Acoustic Boundary Element Method Research Articles

Partitioned solution strategies for coupled BEM–FEM acoustic fluid–structure interaction problems

This paper investigates two FEM–BEM coupling formulations for acoustic fluid–structure interaction (FSI) problems, using the Finite Element Method (FEM) to model the structure and the Boundary Element Method (BEM) to represent a linear acoustic fluid. The coupling methods described interconnect fluid and structure using classical or localized Lagrange multipliers, allowing the connection of non-matching interfaces. First coupling technique is the well known mortar method, that uses classical multipliers and is compared with a new formulation of the method of localized Lagrange multipliers (LLM) for FSI applications with non-matching interfaces. The proposed non-overlapping domain decomposition technique uses a classical non-symmetrical acoustic BEM formulation for the fluid, although a symmetric Galerkin BEM formulation could be used as well. A comparison between the localized methodology and the mortar method in highly non conforming interface meshes is presented. Furthermore, the methodology proposes an iterative preconditioned and projected bi-conjugate gradient solver which presents very good scalability properties in the solution of this kind of problems.

On the instability of time-domain acoustic boundary element method due to the static mode in interior problems

In the analysis of interior acoustic problems, the time domain boundary element method (TBEM) suffers the monotonically increasing instability when using the direct Kirchhoff integral. This instability is related to the non-oscillatory static acoustic mode describing the constant spatial response in an enclosure. In this work, nonphysical natures of non-oscillatory static mode influencing the instability of TBEM calculation are investigated, and a method for stabilization is studied. In TBEM calculation, the static mode is represented by two non-oscillatory eigenmodes with different eigenvalues. The monotonically increasing instability is caused by the unstable poles of non-oscillatory eigenmodes as well as very small, very low frequency noise of an input signal. Interior problems with impedance boundary condition also exhibit the monotonically increasing instability stemming from its pseudo non-oscillatory static mode due to the lack of dissipation at very low frequencies. Calculation of transient sound fields within rigid and lined boxes provides numerical evidences. It is noted that the stabilization effort by modifying the coefficient matrix based on the spectral decomposition can be used only for correcting the unstable pole. The filtering method based on the eigen-analysis must be additionally used to avoid the remaining instability caused by very low frequency noise of input signal.

THE BOUNDARY ELEMENT METHOD

The boundary element method (BEM), along with the finite element and finite difference methods, is commonly used to carry out numerical simulations in a wide variety of subjects in science and engineering. The BEM, rooted in classical mathematics of integral equations, started becoming a useful computational tool around 50 years ago. Many researchers have worked on computational aspects of this method during this time.This paper presents an overview of the BEM and related methods. It has three sections. The first, relatively short section, presents the governing equations for classical applications of the BEM in potential theory, linear elasticity and acoustics. The second describes specialized applications in bodies with thin features including micro-electro-mechanical systems (MEMS). The final section addresses current research. It has three subsections that present the boundary contour, boundary node and fast multipole methods (BCM, BNM and FMM), respectively. Several numerical examples are included in the second and third sections of this paper.

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%.

Flow-induced Noise Calculation of Centrifugal Pumps Based on CFD/CA Method

A hybrid algorithm combination computational fluid dynamics(CFD) with computational acoustics(CA) based on the Lighthill equation theory is adopted to calculate the sound field. Based on Reynolds-averaged equations closed by SST k-ω turbulence model, the three-dimensional unsteady flow is numerically calculated. Acoustic boundary element method is adopted to solve the acoustic radiation of dipole source caused by blade and volute surface pressure. Vibro-acoustic interaction effect on centrifugal pump sound field is analysed and flow induced noise test rig of centrifugal pumps is built to verify the calculation results. The results show that blade passing frequency and multiple are the main frequency of the flow-induced noise, and the pressure fluctuation is strongest near the tongue. It would be improper to ignore the vibro-acoustic interaction influence in sound field simulation especialy at mode frequency. Sound pressure level at pump outlet is higher than inlet, displays largest at blade pass frequency and least at maximum efficiency point. Simulation value tallies the test in the trend, maximum difference 3.1%, which is validated the forecast function of numerical simulation. This provides some useful reference for hydraulic design of centrifugal pump with low noise.

