged pulsating sphere in finite depth

By extending the work of Linton (Linton, C.M., 1991. Radiation and diffraction of waver waves by a submerged sphere in finite depth. Ocean Engineering 18 (1/2), 61–74), the problem of radiation of water waves by a submerged pulsating sphere in finite depth is formulated using the multipole method. As in Linton (1991), this leads to an infinite system of linear equations, which are easily solved numerically. Simple expressions are derived for the hydrodynamic characteristics of such a body. Results showing the effect of varying both the immersion depth and the water depth on the hydrodynamic coefficients of the pulsating sphere are given. The paper resumes the work presented in Lopes (Lopes, D.B.S., 1999. On the study of the Archimedes wave swing device for wave energy utilization (in Portuguese). MSc on the Management and Modelling of the Marine Environment, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa.).

Aeroacoustic Modelling of Low-Speed Flows

A new numerical algorithm for acoustic noise generation is developed. The approach involves two steps comprising an incompressible flow part and an inviscid acoustic part. The acoustic part can be started at any time of the incompressible computation. The formulation can be applied both for isentropic flows and non-isentropic flows. The model is validated for the cases of an isentropic pulsating sphere and non-isentropic flows past a circular cylinder. In the latter case the computations show that the generated acoustic field in addition to the dominant Strouhal frequency, f0, contains a slightly higher frequency, f2, and a modulating lower frequency, f1=f2−f0. Numerical experiments with different interpolation schemes, boundary conditions, etc., show that the appearance of these modes is not an artifact from the numerical discretization.

The full-field equations for acoustic radiation and scattering

The source simulation technique or related approaches like the multipole method, the superposition method, etc. are used for calculating the sound field radiated (or scattered) from complex-shaped structures. However, it is known that these techniques can lead to ill-conditioned systems of equations, and their numerical treatment requires extreme care. A new stabilized variant of the source simulation technique—called the full-field method—has been developed by using the exterior instead of the interior Helmholtz integral formulation or, equivalently, by expanding the sound field into special trial and weighting functions. These functions are chosen in such a way that the resulting matrix becomes more diagonally dominant. The full-field method is applied to the acoustic radiation from a pulsating sphere and to the high-frequency scattering from a cylinder and a nonconvex structure. The numerical results are compared with calculations obtained from other methods. It is shown that the improved method leads to better conditioned sets of equations which can be solved directly without singular-value decomposition, since the associated condition numbers are decreased strongly, in some cases by a few orders of magnitude.

Acoustic shape sensitivity analysis using the boundary integral equation

Boundary integral equations are formulated for the shape sensitivity analysis of the acoustic problems. The concept of the material derivative is employed in deriving the sensitivity equations. Since the equation is derived by the direct differentiation of the boundary integrals containing the field values, it is expected that the sensitivity would be computed more effectively and accurately than the conventional finite difference method. In addition, the equation has the potential to be applied to many complex acoustic problems, because the derived equation is regularized by using the integral identity that incorporates the one-dimensional propagating wave and its material derivative. The validity of the formulations is demonstrated through examples having regular shapes such as the three-dimensional pulsating sphere and the one-dimensional duct, for which the analytical solutions are available. As an example for an irregular domain, the two-dimensional model of an automotive interior cavity is dealt with in the view point of the noise level at the passenger’s ear position. The results show that the present method can be an effective tool for the shape optimization in designing the desired sound field. It is noted that the present method permits accurate sensitivities of the acoustic pressure on the boundary as well as at the field points. The present method is thought to be an alternative to the previous finite difference techniques for computing the shape sensitivity using the boundary element method and the formal derivative method using the finite element method.

