Surface Helmholtz Integral Equation Research Articles

A semi-analytic method for vibro-acoustic analysis of coupled propeller-shaft-hull systems under propeller excitations

A semi-analytic dynamic model is developed for vibro-acoustic analysis of submerged coupled propeller-shaft-hull systems under propeller forces. The propeller, shaft and hull are idealized as a lumped mass, multi-span beam and ring-stiffened shell of revolution, respectively. Stern, middle and thrust bearings are simulated as spring-damping systems. The hull is firstly decomposed into many segments, which are treated as conical shells and analyzed through combining Flügge shell theory and power series method. The shaft is divided into three Timoshenko beams according to the bearings. By expanding acoustic pressure and velocity as Fourier series in circumferential direction, surface Helmholtz integral equation is reduced to line integral, and acoustic pressure is further expressed as displacements of segments after meshing the line to some 3-node isoparameter elements. Boundary conditions and continuity conditions modified by external acoustic pressure are orderly assembled to the dynamic model. As considering the hull as a ring-stiffened conical-cylindrical-spherical shell, vibro-acoustic results of present method are compared with ones of coupled finite element-boundary element method, and the accuracy and efficiency of developed dynamic model are demonstrated. Furthermore, parameter analysis reveals that breathing and beam modes are predominant for vibro-acoustic responses, and sound power is mainly dominated by the cylindrical compartment.

A unified semi-analytic method for vibro-acoustic analysis of submerged shells of revolution

A unified semi-analytic method is developed to predict vibro-acoustic responses of submerged shells of revolution. The shell is firstly decomposed to several narrow shell segments to be treated as conical shells. By employing Flügge shell theory and power series, displacements and forces at any cross-section of conical shells are expressed as eight unknown coefficients. Then, expanding surface pressure and velocity as Fourier series in circumferential direction, the surface Helmholtz integral equation is reduced to the line integral along the generator line, and the pressure is further represented as displacements of all segments through meshing the generator line to some 3-node isoparametric elements and utilizing relationships of velocities and displacements. Last, continuity conditions between adjacent segments, which include modifications introduced by acoustic pressure, and elastic boundary conditions are assembled to the finial vibro-acoustic governing equation of submerged shells of revolution. Through comparing vibro-acoustic results of present method with ones in appropriate references or calculated by FEM/BEM for independent conical, cylindrical and spherical shells and combined spherical-cylindrical-spherical and conical-cylindrical-spherical shells, rapid convergence, wide application and high accuracy of the semi-analytic model are demonstrated. Meanwhile, contributions of different circumferential modes and different compartments of the combined conical-cylindrical-spherical shell to vibro-acoustic responses are investigated.

Evaluation of 2‐D Green's boundary formula and its normal derivative using Legendre polynomials, with an application to acoustic scattering problems

AbstractThis paper presents the non‐singular forms, in a global sense, of two‐dimensional Green's boundary formula and its normal derivative. The main advantage of the modified formulations is that they are amenable to solution by directly applying standard quadrature formulas over the entire integration domain; that is, the proposed element‐free method requires only nodal data. The approach includes expressing the unknown function as a truncated Fourier–Legendre series, together with transforming the integration interval [a, b] to [‐1,1] ; the series coefficients are thus to be determined. The hypersingular integral, interpreted in the Hadamard finite‐part sense, and some weakly singular integrals can be evaluated analytically; the remaining integrals are regular with the limiting values of the integrands defined explicitly when a source point coincides with a field point. The effectiveness of the modified formulations is examined by an elliptic cylinder subject to prescribed boundary conditions.The regularization is further applied to acoustic scattering problems. The well‐known Burton–Miller method, using a linear combination of the surface Helmholtz integral equation and its normal derivative, is adopted to overcome the non‐uniqueness problem. A general non‐singular form of the composite equation is derived. Comparisons with analytical solutions for acoustically soft and hard circular cylinders are made. Copyright © 2001 John Wiley & Sons, Ltd.

