Combined Helmholtz Integral Equation Formulation Research Articles

Automatic CHIEF Point Selection for Finite Element–Boundary Element Acoustic Backscattering

Computing the backscattering of harmonic acoustic waves from underwater elastic targets of arbitrary shapes is a challenging problem of considerable practical significance. The finite element method is well suited for the discretization of the target, while the boundary element method addresses the radiation boundary condition at infinity. A disadvantage of the boundary integral method is that it yields non-unique solutions at certain wavenumbers. This failure is associated with the existence of eigensolutions of the Helmholtz equation in the interior of the complement of the fluid domain (acoustic modes). The combined Helmholtz integral equation formulation (CHIEF) credited to Schenk is employed to combine the surface Helmholtz boundary integral with equations of the interior Helmholtz relation written down at selected points within the cavity of the scatterer (i.e., in the complement of the fluid domain).The difficulty associated with this approach has always been the lack of guidance on the necessary number of interior points and on their locations. The solution to this problem proposed here is to compute the acoustic modes using the finite element method to complement of the fluid domain and to identify locations of the peaks.This novel approach aids the decision as to how many points should be employed and where they should be located. Our numerical experiments demonstrate the robustness of the proposed automatic selection of the CHIEF points’ numbers and locations.

An Optimization Strategy to Position CHIEF Points in Boundary-Element Acoustic Problems

The use of boundary elements in the analysis of exterior acoustic problems poses challenges at specific frequencies, since fictitious eigenfrequencies may arise at the internal resonances of cavities, leading to inaccurate results or even unstable behavior. To filter out these fictitious eigenfrequencies, a scheme based on the combined Helmholtz integral equation formulation (CHIEF) can be used to prevent the so-called non-uniqueness problem, although it requires additional equations and points. The BEM formulation final accuracy will, however, depend on the correct choice of these points. Here, a strategy to help in defining good approximations for the position and number of such points is proposed, based on an optimization process which maximizes the system matrix’s smallest singular value. The accuracy of the method for exterior radiation problems is investigated using different examples. With low computational cost and simple implementation, the two proposed algorithms automatically circumvent the non-uniqueness problem, aiding the implementation of more stable BEM codes.

A Modified Formulation of Singular Boundary Method for Exterior Acoustics

This paper proposes a modified formulation of the singular boundary method (SBM) by introducing the combined Helmholtz integral equation formulation (CHIEF) and the self-regularization technique to exterior acoustics. In the SBM, the concept of the origin intensity factor (OIF) is introduced to avoid the singularities of the fundamental solutions. The SBM belongs to the meshless boundary collocation methods. The additional use of the CHIEF scheme and the self-regularization technique in the SBM guarantees the unique solution of the exterior acoustics accurately and efficiently. Consequently, by using the SBM coupled with the CHIEF scheme and the self-regularization technique, the accuracy of the numerical solution can be improved, especially near the corresponding internal characteristic frequencies. Several numerical examples of two-dimensional and three-dimensional benchmark examples about exterior acoustics are used to verify the effectiveness and accuracy of the proposed method. The proposed numerical results are compared with the analytical solutions and the solutions obtained by the other numerical methods.

Full wavenumber high precision combined coupled double boundary element method for solving external acoustic problems

In this paper, a high-precision combined coupled double boundary element method (CCD-BEM) with a full wavenumber unique solution is proposed by combining the boundary element method (BEM) equation with the virtual indirect BEM (VIBEM) equation and using the equivalent relationship between their coefficient matrices. This method only requires overcoming the non-uniqueness of the VIBEM equation to obtain the unique solution of full-wave number, thus avoiding the failure of interior-point configuration in the combined Helmholtz integral equation formulation (CHIEF) method and the calculation of hyper-singular integral in Burton-Miller method. In addition, the singular matrix in BEM can be calculated indirectly using the equivalent relationship between the BEM and VIBEM coefficient matrix, which completely avoids the direct calculation of weak singular or singular integral. The numerical results of acoustic radiation and scattering show that CCD-BEM has a lower condition number of the coefficient matrix and higher calculation accuracy than BEM.