Numerical simulations of xylophones. II. Time-domain modeling of the resonator and of the radiated sound pressure

This paper presents a time-domain modeling for the sound pressure radiated by a xylophone and, more generally, by mallet percussion instruments such as the marimba and vibraphone, using finite difference methods. The time-domain model used for the one-dimensional (1-D) flexural vibrations of a nonuniform bar has been described in a previous paper by Chaigne and Doutaut [J. Acoust. Soc. Am. 101, 539–557 (1997)] and is now extended to the modeling of the sound-pressure field radiated by the bar coupled with a 1-D tubular resonator. The bar is viewed as a linear array of equivalent oscillating spheres. A fraction of the bar field excites the tubular resonator which, in turn, radiates sound with a certain delay. In the present model, the open end of the resonator is represented by an equivalent pulsating sphere. The total sound field is obtained by summing the respective contributions of the bar and tube. Particular care is given for defining a valid approximation of the radiation impedance, both in continuous and discrete time domain, on the basis of Kreiss’s theory. The model is successful in reproducing the main features of real instruments: sharp attack, tuning of the bar, directivity, tone color, and aftersound due to the bar-resonator coupling.

Solution of a sound field generated by the vibrating surface of a body moving at small Mach number

An approach to calculate the sound field generated by the Vibrating surface of a body with arbitrary shape moving at a small mach number has been developed by using acoustical analogy. A governing equation to calculate the sound pressure on the surface exited by its vibration has been derived on the basis of the united aerodynamics and aeroacoustics approach by Farassat and Long, and the sound field would be determinate. The cases relating to the sound sources of static sphere and moving compact pulsating sphere give a good check for the approach.

A boundary element method based procedure to calculate boundary admittances from measured sound pressures

Owing to the rapid development of computers in the past few years, boundary and finite element methods have become a powerful tool for acoustic calculations of internal and external problems. The calculation of closed domains (i.e. vehicle cabin) is often done using rigid acoustic boundary conditions (Neumann condition) since there is hardly any information about boundary admittance available. The determination of admittances using Kundt's tube does not consider the diffuse sound field. The calculation from the measured reverberation time provides an average admittance of the whole boundary only. This paper presents an application of the boundary element method which can be used to calculate boundary admittances at the nodes of the boundary element mesh from sound pressures measured at mesh points or at internal points. Additionally, this method is suitable for the determination of admittances at openings of the domain, i.e. open windows or the ear canal opening. Herein, one can receive the sound pressure from the calculation of the external acoustic problem or from measurements. The method is validated by consideration of the boundary admittance of a uniformly pulsating sphere.

Shape design sensitivity analysis of acoustic problems using a boundary element method

A boundary integral equation for shape design sensitivity analysis of acoustic problems is presented. In the derivation of the sensitivity equation, the material derivative concept is used. To remove the singularitiess that appear in the conventional sensitivity equation, an integral identity which represents the one-dimensional wave propagation is used. Since the sensitivity equation is regularized, one can obtain the accurate solution using the standard Gaussian scheme. The equation is verified for the pulsating sphere example where an analytical solution is available. The boundary was discretized by a six-node triangular element. As a result, one can obtain the accurate sensitivities of acoustic pressure not only on the boundary but also at the given field points.

A boundary integral approach for acoustic radiation of axisymmetric bodies with arbitrary boundary conditions valid for all wave numbers

A boundary integral method for solving the exterior acoustic radiation problem of axisymmetric bodies with arbitrary boundary conditions is developed. The new formulation derives from the method proposed by Burton and Miller [Proc. R. Soc. London Ser. A 323, 201–210 (1971)], which uses a linear combination of the Helmholtz integral equation and its normal derivative to ensure the uniqueness of the numerical solution at all frequencies. It is shown that the use of tangential operators enables one to easily compute the hypersingular integral without much extra effort. By taking advantage of the properties of axisymmetric geometry, and using the expansion of the boundary conditions and the surface distribution functions in Fourier series with respect to the angle of revolution, the surface integral is reduced to a line integral along the generator of the body, and Fourier integrals of the Green’s function and its derivatives over the circumferential angle. The Fourier integrals with singular kernels are evaluated using recurrence formulas in terms of the complete elliptic integrals. A new approach is then proposed to evaluate the Burton and Miller formulation at a corner by using the left-hand and right-hand limits of normal derivatives. Doing so, isoparametric elements throughout the surface of the body are able to be used. In order to demonstrate the validity and accuracy of the method, numerical results with quadratic isoparametric curvilinear elements are presented for radiation problems of a pulsating sphere, an oscillating sphere, and a finite cylinder with various boundary conditions.

