The perfectly matched layer for the ultra weak variational formulation of the 3D Helmholtz equation

We investigate the feasibility of using the perfectly matched layer (PML) as an absorbing boundary condition for the ultra weak variational formulation (UWVF) of the 3D Helmholtz equation. The PML is derived using complex stretching of the spatial variables. This leads to a modified Helmholtz equation for which the UWVF can be derived. In the standard discrete UWVF, the approximating subspace is constructed from local solutions of the Helmholtz equation. In previous studies plane wave basis functions have been advocated because they simplify the building of the UWVF matrices. For the PML domain we propose a special set of plane wave basis functions which allow fast computations and efficiently reduce spurious numerical reflections. The method is validated by numerical experiments. In comparison to a low‐order absorbing boundary condition, the PML shows superior performance. Copyright © 2004 John Wiley & Sons, Ltd.     

基于拓展单位分解有限元法分析声波在多域场内的响应

基于拓展单位分解有限元方法，将平面波函数和贝塞尔函数作为基函数进行拓展。将亥姆霍兹方程离散，求解时不变情况下多域场内声波的响应，并分析基函数对求解精度的影响。将波动方程的时间导数利用二阶中心差分方法离散，得到方程的隐式表达式，划分时间步迭代求解时变情况下声波在多域场内的响应，分析迭代时间间隔对计算精度的影响，与典型算例的精确解进行比较，验证精确性。结果表明，平面波函数作为拓展基函数，利用二阶中心差分法离散时间导数，分析时不变以及时变情况下多域场内高波数声波的响应问题，具有较高的计算精度和计算效率。     

A full-wave Helmholtz model for continuous-wave ultrasound transmission

A full-wave Helmholtz model of continuous-wave (CW) ultrasound fields may offer several attractive features over widely used partial-wave approximations. For example, many full-wave techniques can be easily adjusted for complex geometries, and multiple reflections of sound are automatically taken into account in the model. To date, however, the full-wave modeling of CW fields in general 3D geometries has been avoided due to the large computational cost associated with the numerical approximation of the Helmholtz equation. Recent developments in computing capacity together with improvements in finite element type modeling techniques are making possible wave simulations in 3D geometries which reach over tens of wavelengths. The aim of this study is to investigate the feasibility of a full-wave solution of the 3D Helmholtz equation for modeling of continuous-wave ultrasound fields in an inhomogeneous medium. The numerical approximation of the Helmholtz equation is computed using the ultraweak variational formulation (UWVF) method. In addition, an inverse problem technique is utilized to reconstruct the velocity distribution on the transducer which is used to model the sound source in the UWVF scheme. The modeling method is verified by comparing simulated and measured fields in the case of transmission of 531 kHz CW fields through layered plastic plates. The comparison shows a reasonable agreement between simulations and measurements at low angles of incidence but, due to mode conversion, the Helmholtz model becomes insufficient for simulating ultrasound fields in plates at large angles of incidence.     

A domain decomposition method for discontinuous Galerkin discretizations of Helmholtz problems with plane waves and Lagrange multipliers

A nonoverlapping domain decomposition (DD) method is proposed for the iterative solution of systems of equations arising from the discretization of Helmholtz problems by the discontinuous enrichment method. This discretization method is a discontinuous Galerkin finite element method with plane wave basis functions for approximating locally the solution and dual Lagrange multipliers for weakly enforcing its continuity over the element interfaces. The primal subdomain degrees of freedom are eliminated by local static condensations to obtain an algebraic system of equations formulated in terms of the interface Lagrange multipliers only. As in the FETI‐H and FETI‐DPH DD methods for continuous Galerkin discretizations, this system of Lagrange multipliers is iteratively solved by a Krylov method equipped with both a local preconditioner based on subdomain data, and a global one using a coarse space. Numerical experiments performed for two‐ and three‐dimensional acoustic scattering problems suggest that the proposed DD‐based iterative solver is scalable with respect to both the size of the global problem and the number of subdomains. Copyright © 2009 John Wiley & Sons, Ltd.     