Acoustic radiation from a sphere pulsating near an impedance plane using a boundary integral equation method

In this study, a boundary integral equation method is proposed for investigating acoustic pressure radiation from a sphere pulsating near a free surface or an impedance plane. The half-space and free-space problems are investigated for the acoustic radiation of pulsating sphere. The effects of free surface and impedance boundaries are introduced into the mathematical model by employing three different half-space Green’s functions, respectively. These Green’s functions are derived, respectively, using the single image-source method, multiple equivalent-source method, and complex equivalent-source method. Green’s functions are implemented into the boundary element (BE) formulation. The surface of the pulsating sphere is discretized with linear and quadratic BEs, and the Combined Helmholtz Integral Equation Formulation (CHIEF) is employed to overcome the non-uniqueness problem. Four different case studies are considered for the sphere pulsating near a free surface or an impedance plane. The first case study involves the sphere pulsating near a free surface (perfectly reflective) and the single image-source method is used in the boundary element method (BEM) formulation. In the second case study, the sphere is assumed as pulsating near a perfectly reflecting and perfectly absorbing impedance planes, respectively. The multiple equivalent-source method is employed for the perfectly reflecting plane, but the multiple equivalent-source method and complex equivalent-source methods for the perfectly absorbing plane. The third case study involves a general impedance plane, and all the methods are employed, respectively, in the BE formulation. The final case study assumes a general impedance plane forming a perpendicular incidence and the complex equivalent-source method is used in this particular case. It is observed that there is a very good comparison between the results obtained from all these methods.

A BEM–FMM approach applied to the combined convected Helmholtz integral formulation for the solution of aeroacoustic problems

Exterior and interior acoustic calculations often exploit the advantages of Boundary Element Method (BEM) numerical scheme. Due to the large dimension of dense matrices arising in real-case applications, direct matrix–vector products become prohibitive, since the computational cost of standard methods increases quadratically. In this work, the black-box Fast Multipole Method (bbFMM), based on the Chebyshev interpolation, is extended to the integral solution of the convective Helmholtz equation stabilized using the Combined Helmholtz Integral Equation Formulation (CHIEF). The CHIEF approach relies on the solution of an over-determined linear system achieved by adding a set of points internal to the scattering body. In order to handle the distribution of the CHIEF points within the body volume, a robust stochastic approach has been also developed. Moreover, an Adaptive bbFMM (AbbFMM) is presented for handling the oscillating kernels, such as those arising in the solution of the convective Helmholtz equation. The proposed approach, which has a complexity of O(NlogN), is interesting for industrial applications, especially in the aeroacoustic field.

Prediction of acoustic radiation from functionally graded shells of revolution in light and heavy fluids

This paper presents a semi-analytical method for the vibro-acoustic analysis of a functionally graded shell of revolution immersed in an infinite light or heavy fluid. The structural model of the shell is formulated on the basis of a modified variational method combined with a multi-segment technique, whereas a spectral Kirchhoff–Helmholtz integral formulation is employed to model the exterior fluid field. The material properties of the shell are estimated by using the Voigt׳s rule of mixture and the Mori–Tanaka׳s homogenization scheme. Displacement and sound pressure variables of each segment are expanded in the form of a mixed series using Fourier series and Chebyshev orthogonal polynomials. A set of collocation nodes distributed over the roots of Chebyshev polynomials are employed to establish the algebraic system of the acoustic integral equations, and the non-uniqueness solution is eliminated using a combined Helmholtz integral equation formulation. Loosely and strongly coupled schemes are implemented for the structure-acoustic interaction problem of a functionally graded shell immersed in a light and heavy fluid, respectively. The present method provides a flexible way to account for the individual contributions of circumferential wave modes to the vibration and acoustic responses of functionally graded shells of revolution in an analytical manner. Numerical tests are presented for sound radiation problems of spherical, cylindrical, conical and coupled shells. The individual contributions of the circumferential modes to the radiated sound pressure and sound power of functionally graded shells are observed. Effects of the material profile on the sound radiation of the shells are also investigated.

Solving the hypersingular boundary integral equation for the Burton and Miller formulation.

This paper presents an easy numerical implementation of the Burton and Miller (BM) formulation, where the hypersingular Helmholtz integral is regularized by identities from the associated Laplace equation and thus needing only the evaluation of weakly singular integrals. The Helmholtz equation and its normal derivative are combined directly with combinations at edge or corner collocation nodes not used when the surface is not smooth. The hypersingular operators arising in this process are regularized and then evaluated by an indirect procedure based on discretized versions of the Calderón identities linking the integral operators for associated Laplace problems. The method is valid for acoustic radiation and scattering problems involving arbitrarily shaped three-dimensional bodies. Unlike other approaches using direct evaluation of hypersingular integrals, collocation points still coincide with mesh nodes, as is usual when using conforming elements. Using higher-order shape functions (with the boundary element method model size kept fixed) reduces the overall numerical integration effort while increasing the solution accuracy. To reduce the condition number of the resulting BM formulation at low frequencies, a regularized version α = ik/(k(2 )+ λ) of the classical BM coupling factor α = i/k is proposed. Comparisons with the combined Helmholtz integral equation Formulation method of Schenck are made for four example configurations, two of them featuring non-smooth surfaces.