A unique boundary integral approach for acoustic radiation of axisymmetric bodies with arbitrary boundary conditions

A boundary integral method for solving the exterior acoustic radiation problem of axisymmetric bodies with arbitrary boundary conditions has been developed. The new formulation derives from the method proposed by Burton and Miller, which uses a linear combination of the Helmholtz integral equation and its normal derivative to ensure the uniqueness of the numerical solution at all frequencies. By taking advantage of the properties of axisymmetric geometry, and using the expansion of the boundary conditions and the surface distribution functions in Fourier series with respect to the angle of revolution, the surface integral is reduced to a line integral along the generator of the body, and Fourier integrals of the Green’s function and its derivatives over the circumferential angle. A main feature of this formulation is that new recurrence formulas of the Fourier coefficients have been developed to evaluate accurately the Fourier integrals with the singular kernels in terms of the complete elliptic integrals. In order to demonstrate the validity and accuracy of the method, numerical results with quadratic isoparametric curvilinear elements are presented for radiation problems of a pulsating sphere, an oscillating sphere, and a finite cylinder with various boundary conditions.

A FRAMEWORK FOR EVALUATING BOUNDARY CONDITIONS

This paper explores solutions to the spherically symmetric Euler equations. Motivated by the work of Hagstrom and Hariharan7 and Geer and Pope,5 we model the effect of a pulsating sphere in a compressible medium. The literature available on this suggests that an accurate numerical solution requires artificial boundary conditions which simulate the propagation of nonlinear waves in open domains. Until recently, the boundary conditions available are in general linear, and based on non-reflection. Exceptions to this are the nonlinear nonreflective conditions of Thompson,11 and the nonlinear reflective condition of Ref. 7. The former is based on the rate of change of the incoming characteristics, while the latter relies on asymptotic analysis and the method of characteristics and accounts for the coupling of incoming and outgoing characteristics. Furthermore in Ref. 7 it was shown in a test situation in which the flow would reach a steady state over a long time, the method proposed in Ref. 11 could lead to an incorrect steady state. The current study considers periodic flows. Moreover, various other types and techniques of boundary conditions are included in this study. The technique recommended by Ref. 7 proved superior to all others considered, and matched the results of asymptotic methods which are valid for low subsonic Mach numbers.

Analytical Solutions for Harmonic Wave Propagation in Poroelastic Media

This paper presents several analytical solutions for problems of harmonic wave propagation in a poroelastic medium. The pressure‐solid displacement form of the harmonic equations of motion for a poroelastic solid are developed from the form of the equations originally presented by Biot. Then these equations are solved for several particular situations. Closed‐form analytical solutions are obtained for several basic problems: independent plane harmonic waves; radiation from a harmonically oscillating plane wall; radiation from a pulsating sphere; and the interior eigenvalue problem for a sphere, for the cases of both a rigid surface and a traction‐free surface. Finally, a series solution is obtained for the case of a plane wave impinging on a spherical inhomogeneity. This inhomogeneity is composed of poroelastic material having different properties from those of the infinite poroelastic medium in which it is embedded, and the incident wave may be composed of any linear combination of Biot “fast” and “slow”...