Stabilization of time domain acoustic boundary element method for the interior problem with impedance boundary conditions

The time domain boundary element method (BEM) is associated with numerical instability that typically stems from the time marching scheme. In this work, a formulation of time domain BEM is derived to deal with all types of boundary conditions adopting a multi-input, multi-output, infinite impulse response structure. The fitted frequency domain impedance data are converted into a time domain expression as a form of an infinite impulse response filter, which can also invoke a modeling error. In the calculation, the response at each time step is projected onto the wave vector space of natural radiation modes, which can be obtained from the eigensolutions of the single iterative matrix. To stabilize the computation, unstable oscillatory modes are nullified, and the same decay rate is used for two nonoscillatory modes. As a test example, a transient sound field within a partially lined, parallelepiped box is used, within which a point source is excited by an octave band impulse. In comparison with the results of the inverse Fourier transform of a frequency domain BEM, the average of relative difference norm in the stabilized time response is found to be 4.4%.

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.

Simulation Analysis on Engine Radiated Noise

In this paper, in order to analyze and predict surface radiated noise, an engine with 6 cylinders is studied by advanced methods. The loads that the engine block is subjected and the modal participation factors can be obtained through multi-body dynamics calculation; by the finite element method, structural modes of the engine can be gotten; by the acoustic boundary element method and the modal acoustic transfer vector method, the acoustic responses on field points can be obtained. According to the simulation results, the acoustic power and the acoustic pressure distributions on field points and noise frequency characteristic can be identified when the engine is operating at different speeds, and the main noise source can be located. These results provide a reference for noise reduction and amelioration of the engine.

Non-cavitating propeller noise source inversion

Ship's propeller cavitation generates major inboard noise and vibration over the aft body surface of a ship. During the last decade, cavitation-induced hull pressure forces have been reduced considerably leading to reduced noise and vibration levels owing to cavitation, in which case the non-cavitating components of propeller excitations have to be considered: pressure fluctuations due to blade loading and blade thickness. In this work, an algorithm to invert for the non-cavitating propeller noise source parameters based on the experimental data in a cavitation tunnel is shown. The fluctuating hull pressure is estimated by forward modeling based on acoustic boundary element method (BEM). The propeller blade loading and thickness noise sources are modeled using rings of dipoles and quadrupoles, respectively. The inversion for these pseudo sources are carried out by matching the data obtained from a cavitation tunnel experiment, where several hydrophone are flush mounted on the ship's surface near the propeller. Proper inversion results are obtained when six or more dipole and quadrupole rings are placed at each blade. Using the inverted source parameters, hull pressures were estimated, which show good agreement with the experiment data.

Application of the fast multipole boundary element method to underwater acoustic scattering

A numerical model is being developed in the MATLAB programming environment to model the acoustic field scattered from a submarine hull. Due to the acoustic impedance properties of water, small particle velocities yield large acoustic pressures, resulting in coupled fluid-structure interactions. Numerical methods can be employed to calculate the scattered acoustic field for complex geometries. Traditionally, these techniques required both significant memory and computational time, limiting their usefulness. Recently, the fast multipole algorithm (FMA) has been used to efficiently calculate the acoustic field on an object’s surface, while the finite element method was used to model the object’s interior. The pressure hull of a submarine can be represented as a piecewise continuous isotropic elastic solid, thus the FMA can also model the submarine interior, with the unknowns expressed on each surface. A possible method for coupling an exterior acoustic model to a structural model, both calculated via the FMA, is outlined here. Some initial acoustic fast multipole boundary element method results are presented.

