Helmholtz Integral Equation Research Articles

Estimating bottom slope from pulse duration.

Employing the Fresnel and Kirchhoff approximations in the Helmholtz integral equation and assuming a Gaussian correlation function, an equation can be derived that predicts the time distribution of the acoustic pressure from a pulse propagated over a rough bottom. A simple expression can be found that relates the half duration time of the pulse to the surface rms slope, grazing angle, and slant range. Applying this expression to a measured time series from a rough Pacific thin sediment area, the computed rms slope is found to be larger than that estimated by spectral means. This increase in slope value can be thought of as an ‘‘effective’’ slope, that is, this computed slope was obtained using a first-order scattering theory, and the increase must therefore be including second-order scattering effects. The size of the increase leads to the conclusion that in thin sediment areas, second-order scattering effects such as multiple scattering, layering resonances, and shadowing must be as important as first-order scattering. [Work supported by ONR-AEAS Program.]

Advances in digital holographic reconstruction of an odd-shaped source.

In order to reconstruct surface vibrations on a source with arbitrary shape from its field measurements, a new computational method, the equivalent surface source (ESS) method, is developed. The ESS method, which is based on the Helmholtz integral equation (HIE), assumed there is an equivalent surface source that is inside the real source surface but may be a different shape and treats this equivalent source as if it is the real source for field outside the actual source. The ESS produces the same field as the actual source for field outside and on the surface of the actual source. The ESS method can be used in both radiation and reconstruction problems. This method is implemented by using the finite-element method with the second-order interpolation function. The singular value decomposition (SVD) method is employed to do the matrix inversion. The SVD method coupled with the finite-element method implementation of HIE enabled the extension of the near-field acoustical holography technique from two- to three-dimensional sources. Test results are given.

Numerical integration of acoustic field data using the exterior Helmholtz integral equation.

Experimental measurements of acoustic pressures and normal pressure gradients over a closed surface surrounding an acoustic source can determine the acoustic field throughout the entire volume of a homogeneous region outside the measurement surface. One means for determining the extended field is to numerically integrate the exterior Helmholtz integral equation over the measurement surface. An implementation of this approach utilizing a finite-difference approximation to the exact Helmholtz equation integrated over a generalized cylindrical measurement surface has been developed to study acoustic fields radiating from large, immersed structures. In this talk the numerical algorithm will be described, and examples based on both simulated and actual measurement data will be presented. The results show pressure, velocity, and intensity distributions in the evanescent wave region immediately adjacent to localized and distributed acoustic sources, in the geometric near field and in the acoustic far field. Effects of errors due to finite spacing of measurement points over the measurement surface will be illustrated and evaluated.

A weighted residual formulation for the CHIEF method in acoustics

In this article, the combined Helmholtz integral equation formulation (CHIEF) proposed by Schenck is greatly improved so that it will alleviate the annoying nodal surface problem. The constraint provided by each CHIEF equation and its first-order derivatives is enforced in a weighted residual sense over a small interior region rather than only at a single point. Consequently, ‘‘CHIEF blocks’’ rather than CHIEF points are used in the numerical implementation. Although the corresponding interior problem may have nodal surfaces at characteristic frequencies, it has no such things as ‘‘nodal blocks.’’ Therefore, the ‘‘CHIEF-block’’ (CHIEF-B) method is expected to work even if all of the CHIEF blocks fall on the intersection of several nodal surfaces. This new formulation is implemented in an isoparametric element environment. Numerical examples in spherical radiation and scattering at intermediate frequencies are given to test the new formulation.

A boundary element formulation for acoustic shape sensitivity analysis

The Helmholtz integral equation forms a conventional basis for acoustic boundary element analysis (BEA). Implicit differentiation of the discretized Helmholtz integral equation is shown to yield an effective approach for the computation of rates of change (sensitivities) of acoustic response quantities with respect to changes in the shape of an acoustic model. A theoretical formulation is presented that allows for the reuse of the factorization of the overall BEA system left-hand side matrix formed in a previous analysis, thus obviating the need to factor perturbed matrices in the sensitivity analysis process. The singularity strengths of the new kernel functions utilized to compute sensitivities of matrix coefficients required by this approach are shown to be the same as those present in ordinary BEA. An indirect approach for the computation of the diagonal contributions to sensitivity matrices associated with these new kernels is also discussed. The sensitivity analysis formulation includes surface pressures and normal gradients (velocities), surface tangential pressure gradients, and pressure and pressure gradients at arbitrary domain sample points. It is concluded that the overall efficiency of implementations of these formulations is significant. Numerical results for a series of example problems are presented to quantify the accuracy and efficiency of this approach.

