Enriched Constant Elements in the Boundary Element Method for Solving 2D Acoustic Problems at Higher Frequencies

The boundary element method(BEM)is a popular method for solving acoustic wave propagation problems,especially those in exterior domains,owing to its ease in handling radiation conditions at infinity.However,BEM models must meet the requirement of 6–10 elements per wavelength,using the conventional constant,linear,or quadratic elements.Therefore,a large storage size of memory and long solution time are often needed in solving higher-frequency problems.In this work,we propose two new types of enriched elements based on conventional constant boundary elements to improve the computational efficiency of the 2D acoustic BEM.The first one uses a plane wave expansion,which can be used to model scattering problems.The second one uses a special plane wave expansion,which can be used tomodel radiation problems.Five examples are investigated to showthe advantages of the enriched elements.Compared with the conventional constant elements,the new enriched elements can deliver results with the same accuracy and in less computational time.This improvement in the computational efficiency is more evident at higher frequencies(with the nondimensional wave numbers exceeding 100).The paper concludes with the potential of our proposed enriched elements and plans for their further improvement.

Diagonal form fast multipole boundary element method for 2D acoustic problems based on Burton-Miller boundary integral equation formulation and its applications

This paper describes formulation and implementation of the fast multipole boundary element method （FMBEM） for 2D acoustic problems. The kernel function expansion theory is summarized, and four building blocks of the FMBEM are described in details. They are moment calculation, moment to moment translation, moment to local translation, and local to local translation. A data structure for the quad-tree construction is proposed which can facilitate implementation. An analytical moment expression is derived, which is more accurate, stable, and efficient than direct numerical computation. Numerical examples are presented to demonstrate the accuracy and efficiency of the FMBEM, and radiation of a 2D vibration rail mode is simulated using the FMBEM.

Numerical quadrature for singular and near-singular integrals of boundary element method and its applications in large-scale acoustic problems

The numerical quadrature methods for dealing with the problems of singular and near-singular integrals caused by Burton-Miller method are proposed, by which the conventional and fast multipole BEMs （boundary element methods） for 3D acoustic problems based on constant elements are improved. To solve the problem of singular integrals, a Hadamard finite-part integral method is presented, which is a simplified combination of the methods proposed by Kirkup and Wolf. The problem of near-singular integrals is overcome by the simple method of polar transformation and the more complex method of PART （Projection and Angular ＆ Radial Transformation）. The effectiveness of these methods for solving the singular and near-singular problems is validated through comparing with the results computed by the analytical method and/or the commercial software LMS Virtual.Lab. In addition, the influence of the near-singular integral problem on the computational precisions is analyzed by computing the errors relative to the exact solution. The computational complexities of the conventional and fast multipole BEM are analyzed and compared through numerical computations. A large-scale acoustic scattering problem, whose degree of freedoms is about 340,000, is implemented successfully. The results show that, the near singularity is primarily introduced by the hyper-singular kernel, and has great influences on the precision of the solution. The precision of fast multipole BEM is the same as conventional BEM, but the computational complexities are much lower.

A Global Uniqueness Result in Inverse Acoustic Obstacle Scattering Problem

It is proved that a sound-soft scatterer in R^N （N = 2, 3） is uniquely determined by a finite number of acoustic far-field measurements. The admissible scatterer possibly consists of finitely many solid obstacles and subsets of （N - 1）- dimensional hyperplanes.

Comparisons between Isotropic and Anisotropic TV Regularizations in Inverse Acoustic Scattering

This article compares the isotropic and anisotropic TV regularizations used in inverse acoustic scattering. It is observed that compared with the traditional Tikhonov regularization, isotropic and anisotropic TV regularizations perform better in the sense of edge preserving. While anisotropic TV regularization will cause distortions along axes. To minimize the energy function with isotropic and anisotropic regularization terms, we use split Bregman scheme. We do several 2D numerical experiments to validate the above arguments.