Numerical study of strongly nonlinear acoustic waves, shock waves, and streaming caused by a harmonically pulsating sphere

The nonlinear propagation of spherical waves emitted from a sphere executing harmonic pulsations in an unbounded ideal gas is numerically studied without the restriction of small amplitude, namely, the weak nonlinearity. The energy dissipation due to viscosity and thermal conductivity is supposed to be negligible everywhere except for the discontinuous shock front. By the numerical analysis based on a high-resolution upwind finite difference scheme, the wave phenomena caused by the strongly nonlinear effect are revealed, which form several striking contrasts to the known results for the weakly nonlinear spherical waves. The profile of the emitted wave is rapidly distorted by the strongly nonlinear effect and shock waves are formed near the sphere. The most remarkable phenomenon is the occurrence of acoustic streaming (mean mass outflow), which gradually reduces the density of the gas near the sphere as time proceeds. In the case that the sphere pulsates with a small amplitude compared with its mean radius, beyond the shock formation distance, the wave profile is immediately transformed into an asymmetrical sawtooth-like profile. In the case that the amplitude of pulsation is comparable with the mean radius, two shocks can be formed in each wave cycle and in due course these two shocks merge into one shock.

A dual-reciprocity method for acoustic radiation in a subsonic non-uniform flow

In this paper, the dual-reciprocity boundary-element method is used to model acoustic radiation in a subsonic non-uniform-flow field. The boundary-integral formulation is based on a direct boundary-integral equation developed very recently by the authors for acoustic radiation in a subsonic uniform flow. All the terms due to the non-uniform-flow effect are taken to the right-hand side and treated as source terms. The source terms result in a domain integral in the standard boundary-integral formulation. The dual-reciprocity method is then used to transform the resulting domain integral into boundary integrals. Numerical tests show reasonably good agreement with an analytical soution for a pulsating sphere submerged in a potential-flow field.

On The Mathematics Of Estimating Sound Power By The Scanning Method

A mathematical model is developed estimating sound power P by use of the scanning method. The scanning is assumed to be slow and for the purposes of the analysis, only simple paths are used. The paper shows that scanning along such paths can be very accurate. The paper discusses the existence of a sequence of paths C r the scans of which converge to the true value of P. It is found that, in general, the scan has to take into account a weighting function, a modification that is easy to incorporate in practice. It is argued that what is relevant is the percentage accuracy of the r th scan estimate, and we illustrate this by considering the case of a finite set of uncorrelated small pulsating spheres inside a sphere.

Application of BEM (boundary element method)-based acoustic holography to radiation analysis of sound sources with arbitrarily shaped geometries

A method employing holograms that conform with arbitrarily shaped sources has been developed for enhancing conventional near-field acoustic holography, which has been limited to sources with simple geometries, e.g., planar or cylindrical surfaces. Four holography transformation algorithms have been developed, based on acoustic holography theory and the boundary element method (BEM). Singular value decomposition (SVD) has been incorporated into the algorithms in order to alleviate the ill-posed nature frequently encountered in backward reconstruction of a field. A pulsating sphere, a cylinder with spherical endcaps, and a vibrating piston set in a rigid sphere have been adopted in a numerical simulation for verifying the algorithms. Satisfactory agreement has been achieved between the holographically transformed results and the analytical solutions.

Computation of Acoustic Shape Design Sensitivity Using a Boundary Element Method

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.

Weakly nonlinear waves generated by vibration of a spherical body

The nonlinear propagation of directional spherical waves generated in an unbounded inviscid ideal gas by vibratory motions with small but finite amplitude and moderate frequency of a spherical body is considered. Starting with a regular perturbation expansion for a velocity potential in the near field, a higher-order problem is investigated in the far field up to the shock formation distance. It is thereby shown that, in the far field concerned, a well-known simple far-field equation remains valid for the radial velocity u*r including higher-order corrections up to O(εN/r) (N<−1/ε ln ε), where r is a nondimensional radial coordinate and ε(≪1) is the expansion parameter of the expansion. A boundary condition appropriate to the equation, which ensures the matching of a far-field solution with a near-field solution, can be determined from the near-field solution obtained by the regular perturbation procedure. As an application of the theory, the third-order problem is solved for weakly nonlinear acoustic waves radiated by a pulsating sphere. It is further shown that, for weakly nonlinear cylindrical waves with moderate frequency, a similar far-field equation becomes invalid at the third approximation in the far field up to the shock formation distance.