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。      

Frequency domain finite-element and spectral-element acoustic wave modeling using absorbing boundaries and perfectly matched layer

Wave propagation modeling as a vital tool in seismology can be done via several different numerical methods among them are finite-difference, finite-element, and spectral-element methods (FDM, FEM and SEM). Some advanced applications in seismic exploration benefit the frequency domain modeling. Regarding flexibility in complex geological models and dealing with the free surface boundary condition, we studied the frequency domain acoustic wave equation using FEM and SEM. The results demonstrated that the frequency domain FEM and SEM have a good accuracy and numerical efficiency with the second order interpolation polynomials. Furthermore, we developed the second order Clayton and Engquist absorbing boundary condition (CE-ABC2) and compared it with the perfectly matched layer (PML) for the frequency domain FEM and SEM. In spite of PML method, CE-ABC2 does not add any additional computational cost to the modeling except assembling boundary matrices. As a result, considering CE-ABC2 is more efficient than PML for the frequency domain acoustic wave propagation modeling especially when computational cost is high and high-level absorbing performance is unnecessary.     

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.     

基于Z变换的高阶完全匹配层实现

基于拉伸坐标完全匹配层(SC-PML)公式和Z变换方法,提出以非分裂场形式来实现具有多极点的高阶完全匹配层的高效算法,来截断时域有限差分(FDTD)网格.在吸收性能方面,高阶PML同时具有传统PML和复频率偏移完全匹配层(CFS-PML)二者的优点.提供的数值算例是二维TE极化电磁波与无限长且有限宽度的理想电导体(PEC)薄片的相互作用.仿真结果显示,高阶PML公式在衰减低频行波和隐失波及减少后期反射方面效果好,比传统SC-PML和复频率偏移的卷积完全匹配层(CPML)算法有更好的吸收性能.     

卷积完全匹配层截断3维金属矩形波导的应用研究

 讨论了高功率微波源模拟中波导开放边界截断的需求，分析了不同类型完全匹配层(PML)的特点，选用卷积形式PML截断色散波导器件的开放边界。从自由空间电磁波的平面波解和分裂形式的PML出发，构造了未分裂形式的PML，用傅里叶变换的卷积定理，导出了直角坐标系下卷积完全匹配层(CPML)介质中电磁场的迭代形式的离散方程。在不同频率和模式激励源作用下，模拟计算了CPML截断矩形波导开放边界的性能，数值结果表明最大相对误差都小于-70 dB，远好于Mur吸收边界的截断效果。     

CPML在空中核爆电磁脉冲三维数值模拟中的应用

在分析不同类型完全匹配层(PML,perfectl ymatched layer)特点的基础上,选用了卷积形式PML(CPML,convolutional perfectly matched layer)截断空中核爆电磁脉冲(NEMP,nuclear electromagnetic pulse)数值模拟的开放边界.从自由空间中电磁波的平面波解和分裂形式的PML出发,构造了未分裂形式的PML,应用付里叶变换的卷积定理,推导出三维旋转椭球-双曲正交坐标系下CPML介质中电磁场的迭代形式的离散方程.获得了很好的截断效果,展示了CPML在NEMP数值模拟中的应用前景.     

卷积完全匹配层在空爆电磁脉冲2维模拟中的应用

 在分析不同类型完全匹配层特点的基础上，选用了卷积形式完全匹配层（CPML）截断空中核爆电磁脉冲数值模拟的开放边界。从自由空间中电磁波的平面波解和分裂形式的完全匹配层出发，构造了未分裂形式的完全匹配层，应用傅里叶变换的卷积定理，推导出2维旋转椭球-双曲正交坐标系下卷积形式完全匹配层介质中电磁场的迭代形式的离散方程。计算表明，采用CPML吸收边界方法使得截断的外边界处的场的计算误差大大减少。     

Unconditionally Stable Complex Envelope Wave Equation PML Algorithm for Band Limited FDTD Simulations

Unconditionally stable complex envelope (CE) perfectly matched layer (PML) absorbing boundary conditions (ABCs) are presented for truncating the scalar wave-equation finite difference time domain (WE-FDTD) grids. The formulations are based on incorporating the alternating direction implicit (ADI) scheme into the CE FDTD implementations of the scalar wave-equation derived in the PML region at the domain boundaries. Numerical example carried out in two dimensional domain shows that the proposed formulations are more accurate than the classical ADI scalar wave equation PML formulations when it is used for modelling band limited electromagnetic applications.     

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。     

全局稳定性一致的共形卷积完全匹配层

在正交网格体系中建立物理模型共形描述的基础上,针对采用扩展元胞技术的共形时域有限差分(ECT-CFDTD)方法模拟计算波导器件遇到的开放端口截断问题,给出了积分形式的共形卷积完全匹配层方法,算法具有与ECT-CFDTD相同的数值稳定性。设置不同的完全匹配层的控制参数,对波导中有消逝波存在的情况进行长时间模拟计算,分析共形卷积完全匹配层对消逝波的长效截断能力,分析卷积完全匹配层的截断误差。计算结果显示:积分形式的共形卷积完全匹配层可有效截断波导器件的开放端口。     

一种适用于任意阶空间差分时域有限差分方法的色散介质通用吸收边界条件算法

基于伸展坐标完全匹配层方程和辅助微分方程 方法, 给出了一种在时域有限差分(FDTD) 计算中适用于常见色散介质的通用边界条件算法. 该算法适用于任意阶的FDTD空间差分, 并且由于所采用的D-H方程独立于计算区域, 所以可以被用于截断任意电介质. 数值试验结果表明, 与卷积完全匹配层 算法相比较, 所提出的吸收边界条件算法不仅通用性强、 计算复杂度低、 计算时间短, 并且吸收效果有明显的提高.     

