Improvements for the ultra weak variational formulation

In this paper, we investigate strategies to improve the accuracy and efficiency of the ultra weak variational formulation (UWVF) of the Helmholtz equation. The UWVF is a Trefftz type, nonpolynomial method using basis functions derived from solutions of the adjoint Helmholtz equation. We shall consider three choices of basis function: propagating plane waves (original choice), Bessel basis functions, and evanescent wave basis functions. Traditionally, two‐dimensional triangular elements are used to discretize the computational domain. However, the element shapes affect the conditioning of the UWVF. Hence, we investigate the use of different element shapes aiming to lower the condition number and number of degrees of freedom. Our results include the first tests of a plane wave method on meshes of mixed element types. In many modeling problems, evanescent waves occur naturally and are challenging to model. Therefore, we introduce evanescent wave basis functions for the first time in the UWVF to tackle rapidly decaying wave modes. The advantages of an evanescent wave basis are verified by numerical simulations on domains including curved interfaces.Copyright © 2013 John Wiley & Sons, Ltd.     

Fast direct solution of the Helmholtz equation with a perfectly matched layer or an absorbing boundary condition

We consider the efficient numerical solution of the Helmholtz equation in a rectangular domain with a perfectly matched layer (PML) or an absorbing boundary condition (ABC). Standard bilinear (trilinear) finite‐element discretization on an orthogonal mesh leads to a separable system of linear equations for which we describe a cyclic reduction‐type fast direct solver. We present numerical studies to estimate the reflection of waves caused by an absorbing boundary and a PML, and we optimize certain parameters of the layer to minimize the reflection. Copyright © 2003 John Wiley & Sons, Ltd.     

Formulation and validation of Berenger's PML absorbing boundary forthe FDTD simulation of acoustic scattering

Berenger's perfectly matched layer (PML) absorbing boundary condition for electromagnetic (EM) waves is derived to absorb 2-D and 3-D acoustic waves in finite difference time domain (FDTD) simulation of acoustic wave propagation and scattering. A PML medium suitable for acoustic waves is constructed. Plane wave propagation in the PML medium is solved for both 2-D and 3-D cases and explicit FDTD boundary conditions are derived. The equations show that a matched PML medium is a perfect simulation of free space in that a plane wave does not change its direction of propagation or its speed when it propagates from free space into a matched PML medium. FDTD simulation of a pulsed point source propagating in two dimensions is carried out to test the performance of the PML boundary for acoustic waves. Results show that an eight layer PML boundary condition reduces the reflected error 40 dB over Mur's second order boundary condition     

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.     

Perfectly matched layers for transient elastodynamics of unbounded domains

One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outward from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs, almost perfectly, outgoing waves of all non‐tangential angles‐of‐incidence and of all non‐zero frequencies. In a recent work [Computer Methods in Applied Mechanics and Engineering 2003; 192: 1337–1375], the authors presented, inter alia, time‐harmonic governing equations of PMLs for anti‐plane and for plane‐strain motion of (visco‐) elastic media. This paper presents (a) corresponding time‐domain, displacement‐based governing equations of these PMLs and (b) displacement‐based finite element implementations of these equations, suitable for direct transient analysis. The finite element implementation of the anti‐plane PML is found to be symmetric, whereas that of the plane‐strain PML is not. Numerical results are presented for the anti‐plane motion of a semi‐infinite layer on a rigid base, and for the classical soil–structure interaction problems of a rigid strip‐footing on (i) a half‐plane, (ii) a layer on a half‐plane, and (iii) a layer on a rigid base. These results demonstrate the high accuracy achievable by PML models even with small bounded domains. 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。     

An efficient finite element time‐domain formulation for the elastic second‐order wave equation: A non‐split complex frequency shifted convolutional PML

The perfectly matched layer (PML) technique has demonstrated very high efficiency as absorbing boundary condition for the elastic wave equation recast as a first‐order system in velocity and stress in attenuating non‐grazing bulk and surface waves. This paper develops a novel convolutional PML formulation based on the second‐order wave equation with displacements as the only unknowns to annihilate spurious reflections from near‐grazing waves. The derived variational form allows for the use of e.g. finite element and the spectral element methods as spatial discretization schemes. A recursive convolution update scheme of second‐order accuracy is employed such that highly stable, effective time integration with the Newmark‐beta (implicit and explicit with mass lumping) method is achieved. The implementation requires minor modifications of existing displacement‐based finite element software, and the stability and efficiency of the proposed formulation is verified by relevant two‐dimensional benchmarks that accommodate bulk and surface waves. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