On the numerical implementation of a Cauchy principal value integral to insure a unique solution for acoustic radiation and scattering

In this paper, the boundary integral formulation proposed by Burton and Miller [Proc. R. Soc. London Ser. A 323, 201–210 (1971)] to insure a unique solution for all frequencies is implemented in an isoparametric element environment. A regularized normal derivative integral equation, originally derived by Maue [Z. Phys. 126, 601–618 (1949)], is used in the formulation to form a linear combination with the conventional Helmholtz integral equation. This regularized normal derivative integral equation converges in the Cauchy principal value sense rather than only in the finite-part sense. The Cauchy principal value integral can be further transformed into an integral that converges in the normal sense. The C° continuous isoparametric elements are used in the formulation. Collocation points are placed inside each element to insure a unique normal direction and continuity of tangential derivatives of the acoustic pressure. Through a systematic collocation point generation scheme, the number of collocation points is always greater than the number of nodal points. The overdetermined system is then solved by a least-squares procedure. Numerical examples are given for several radiation and scattering problems.

A Boundary Element Approach to Optimization of Active Noise Control Sources on Three-Dimensional Structures

This paper presents the theoretical development of an approach to active noise control (ANC) applicable to three-dimensional radiators. The active noise control technique, termed ANC Optimization Analysis, is based on minimizing the total radiated power by adding secondary acoustic sources on the primary noise source. ANC Optimization Analysis determines the optimum magnitude and phase at which to drive the secondary control sources in order to achieve the best possible reduction in the total radiated power from the noise source/control source combination. For example, ANC Optimization Analysis predicts a 20 dB reduction in the total power radiated from a sphere of radius α at a dimensionless wavenumber ka of 0.125, for a single control source representing 2.5 percent of the total area of the sphere. ANC Optimization Analysis is based on a boundary element formulation of the Helmholtz Integral Equation, and thus, the optimization analysis applies to a single frequency, while multiple frequencies can be treated through repeated analyses.

A comparison between the boundary element method and the wave superposition approach for the analysis of the scattered fields from rigid bodies and elastic shells

The steady state analysis of the scattering of plane acoustic waves from submerged rigid and elastic bodies using two approaches is presented. The first approach uses a combined finite element/boundary element (FE/BE) methodology. The NASA structural analysis (NASTRAN) program is used to formulate the structural matrices based on the finite element method (FEM). The surface integral equation radiation and scattering (SIERRAS) program creates the fluid matrices based on the boundary element method (BEM) and solves the coupled fluid-structure interaction problem. A superparametric boundary element (BE) with nine nodes is employed. The combined Helmholtz integral equation formulation (CHIEF) is employed to provide a unique solution for all frequencies. In the second approach, the superposition method (SUP) is used for modeling the fluid. The SUP method is an off-boundary approach that employs a number of point sources moved inside the body to represent the fluid response at the surface. This allows the fluid matrices to be formed without surface integration. Formulations for the SUP method are introduced for both the radiation and scattering problems. The program SUPER reads the NASTRAN structural matrices and solves the combined FE/SUP fluid-structure equations. The FE/BEM and FE/SUP methods are applied to the scattering of an infinite set of plane waves from a submerged rigid sphere, rigid prolate spheroid, right cylinder, and from an elastic cylindrical shell with hemispherical endcaps. The SUP method is found to be easier to implement and to provide an accurate result at internal resonance frequencies where the BEM requires additional equations to ensure an accurate solution.