A Burton–Miller formulation of the boundary element method for baffle problems in acoustics and the BEM/FEM coupling

Baffle problems, i.e. radiation problems from objects mounted behind a hole of an infinite hard reflecting wall, can be simulated as a multi domain problem consisting of a finite interior domain around the object, and two infinite half spaces in front and behind the baffle plane. A formulation of such problems is presented in the context of the Burton–Miller boundary element method. Additionally, the coupling of the acoustic boundary element method and the structural finite element method in the context of the Burton–Miller-formulation of the baffle problem is discussed.

Energy flow model considering near field energy for predictions of acoustic energy in low damping medium

The Acoustic Energy Flow Boundary Element Method (AEFBEM) is developed to predict the acoustic energy density and intensity of an engineering system. Up to now, the acoustic energy flow model has been used only for analysis of high frequencies or radiation noise because of plane wave and far-field assumptions. In this research, a new energy flow governing equation that can consider the near field acoustic energy term and spherical wave characteristics is derived successfully to predict the acoustic energy density and intensity of a system in the medium-to-high frequency range. A near field term of acoustic energy in spherical coordinate is added to the relationship between energy density and energy flow. But with the far-field assumption, this term vanishes, so the relationship between energy density and energy flow becomes the same as that of the plane wave. By considering the near field energy term without far-field assumption, the energy density at medium frequencies can be estimated. However, the governing equation has to be numerically manipulated for use in the analysis of complex structures; therefore, the Boundary Element Method (BEM) is implemented. AEFBEM is a numerical analysis method formulated by applying the boundary element method to an acoustic energy flow governing equation. It is very powerful in predicting the acoustic energy density and intensity of complex structures in medium-to-high frequency ranges, and can analyze interior noise and radiating sound. To verify its validity, several numerical results are provided. BEM and AEFBEM were compared with respect to energy density, and the results from both methods were similar.

Simulation of reinforcement-corrosion-induced crack propagation in concrete by acoustic emission technique and boundary element method analysis

Corrosion of reinforcement is one of the major causes of deterioration in reinforced concrete structures. Various crack patterns are nucleated around reinforcement in concrete due to expansion of corrosion products. Crack kinematics of locations, types, and orientations are quantitatively determined by the acoustic emission (AE) – simplified Green’s functions for moment tensor analysis (SiGMA) procedure in association with laboratory tests conducted on concrete specimens simulating corrosion-induced damage. These kinematic outcomes are obtained as three-dimensional (3-D) locations and vectors, and are thus visualized in 3-D by using virtual reality modeling language (VRML). Numerical analysis is conducted by the boundary element method (BEM) based on the concept of linear elastic fracture mechanics (LEFM) to clarify the mechanisms of corrosion-induced crack extension. Relationships between dimensionless stress intensity factors and cracking types are studied by BEM. Contributions of mode I and mode II failures are dependent on the cracking types. It is found that the process of crack propagation due to corrosion of reinforcement in concrete is mostly a mode I fracture with mixed-mode and, in a few cases, mode II fracture.Key words: acoustic emission, moment tensor, corrosion cracking, stress intensity factor.

Sensitivity Analysis and Optimization Using Energy Finite Element and Boundary Element Methods

DOI: 10.2514/1.20811 This paper presents a continuum-based design sensitivity analysis and optimization of high-frequency radiation problemsusingtheenergy finiteelementmethodandenergyboundaryelementmethod.Thenoiseradiatedfromthe vibrating structure at a high-frequency range is obtained through a sequential procedure. The structural energy finite element method calculates structural energy distribution, which is then used as the boundary condition for the energy boundary element method to calculate the energy density at a far-field observation point. For design sensitivity analysis, the direct differentiation method calculates the sensitivity of the exterior noise through the sensitivity of the structural energy density obtained from the energy finite element method. The adjoint variable method calculates the adjoint load from an acoustic energy boundary element method reanalysis, and the adjoint response is obtained from a structural energy finite element method reanalysis. The sensitivity information is obtained by carrying out numerical integration only on the structural finite element part. The proposed design sensitivity analysisapproach hasbeen applied in the design ofautomotive and navalstructures to search for thebest material layout to achieve the lowest noise level at high frequency.

