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.     

有耗介质空间完全匹配层吸收边界条件及其应用

 针对Gedney提出的完全匹配层(PML)无法用于有耗各向同性计算域的截断及其对倏逝波的衰减不理想等问题,提出了一种扩展方法。扩展的PML的主要思想是在各向异性的PML中引入与有耗介质空间相一致的复介电常数和复磁导率,使之可以与有耗介质计算域相匹配。通过给PML的张量介电常数、张量磁导率增加衰减因子以加速倏逝波的衰减。构造了PML吸收效果验证模型,数值结果证明了扩展的PML在处理有耗介质计算域截断问题中的有效性。利用该吸收边界条件,采用时域有限差分法计算了电磁脉冲作用下地面铺设电缆的电磁脉冲响应,计算结果和试验结果取得了较好的一致。     

二阶声场波动方程的非分裂完全匹配层算法

在声场仿真中,完全匹配层(Perfectly Matched Layer,PML)是一种十分有效的吸收边界并得到广泛应用。为了解决基于二阶声场波动方程数值仿真中的吸收边界问题,提出了一种非分裂PML算法。首先,基于伸缩坐标变换,推导了PML算法的频域表达式。然后,通过构造辅助微分方程,得到了非分裂PML的时域表达式。最后,进行了相关理论分析和数值仿真,结果表明:相对于已有的声场分裂PML算法,该算法在保持相同的吸收效率的同时,能较大地节约存储空间,提高计算效率,且更易于实现。      

Application of convolution perfectly matched layer in finite element method calculation for 2D acoustic wave

A method was presented to extend the Convolution Perfectly Matched Layer(CPML), which bases on the complex coordinates transformation and complex frequency shifted stretched-coordinate metrics,to the 2D acoustic equation calculated with the method of Finite Element Method(FEM).This non-physical layer is used at the computational edge of a FEM as an Absorbing Boundary Condition(ABC) to truncate unbounded media.In this paper,the CPML equations have been presented in frequency domain and in time domain,respectively,and the calculations have been realized in the FEM software of COMSOL.The main advantage of CPML over the classical PML layer is that it is based on the unsplit components of the wave field leading to a more stable,highly effective absorption and a more facility to realize.The results of numerical simulation demonstrate that CPML has better absorbability than PML and it absorbs the outgoing energy more effectively.     

卷积完全匹配层在两维声波有限元计算中的应用

将基于复坐标变换和复频移扩展坐标变量的卷积完全匹配层Convolution Perfectly Machted Layer(CPML)引入到两维声波方程的有限元(FEM)计算中,该匹配层作为一种吸收边界条件Absorbing Boundary Condition(ABC)应用在有限元计算的边界截断上。文中分别给出了频域和时域的CPML方程的表达形式,并在有限元计算软件COMSOL中完成数值计算。相对于经典的PML,CPML最大的优势在于它不需要把场分裂开,这使其具有更好的稳定性和更高的吸收性能,且更易于实现。数值计算结果表明,CPML边界层有着比PML更好的吸收效果,其更有效的吸收了进入其中的声场能量。      

双相各向同性介质BISQ模型弹性波方程数值模拟方法

本文从Taylor级数展开式出发,推导出了交错网格一阶空间导数的任意偶数阶精度展开式和相应差分系数计算式;从本构方程和运动方程推导出了BISQ模型双相介质一阶双曲型应力-速度弹性波方程交错网格任意偶数阶精度差分格式以及推导出二维双相各向同性介质完全吸收层边界条件公式和相应的高阶交错网格差分格式。通过数值模拟研究表明,该方法边界吸收效果好,稳定性好,能够高精度模拟双相介质中地震弹性波场,且计算效率也高。     

A staggered-grid finite-difference method with perfectly matched layers for poroelastic wave equations.

