A Solver for Helmholtz System Generated by the Discretization of Wave Shape Functions

An interesting discretization method for Helmholtz equations was introduced in B. Després [1]. This method is based on the ultra weak variational formulation (UWVF) and the wave shape functions, which are exact solutions of the governing Helmholtz equation. In this paper we are concerned with fast solver for the system generated by the method in [1]. We propose a new preconditioner for such system, which can be viewed as a combination between a coarse solver and the block diagonal preconditioner introduced in [13]. In our numerical experiments, this preconditioner is applied to solve both two-dimensional and three-dimensional Helmholtz equations, and the numerical results illustrate that the new preconditioner is much more efficient than the original block diagonal preconditioner.     

Near-field performance analysis of locally-conformal perfectly matched absorbers via Monte Carlo simulations

In the numerical solution of some boundary value problems by the finite element method (FEM), the unbounded domain must be truncated by an artificial absorbing boundary or layer to have a bounded computational domain. The perfectly matched layer (PML) approach is based on the truncation of the computational domain by a reflectionless artificial layer which absorbs outgoing waves regardless of their frequency and angle of incidence. In this paper, we present the near-field numerical performance analysis of our new PML approach, which we call as locally-conformal PML, using Monte Carlo simulations. The locally-conformal PML method is an easily implementable conformal PML implementation, to the problem of mesh truncation in the FEM. The most distinguished feature of the method is its simplicity and flexibility to design conformal PMLs over challenging geometries, especially those with curvature discontinuities, in a straightforward way without using artificial absorbers. The method is based on a special complex coordinate transformation which is ‘locally-defined’ for each point inside the PML region. The method can be implemented in an existing FEM software by just replacing the nodal coordinates inside the PML region by their complex counterparts obtained via complex coordinate transformation. We first introduce the analytical derivation of the locally-conformal PML method for the FEM solution of the two-dimensional scalar Helmholtz equation arising in the mathematical modeling of various steady-state (or, time-harmonic) wave phenomena. Then, we carry out its numerical performance analysis by means of some Monte Carlo simulations which consider both the problem of constructing the two-dimensional Green’s function, and some specific cases of electromagnetic scattering.     

Near-field performance analysis of locally-conformal perfectly matched absorbers via Monte Carlo simulations

In the numerical solution of some boundary value problems by the finite element method (FEM), the unbounded domain must be truncated by an artificial absorbing boundary or layer to have a bounded computational domain. The perfectly matched layer (PML) approach is based on the truncation of the computational domain by a reflectionless artificial layer which absorbs outgoing waves regardless of their frequency and angle of incidence. In this paper, we present the near-field numerical performance analysis of our new PML approach, which we call as locally-conformal PML, using Monte Carlo simulations. The locally-conformal PML method is an easily implementable conformal PML implementation, to the problem of mesh truncation in the FEM. The most distinguished feature of the method is its simplicity and flexibility to design conformal PMLs over challenging geometries, especially those with curvature discontinuities, in a straightforward way without using artificial absorbers. The method is based on a special complex coordinate transformation which is ‘locally-defined’ for each point inside the PML region. The method can be implemented in an existing FEM software by just replacing the nodal coordinates inside the PML region by their complex counterparts obtained via complex coordinate transformation. We first introduce the analytical derivation of the locally-conformal PML method for the FEM solution of the two-dimensional scalar Helmholtz equation arising in the mathematical modeling of various steady-state (or, time-harmonic) wave phenomena. Then, we carry out its numerical performance analysis by means of some Monte Carlo simulations which consider both the problem of constructing the two-dimensional Green’s function, and some specific cases of electromagnetic scattering.     