Meshless acoustic analysis using a weakly singular Burton-Miller boundary integral formulation

The Burton-Miller boundary integral formulation is solved by a complex variable boundary element-free method(CVBEFM)for the boundary-only meshless analysis of acoustic problems with arbitrary wavenumbers.To regularize both strongly singular and hypersingular integrals and to avoid the computation of the solid angle and its normal derivative,a weakly singular Burton-Miller formulation is derived by considering the normal derivative of the solid angle and adopting the singularity subtraction procedures.To facilitate the implementation of the CVBEFM and the approximation of gradients of the boundary variables,a stabilized complex variable moving least-square approximation is selected in the meshless discretization procedure.The results show the accuracy and efficiency of the present CVBEFM and reveal that the method can produce satisfactory results for all wavenumbers,even for extremely large wavenumbers such as k=10000.

An Adaptive Uniaxial Perfectly Matched Layer Method for Time-Harmonic Scattering Problems

The uniaxial perfectly matched layer (PML) method uses rectangular domain to define the PML problem and thus provides greater flexibility and efficiency in deal- ing with problems involving anisotropic scatterers.In this paper an adaptive uniaxial PML technique for solving the time harmonic Helmholtz scattering problem is devel- oped.The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates.The adaptive finite element method based on a posteriori error estimate is proposed to solve the PML equa- tion which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the thickness of the PML absorb- ing layer.Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.In particular,it is demonstrated that the PML layer can be chosen as close to one wave-length from the scatterer and still yields good accuracy and efficiency in approximating the far fields.

ACOUSTIC PROPAGATION IN SHEARED MEAN FLOW USING COMPUTATIONAL AEROACOUSTICS

Acoustic propagation problems in the sheared mean flow are numerically investigated using different acoustic propagation equations , including linearized Euler equations ( LEE ) and acoustic perturbation equations ( APE ) .The resulted acoustic pressure is compared for the cases of uniform mean flow and sheared mean flow using both APE and LEE.Numerical results show that interactions between acoustics and mean flow should be properly considered to better understand noise propagation problems , and the suitable option of the different acoustic equations is indicated by the present comparisons.Moreover , the ability of APE to predict acoustic propagation is validated.APE can replace LEE when the 3-D flow-induced noise problem is solved , thus computational cost can decrease.

RECONSTRUCTION STABILITY OF NEARFIELD ACOUSTIC HOLOGRAPHY

The distributed source boundary point method （DSBPM） is used as the spatial transform algorithm for realizing nearfield acoustic holography （NAH）, the sensitivity of the reconstructed solution to the measurement errors is analyzed, and the regularization method is proposed to stabilize the reconstruction process, control the influence of the measurement errors and get a better approximate solution. An oscillating sphere is investigated as a numerical example, the influence of the measurement errors on the reconstruction solution is demonstrated, and the feasibility and validity of the regularization method are validated. Key words： Acoustic holography Boundary point method Inverse problem Regularization

Fast multipole accelerated boundary element method for the Helmholtz equation in acoustic scattering problems

We apply the fast multipole method (FMM) accelerated boundary element method (BEM) for the three-dimensional (3D) Helmholtz equation, and as a result, large-scale acoustic scattering problems involving 400000 elements are solved efficiently. This is an extension of the fast multipole BEM for two-dimensional (2D) acoustic problems developed by authors recently. Some new improvements are obtained. In this new technique, the improved Burton-Miller formulation is employed to over-come non-uniqueness difficulties in the conventional BEM for exterior acoustic problems. The computational efficiency is further improved by adopting the FMM and the block diagonal preconditioner used in the generalized minimum residual method (GMRES) iterative solver to solve the system matrix equation. Numerical results clearly demonstrate the complete reliability and efficiency of the proposed algorithm. It is potentially useful for solving large-scale engineering acoustic scattering problems.

Localized MFS for three‐dimensional acoustic inverse problems on complicated domains

This paper proposes a semi‐analytical and local meshless collocation method,the loca-lized method of fundamental solutions(LMFS),to address three‐dimensional(3D)acoustic inverse problems in complex domains.The proposed approach is a recently developed numerical scheme with the potential of being mathematically simple,nu-merically accurate,and requiring less computational time and storage.In LMFS,an overdetermined sparse linear system is constructed by using the known data at the nodes on the accessible boundary and by making the remaining nodes satisfy the governing equation.In the numerical procedure,the pseudoinverse of a matrix is solved via the truncated singular value decomposition,and thus the regularization techniques are not needed in solving the resulting linear system with a well‐conditioned matrix.Numerical experiments,involving complicated geometry and the high noise level,confirm the ef-fectiveness and performance of the LMFS for solving 3D acoustic inverse problems.