A particle velocity-strain, finite-difference (FD) method with a perfectly matched layer (PML) absorbing boundary condition is developed for the simulation of elastic wave propagation in multidimensional heterogeneous poroelastic media. Instead of the widely used second-order differential equations, a first-order hyperbolic leap-frog system is obtained from Biot's equations. To achieve a high accuracy, the first-order hyperbolic system is discretized on a staggered grid both in time and space. The perfectly matched layer is used at the computational edge to absorb the outgoing waves. The performance of the PML is investigated by calculating the reflection from the boundary. The numerical method is validated by analytical solutions. This FD algorithm is used to study the interaction of elastic waves with a buried land mine. Three cases are simulated for a mine-like object buried in "sand," in purely dry "sand" and in "mud." The results show that the wave responses are significantly different in these cases. The target can be detected by using acoustic measurements after processing.     

一维非线性声波传播特性

针对一维非线性声波的传播问题进行了有限元仿真和实验研究. 首先推导了一维非线性声波方程的有限元形式, 含有高阶矩阵的非线性项导致声波具有波形畸变、谐波滋生、基频信号能量向高次谐波传递等非线性特性. 编制有限元程序对一维非线性声波进行了计算并对仿真得到的畸变非线性声波信号进行处理, 分析其传播性质和物理意义. 为验证有限元计算结果, 开展了水中的非线性声波传播的实验研究, 得到了不同输入信号幅度激励下和不同传播距离的畸变非线性声波信号. 然后对基波和二次谐波的传播性质进行详细讨论, 分析了二次谐波幅度与传播距离和输入信号幅度的变化关系及其意义, 拟合出二次谐波幅度随传播距离变化的方程并阐述了拟合方程的物理意义. 结果表明, 数值仿真信号及其频谱均与实验结果有较好的一致性, 证实计算方法和结果的正确性, 并提出了具有一定物理意义的二次谐波随传播距离变化的简单数学关系. 最后还对固体中的非线性声波传播性质进行了初步探讨. 本研究工作可为流体介质中的非线性声传播问题提供理论和实验依据.     

Transient acoustic radiation from impulsively accelerated bodies by the finite element method

Bodies under impulsive motion, immersed in an infinite acoustic fluid, severely put to test any numerical method for the transient exterior acoustic problem. Such problems, in the context of the finite element method (FEM), are not well studied. FE modeling of such problems requires truncation of the infinite fluid domain at a certain distance from the structure. The volume of computation depends upon the extent of this domain as well as the mesh density. The modeling of the fluid truncation boundary is crucial to the economy and accuracy of solution and various boundary dampers have been proposed in the literature for this purpose. The second order damper leads to unsymmetric boundary matrices and this necessitates the use of an unsymmetric equation solver for large problems. The present paper demonstrates the use of FEM with zeroth, first and second order boundary dampers in conjunction with an unsymmetric, out of core, banded equation solver for impulsive motion problems of rigid bodies in an acoustic fluid. The results compare well with those obtained from analytical methods.     

A novel unsplit perfectly matched layer for the second-order acoustic wave equation

When solving acoustic field equations by using numerical approximation technique, absorbing boundary conditions (ABCs) are widely used to truncate the simulation to a finite space. The perfectly matched layer (PML) technique has exhibited excellent absorbing efficiency as an ABC for the acoustic wave equation formulated as a first-order system. However, as the PML was originally designed for the first-order equation system, it cannot be applied to the second-order equation system directly. In this article, we aim to extend the unsplit PML to the second-order equation system. We developed an efficient unsplit implementation of PML for the second-order acoustic wave equation based on an auxiliary-differential-equation (ADE) scheme. The proposed method can benefit to the use of PML in simulations based on second-order equations. Compared with the existing PMLs, it has simpler implementation and requires less extra storage. Numerical results from finite-difference time-domain models are provided to illustrate the validity of the approach.     