Reflection coefficient measurement techniques for underwater acoustic panels

A new measurement method for evaluating underwater acoustic materials is proposed. It is based on a two-hydrophones technique and allows the determination of the complex reflection coefficient for size-limited acoustical panels. The results are shown to be independent of the incident plane wave amplitude and the distance between the hydrophones and the panel. Therefore, this technique involves direct and quick measurements so that it is possible to reduce the effects of scattered waves from the edges of the sample by averaging. Simulations have been made to check the stability of the method versus noise. The experiment covered the frequency range 1–12 kHz. Techniques (intensity, impedance, and pressure measurements) tested at G.E.R.D.S.M. (Groupe d'Etudes et de Recherches en Détection Sous-Marine) using surfacic sensors are also presented. Their accuracies are discussed through a mathematical model based on Helmholtz integral equation. Results obtained by the various methods on finite panels are compared.

The thin‐shape breakdown (TSB) of the Helmholtz integral equation

Many numerical implementations of the Helmholtz integral equation exist today that can predict routinely the field scattered by a volume‐holding body. One such object is the ellipsoidal ‘‘core’’ of a typical airborne or submerged vehicle stripped of its thin appendages, i.e., stripped of its finlike control surfaces, etc. The reason for these omissions has often been an inherent limitation of the cited modeling tools rather than a rational dismissal of the potential effect of these neglected appendages on the complete body’s expected scattering cross section. The limitation of existing techniques is this: The standard form of Gauss’ theorem on which they are based, that leads to the Helmholtz integral, becomes meaningless when the thickness of the shape addressed tapers down to zero even over only part of the structure. This paper exposes analytically the fundamental cause of this thin‐shape breakdown (TSB), and in the process derives an alternate boundary element formulation for its cure.

Graphing capabilities for CHIEF (combined Helmholtz integral equation formulation)

The computer program CHIEF [G. W. Benthien, D. Barach, and D. Gillette, “CHIEF Users Manual,” Naval Ocean Systems Center Report 920 (Sept. 1988)] is designed to obtain approximate solutions to exterior steady‐state acoustic radiation problems for surfaces of arbitrary shapes vibrating with a prescribed normal velocity. Being unable to view the shapes defined by CHIEF can lead to many difficult debugging problems as well as incorrect solutions. DISPLAY3D provides points as though the data were locations on a three‐dimensional surface on a solid and a description of how these points are connected. DISPLAY3D has several options that make it a versatile package. The code permits the user to change the orientation from which the data are viewed. DISPLAY3D also provides the option of viewing the direction of the normal velocity vector as defined by CHIEF. These options, as well as others, will be explained in detail. [Work supported by ONT.]

The prediction of the sound field due to an arbitrary vibrating body in a rectangular enclosure

A numerical technique is presented for the prediction of the sound field due to an arbitrary vibrating body in a rectangular enclosure. The method, based on the Helmholtz integral equation, is claimed to have the potential for designing a realistic machine structure in enclosed surroundings. It is assumed that the vibration of the machine surface is known either from experiment or from a finite‐element calculation. The accuracy of the technique is verified by using it to predict the sound field from a vibrating sphere in a rectangular enclosure for which the analytical solution is known. The potential of its application to realistic problems is shown by comparing its predictions of the radiation from a vibrating box in a commonly encountered rectangular room with experimental results.

A new integral equation formulation for an axisymmetric structure

A new method is developed to compute the acoustic field outside an axisymmetric structure from the normal velocity values on the surface. Surface pressure and normal velocity are expanded in a series of functions that are orthonormal on the surface of the structure and have a constant ratio of pressure to normal derivative of pressure at vanishing frequency. The Helmholtz integral equation is next used to compute the field everywhere outside the structure. The method is tested by applying it to scattering from a rigid cylinder with hemispherical endcaps. The series is shown to converge very rapidly.

An in vacuo modal method to analyze fluid-loaded shells of revolution

A general approach is presented to evaluate the acoustic field and vibratory response of harmonically excited fluid-loaded shells of revolution. The excitation force and velocity of shells of revolution are expressed as a modal sum of the in vacuo eigenvectors. The eigenvectors for shells of interest are evaluated by the finite element method. A new method is presented to determine the acoustic loading on the shells. This method is based on the use of internal source distributions to obtain acoustic radiation impedances. In the present study, the modal velocity coefficients are obtained from a set of algebraic equations and the Helmholtz integral equation is used to evaluate the acoustic pressure in the fluid. Numerical results for spherical and spheroidal shells will be presented and discussed. [Work supported by ONR.]