传播波与倏逝波对全息重建影响的研究

通过分析近场声全息中传播波和倏逝波的声场分布与声源大小、辐射频率及测量距离的关系,得出了传播波和倏逝波在空间声场中的变化规律、影响因素以及两者之间的对应关系,确定了可利用的倏逝波传播距离。在此基础上,提出了一种改进的远场声全息方法：首先利用传统远场声全息方法重建声源,然后通过传播波与倏逝波的关系,得出声源面上相应的倏逝波成分,两者叠加获取接近于近场声全息的重建结果。最后,通过仿真和实验验证了这一结果。     

A comparison of wave‐based discontinuous Galerkin,ultra‐weak and least‐square methods for wave problems

Several numerical methods using non‐polynomial interpolation have been proposed for wave propagation problems at high frequencies. The common feature of these methods is that in each element, the solution is approximated by a set of local solutions. They can provide very accurate solutions with a much smaller number of degrees of freedom compared to polynomial interpolation. There are however significant differences in the way the matching conditions enforcing the continuity of the solution between elements can be formulated. The similarities and discrepancies between several non‐polynomial numerical methods are discussed in the context of the Helmholtz equation. The present comparison is concerned with the ultra‐weak variational formulation (UWVF), the least‐squares method (LSM) and the discontinuous Galerkin method with numerical flux (DGM). An analysis in terms of Trefftz methods provides an interesting insight into the properties of these methods. Second, the UWVF and the LSM are reformulated in a similar fashion to that of the DGM. This offers a unified framework to understand the properties of several non‐polynomial methods. Numerical results are also presented to put in perspective the relative accuracy of the methods. The numerical accuracies of the methods are compared with the interpolation errors of the wave bases. Copyright © 2010 John Wiley & Sons, Ltd.     

Singularity extraction technique for integral equation methods with higher order basis functions on plane triangles and tetrahedra

A numerical solution of integral equations typically requires calculation of integrals with singular kernels. The integration of singular terms can be considered either by purely numerical techniques, e.g. Duffy's method, polar co‐ordinate transformation, or by singularity extraction. In the latter method the extracted singular integral is calculated in closed form and the remaining integral is calculated numerically. This method has been well established for linear and constant shape functions. In this paper we extend the method for polynomial shape functions of arbitrary order. We present recursive formulas by which we can extract any number of terms from the singular kernel defined by the fundamental solution of the Helmholtz equation, or its gradient, and integrate the extracted terms times a polynomial shape function in closed form over plane triangles or tetrahedra. The presented formulas generalize the singularity extraction technique for surface and volume integral equation methods with high‐order basis functions. Numerical experiments show that the developed method leads to a more accurate and robust integration scheme, and in many cases also a faster method than, for example, Duffy's transformation. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical comparison of two meshfree methods for acoustic wave scattering

Density results using an infinite number of acoustic waves allow us to derive meshless methods for solving the homogeneous and the inhomogeneous Helmholtz equation. In this paper we consider the numerical simulation of acoustic scattering problems in a bounded domain using the plane waves method and the method of fundamental solutions. We establish a link between the two methods, namely the plane waves method may be seen as the asymptotic case of the method of fundamental solutions for distant source points. Several numerical tests comparing these methods are presented.     

Numerical resolution of the boundary integral equations for elastic scattering by a plane crack

The problem of wave scattering by a plane crack is solved, either in the case of acoustic waves or in the case of elastic waves incidence using the boundary integral equation method. A collocation method is often used to solve that equation, but here we will use a variational method, first writing the problem of Fourier variables, and then writing the associated integrals in the sesquilinear form with weak singularity kernels. This representation is used in the numerical approach, made with a finite element method in the surface of the crack. Numerical tests were made with circular and elliptical cracks, but this method can be extended to other shapes, with the same convergence profiles. Extensive results are given concerning the crack opening displacement, the scattering cross-section, the back-scattered amplitude and far-field patterns.     

Modelling of progressive short waves using wave envelopes

We consider progressive waves such that the time independent potential satisfies the Helmholtz equation, for example, the travelling wave diffracted from a body. In order to model the wave potential using finite elements it is usual to discretize the domain such that there are about ten nodal points per wavelength. However, such a procedure is computationally expensive and impractical if the waves are short. The goal is to be able to model accurately with few elements problems such as sonar and radar. Therefore we seek a new method in which the discretization of the domain is more economical. To do so, we express the complex potential ϕ in terms of the real wave envelope A and the real phase p such that ϕ=Ae^ip, and expect that in most regions the functions A and p vary much more gradually over the domain than does the oscillatory potential ϕ. Therefore instead of modelling the potential we model the wave envelope and the phase. The usual approach then uses the well known geometrical optics approximation (see p. 109 of Reference 1) : if the wave number k is large then the potential can be expanded in decreasing powers of k. The first two terms give the eikonal equation for the phase and the transport equation for the wave envelope respectively (see p. 149 of Reference 2). However, using the geometrical optics approximation (or ray theory) gives no diffraction effects. This approach shall therefore not be considered. (We note though that Keller's theory of geometrical diffraction, an extension to geometrical optics, does allow for diffraction effects and this may be considered at a later date.) We shall consider a new method which shall be described in the present paper and apply it to two-dimensional problems, although the method is equally valid for arbitary three-dimensional problems. (The method has already been validated for the case of one-dimensional problems.) An iterative procedure is described whereby an estimate of the phase is first given and from the resulting finite element calculation for the wave envelope a better estimate for the phase is obtained. The iterated values for the phase and wave envelope converge to the expected values for the test progressive wave examples considered. Even if a very poor estimate for the phase is first given the iterated values converge to the exact values but very slowly. © 1997 John Wiley & Sons, Ltd.     