Prediction of radiative heat transfer in 2D irregular geometries using the collocation spectral method based on body-fitted coordinates

Recently, an efficient numerical method, which is called the collocation spectral method (CSM), for radiative heat transfer problems, has been proposed by the present authors. In this numerical method there exists the exponential convergence rate, which can obtain a very high accuracy even using a small number of grids. In this article, the CSM based on body-fitted coordinates (BFC) is extended to simulate radiative heat transfer problems in participating medium confined in 2D complex geometries. This numerical method makes simultaneously the use of the merits of both the CSM and BFC. In this numerical approach, the radiative transfer equation (RTE) in orthogonal Cartesian coordinates should be transformed into the equation in body-fitted nonorthogonal curvilinear coordinates. In order to test the efficiency of the developed method, several 2D complex irregular enclosures with curved boundaries and containing an absorbing, emitting and scattering medium are examined. The results obtained by the CSM are assessed by comparing the predictions with those in references. These comparisons indicate that the CSM based on BFC can be recommended as a good option to solve radiative heat transfer problems in complex geometries.     

Complex Envelope Crank-Nicolson Nearly PML Algorithm for the Left-handed Material FDTD Simulations

Unconditionally stable complex envelope (CE) absorbing boundary conditions (ABCs) are presented for truncating left handed material (LHM) domains. The proposed algorithm is based on incorporating the Crank Nicolson (CN) scheme into the CE finite difference time domain (FDTD) implementations of the nearly perfectly matched layer (NPML) formulations. The validity of the formulations is shown through numerical example carried out in one dimensional Lorentzian type LHM FDTD domain.     

Helmholtz and parabolic equation solutions to a benchmark problem in ocean acoustics

The Helmholtz equation (HE) describes wave propagation in applications such as acoustics and electromagnetics. For realistic problems, solving the HE is often too expensive. Instead, approximations like the parabolic wave equation (PE) are used. For low-frequency shallow-water environments, one persistent problem is to assess the accuracy of the PE model. In this work, a recently developed HE solver that can handle a smoothly varying bathymetry, variable material properties, and layered materials, is used for an investigation of the errors in PE solutions. In the HE solver, a preconditioned Krylov subspace method is applied to the discretized equations. The preconditioner combines domain decomposition and fast transform techniques. A benchmark problem with upslope-downslope propagation over a penetrable lossy seamount is solved. The numerical experiments show that, for the same bathymetry, a soft and slow bottom gives very similar HE and PE solutions, whereas the PE model is far from accurate for a hard and fast bottom. A first attempt to estimate the error is made by computing the relative deviation from the energy balance for the PE solution. This measure gives an indication of the magnitude of the error, but cannot be used as a strict error bound.     

含噪声包裹相位图的加权最小二乘相位展开算法研究

二维相位展开广泛应用在精密光学测量、自适应光学、合成孔径雷达、图像处理等领域中。为处理含噪声包裹相位图,以预条件共轭斜量法求解权重最小二乘相位展开方程。引入非加权二维离散余弦变换求解泊松方程得到的最小二乘相位解作为共轭斜量法的初始解,从而加快了收敛速度,同时提出一种新质量图确定算法求解过程中的权重项。计算机模拟和试验表明算法计算速度快,能有效地消除传统路径积分法在处理信噪比低包裹相位图时的"拉线"现象,是一种有效的相位展开方法。     

二维声学多层快速多极子边界元及其应用

与模型自由度的平方成正比的存储量和计算量,使传统边界元无法应用到大型模型的计算。为此,发展了一种二维声学多层快速多极子边界元算法。通过二维Helmholtz核函数展开理论的简要介绍,推导了源点矩计算、源点矩转移、源点矩至本地展开转移、本地展开转移公式,并详细描述了二维声学快速多极子边界元算法的具体实现步骤。使用快速傅里叶插值进行源点矩和本地展开系数的多层传递。采用对角左预处理方法,改善边界方程的条件数,减少迭代求解次数。最后通过数值算例,验证了所发展的二维声学快速多极子算法的正确性和高效性。      

A theoretical and numerical resolution of an acoustic multiple scattering problem in three-dimensional case

This paper develops an analytical solution for sound, electromagnetic or any other wave propagation described by the Helmholtz equation in three-dimensional case. First, a theoretical investigation based on multipole expansion method and spherical wave functions was established, through which we show that the resolution of the problem is reduced to solving an infinite, complex and large linear system. Second, we explain how to suitably truncate the last infinite dimensional system to get an accurate stable and fast numerical solution of the problem. Then, we evaluate numerically the theoretical solution of scattering problem by multiple ideal rigid spheres. Finally, we made a numerical study to present the “Head related transfer function” with respect to different physical and geometrical parameters of the problem.     