Regularized meshless method analysis of the problem of obliquely incident water wave

In this paper, an application of regularized meshless method (RMM) for solving the problem of obliquely incident water wave passing a submerged breakwater is presented. By using desingularization technique to regularize the singularity and hypersingularity of the kernel functions, source points can be located on the physical boundary of an arbitrary domain. To verify the practicability and validity of the RMM, simulations for observing the propagation of oblique incident wave through a barrier are presented where the modified Helmholtz equation is satisfied. Finally, three examples are given to show the effects of breakwater with rigid and absorbing boundary conditions to energy dissipation caused by existence of a barrier. After comparing such analytical solution with the corresponding boundary element method (BEM) solutions, they are shown to be in good agreement.     

An efficient complex‐frequency shifted‐perfectly matched layer for second‐order acoustic wave equation

In the context of simulations of wave propagations in unbounded domain, absorbing boundary conditions are often used to truncate the simulation domain to a finite space. Perfectly matched layer (PML) has proven to be an excellent absorbing boundary conditions. However, as this technique was primarily designed for the first‐order equation system, it cannot be applied to the second‐order equation system directly. In this paper, based on a complex‐coordinate stretching technique, we developed a novel, efficient auxiliary‐differential equation form of the complex‐frequency shifted‐PML for the second‐order equation system. This facilitates the use of complex‐frequency shifted‐PML in acoustic simulations based upon wave equations of second‐order form. Compared with previous state‐of‐the‐art methods, the proposed one has the advantage of simpler implementation. It is an unsplit‐field scheme that can be extended to higher‐order discretization schemes conveniently. Numerical results from both homogeneous and heterogeneous computational domains are provided to illustrate the validity of the method. Copyright © 2013 John Wiley & Sons, Ltd.     

Multifrequency analysis for the Helmholtz equation

In this paper a new numerical method for the multifrequency analysis of the three-dimensional Helmholtz equation is introduced. The collocation boundary element method (BEM) is used for the discretisation of the problem. The identity of the Fourier transform with respect to the wave number is applied to the matrix of the resulting linear system. The analytical form and some important properties are derived. Some numerical examples for the solution are presented.     

Perfectly matched layers for frequency-domain integral equation acoustic scattering problems

Simulations of acoustic wavefields in inhomogeneous media are always performed on finite numerical domains. If contrasts actually extend over the domain boundaries of the numerical volume, unwanted, non-physical reflections from the boundaries will occur. One technique to suppress these reflections is to attenuate them in a locally reflectionless absorbing boundary layer enclosing the spatial computational domain, a perfectly matched layer (PML). This technique is commonly applied in time-domain simulation methods like finite element methods or finite-difference time-domain, but has not been applied to the integral equation method. In this paper, a PML formulation for the three-dimensional frequency-domain integral-equation-based acoustic scattering problem is derived. Three-dimensional acoustic scattering configurations are used to test the PML formulation. The results demonstrate that strong attenuation (a factor of 200 in amplitude) of the scattered pressure field is achieved for thin layers with a thickness of less than a wavelength, and that the PMLs themselves are virtually reflectionless. In addition, it is shown that the integral equation method, both with and without PMLs, accurately reproduces pressure fields by comparing the obtained results with analytical solutions.     

Solution of Inverse Boundary Optimization Problem by Trefftz Method and Exponentially Convergent Scalar Homotopy Algorithm

The inverse boundary optimization problem, governed by the Helmholtz equation, is analyzed by the Trefftz method (TM) and the exponentially convergent scalar homotopy algorithm (ECSHA). In the inverse boundary optimization problem, the position for part of boundary with given boundary condition is unknown, and the position for the rest of boundary with additionally specified boundary conditions is given. Therefore, it is very difficult to handle the boundary optimization problem by any numerical scheme. In order to stably solve the boundary optimization problem, the TM, one kind of boundary-type meshless methods, is adopted in this study, since it can avoid the generation of mesh grid and numerical integration. In the boundary optimization problem governed by the Helmholtz equation, the numerical solution of TM is expressed as linear combination of the T-complete functions. When this problem is considered by TM, a system of nonlinear algebraic equations will be formed and solved by ECSHA which will converge exponentially. The evolutionary process of ECSHA can acquire the unknown coefficients in TM and the spatial position of the unknown boundary simultaneously. Some numerical examples will be provided to demonstrate the ability and accuracy of the proposed scheme. Besides, the stability of the proposed meshless method will be validated by adding some noise into the boundary conditions.     