An Adaptive Finite Element PML Method for the Acoustic Scattering Problems in Layered Media

The paper concerns the numerical solution for the acoustic scattering problems in a two-layer medium.The perfectly matched layer(PML)technique is adopted to truncate the unbounded physical domain into a bounded computational domain.An a posteriori error estimate based adaptive finite element method is developed to solve the scattering problem.Numerical experiments are included to demonstrate the efficiency of the proposed method.

FINITE ELEMENT APPROXIMATION OF EIGENVALUE PROBLEM FOR A COUPLED VIBRATION BETWEEN ACOUSTIC FIELD AND PLATE

We formulate a coupled vibration between plate and acoustic field in mathematically rigorous fashion. It leads to a non-standard eigenvalue problem. A finite element approximation is considered in an abstract way, and the approximate eigenvalue problem is written in an operator form by means of some Ritz projections. The order of convergence is proved based on the result of Babugka and Osborn. Some numerical example is shown for the problem for which the exact analytical solutions are calculated. The results shows that the convergence order is consistent with the one by the numerical analysis.

The enhanced volume source boundary point method for the calculation of acoustic radiation problem

The Volume Source Boundary Point Method (VSBPM) is greatly improved so that it will speed up the VSBPM's solution of the acoustic radiation problem caused by the vibrating body. The fundamental solution provided by Helmholtz equation is enforced in a weighted residual sense over a tetrahedron located on the normal line of the boundary node to replace the coefficient matrices of the system equation. Through the enhanced volume source boundary point analysis of various examples and the sound field of a vibrating rectangular box in a semi-anechoic chamber, it has revealed that the calculating speed of the EVSBPM is more than 10 times faster than that of the VSBPM while it works on the aspects of its calculating precision and stability, adaptation to geometric shape of vibrating body as well as its ability to overcome the non-uniqueness problem.

An application of radiation boundary condition to scattering problem of acoustic waves in fluids

A numerical method of solving acoustic wave scattering pnblem in fluids is described. Radiation boundary condition (RBC) obtained by factorization method of Helmholtz equation is applied to transforming the exterior boundary value problem in unbounded region into one in a finite region. Combined with RBC and scatterer surface boundary condition, Helmholtz equation is solved numerically by the finite difference method. Computational results for sphere and prolate spheroidal scatterers are in excellent agreement with eigenfunction solutions and much better than the results of OSRC method.

Recent Advances and Emerging Applications of the Singular Boundary Method for Large-Scale and High-Frequency Computational Acoustics

With the rapid development of computer technology,numerical simulation has become the third scientific research tool besides theoretical analysis and experi-mental research.As the core of numerical simulation,constructing efficient,accurate and stable numerical methods to simulate complex scientific and engineering prob-lems has become a key issue in computational mechanics.The article outlines the ap-plication of singular boundary method to the large-scale and high-frequency acoustic problems.In practical application,the key issue is to construct efficient and accurate numerical methodology to calculate the large-scale and high-frequency soundfield.This article focuses on the following two research areas.They are how to discretize partial differential equations into more appropriate linear equations,and how to solve linear equations more efficiently.The bottle neck problems encountered in the compu-tational acoustics are used as the technical routes,i.e.,efficient solution of dense linear system composed of ill-conditioned matrix and stable simulation of wave propagation at low sampling frequencies.The article reviews recent advances in emerging appli-cations of the singular boundary method for computational acoustics.This collection can provide a reference for simulating other more complex wave propagation.

FullApertureReconstruction of theAcoustic Far-Field Pattern from Few Measurements

We propose a numerical procedure to extend to full aperture the acoustic farfield pattern(FFP)when measured in only few observation angles.The reconstruction procedure is a multi-step technique that combines a total variation regularized iterative method with the standard Tikhonov regularized pseudo-inversion.The proposed approach distinguishes itself from existing solution methodologies by using an exact representation of the total variation which is crucial for the stability and robustness of Newton algorithms.We present numerical results in the case of two-dimensional acoustic scattering problems to illustrate the potential of the proposed procedure for reconstructing the full aperture of the FFP from very few noisy data such as backscattering synthetic measurements.