A dispersion minimizing finite difference scheme and preconditioned solver for the 3D Helmholtz equation

In this paper, a new 27-point finite difference method is presented for solving the 3D Helmholtz equation with perfectly matched layer (PML), which is a second order scheme and pointwise consistent with the equation. An error analysis is made between the numerical wavenumber and the exact wavenumber, and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters. A full-coarsening multigrid-based preconditioned Bi-CGSTAB method is developed for solving the linear system stemming from the Helmholtz equation with PML by the finite difference scheme. The shifted-Laplacian is extended to precondition the 3D Helmholtz equation, and a spectral analysis is given. The discrete preconditioned system is solved by the Bi-CGSTAB method, with a multigrid method used to invert the preconditioner approximately. Full-coarsening multigrid is employed, and a new matrix-based prolongation operator is constructed accordingly. Numerical results are presented to demonstrate the efficiency of both the new 27-point finite difference scheme with refined parameters, and the preconditioned Bi-CGSTAB method with the 3D full-coarsening multigrid.     

The equilibrium limit of a constitutive model for two-phase granular mixtures and its numerical approximation

In this paper we analyze the equilibrium limit of the constitutive model for two-phase granular mixtures introduced in Papalexandris (2004) [13], and develop an algorithm for its numerical approximation. At, equilibrium, the constitutive model reduces to a strongly coupled, overdetermined system of quasilinear elliptic partial differential equations with respect to the pressure and the volume fraction of the solid granular phase. First we carry a perturbation analysis based on standard hydrostatic-type scaling arguments which reduces the complexity of the coupling of the equations. The perturbed system is then supplemented by an appropriate compatibility condition which arises from the properties of the gradient operator. Further, based on the Helmholtz decomposition and Ladyzhenskaya’s decomposition theorem, we develop a projection-type, Successive-Over-Relaxation numerical method. This method is general enough and can be applied to a variety of continuum models of complex mixtures and mixtures with micro-structure. We also prove that this method is both stable and consistent hence, under standard assumptions, convergent. The paper concludes with the presentation of representative numerical results.     

Preconditioning bandgap eigenvalue problems in three-dimensional photonic crystals simulations

To explore band structures of three-dimensional photonic crystals numerically, we need to solve the eigenvalue problems derived from the governing Maxwell equations. The solutions of these eigenvalue problems cannot be computed effectively unless a suitable combination of eigenvalue solver and preconditioner is chosen. Taking eigenvalue problems due to Yee’s scheme as examples, we propose using Krylov–Schur method and Jacobi–Davidson method to solve the resulting eigenvalue problems. For preconditioning, we derive several novel preconditioning schemes based on various preconditioners, including a preconditioner that can be solved by Fast Fourier Transform efficiently. We then conduct intensive numerical experiments for various combinations of eigenvalue solvers and preconditioning schemes. We find that the Krylov–Schur method associated with the Fast Fourier Transform based preconditioner is very efficient. It remarkably outperforms all other eigenvalue solvers with common preconditioners like Jacobi, Symmetric Successive Over Relaxation, and incomplete factorizations. This promising solver can benefit applications like photonic crystal structure optimization.     

Application of the complex point source method to the Schrödinger equation

The paraxial wave equation is a reduced form of the Helmholtz equation. Its solutions can be directly obtained from the solutions of the Helmholtz equation by using the method of complex point source. We applied the same logic to quantum mechanics, because the Schrödinger equation is parabolic in nature as the paraxial wave equation. We defined a differential equation, which is analogous to the Helmholtz equation for quantum mechanics and derived the solutions of the Schrödinger equation by taking into account the solutions of this equation with the method of complex point source. The method is applied to the problem of diffraction of matter waves by a shutter.     