Studies on surface discretization in boundary element method applications

The implementation of the boundary element method (BEM) to sound radiation problems consists of discretizing the vibrating surface into a number of elemental radiators and then applying the Helmholtz integral equation for predicting the acoustic field. As with other numerical techniques such as the finite element method, the accuracy of the model is dependent upon, among other factors, the discretization scheme. Creating an efficient mesh is important in order to insure accurate results with a reasonable amount of computational effort. Results of work conducted comparing the use of planar as well as linear and quadratic isoparametric elements are shown. The case studies involve modeling the acoustic field and radiation efficiency of a pulsating sphere at various wavenumbers and modeling the directivity pattern of a vibrating finite cylinder. Procedures for checking accuracy of solution are compared such as matrix condition number and velocity potential of interior points.

A banded matrix boundary element method

Boundary element methods (BEMs) for the Helmholtz integral equation at frequency f result in matrices of order f4 elements requiring of f6 operations for the solution. Scattering or radiation problems for objects of more than a few ka in size result in prohibitively large computational problems. A method is developed for estimating the contribution of matrix elements corresponding to points more than 2–3 wavelengths apart in terms of matrix elements of points closer than the 2–3 wavelengths. The result is a banded matrix with a fixed bandwidth independent of f, of order f2 elements, and requiring of order f2 operations for the solution. The estimation for the points farther apart than 2–3 wavelengths is constructed in terms of the partial sums of closer points. For many objects of interest the bandwidth is approximately 1000, making the problems tractable. Combining the banded BEM with a finite element model of the object has the effect of increasing the bandwidths of the matrices by a relatively small amount. [Work supported by ONR.]

Scattering of plane waves from objects having surface impedance discontinuities

The scattering of acoustic plane waves from partially coated submerged objects is investigated. On the boundary between the uncoated and coated portions of the surface, an impedance discontinuity exists giving rise to sharply varying surface pressures and normal velocities. In the limiting case in which the uncoated surface has infinite impedance and the coated surface has zero impedance, it can be shown that a singularity in surface normal velocity exists. When finite and nonzero impedances are considered for the hard and soft boundary conditions, respectively, the singularity in normal velocity apparently disappears, but the rapid variation in surface pressures and normal velocities still presents numerical difficulties. Two specific systems are investigated: an infinite cylinder and a finite cylinder. Both systems are coated over two unconnected regions on their surfaces. The infinite cylinder is analyzed using a spectral expansion with the series coefficients evaluated using an overdetermined collocation technique. The finite cylinder is analyzed using a boundary element approach based on the standard Helmholtz integral equation. CHIEF points are used to avoid ill‐conditioning at problem frequencies. [Work supported by ONR.]

A new boundary element method for resonance of an acoustic enclosure

Abstract This paper presents a new boundary element formulation in which the eigenvalue appears outside the integral operator, which distinguishes it from the Helmholtz integral equation. Thus, the formation of global matrices need only be assembled once. Since the kernel of the operator used in the new formulation is real‐valued, all calculations can be carried out in a much simpler way in the real domain. The complex acoustic pressure amplitude is considered herein to deivate by a certain amount from a harmonic function. It is an important contribution that an exact relation between the deviator and the complex acoustic pressure amplitude is constructed locally and thus no more approximations are introduced except conventional boundary discretizations. Several examples are given to illustrate the feasibility of an accurate, effective prediction of resonance.

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.

PRISE EN COMPTE DU RAYONNEMENT ACOUSTIQUE EN FLUIDE LOURD PAR ÉQUATIONS INTÉGRALES DANS LA MODÉLISATION DES TRANSDUCTEURS PIÉZOÉLECTRIQUES BIDIMENSIONNELS PAR ÉLÉMENTS FINIS

A piezo-elastic problem including fluid-structure interaction is adressed using a single variationnal formulation based on Helmholtz integral equations in the continuous wave regime. Internal losses are taken into account by introducing complex material constants. This formulation has been implemented in the numerical code E.T.BID.M which is a plane strain bidimensionnal finite elements program. The response of transducers having large transversal dimensions can be accurately computed.