A comparison of techniques for overcoming non‐uniqueness of boundary integral equations for the collocation partition of unity method in two‐dimensional acoustic scattering

SUMMARYThe Partition of Unity Method has become an attractive approach for extending the allowable frequency range for wave simulations beyond that available using piecewise polynomial elements. The non‐uniqueness of solution obtained from the conventional boundary integral equation (CBIE) is well known. The CBIE derived through Green's identities suffers from a problem of non‐uniqueness at certain characteristic frequencies. Two of the standard methods of overcoming this problem are the so‐called Combined Helmholtz Integral Equation Formulation (CHIEF) method and that of Burton and Miller. The latter method introduces a hypersingular integral, which may be treated in various ways. In this paper, we present the collocation partition of unity boundary element method (PUBEM) for the Helmholtz problem and compare the performance of CHIEF against a Burton–Miller formulation regularised using the approach of Li and Huang. Copyright © 2013 John Wiley & Sons, Ltd.

Stabilization of time domain acoustic boundary element method for the exterior problem avoiding the nonuniqueness

The time domain boundary element method (TBEM) to calculate the exterior sound field using the Kirchhoff integral has difficulties in non-uniqueness and exponential divergence. In this work, a method to stabilize TBEM calculation for the exterior problem is suggested. The time domain CHIEF (Combined Helmholtz Integral Equation Formulation) method is newly formulated to suppress low order fictitious internal modes. This method constrains the surface Kirchhoff integral by forcing the pressures at the additional interior points to be zero when the shortest retarded time between boundary nodes and an interior point elapses. However, even after using the CHIEF method, the TBEM calculation suffers the exponential divergence due to the remaining unstable high order fictitious modes at frequencies higher than the frequency limit of the boundary element model. For complete stabilization, such troublesome modes are selectively adjusted by projecting the time response onto the eigenspace. In a test example for a transiently pulsating sphere, the final average error norm of the stabilized response compared to the analytic solution is 2.5%.

Stabilization of time-domain acoustic boundary element method for sound radiation problems

Time-domain acoustic solution from the Kirchhoff integral equation for the exterior problem is not unique due to the presence of fictitious internal modes and also suffers from the instability that stems from the time marching scheme. In this work, methods to stabilize the time-domain acoustic boundary element calculation were suggested. Low-order fictitious internal modes within the effective frequency range of boundary element calculation were suppressed by the newly formulated time-domain CHIEF (Combined Helmholtz Integral Equation Formulation) method. Additional interior points were included, similar to frequency-domain problems, to satisfy the zero pressure constraint considering the shortest retarded time between boundary nodes and interior points. However, the calculation was yet unstable due to remaining unstable high-order modes beyond the effective frequency limit. To further stabilize the computation, unstable high-order internal modes were nullified using the wave vector filtering method. In comparison with the time-domain Burton-Miller formulation, the proposed method has no hyper-singular integral and the monotonically increasing instability was not observed. As a test example, sound radiation from a pulsating sphere was used and a good stabilization performance was shown. Average relative-difference norm of the stabilized time response from the analytic solution was 2.7%. (This work was partially supported by BK21 project)