Absorbing boundary conditions for scalar waves in anisotropic media. Part 1: Time harmonic modeling

With the ultimate goal of devising effective absorbing boundary conditions (ABCs) for general anisotropic media, we investigate the accuracy aspects of local ABCs designed for the scalar anisotropic wave equation in the frequency domain (time harmonic case). The ABC analyzed in this paper is the perfectly matched discrete layers (PMDL). PMDL is a simple variant of perfectly matched layers (PML) and is equivalent to rational approximation-based local ABCs. Specifically, we derive a sufficient condition for PMDL to accurately absorb wave modes with outgoing group velocities and this condition turns out to be a simple bound on the PMDL parameters. The reflection coefficient derived in this paper clearly reveals that the PMDL absorption is based on group velocities, and not phase velocities, and hence a PMDL can be designed to correctly identify and accurately absorb all outgoing wave modes (even those with opposing signs of phase and group velocities). The validity of the sufficient condition is demonstrated through a series of frequency domain simulations. In part 2 of this paper [S. Savadatti, M.N. Guddati, Absorbing boundary conditions for scalar waves in anisotropic media. Part 2: Time-dependent modeling, J. Comput. Phys. (2010), http://dx.doi.org/10.1016/j.jcp.2010.05.017], the accuracy condition presented here is shown to govern both the well-posedness and accuracy aspects of PMDL designed for transient (time-dependent) modeling of scalar waves in anisotropic media.     

基于广角FD-BPM的PML边界处理方法

本文提出了基于三阶Padé近似广角有限差分光束传输法(FD-BPM)的简单有效的完美匹配层(PML)边界处理方法.分别给出了该方法应用于斜波导和多模干涉(MMI)数值模拟的结果,并与用完全透明边界条件处理的模拟计算结果进行了比较.本文最后对基于广角FD-BPM的PML边界处理方法进行了优化.     

The Improvement of PML Approach for the Calculation of Microwave and Millimeter Waves Circuit

The improved PML absorbing boundary condition have been introduced in this paper. The reflection coefficient of the absorbing boundary condition is less than the reflection coefficient of Berenger's PML algorithm. And in order to validate the theory and its effectiveness, We have applied the absorbing boundary condition to the calculation of microwave and millimeter waves circuit and get its reflection coefficient S ₁₁. The numerical result is shown in this paper.     

双重弹性波波场分离数值模拟及相关理论证明

提出一种等效的双重弹性波波场分离数值模拟方法,用于模拟纯纵波和纯横波分离模式的质点振动速度、位移以及散度场和旋度场,并将该方法应用于全弹性波波动方程数值模拟中.同时,详细推导双重弹性波波场分离波动方程的高阶交错网格有限差分数值计算公式及其稳定性条件、数值频散关系和完全匹配层(PML)吸收边界条件.理论分析和数值计算均表明,该方法可以实现高精度双重弹性波波场分离数值模拟,且纯纵波和纯横波得到完全分离,边界吸收效果较好.与前人工作相比,存储量和计算时间均得到有效改善,数值计算结果进一步验证了该方法的优越性.     

An Efficient Algorithm for Truncating Spatial Domain in Modeling Light Scattering by Finite-Difference Technique

The finite-difference time domain technique is one of the most robust and accurate numerical methods for the solution of light scattering by small particles with arbitrary composition and geometry. In practice, this method requires that the spatial domain for the computation of near-field be truncated. An absorbing boundary condition must be imposed in conjunction with this truncation. The performance of this boundary condition is essential to the stability of numerical computations and the reliability of results. In the present study, a new boundary condition, referred to as the mixed T algorithm, has been developed, which is a generalization of the transmitting boundary condition originally developed by Liao and co-workers. The present algorithm does not require spatial interpolation for wave values at interior grid points. In addition, it produces two minima of spurious reflections at small and large incident angles, allowing efficient absorption of the scattered waves at the boundary for large incident angles. When the third-order mixed T algorithm is used, the reflection coefficient of the boundary is less than 1% for incident angles from 0° to about 70°. We find that the numerical instability associated with the transmitting boundary condition is caused by the location-dependent amplitude of outgoing waves in the vicinity of the boundary. For this reason, the mixed T algorithm is stabilized by consistently introducing diffusive coefficients into the boundary equation. When the stabilized algorithm is applied, the near-field within the truncated domain can be computed by using single-precision arithmetic without overflows for more than 10⁵steps in the time-marching iteration. Finally, the new absorbing boundary condition is validated by carrying out numerical experiments involving the propagation of a TM wave excited by a sinusoidal point source, simultaneous simulation of the wave propagation in small and large domains, and the scattering of a TM wave by an infinite circular cylinder.     