Overview of the discontinuous enrichment method,the ultra‐weak variational formulation,and the partition of unity method for acoustic scattering in the medium frequency regime and performance comparisons

The efficient finite element discretization of the Helmholtz equation becomes challenging in the medium frequency regime because of numerical dispersion, or what is often referred to in the literature as the pollution effect. A number of FEMs with plane wave basis functions have been proposed to alleviate this effect, and improve on the unsatisfactory preasymptotic convergence of the polynomial FEM. These include the partition of unity method, the ultra‐weak variational formulation, and the discontinuous enrichment method. A previous comparative study of the performance of such methods focused on the first two aforementioned methods only. By contrast, this paper provides an overview of all three methods and compares several aspects of their performance for an acoustic scattering benchmark problem in the medium frequency regime. It is found that the discontinuous enrichment method outperforms both the partition of unity method and the ultra‐weak variational formulation by a significant margin. Copyright © 2011 John Wiley & Sons, Ltd.     

Modelling of short wave diffraction problems using approximating systems of plane waves

This paper describes a finite element model for the solution of Helmholtz problems at higher frequencies that offers the possibility of computing many wavelengths in a single finite element. The approach is based on partition of unity isoparametric elements. At each finite element node the potential is expanded in a discrete series of planar waves, each propagating at a specified angle. These angles can be uniformly distributed or may be carefully chosen. They can also be the same for all nodes of the studied mesh or may vary from one node to another. The implemented approach is used to solve a few practical problems such as the diffraction of plane waves by cylinders and spheres. The wave number is increased and the mesh remains unchanged until a single finite element contains many wavelengths in each spatial direction and therefore the dimension of the whole problem is greatly reduced. Issues related to the integration and the conditioning are also discussed. Copyright © 2002 John Wiley & Sons, Ltd.     

Three‐dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid‐frequency Helmholtz problems

Recently, a discontinuous Galerkin finite element method with plane wave basis functions and Lagrange multiplier degrees of freedom was proposed for the efficient solution in two dimensions of Helmholtz problems in the mid‐frequency regime. In this paper, this method is extended to three dimensions and several new elements are proposed. Computational results obtained for several wave guide and acoustic scattering model problems demonstrate one to two orders of magnitude solution time improvement over the higher‐order Galerkin method. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical modelling of elastic wave scattering in frequency domain by the partition of unity finite element method

In this paper, we investigate a numerical approach based on the partition of unity finite element method, for the time‐harmonic elastic wave equations. The aim of the proposed work is to accurately model two‐dimensional elastic wave problems with fewer elements, capable of containing many wavelengths per nodal spacing, and without refining the mesh at each frequency. The approximation of the displacement field is performed via the standard finite element shape functions, enriched by superimposing pressure and shear plane wave basis, which incorporate knowledge of the wave propagation. A variational framework able to handle mixed boundary conditions is described. Numerical examples dealing with the radiation and the scattering of elastic waves by a circular body are presented. The results show the performance of the proposed method in both accuracy and efficiency. Copyright © 2008 John Wiley & Sons, Ltd.     

A space–time discontinuous Galerkin method for the solution of the wave equation in the time domain

In recent years, the focus of research in the field of computational acoustics has shifted to the medium frequency regime and multiscale wave propagation. This has led to the development of new concepts including the discontinuous enrichment method. Its basic principle is the incorporation of features of the governing partial differential equation in the approximation. In this contribution, this concept is adapted for the simulation of transient problems governed by the wave equation. We present a space–time discontinuous Galerkin method with Lagrange multipliers, where the shape approximation in space and time is based on solutions of the homogeneous wave equation. The use of hierarchical wave‐like basis functions is enabled by means of a variational formulation that allows for discontinuities in both the spatial and the temporal discretizations. Numerical examples in one space dimension demonstrate the outstanding performance of the proposed method compared with conventional space–time finite element methods. Copyright © 2008 John Wiley & Sons, Ltd.     