Perfectly matched layer in numerical wave propagation: factors that affect its performance

The perfectly matched layer (PML) boundary condition is generally employed to prevent spurious reflections from numerical boundaries in wave propagation methods. However, PML requires additional computational resources. We have examined the performance of the PML by changing the distribution of sampling points and the PML's absorption profile with a view to optimizing the PML's efficiency. We used the collocation method in our study. We found that equally spaced field sampling points give better absorption of beams under both optimal and nonoptimal conditions for low PML widths. At high PML widths, unequally spaced basis points may be equally efficient. The efficiency of various PML absorption profiles, including new ones, has been studied, and we conclude that for better numerical efficiency it is important to choose an appropriate profile.     

A Galerkin least-squares finite element method for the two-dimensional Helmholtz equation

In this paper a Galerkin least-squares (GLS) finite element method, in which residuals in least-squares form are added to the standard Galerkin variational equation, is developed to solve the Helmholtz equation in two dimensions. An important feature of GLS methods is the introduction of a local mesh parameter that may be designed to provide accurate solutions with relatively coarse meshes. Previous work has accomplished this for the one-dimensional Helmholtz equation using dispersion analysis. In this paper, the selection of the GLS mesh parameter for two dimensions is considered, and leads to elements that exhibit improved phase accuracy. For any given direction of wave propagation, an optimal GLS mesh parameter is determined using two-dimensional Fourier analysis. In general problems, the direction of wave propagation will not be known a priori. In this case, an optimal GLS parameter is found which reduces phase error for all possible wave vector orientations over elements. The optimal GLS parameters are derived for both consistent and lumped mass approximations. Several numerical examples are given and the results compared with those obtained from the Galerkin method. The extension of GLS to higher-order quadratic interpolations is also presented.     

Trefftz Methods for Time Dependent Partial Differential Equations

In this paper we present a mesh-free approach to numerically solving a class of second order time dependent partial differential equations which include equations of parabolic, hyperbolic and parabolic-hyperbolic types. For numerical purposes, a variety of transformations is used to convert these equations to standard reaction-diffusion and wave equation forms. To solve initial boundary value problems for these equations, the time dependence is removed by either the Laplace or the Laguerre transform or time differencing, which converts the problem into one of solving a sequence of boundary value problems for inhomogeneous modified Helmholtz equations. These boundary value problems are then solved by a combination of the method of particular solutions and Trefftz methods. To do this, a variety of techniques is proposed for numerically computing a particular solution for the inhomogeneous modified Helmholtz equation. Here, we focus on the Dual Reciprocity Method where the source term is approximated by radial basis functions, polynomial or trigonometric functions. Analytic particular solutions are presented for each of these approximations. The Trefftz method is then used to solve the resulting homogenous equation obtained after the approximate particular solution is subtracted off. Two types of Trefftz bases are considered, F-Trefftz bases based on the fundamental solution of the modified Helmholtz equation, and T-Trefftz bases based on separation of variables solutions. Various techniques for satisfying the boundary conditions are considered, and a discussion is given of techniques for mitigating the ill-conditioning of the resulting linear systems. Finally, some numerical results are presented illustrating the accuracy and efficacy of this methodology.     

下卧基岩饱和地基在移动荷载作用下的动力响应

基于Biot多孔弹性介质波动理论,在平面应变条件下研究了下卧基岩饱和地基在移动线荷载作用下的动力响应。通过引入势函数并利用Helmholtz原理,再经过Fourier变换和逆变换,获得了移动线荷载作用下饱和地基的位移、应力、孔隙水压力解答。最后通过快速逆傅里叶变化（IFFT）得到数值计算结果,详细分析了土颗粒的压缩性、孔隙水的压缩性、饱和土的剪切模量、孔隙率、渗透性、移动荷载速度和饱和土层厚度等参数对动力响应的影响     

On approximate cardinal preconditioning methods for solving PDEs with radial basis functions

The approximate cardinal basis function (ACBF) preconditioning technique has been used to solve partial differential equations (PDEs) with radial basis functions (RBFs). In [Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Adv Comput Math; in press], a preconditioning scheme that is based upon constructing the least-squares approximate cardinal basis function from linear combinations of the RBF–PDE matrix elements has shown very attractive numerical results. This preconditioning technique is sufficiently general that it can be easily applied to many differential operators.In this paper, we review the ACBF preconditioning techniques previously used for interpolation problems and investigate a class of preconditioners based on the one proposed in [Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Adv Comput Math; in press] when a cardinality condition is enforced on different subsets. We numerically compare the ACBF preconditioners on several numerical examples of Poisson's, modified Helmholtz and Helmholtz equations, as well as a diffusion equation and discuss their performance.     

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.     

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

The 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.