A robust formulation for solving fluid-structure interaction problems

In this article, a coupled finite element and boundary element approach for the acoustic radiation and scattering from submerged elastic bodies of arbitrary shape is presented. An alternative to the direct boundary element method for acoustics is proposed. By taking an auxiliary source surface inside the radiating boundary and following the usual discretization and integration procedures employed in the boundary element method, both the singularities of the integrands and the nonuniqueness problems do not arise. In addition, the difficulty of slope discontinuity also can be overcome. This procedure is formulated in a similar fashion of wave superposition method, except that the direct boundary integral equations are adopted. The proposed formulation employ the surface Helmholtz integral equation and its normal gradient like that adopted in the Burton–Miller approach, but do not employ any coupling constant. Typical examples are presented that demonstrate the efficiency of the proposed technique.

SOUND ATTENUATION CHARACTERISTICS OF RIGHT-ANGLE PIPE BENDS

The numerical solution of the interior surface Helmholtz integral equation for toroidal pipes is presented. Acoustic surface pressures of various pipes in the shape of a quarter torus were calculated. Sound attenuation spectra of pipes possessing different bend sharpnesses are presented. A lower limit for the bend sharpness is determined for the attainment of a considerable level of sound attenuation.

A three-dimensional acoustic infinite element based on a prolate spheroidal multipole expansion

This paper describes a new three-dimensional (3-D) time-harmonic acoustic infinite element for modeling acoustic fields in exterior domains, typically surrounding a structure. This ‘‘prolate spheroidal infinite element’’ is based on a new multipole expansion that is the exact solution for arbitrary scattered and/or radiated fields exterior to a prolate spheroid of any eccentricity. A combination of both prolate and oblate spheroidal infinite elements (the latter to be published separately) provides a capability for very efficiently modeling acoustic fields surrounding structures of virtually any practical shape. This new prolate element has symmetric matrices that are as cheap to generate as for 2-D elements because only 2-D integrals need to be numerically evaluated. The prolate element (along with a symmetric-matrix fluid–structure coupling element, also to be published separately) fits naturally into purely structural finite element codes, thereby providing a structural acoustics capability. For the class of coupled structural acoustic problems studied so far, this approach is two to three orders of magnitude faster, for the same accuracy, than the boundary element method (BEM), which is based on the surface Helmholtz integral equation. Experience with developing and using both BEM and infinite-element codes has revealed that the severe computational inefficiency of the BEM limits practical 3-D structural acoustic problems to low frequencies and simple structures, whereas this new infinite-element approach can handle the full range of frequencies and structural complexities that are encountered in purely structural analyses, at little additional cost.

A Numerical Study Of The Coefficient Matrix Of The Boundary Element Method Near Characteristic Frequencies

In this paper the well-known failure in numerical computations of the exterior surface Helmholtz integral equation at certain characteristic frequencies is investigated. The problem of characteristic frequencies is numerically reflected in an ill-conditioning of the coefficient matrix arising from the transformation of the integral equation to a linear set of equations. By means of a rank revealing factorization the rank deficiency of the coefficient matrix is determined, and thereby the necessary number of equations to be added to the original system of equations can be found. One of the most popular methods to overcome this "non-uniqueness" problem is the Combined Helmholtz Integral Equation Formulation (CHIEF) proposed by Schenck [1]. However, CHIEF points placed on or near a nodal surface of the corresponding interior problem do not provide a linearly independent constraint, and the problem of selecting "good" CHIEF points is still a topic of considerable interest. In this paper the CHIEF is used with a rank revealing factorization, and it is shown that the necessary number of "good" CHIEF points can be predicted; furthermore, a method of determining whether a CHIEF point is "good" is presented. The new approach has been tested on an axisymmetric BEM formulation with one or several CHIEF points.

Structural acoustic modeling using a new high accuracy acoustic infinite element

This new 3-D time-harmonic acoustic infinite element uses a truncated form of the exact asymptotic series solution for general scattered and/or radiated fields, providing an accurateapproximationallthewaytoinfinity. The infinite element and a companion fluid–structure coupling element have symmetric matrices that fit seamlessly into any structural finite element code, providing a structural acoustic capability as versatile and powerful as the structural code itself. Most important, this approach is severalordersofmagnitude faster, for the same accuracy, than the traditional and almost universally used boundary element method (BEM) for the acoustics, i.e., the surface Helmholtz integral equation, which the author also developed and used for several years. The reason is that the bandwidth of the coupled system equations is much smaller for infinite elements (being a domain method) than for the BEM. Experience with both methods has revealed that the severe computational inefficiency of the BEM limits practical problems to low frequencies and simple structures, whereas this infinite element approach can handle the full range of frequencies and structural complexities that are encountered in purely structural analyses, at little additional cost.