On the governing equations for the compressing process and its coupling with other processes

As a continuation of a recent linear analysis by Mao et al.(Acta Mech Sin,2010,26:355),in this paper we propose a general theoretical formulation for the compressing process in complex Newtonian fluid flows,which covers gas dynamics,aeroacoustics,nonlinear thermoviscous acoustics,viscous shock layer,etc.,as its special branches.The principle on which our formulation is based is the maximally natural and dynamic Helmholtz decomposition of the Navier-Stokes equation,along with the kinematic Helmholtz decompos...     

Structural-acoustic coupling in complex shaped cavities

Structural-acoustic interaction is a topic of increasing interest in the study of vehicle noise control. During recent years particular attention has been devoted towards analytical and numerical methods for the analysis of acoustic problems of complex shaped cavities. Among them non-local methods are very appropriate for the low and medium frequency range. The integral formulation of the Helmholtz equation allows the sound pressure level, radiated by a vibrating panel, to be determined. The method may be used to account for the acoustic structural coupling and can be straightforwardly extended to consider interconnected cavities, internal vibrating panels, internal barriers and absorbing materials. Comparisons with experimental results as well as with modal methods are presented which show satisfactory agreement.     

High order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces

This work overcomes the difficulty of the previous matched interface and boundary (MIB) method in dealing with interfaces with non-constant curvatures for optical waveguide analysis. This difficulty is essentially bypassed by avoiding the use of local cylindrical coordinates in the improved MIB method. Instead, novel jump conditions are derived along global Cartesian directions for the transverse magnetic field components. Effective interface treatments are proposed to rigorously impose jump conditions across arbitrarily curved interfaces based on a simple Cartesian grid. Even though each field component satisfies the scalar Helmholtz equation, the enforcement of jump conditions couples two transverse magnetic field components, so that the resulting MIB method is a full-vectorial approach for the modal analysis of optical waveguides. The numerical performance of the proposed MIB method is investigated by considering interface problems with both constant and general curvatures. The MIB method is shown to be able to deliver a fourth order of accuracy in all cases, even when a high frequency solution is involved.     

Helmholtz equation and non-singular boundary elements applied to multi-disciplinary physical problems

The famous scientist Hermann von Helmholtz was born 200 years ago. Many complex physical wave phenomena in engineering can effectively be described using one or a set of equations named after him: the Helmholtz equation. Although this has been known for a long time, from a theoretical point of view, the actual numerical implementation has often been hindered by divergence-free and/or curl-free constraints. There is further a need for a numerical method that is accurate, reliable and takes into account radiation conditions at infinity. The classical boundary element method satisfies the last condition, yet one has to deal with singularities in the implementation. We review here how a recently developed singularity-free three-dimensional boundary element framework with superior accuracy can be used to tackle such problems only using one or a few Helmholtz equations with higher order (quadratic) elements which can tackle complex curved shapes. Examples are given for acoustics (a Helmholtz resonator among others) and electromagnetic scattering.     

三维问题的局部块分解预条件

利用块三对角矩阵的嵌套局部块分解构造了一个不完全分解预条件子,并考虑了其修正型变种,分析了两者的存在性及若干性质.针对标准七点差分矩阵,给出了预条件后的实际条件数.结果表明,采用局部块分解预条件时条件数与矩阵阶数的2/3次幂成正比,而采用修正型预条件时条件数与矩阵阶数的立方根成正比.最后考虑了预条件的高效实现并在主频为550MHz、内存为256M的微机上作了若干数值实验,并与其它较有效的预条件方法进行了比较.     

二维三温辐射扩散方程组两层预条件子的自适应求解

针对实际应用中若干典型三温线性系统,分析求解二维三温辐射扩散方程离散线性系统的代数两层预条件子(PCTL)的算法效率.结果表明,PCTL的算法效率与三个温度之间的耦合强度以及单温子系统对角占优性强弱程度有很大关系.为此,通过刻画三温线性系统的耦合强度和单温子系统对角占优性特征,提出一种PCTL中子系统的自适应求解算法.数值结果表明,可以显著改善PCTL的算法效率.对于实际数值模拟应用中37个典型三温线性系统,相对于经典AMG算法,算法整体加速2.5倍.数值实验表明算法具有很强的鲁棒性.