声场中倏逝波特性及改进全息重建方法的研究

近场声全息是通过在紧靠被测声源物体的近场测量面上记录全息数据,不但可以记录传播波成分,还可以获得反映声场高空间特性的倏逝波成分。通过分析传播波和倏逝波在声场中的分布情况与声场空间位置、声源大小及频率的关系,得出了传播波和倏逝波在空间声场中的变化规律、影响因素以及二者之间的对应关系。通过对不同背景噪声下倏逝波的传播情况确定了可利用倏逝波传播距离。在此基础上,提出了一种改进的声全息方法,通过用这种方法,可得出声源面上传播波与倏逝波成分,获取等效于近场声全息的重建结果。最后,通过仿真验证了这一结果。     

A hybrid discontinuous in space and time Galerkin method for wave propagation problems

Medium‐frequency regime and multi‐scale wave propagation problems have been a subject of active research in computational acoustics recently. New techniques have attempted to overcome the limitations of existing discretization methods that tend to suffer from dispersion. One such technique, the discontinuous enrichment method, incorporates features of the governing partial differential equation in the approximation, in particular, the solutions of the homogeneous form of the equation. Here, based on this concept and by extension of a conventional space–time finite element method, a hybrid discontinuous Galerkin method (DGM) for the numerical solution of transient problems governed by the wave equation in two and three spatial dimensions is described. The discontinuous formulation in both space and time enables the use of solutions to the homogeneous wave equation in the approximation. In this contribution, within each finite element, the solutions in the form of polynomial waves are employed. The continuity of these polynomial waves is weakly enforced through suitably chosen Lagrange multipliers. Results for two‐dimensional and three‐dimensional problems, in both low‐frequency and medium‐frequency regimes, show that the proposed DGM outperforms the conventional space–time finite element method. Copyright © 2014 John Wiley & Sons, Ltd.     

Singular enrichment functions for Helmholtz scattering at corner locations using the boundary element method

In this paper, we use an enriched approximation space for the efficient and accurate solution of the Helmholtz equation to solve problems of wave scattering by polygonal obstacles. This is implemented in both boundary element method (BEM) and partition of unity BEM (PUBEM) settings. The enrichment draws upon the asymptotic singular behaviour of scattered fields at sharp corners, leading to a choice of fractional-order Bessel functions that complement the existing Lagrangian (BEM) or plane wave (PUBEM) approximation spaces. Numerical examples consider configurations of scattering objects, subject to the Neumann “sound hard” boundary conditions, demonstrating that the approach is a suitable choice for both convex scatterers and also for multiple scattering objects that give rise to multiple reflections. Substantial improvements are observed, significantly reducing the number of degrees of freedom required to achieve a prescribed accuracy in the vicinity of a sharp corner.     

Reflection and refraction of plane waves at the interface between piezoelectric and piezomagnetic media

The paper analyzes the reflection and refraction of a plane wave incidence obliquely at the interface between piezoelectric and piezomagnetic media. The materials are assumed to be transversely isotropic. Numerical calculations are performed for BaTiO₃/CoFe₂O₄ material combination. Four cases, incidence of the coupled quasi-pressure (QP) and quasi-shear vertical (QSV) wave from BaTiO₃ or CoFe₂O₄ media, are discussed. The reflection and transmission coefficients and energy coefficients varying with the incident angle are examined. Calculated results are verified by considering the energy conservation. Results show that the reflected and transmitted wave fields in the sagittal plane consist of six kinds of waves, i.e. the coupled QP and QSV waves, evanescent electroacoustic (EA) and magnetic potential (MP) waves in the piezoelectric medium (BaTiO₃), evanescent magnetoacoustic (MA) and electric potential (EP) waves in the piezomagnetic medium (CoFe₂O₄), among which the EA, MA, MP and EP waves propagate along the interface. The most amount of the incident energy goes with the waves that are the same type as the incident wave, while the energy arising from wave mode conversion occupies a less part of the incident energy. The electric energy in BaTiO₃ is higher than the magnetic energy in CoFe₂O₄; they both attain their maximum values at/before the critical angle. Critical angles have little effect on evanescent waves except when the total reflection takes place. These results would provide useful complementary information for magnetoelectric composite materials.     

Boundary type finite element method for surface wave motion based on trigonometric function interpolation

There are many physical phenomena which can be handled by the Helmholtz equation. The equation explains certain phenomena of wave propagation. This paper presents a new finite element method to analyse surface wave motion. The characteristic point of this method is that the interpolation equation is chosen to satisfy the governing Helmholtz equation using trigonometric functions. This follows that the variational functional to be minimized can be formulated such that the integration is limited to the boundary of the element. The numerical solutions obtained are compared with analytical and experimental solutions. From these comparative studies, it is concluded that the present method provides a useful tool for the analysis of surface wave motion.