시간 영역 음향 경계요소법에서의 비유일성 문제 해결을 위한 방법에 관하여

Kirchhoff 적분식을 이용하여 외부 음향 문제의 시간 영역 응답을 계산하는 경우, 주파수영역 해석과 마찬가지로 가상적인 내부 음향 모드에 기인한 비유일성 문제가 발생한다. 이를 해결하는 방법들 중의 하나로서 CHIEF(Combined Helmholtz Integral Equation Formulation) 방법이 쓰이는데, 이는 몇몇 내부 수음점의 응답을 0으로 추가하여 구속하는 조건을 부가하는 기법이다. 이 기법은 주파수 영역 경계요소법에서는 간편한 수식 때문에 많이 사용되고 있지만, 시간 영역에서는 사용된 예가 없다. 본 연구에서는 대상체 내부의 가상 수음점과 경계 표면의 절점들간의 최소 거리에 대한 지연시간을 고려하여, 계산하고자 하는 미지수인 현재 시간의 경계 표면 음장을 구속함으로써, 시간 영역 해석에 적합하도록 CHIEF 방법을 수식화하였다. 예제로서, 반지름 방향으로 진동하는 구의 음향 방사 문제를 다루었다. CHIEF 방법을 적용함에 따라 저차의 내부 음향 모드에 기인한 비유일성 문제를 해결할 수 있었고, 비요동 모드에 의한 수치적 불안정성을 피할 수 있었다. 그러나, 유효주파수 밖에 남은 내부 음향의 고차모드들에 의한 수치적 불안정성은 증가하였다. The time-domain solution from the Kirchhoff integral equation for an exterior problem is not unique at certain eigen-frequencies associated with the fictitious internal modes as happening in frequency-domain analysis. One of the solution methods is the CHIEF (Combined Helmholtz Integral Equation Formulation) approach, which is based on employing additional zero-pressure constraints at some interior points inside the body. Although this method has been widely used in frequency-domain boundary element method due to its simplicity, it was not used in time-domain analysis. In this work, the CHIEF approach is formulated appropriately for time-domain acoustic boundary element method by constraining the unknown surface pressure distribution at the current time, which was obtained by setting the pressure at the interior point to be zero considering the shortest retarded time between boundary nodes and interior point. Sound radiation of a pulsating sphere was used as a test example. By applying the CHIEF method, the low-order fictitious modes could be damped down satisfactorily, thus solving the non-uniqueness problem. However, it was observed that the instability due to high-order fictitious modes, which were beyond the effective frequency, was increased.

Analytic Method on Vibrate Induced Noise of Engine Block

Numerical prediction of the acoustic radiation of vibrating engine block is conventionally based on the boundary element method (BEM). This approach yields good predictions, provided that the dynamic behavior of engine block is well established. However, interior resonances may lead to singular matrices and nonunique solution. To avoid the problems with interior resonances and singular matrices, this paper introduced the combined Helmholtz integral equation formulation method (CHIEF METHOD). The numerical experiment shows the performance of CHIEF METHOD and then we use this method to estimate the acoustic radiation of engine block. Based on spatially mean square velocity and acoustic radiation efficiency of vibrating engine block, a simplified analytic method is presented in this paper, by which the vibrate induced noise can be predicted in the early design process of engine development.

An efficient method for solving the nonuniqueness problem in acoustic scattering

The problem of acoustic wave scattering by closed objects via second kind integral equations, is considered. Based on, combined Helmholtz integral equation formulation (CHIEF) method, an efficient method for choosing and utilizing interior field relations is suggested for solving the non- uniqueness problem at the characteristic frequencies. The implementation of the algorithm fully utilizes previous computation and thus significantly reduces the CPU time compared to the usual least-squares treatment. The method is tested for acoustic wave scattering by both acoustically hard and soft spheres. Accurate results compared to the known exact solutions are obtained. Copyright © 2006 John Wiley & Sons, Ltd.

Consideration of Radiation Impedance Calculation Methods for Cylindrical Array Sound Source

In a phased array, the interval between the centers of the sound source elements which adjoin each other is set below half the wavelength in order to suppress the gratinglobe. At this interval, the array sound source can gain strong radiation power. The estimation of radiation impedance is important in the analysis of the radiation power. The Greenspon's method which is a radiation impedance calculation method of rectangular pistons on a cylinder is explained, and some defects of this method are clarified. For the evaluation of Greenspon's method, we introduce the combined Helmholtz integral equation formulation (CHIEF) method. The radiation impedance is calculated using both Greenspon's method and the CHIEF method, and the results are compared. On the basis of comparison of the radiation power measured while driving the cylindrical phased array with the theoretical value of radiation power determined using the caluculated value of radiation impedance, it is considered that Greenspon's method and the CHIEF method give accurate estimations.

Prediction of the acoustic field produced by a foetal heart monitoring transducerusing the finite element and boundary element methods

The aim of this work is to predict the acoustic pressure field produced by a seven-element foetal heartmonitoring transducer for a given sinusoidal voltage input. As each element is nominally identical, anaxisymmetric model of the single-element transducers is first produced. The orthotropic andpiezoelectric properties of the crystal are accounted for. The fluid loading is modelled using fluidfinite elements (FEs) together with a boundary element (BE). The latter enables the modelling of asemi-infinite region of fluid. A combined Helmholtz integral equation formulation with sixcollocation points is used to solve the BE problem. The acoustic field within the BE, may then bepredicted. Based on the assumption that the mechanical vibration of one element does not affect thatof the other, the complete acoustic field generated by the seven elements is obtained by superposition.Measurements show that the trends produced by this type of transducer are successfully predictedusing the FE/BE approach.