基于辅助差分方程的完全匹配层在时域多分辨率分析算法中的应用与性能分析

将基于辅助微分方程的完全匹配层(ADE-PML)吸收边界条件引入到基于Daubechies尺度函数的时域多分辨率分析算法中. 与目前广泛应用的Berenger完全匹配层(PML)和各向异性介质完全匹配层(APML) 相比, 该吸收边界条件的实现更加容易且更节省内存. 数值结果表明, ADE-PML在吸收传播模和低频凋落模方面均优于PML和APML.     

An improved perfectly matched layer for the eigenmode expansion technique

When performing optical simulations for rotationally symmetric geometries using the eigenmode expansion technique, it is necessary to place the geometry under investigation inside a cylinder with perfectly conducting walls. The parasitic reflections at the boundary of the computational domain can be suppressed by introducing a perfectly matched layer (PML) using e.g. complex coordinate stretching of the cylinder radius. However, the traditional PML suffers from an artificial field divergence limiting its usefulness. We show that the choice of a constant cylinder radius leads to mode profiles with exponentially increasing field amplitudes resulting in numerical instability. As a remedy we propose an improved PML based on a mode-dependent cylinder radius and mode profiles with stable field amplitudes. The new PML formulation eliminates the artificial field divergence and ensures numerical stability.     

Independent control of birefringence and chromatic dispersion in a photonic crystal fiber using two hollow ring defects

We propose a new type of birefringent index guiding photonic crystal fiber (PCF) with two hollow GeO₂-doped silica ring defects imbedded in a hexagonal hole arrays. Rigorous finite element method (FEM) using the perfectly matched layer (PML) was applied to calculate complex propagation constant and analyze optical transmission characteristics. Birefringence was flexibly controlled independent of chromatic dispersion, and improvement in confinement loss was achieved by optimizing hollow ring defect parameters.     

色散媒质中的理想匹配层吸收边界及其误差分析

根据色散媒质的性质及电磁波在不同媒质交界面上的反射的机理，证明并分析了在理想匹配层吸收边界中引入色散可实现有效吸收入射到吸收边界上的电磁波，从而达到减少时域有限差分法计算区域和计算误差的目的。     

FDTD方法分析高功率微波粗糙地面散射特性

采用时域有限差分方法(FDTD)对高功率微波(HPM)近地面传输进行了仿真,其中提出的一种完全匹配层-广义完全匹配层混合吸收边界,较好地抑制了常规FDTD方法在计算低掠射角入射时产生的边缘绕射现象。将理想水平地面数值计算结果与解析法结果对比验证计算模型的正确性,通过数值计算结果分析高功率微波粗糙地面的散射特性。理论分析和仿真结果表明:粗糙地面对散射系数变化影响较明显,从宏观角度上看,散射系数曲线同理想水平地面散射系数曲线相似,即随掠射角的增大散射系数先减小后增大;从微观角度上看,由于粗糙地面的作用,其散射系数曲线并不存在布鲁斯特角,且在某一区域并非单调增加或减小。     

A study of the propagation of electromagnetic pulses through photonic crystal structures

A mathematical model for the study of two- and three-dimensional photonic crystals and of waveguide systems based on them was designed and applied. The numerical algorithm for the construction of the solution was built using such numerical methods as the method of finite differences in the time domain (FDTD), the TS/SF method, and the method of a perfectly matched layer (PML), which makes it possible to apply the algorithm under consideration to the study of other waveguide systems as well. The results of the simulation of particular waveguide systems are presented.     

An FDTD analysis of photonic crystal waveguides comprising third-order nonlinear materials

A finite difference time domain (FDTD) study of two-dimensional photonic crystals containing nonlinear materials is presented in this paper. An appropriate Z-transform oriented formulation of the FDTD method for the simulation of third-order nonlinear Kerr- and Raman-type media is analyzed and applied to model nonlinear photonic crystal waveguide structures. For their reflectionless termination a novel perfectly matched layer (PML) is proposed and evaluated comparatively to other periodic and inhomogeneous absorbers. Furthermore, the absorbing efficiency of the proposed PML is investigated varying its parameters.     

Uniform stable conformal convolutional perfectly matched layer for enlarged cell technique conformal finite-difference time-domain method

Based on conformal construction of physical model in a three-dimensional Cartesian grid,an integral-based conformal convolutional perfectly matched layer(CPML) is given for solving the truncation problem of the open port when the enlarged cell technique conformal finite-difference time-domain(ECT-CFDTD) method is used to simulate the wave propagation inside a perfect electric conductor(PEC) waveguide.The algorithm has the same numerical stability as the ECT-CFDTD method.For the long-time propagation problems of an evanescent wave in a waveguide,several numerical simulations are performed to analyze the reflection error by sweeping the constitutive parameters of the integral-based conformal CPML.Our numerical results show that the integral-based conformal CPML can be used to efficiently truncate the open port of the waveguide.