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.     

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.     

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.     

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。     

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。      

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。     

Simulation of wireline sonic logging measurements acquired with Borehole–Eccentered tools using a high-order adaptive finite-element method

The paper introduces a high-order, adaptive finite-element method for simulation of sonic measurements acquired with borehole-eccentered logging instruments. The resulting frequency-domain based algorithm combines a Fourier series expansion in one spatial dimension with a two-dimensional high-order adaptive finite-element method (FEM), and incorporates a perfectly matched layer (PML) for truncation of the computational domain. The simulation method was verified for various model problems, including a comparison to a semi-analytical solution developed specifically for this purpose. Numerical results indicate that for a wireline sonic tool operating in a fast formation, the main propagation modes are insensitive to the distance from the center of the tool to the center of the borehole (eccentricity distance). However, new flexural modes arise with an increase in eccentricity distance. In soft formations, we identify a new dipole tool mode which arises as a result of tool eccentricity.     

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.     

快速多极边界元法用于扩散光学断层成像研究

在求解扩散光学断层成像中的正向问题时, 目前普遍采用有限元法, 但是随着实际模型规模的增大, 有限元法的计算量问题日益显著, 而边界元法则由于可以降低计算维度使计算量减少而备受关注. 本文以均匀的高散射介质为模型, 研究了将快速多极边界元法用于扩散光学断层成像的正向问题. 快速多极边界元法利用核函数的多极展开, 将常规边界元法中系数矩阵和迭代矢量的乘积项等价为相应四叉树结构的一次递归, 再结合广义最小残量法进行迭代求解. 将计算结果和蒙特卡罗法的模拟结果进行了比较, 表明利用快速多极边界元法的模拟结果和蒙特卡罗法的结果有很好的一致性. 研究结果验证了快速多极边界元法可以用于扩散光学断层成像, 为其大规模和实时成像带来可观的前景. 关键词： 扩散光学断层成像 边界元法 快速多极边界元法     

Direct simulation Monte Carlo study of metal evaporation with collimator in e-beam physical vapor deposition

The flow properties and substrate deposition rate profile, which are the important parameters in electron beam physical vapor deposition, are investigated computationally in this article.Collimators are used to achieve the desired vapor beam and deposition rate profile in some applications.This increases the difficulty measuring boundary conditions and the size of the liquid metal pool inside the collimators.It is accordingly hard to obtain accurate results from numerical calculations.In this article, two-dimensional direct simulation Monte Carlo(DSMC) codes are executed to quantify the influence of uncertainties of boundary conditions and pool sizes.Then, three-dimensional DSMC simulations are established to simulate cerium and neodymium evaporation with the collimator.Experimental and computational results of substrate deposition rate profile are in excellent agreement at various evaporation rates and substrate heights.The results show that the DSMC method can assist in metal evaporation with a collimator.     

Domain compression via anisotropic metamaterials designed by coordinate transformations

We introduce a spatial coordinate transformation technique to compress the excessive white space (i.e. free-space) in the computational domain of finite methods. This approach is based on the form-invariance property of Maxwell’s equations under coordinate transformations. Clearly, Maxwell’s equations are still satisfied inside the transformed space, but the medium turns into an anisotropic medium whose constitutive parameters are determined by the coordinate transformation. The proposed technique can be employed to reduce the number of unknowns especially in high-frequency applications wherein a finite method requires an electrically-large computational domain. After developing the analytical background of this technique, we report some numerical results for finite element simulations of electromagnetic scattering problems.     

A low diffusion particle method for simulating compressible inviscid flows

A new particle method is presented for the numerical simulation of compressible inviscid gas flows, through procedures which involve relatively small modifications to an existing direct simulation Monte Carlo (DSMC) algorithm. Implementation steps are outlined for simulations involving various grid geometries and for gas mixtures comprising an arbitrary number of species. The proposed method is compared with other numerical schemes through a series of one-dimensional and two-dimensional test cases, and is shown to provide a significant reduction in both artificial diffusion and statistical scatter effects relative to existing DSMC-based equilibrium particle methods.     

3-D Quantum Transport Solver Based on the Perfectly Matched Layer and Spectral Element Methods for the Simulation of Semiconductor Nanodevices

A 3-D quantum transport solver based on the spectral element method (SEM) and perfectly matched layer (PML) is introduced to solve the 3-D Schr?dinger equation with a tensor effective mass. In this solver, the influence of the environment is replaced with the artificial PML open boundary extended beyond the contact regions of the device. These contact regions are treated as waveguides with known incident waves from waveguide mode solutions. As the transmitted wave function is treated as a total wave, there is no need to decompose it into waveguide modes, thus significantly simplifying the problem in comparison with conventional open boundary conditions. The spectral element method leads to an exponentially improving accuracy with the increase in the polynomial order and sampling points. The PML region can be designed such that less than -100 dB outgoing waves are reflected by this artificial material. The computational efficiency of the SEM solver is demonstrated by comparing the numerical and analytical results from waveguide and plane-wave examples, and its utility is illustrated by multiple-terminal devices and semiconductor nanotube devices.     

Perfectly matched layers for modelling seismic oceanography experiments

Seismic oceanography techniques are able to provide oceanographic properties of the water masses by processing seismic reflection data. These techniques have reported reflected waves due to the fine structure in the ocean, whose order of magnitude is as weak as −80 dB. Thus, if we focus our attention on numerical simulation of this kind of oceanography experiments, the numerical performance of the method should allow obtaining accurate results, where the spurious reflections from the artificial boundaries of the computational grid are, at least, one order of magnitude smaller than the physical phenomena. This can be achieved by introducing perfectly matched layers (PML), which simulate non-reflecting boundaries. The aim of this work is to propose a numerical underwater propagation method, which combines a second-order finite-difference scheme in the physical region of interest with a first-order pressure/velocity discretization in the PML domain. This numerical method provides a low-cost computational algorithm with an accuracy, which allows recovering the reflected phenomena from the ocean fine structure, and moreover, with a spurious error of order −100 dB from the PML domain.     