Spurious and true eigensolutions of Helmholtz BIEs and BEMs for a multiply connected problem

The spurious eigenvalues of an annular domain have been verified for the singular and hypersingular boundary‐element methods (BEMs) and circumvented by using the Burton‐Miller approach. Do they also occur in other formulations: continuous formulations such as the singular and hypersingular boundary integral equations (BIEs), the null‐field BIEs and the fictitious BIEs, or such discrete formulations as the null‐field BEMs and the fictitious BEMs? For the ten formulations of the multiply connected problem the study of otherwise the same issues is continued in the present paper. By using the degenerate kernels and the Fourier series, it is demonstrated analytically for the six continuous formulations of BIEs that spurious eigensolutions depend on the geometry of the inner boundary but not on that of the outer boundary. This conclusion can be extended to the six discrete formulations of BEMs. To filter out the spurious eigenvalues, the CHIEF (combined Helmholtz integral equation formulation) method is used here instead of the Burton‐Miller approach. The optimum number and appropriate positions of the CHIEF points are also addressed. It is then shown that, in the null‐field and fictitious BEMs, the spurious and true eigenvalues can be detected and distinguished by using the singular‐value‐decomposition‐updating techniques in conjunction with the Fredholm alternative theorem. Illustrative examples show the validity of the proposed methodologies.

Implementation of CHIEF in MATLAB for prediction of sound radiated from arbitrarily shaped vibrating bodies

The Boundary Element Method can be used to predict sound pressure levels radiated from an arbitrarily shaped vibrating body. Using the direct, indirect, and approximation formulation one can solve such an exterior acoustic problem. In this work, the direct formulation is chosen. The two major methods employed for implementation of this formulation are the Combined Helmholtz Integral Equation Formulation (CHIEF) and the Burton–Miller Method. It was decided that the CHIEF method would be the focus of this research. Using the CHIEF method as a guide, a computer simulation was developed that incorporates quadratic elements, calculates the sound pressure and sound power, is able to import surface velocities, and uses the program ideas as a preprocessor. This program was verified using theoretical models as well as experimental measurements conducted in the Western Michigan University Noise and Vibration Laboratory. This program was written in Matlab with the understanding that it can be processed on a personal computer.

ANALYTICAL STUDY AND NUMERICAL EXPERIMENTS FOR RADIATION AND SCATTERING PROBLEMS USING THE CHIEF METHOD

In this paper the well-known failure in numerical computations of the exterior Helmholtz integral equation at certain characteristic frequencies is investigated analytically and numerically. The combined Helmholtz integral equation formulation (CHIEF) method is a very popular technique that can overcome the non-unique problem, but this method breaks down if the interior points are not properly chosen. The CHIEF method in conjunction with the singular value decomposition (SVD) technique is an easy and efficient method to ensure a unique solution for the exterior problem. Based on the circulant properties and degenerate kernels, an analytical scheme in discrete system of a cylinder is achieved. The optimum numbers and proper positions for the collocation points in the interior domain are analytically studied and suggested in the numerical scheme. Numerical experiments are designed to verify the analytical results. One analytical example for a circular cylinder and one numerical example for a square rod are demonstrated to show the validity of the CHIEF method.

Radiation Field Prediction System Using Solutions of Exterior Steady-State Radiation Problem

We have been studying a radiation field prediction system for an acoustic source that has an arbitrary shape. First, the nearfield sound pressure distribution around the source is measured and the normal velocity distribution over the surface of the source is predicted using the equivalent phased array method (EPAM). Then, radiation characteristics of the source, such as the pressure field, are predicted using the combined Helmholtz integral equation formulation(CHIEF). First in this paper, EPAM based on solutions for the exterior steady-state acoustic radiation problem is described. This method is applicable for predicting the normal surface velocity for sources that have an arbitrary shape. Then, an experimental example using a cylindrical source is presented. The predicted result of the pressure field is shown, and then the validity of this radiation field prediction system is confirmed by comparing the predicted pressure field with the actual field. The influence of the aperture restriction for the measurement contour on the surface of normal velocity predicted by EPAM is discussed. It is found that the error between the predicted and the ideal velocity distribution becomes smaller as the measurement contour covers the cylinder.