Acoustic radiation from cylinders with a plane of symmetry using internal multipole line source distributions. I

A new method is proposed to address the two-dimensional exterior acoustic radiation problem from infinite cylinders with a plane of symmetry. This approach, which has its roots in fluid mechanics, is based on the use of internal monopole and dipole line source distributions within the surface. The monopole and dipole line source distributions are determined by prescribed normal velocity conditions over the surface of the shell and are subsequently used to determine the associated surface pressure. The exterior acoustic field is then simply calculated using the surface velocity and pressure via the usual surface Helmholtz integral equation. In contrast to earlier integral equation formulations of the problem the present method does not suffer from either nonuniqueness or result in hypersingular integral equations. Numerical results are presented to illustrate the accuracy of the new internal source density method for the case of a circular cylinder subjected to various specified normal velocity boundary conditions and frequencies. Additional numerical results are also presented for the exterior acoustic radiation problem for a strip on an elliptical cylinder. These results show the characteristics of the surface pressure as a function of the surface velocity and strip width.

Uniqueness of Solutions to Variationally Formulated Acoustic Radiation Problems

Determination of the surface acoustic pressure given the surface velocity of a vibrating body can be formulated in various ways. However, for some such formulations such as the surface Helmholtz integral equation, solutions are not unique at certain discrete frequencies. Such uniqueness problems can also be present for variational formulations of the problem, but the variational formulation based on the normal derivative of the Kirchhoff integral theorem has unique solutions for vibrating disks and plate-like bodies. For bodies of finite volume, but for which each surface point is vibrating in phase, the total radiated acoustic power is always unique, even though the pressure may not be. The latter conclusion is supported by numerical calculations based on the Rayleigh-Ritz technique for the case of a finite cylinder vibrating as a rigid body in the axial direction.

Stationary variational expressions for radiated and scattered acoustic power and related quantities

The construction of stationary expressions for quantities of physical interest such as radiated power and target strength is discussed broadly for acoustic problems involving radiation or scattering from finite objects of arbitrary shape. The Kirchhoff-Helmholtz integral corollaries of the wave equation, which express acoustic pressure at either interior or exterior points in terms of pressure and its normal derivative over any closed surface, yield for both interior and exterior problems two mathematically dissimilar but related functional relations between surface field quantities. One of these is the better known surface Helmholtz integral equation; the other is a differential-integral relation which involves the tangential derivatives of pressure on the surface. The four linear operators involved in these functional relations are studied and it is found that two are self-adjoint, while the other two are an adjoint pair. A general technique for constructing variational expressions recently developed by Gerjuoy et al. [28] is adapted to acoustic radiation and scattering problems with the functional relations taken as the primary governing relations. Included examples are stationary expressions for the radiated power when either the normal velocity or the pressure are specified on the surface (the other quantity being unknown) and the target strength for scattering from a rigid object. The adjoint relations allow simple physical interpretations for the Lagrange multipliers that arise in the theory, such that the guesses for good trial functions can take advantage of existing physical insight. It is demonstrated with a specific example (transversely vibrating disk) that the resulting estimate for radiated power is substantially more accurate than that of the trial function for surface pressure which was inserted into the stationary expression.

Holographic reconstruction of odd‐shaped 3‐D sources

As previously described [J. D. Maynard, J. Acoust. Soc. Am. Suppl. 174, S37 (1983)], nearfield acoustical holography can reconstruct the surface velocity and pressure over a level surface in a separable coordinate system from measusrements of the radiated pressure field over a parallel level surface. This study investigates the feasability of reconstructing the surface pressure and normal velocity over surfaces which do not conform to a level surface. Such a reconstruction involves numerically determining a distribution of surface pressure and velocity which satisfies the surface Helmholtz integral equation and which is consistent with the Helmholtz integral equation for the pressure field measured on the level surface. Details of the algorithm and results of computer simulations for a uniformly pulsating sphere and a piston set in a sphere (measurements simulated over a planar surface) are presented. [Work supported by ONR and NASA.]