An efficient locally one-dimensional finite-difference time-domain method based on the conformal scheme

An efficient conformal locally one-dimensional finite-difference time-domain(LOD-CFDTD) method is presented for solving two-dimensional(2D) electromagnetic(EM) scattering problems. The formulation for the 2D transverse-electric(TE) case is presented and its stability property and numerical dispersion relationship are theoretically investigated. It is shown that the introduction of irregular grids will not damage the numerical stability. Instead of the staircasing approximation, the conformal scheme is only employed to model the curve boundaries, whereas the standard Yee grids are used for the remaining regions. As the irregular grids account for a very small percentage of the total space grids, the conformal scheme has little effect on the numerical dispersion. Moreover, the proposed method, which requires fewer arithmetic operations than the alternating-direction-implicit(ADI) CFDTD method, leads to a further reduction of the CPU time. With the total-field/scattered-field(TF/SF) boundary and the perfectly matched layer(PML), the radar cross section(RCS) of two2 D structures is calculated. The numerical examples verify the accuracy and efficiency of the proposed method.     

Vector-FEM Analysis of Millimeter Wave Circulator with SOC Truncation Technique

In this paper, the application of the edge-based vector finite element method combined with the short-open calibration (SOC) technique to waveguide junction circulators was presented. By applying permittivity and permeability tensors, the functional formula for tetrahedral vector elements are analytically derived in terms of the electric field strength. And the SOC technique is directly accommodated in the FEM algorithm and used to truncate the computational domain. The SOC technique removes or separates unwanted parasitics brought by the approximation of the impressed voltage source and also the problem of resulting inconsistency between different simulations. Truncation of computational domain by SOC technique makes the iterative solvers for large-sparse linear matrix equations from FEM converge much faster than by perfectly matching layers (PML). The analysis of three 2-port networks is sufficient to form the admittance matrix of the corresponding three-port waveguide junction circulators. To validate the theory, typical three-dimensional numerical results of isotropic dielectric-filled discontinuity are first presented and compared with other hybrid numerical techniques. Further, the ferrite filled waveguide junction circulators are analysed and compared with the measured and available publications. The comparison shows that good agreement is obtained.     

Probabilistically induced domain decomposition methods for elliptic boundary-value problems

Monte Carlo as well as quasi-Monte Carlo methods are used to generate only few interfacial values in two-dimensional domains where boundary-value elliptic problems are formulated. This allows for a domain decomposition of the domain. A continuous approximation of the solution is obtained interpolating on such interfaces, and then used as boundary data to split the original problem into fully decoupled subproblems. The numerical treatment can then be continued, implementing any deterministic algorithm on each subdomain. Both, Monte Carlo (or quasi-Monte Carlo) simulations and the domain decomposition strategy allow for exploiting parallel architectures. Scalability and natural fault tolerance are peculiarities of the present algorithm. Examples concern Helmholtz and Poisson equations, whose probabilistic treatment presents additional complications with respect to the case of homogeneous elliptic problems without any potential term and source.     

3D quantum transport solver based on the perfectly matched layer and spectral element methods for the simulation of semiconductor nanodevices

A 3D quantum transport solver based on the spectral element method (SEM) and perfectly matched layer (PML) is introduced to solve the 3D Schrödinger equation with a tensor effective mass. In this solver, the influence of the environment is replaced with the artificial PML open boundary extended beyond the contact regions of the device. These contact regions are treated as waveguides with known incident waves from waveguide mode solutions. As the transmitted wave function is treated as a total wave, there is no need to decompose it into waveguide modes, thus significantly simplifying the problem in comparison with conventional open boundary conditions. The spectral element method leads to an exponentially improving accuracy with the increase in the polynomial order and sampling points. The PML region can be designed such that less than −100 dB outgoing waves are reflected by this artificial material. The computational efficiency of the SEM solver is demonstrated by comparing the numerical and analytical results from waveguide and plane-wave examples and its utility is illustrated by multiple-terminal devices and semiconductor nanotube devices.     

Monte Carlo simulation of quantum statistical lattice models

In this article we review recent developments in computational methods for quantum statistical lattice problems. We begin by giving the necessary mathematical basis, the generalized Trotter formula, and discuss the computational tools, exact summations and Monte Carlo simulation, that will be used to examine explicit examples. To illustrate the general strategy, the method is applied to an analytically solvable, non-trivial, model: the one-dimensional Ising model in a transverse field. Next it is shown how to generalized Trotter formula most naturally leads to different path-integral representations of the partition function by considering one-dimensional fermion lattice models. We show how to analyze the different representations and discuss Monte Carlo simulation results for one-dimensional fermions. Then Monte Carlo work on one- and two-dimensional spin-$\text{1}\text{2}$ models based upon the Trotter formula approach is reviewed and the more dedicated Handscomb Monte Carlo method is discussed. We consider electron-phonon models and discuss Monte Carlo simulation data on the Molecular Crystal Model in one, two and three dimensions and related one-dimensional polaron models. Exact numerical results are presented for free fermions and free bosons in the canonical ensemble. We address the main problem of Monte Carlo simulations of fermions in more than one dimension: the cancellation of large contributions. Free bosons on a lattice are compared with bosons in a box and the effects of finite size on Bose-Einstein condensation are discussed.