Research on acoustic-structure sensitivity using FEM and BEM

Acoustic-structure sensitivity is used to predict the change of acoustic pressure when a structure design variable is changed. The sensitivity is significant for reducing noise of structure. Using FEM (finite element method) and BEM (boundary element method) acoustic-structure sensitivity was formulated and presented. The dynamic response and response velocity sensitivity with respect to structure design variable were carried out by using structural FEM, the acoustic response and acoustic pressure sensitivity with respect to structure velocity were carried out by using acoustic BEM. Then, acoustic-structure sensitivity was computed by linking velocity sensitivity in FEM and acoustic sensitivity in BEM. This method was applied to an empty box as an example. Acoustic pressure sensitivity with respect to structure thickness achieved in frequency ranges 1–100 Hz, and its change rule along with stimulating frequency and design variable were analyzed. Results show that acoustic-structure sensitivity method linked with FEM and BEM is effective and correct.

An efficient Green's function for acoustic waveguide problems

AbstractEfficient implementations of the boundary element method for underwater acoustics should employ Green's functions which directly satisfy the boundary conditions on the free surface and the horizontal parts of the bottom boundary. In the present work, these Green's functions are constructed by using either eigenfunction expansions or Ewald's method. This method is discussed in detail, including an attempt to optimize the value of the parameter b, which splits the integral employed in Ewald's representation.A numerical analysis of the infinite series used in the two‐dimensional acoustic waveguide problem is described, considering the following simplifications: the source of acoustic disturbance is time‐harmonic, the velocity of sound is constant and the medium in the absence of perturbations is quiescent. Copyright © 2006 John Wiley & Sons, Ltd.

An efficient stabilized boundary element formulation for 2D time-domain acoustics and elastodynamics

The present paper describes a procedure that improves efficiency, stability and reduces artificial energy dissipation of the standard time-domain direct boundary element method (BEM) for acoustics and elastodynamics. Basically, the developed procedure modifies the boundary element convolution-related vector, being very easy to implement into existing codes. A stabilization parameter is introduced into the recent-in-time convolution operations and the operations related to the distant-in-time convolution contributions are approximated by matrix interpolations. As it is shown in the numerical examples presented at the end of the text, the proposed formulation substantially reduces the BEM computational cost, as well as its numerical instabilities.

Treatment of Sharp Edges & Corners in the Acoustic Boundary Element Method under Neumann Boundary Condition

Boundary element method in acoustics for Neumann boundary condition problems including sharp edges & corners is investigated. In previous acoustic boundary element method, acoustic pressure and normal velocity are the two variables at sharp edges & corners. However, the normal velocity at sharp edges & corners is discontinuous due to the indefinite normal vector. To avoid the indefinite normal vector and the discontinuous normal velocity at sharp edges & corners, normal vector of elemental node is defined and applied in the numer- ical implementation. Then the normal velocity is trans- formed to velocity which is unique even at sharp edges & corners. Such treatments make sure that all variables in the acoustic boundary element method are definite. Computational efficiency and accuracy of the new model are demonstrated by threecases of interioracousticprob- lems for which analytical solutionscan be found and one classical exterior acoustic radiation problem. Curvilin- ear quadrilateral isoparametric elements are applied in the computation. It is found that the numerical results agree with corresponding analytical solutions quite well. keyword: Boundary element method, Sharp edge, Sharp corner, Normal velocity, Normal vector, Helmholtz integral equation

APPLICATION OF THE FAST MULTIPOLE BEM FOR STRUCTURAL–ACOUSTIC SIMULATIONS

The fast multipole method is employed to accelerate the Galerkin boundary element method for acoustics. Important parameters as the nearfield size and the expansion length are discussed and numerical examples are provided. An approximate inverse preconditioner and a preconditioner based on spectral properties of the boundary integral operators are presented. For the example of sound radiation from a brake disk, both approaches prove suitable to reduce the iteration count and to improve the efficiency of the method.