Solution of radiation problems by collocation of integral formulations in terms of single and double layer potentials

The purpose of this paper is the solution of the exterior steady state acoustic radiation problem for an arbitrary surface of which the normal velocity is specified, by using integral formulations. To obtain integral formulations, both a direct method, based on the surface Helmholtz integral equation, and an indirect method, based on single and double layer potentials, have been used. Filippi's method, with a hybrid potential, ensures uniqueness for any wavenumber. Three different surface elements have been used as a basis for obtaining approximate solutions by collocation of the integral equations. A comparison of the rates of convergence for the three surface elements is presented, based on a suitably defined overall error. There is no overall convergence for the double layer representation when a constant approximate solution is used for each surface element in spite of the good local convergence at the centers of the elements. The rate of convergence also appears to be highly dependent on the normal interpolation technique. Special care has been taken in the calculation of the singular integrals which are the terms of self-influence. The numerical implementation of the method, which has been tested for special cases, is applicable to general three dimensional sound radiation problems, since no special symmetry conditions are involved.

Numerical investigation of an acoustic slow waveguide

The effect of a cylinder of material whose sound speed is less than that of water on the radiation beam pattern of an omnidirectional sound source is investigated. Specifically a point source is placed near one end of the cylinder. The cylinder serves as an acoustic waveguide and enhances the radiation from the source in the direction of the waveguide. The waveguide is mathematically modeled by use of the surface Helmholtz integral equation together with the requirement of continuity of acoustic pressure and normal velocity on the surface common to the waveguide and surrounding water as reported by P. H. Rogers and . J. Trott [J. Acoust. Soc. Am. 56, 1111–1117 (1974)]. The integral equations are solved numerically. Calculated farfield patterns and directivity gains for waveguides of various length-to-diameter ratios are presented. Directivity gains greater than 14 dB together with farfield patterns having side lobes more than 10 dB down are predicted over fairly broad bandwidths for several waveguides. Experimental results were obtained and are in good agreement with the numerical results.

Numerical investigation of an acoustic slow waveguide

The effect of a cylinder of material whose sound speed is less than that of water on the radiation beam pattern of an omnidirectional sound source is investigated. Specifically a point source is placed near one end of the cylinder. The cylinder serves as an acoustic waveguide and enhances the radiation from the source in the direction of the waveguide. The waveguide is mathematically modeled by use of the surface Helmholtz integral equation together with the requirement of continuity of acoustic pressure and normal velocity on the surface common to the waveguide and surrounding water as reported by P. H. Rogers and . J. Trott [J. Acoust. Soc. Am. 56, 1111–1117 (1974)]. The integral equations are solved numerically. Calculated farfield patterns and directivity gains for waveguides of various length‐to‐diameter ratios are presented. Directivity gains greater than 14 dB together with farfield patterns having side lobes more than 10 dB down are predicted over fairly broad bandwidths for several waveguides. Experimental results were obtained and are in good agreement with the numerical results.

Radiation of difference-frequency sound generated by nonlinear interaction in a silicone rubber cylinder

In this paper we describe an experimental and theoretical investigation of the parametric signal generated in one medium, a silicone rubber cylinder, and radiated into a second medium, water. The primary waves (∠1.4 MHz) are confined to a narrow region around the axis of the cylinder and, because of the relatively high absorption at these frequencies, are largely attenuated before reaching the end of the cylinder. The silicone rubber cylinder serves two purposes. First, the low sound speed (∠1000 m/sec) and high parameter of nonlinearity (1+B/2A?5) of the silicone rubber makes the virtual sources more than seven times as strong as they would be in water. Second, because the sound speed in silicone rubber is less than that of the surrounding water, the silicone rubber cylinder acts as a slow waveguide antenna for the difference-frequency waves. The system is analyzed by numerically solving the system of equations obtained by coupling the surface Helmholtz integral equation for the interior of the silicone cylinder (which includes the virtual sourcesI with the integral equation for the region exterior to the cylinder. Experimental results obtained for difference frequencies between 10 and 20 kHz compare favorably with theory. Subject Classification: [43]25.35, [43]25.15; [43]30.75.