Simulation of laser-generated ultrasonic wave propagation in solid media and air with application to NDE

Ultrasonic methods are well known as powerful and reliable tool for defect detection. In the previous decades focus and interest have been directed to non-contact sensors and methods, showing many advantages over contact techniques where inspection depends on contact conditions (pressure, coupling medium, contact area). The non-contact hybrid ultrasonic method described here is of interest for many applications, requiring periodic inspection in service or after manufacturing. Despite the potential impact of laser-generated ultrasound in many areas of industry, robust tools for studying the phenomenon are lacking and thus limit the design and optimization of non-destructive testing and evaluation techniques. Here a specific numerical method is presented to efficiently and accurately solve ultrasound wave propagation problems with frequencies in the MHz range traveling in relatively large bodies and through air. This work improves a previous numerical model where propagation of the acoustic waves through air had not been considered, allowing us to simulate the presence of a non-contact transducer in reception in order to simulate numerically the complete experimental setup. It is very important to limit the amount of air to be considered in the FE analyses; otherwise the computational cost would often exceed the resources available. A way to solve the problem is to implement non-reflecting boundary conditions. A non-reflecting boundary condition allows all outgoing waves to exit the domain at the boundary where they have been imposed without reflection; thus, it is possible to model only the portion of air between the non-contact transducer and the solid under testing. Several numerical and experimental analyses were conducted on a 136 lb AREMA rail; here we study in detail two fully non-contact testing configurations for the rail head and web. The information that can be acquired is very valuable for choosing the right setup and configuration when performing non-contact hybrid ultrasonic inspection.     

粒子模拟软件吸收边界的研究

 实现了时域有限差分法的Gedney完全匹配层吸收边界条件，并进行了数值验证和参数优化。从结果可以看出其有较低的反射，性能优良；对吸收媒质厚度和电导率分布阶数的参数优化结果与Gedney经验值基本一致。将此吸收边界模块加入到国产粒子模拟软件CHIPIC中，同美国商业粒子模拟软件MAGIC相比有更好的吸收效果。通过对2维极坐标下磁控管的模拟，证明了此吸收边界具有实际应用价值。     

声表面波在圆弧处反射及透射的数值研究*

为了研究直达表面波在圆弧过渡面处传播性能的变化,采用有限元方法模拟了热弹机制下,线性脉冲激光辐照金属铝块表面时激发的表面波在近表面传播过程中,在不同曲率半径的圆弧处发生的反射及透射现象,建立了圆弧半径与反射表面波以及透射表面波时域信号特征之间的联系。计算结果表明:曲率半径与表面波中心波长的数值关系对表面波在圆弧处的传播有显著的影响;同时证明了根据透射表面波信号的到达时间可以反演圆弧半径的大小,为之后利用表面波信号定量检测材料表面圆弧凹痕的深度提供了理论依据。     

时域有限差分法在建筑声学中的应用及前景

介绍了声波方程的基本差分格式及稳定条件、数值色散、吸收边界条件等数值计算理论，例举了前人用时域有限差分法对噪声传播过程的模拟和室内声学中座椅吸声低谷效应模拟的模拟结果。本文指出，由于时域有限差分法的特点使其具有在模拟脉冲响应方而的特别优势，因而，应用这一技术研究厅堂的声学特性，尤其是低频特性，将有重要意义。