Acoustic slow waveguide antenna

The acoustic analog of the dielectric rod antenna for electromagnetic radiation is investigated both theoretically and experimentally. In acoustics, a cylinder of silicone rubber (whose bulk sound speed is about two-thirds that of water) is analogous to the dielectric rod. An acoustic wave in the direction parallel to the axis of the waveguide is concentrated within the guide by progressive refraction. A hydrophone (whose free-field sensitivity is omnidirectional) is attached near one end of the waveguide and detects this concentrated radiation as a gain in signal for this direction relative to any other direction. Similarly, if the slow waveguide is attached to an acoustic source, it will enhance the source level in the direction of the waveguide. The antenna is mathematically modeled by coupling the surface Helmholtz integral equation appropriate to the region within the silicone waveguide to a similar equation for the water region exterior to the cylinder (which includes the point hydrophone) by requiring continuity of acoustic pressure and normal velocity on the common surface. The coupled integral equations are solved numerically. Predicted beam patterns and directivity gains are in excellent agreement with experiment. Broad-band directivity gains of over 9 dB are both predicted and observed for two test antennas.

Acoustic slow waveguide antenna

The acoustic analog of the dielectric rod antenna for electromagnetic radiation is investigated both theoretically and experimentally. In acoustics, a cylinder of silicone rubber (whose bulk sound speed is about two-thirds that of water) is analogous to the dielectric rod. An acoustic wave in the direction parallel to the axis of the waveguide is concentrated within the guide by progressive refraction. A hydrophone (whose free-field sensitivity is omnidirectional) is attached near one end of the waveguide and detects this concentrated radiation as a gain in signal for this direction relative to any other direction. Similarly, if the slow waveguide is attached to an acoustic source, it will enhance the source level in the direction of the waveguide. The antenna is mathematically modeled by coupling the surface Helmholtz integral equation appropriate to the region within the silicone waveguide to a similar equation for the water region exterior to the cylinder (which includes the point hydrophone) by requiring continuity of acoustic pressure and normal velocity on the common surface. The coupled integral equations are solved numerically. Predicted beam patterns and directivity gains are in excellent agreement with experiment.

Formal solution of the surface Helmholtz integral equation at a nondegenerate characteristic frequency

The surface Helmholtz integral equation (SHIE) for the acoustic pressure on an object whose surface S vibrates harmonically with a given normal velocity, is known to fail by nonuniqueness at certain “characteristic” frequencies. These frequencies are identical to the eigenfrequencies of the homogeneous Dirichlet problem for a volume of fluid enclosed by the surface S. This paper deals with nondegenerate characteristic frequencies, that is, frequencies at which exactly one independent solution of the interior homogeneous Dirichlet problem exists but no nontrivial solutions of the interior homogeneous Neuman problem exist. The problem is reformulated in terms of operators on the Hilbert space of functions which are square integrable over S. The eigenvectors of the (skew-Hermitian) Green's function operator for the interior Neumann problem provide an orthonormal basis for the space and, hence, the unknown Green's function operator for the exterior Dirichlet problem can be completely specified by its matrix elements with respect to these eigenvectors. The nonuniqueness failure of the SHIE is found to result from the indeterminacy of a single element of this infinite matrix. An expression for the indeterminate matrix element is obtained directly from the SHIE by a perturbation technique yielding a formal solution of the problem.

Finite element analysis of acoustically radiating structures with applications to sonar transducers

A numerical technique for mathematically modeling the vibrational response of a complex structure immersed in an infinite acoustic medium is presented. The elastic response of the structure is modeled using the finite element method, and the acoustic radiation loading on the structure is modeled by approximating the surface Helmholtz integral equation formulation of the acoustic radiation problem. Arbitrary (and distinct) nodal point distributions and interpolation functions can be used in the finite element and acoustic radiation models. A technique defined in terms of these nodes and interpolation functions is presented for combining the results of these models into a combined equation of motion for the acoustically damped structure. The application of this technique to sonar transducers is discussed, including the modeling of piezoelectric material. The problem of obtaining reliable piezoelectric material parameters is discussed. Mathematical models are given for a piezoelectric sphere, a piezoelectric free-flooded cylinder, and a coaxial array of two cylinders. In general, comparison of numerical and experimental results (electric impedance plot, transmitting voltage response, beam patterns) for these devices agree within 5%. Results for the coaxial array exhibit somewhat more error.