A Galerkin Method for the Simulation of the Transient 2-D/2-D and 3-D/3-D Linear Boltzmann Equation

Many production steps used in the manufacturing of integrated circuits involve the deposition of material from the gas phase onto wafers. Models for these processes should account for gaseous transport in a range of flow regimes, from continuum flow to free molecular or Knudsen flow, and for chemical reactions at the wafer surface. We develop a kinetic transport and reaction model whose mathematical representation is a system of transient linear Boltzmann equations. In addition to time, a deterministic numerical solution of this system of kinetic equations requires the discretization of both position and velocity spaces, each two-dimensional for 2-D/2-D or each three-dimensional for 3-D/3-D simulations. Discretizing the velocity space by a spectral Galerkin method approximates each Boltzmann equation by a system of transient linear hyperbolic conservation laws. The classical choice of basis functions based on Hermite polynomials leads to dense coefficient matrices in this system. We use a collocation basis instead that directly yields diagonal coefficient matrices, allowing for more convenient simulations in higher dimensions. The systems of conservation laws are solved using the discontinuous Galerkin finite element method. First, we simulate chemical vapor deposition in both two and three dimensions in typical micron scale features as application example. Second, stability and convergence of the numerical method are demonstrated numerically in two and three dimensions. Third, we present parallel performance results which indicate that the implementation of the method possesses very good scalability on a distributed-memory cluster with a high-performance Myrinet interconnect.     

Hybrid Fourier–Vlasov simulation in non-inertial reference frames

We present some useful extensions of the spectral Fourier–Vlasov algorithm for simulations of interactions of collisionless plasmas with ion beams. For many practical applications the relative drifts of various particle populations require high resolution of particle distribution functions (PDFs) or the use of large phase space domain, which makes the simulations extremely memory- and time-consuming. We propose using non-inertial reference frames moving in the velocity dimensions for the beam particle distribution functions. As a result, it is possible to simulate plasma–beam interactions at a much lower resolution. This method is particularly suitable for simulations of fast particle beams and plasmas with heavy ion species or cold particle populations. In addition, for simulations of strongly nonlinear instabilities which cause strong plasma heating, the adaptive mesh refinement and phase space reduction are proposed. Contrary to the Vlasov simulation in the real velocity space, plasma heating in the Fourier-inverted space leads to the PDF profile shrinking. Thus, instead of having to extrapolate the PDF into the regions where it was previously undefined, in the Fourier-space, it is sufficient to interpolate the pre-existing solution.     

Multiple almost periodic solutions in nonautonomous delayed neural networks

A general methodology that involves geometric configuration of the network structure for studying multistability and multiperiodicity is developed. We consider a general class of nonautonomous neural networks with delays and various activation functions. A geometrical formulation that leads to a decomposition of the phase space into invariant regions is employed. We further derive criteria under which the n-neuron network admits 2n exponentially stable sets. In addition, we establish the existence of 2n exponentially stable almost periodic solutions for the system, when the connection strengths, time lags, and external bias are almost periodic functions of time, through applying the contraction mapping principle. Finally, three numerical simulations are presented to illustrate our theory.     

弱非线性耦合二维各向异性谐振子的动力学行为

研究了弱非线性耦合二维各向异性谐振子的奇点稳定性及其在相空间中的轨迹.首先,求得弱非线性耦合二维各向异性谐振子的奇点;其次,分别利用Lyapunov间接法和梯度系统方法讨论该系统的平衡点稳定性;最后,用Matlab方法对系统进行数值模拟,并运用庞加赖截面观察系统在相空间的运动轨迹,发现随着能量的增加系统经历规则运动、规则运动与混沌并存等阶段,最后出现了混沌现象.     

Virtual functions of the space–time finite element method in moving mass problems

Classical time integration schemes fail in vibration analysis of complex problems with moving concentrated parameters. Moving mass problems and moving support problems belong to this group. Commercial systems of dynamic simulations do not support such an analysis. Moreover, the classical finite element method with the Newmark-type time integration method does not allow us to obtain convergent results at all. The reason lies in the impossibility of full mathematical consideration of the time integration stage and the analysis of inertial terms of a travelling mass. Both of them, unfortunately, are decoupled. In this paper we propose an efficient and exact numerical approach to the problem by using the space–time finite element method. We derive characteristic matrices of the discrete element of the string and the Bernoulli–Euler beam that carry the concentrated mass. We present four types of virtual functions in time and we apply two of them to the practical analysis. Displacements in time obtained numerically are compared with semi-analytical results. Almost perfect coincidence proves the efficiency of the approach.     

Outflow Boundary Conditions for the Fourier Transformed One-Dimensional Vlasov–Poisson System

In order to facilitate numerical simulations of plasma phenomena where kinetic processes are important, we have studied the technique of Fourier transforming the Vlasov equation analytically in the velocity space, and solving the resulting equation numerically. Special attention has been paid to the boundary conditions of the Fourier transformed system. By using outgoing wave boundary conditions in the Fourier transformed space, small-scale information in velocity space is carried outside the computational domain and is lost. Thereby the so-called recurrence phenomenon is reduced. This method is an alternative to using numerical dissipation or smoothing operators in velocity space. Different high-order methods are used for computing derivatives as well as for the time-stepping, leading to an over-all fourth-order method.     

About frame estimation of growth functions and robust prediction in bioprocess modeling

We address the problem of determining functional framing from experimental data points in view of robust time-varying predictions, which is of crucial importance in bioprocess monitoring. We propose a method that provides guaranteed functional bounds, instead of sets of parameters values for growth functions such as the classical Monod or Haldane functions commonly used in bioprocess modeling. We illustrate the applicability of the method with bioreactor simulations in batch and continuous mode, as well as on real data. We also present two extensions of the method adding flexibility in its application, and discuss its efficiency in providing guaranteed state estimations.     

Low phase noise design of microwave oscillators (invited article)

Time domain methods for numerical nonlinear analysis of the steady state solution as well as the spectral behavior of microwave oscillators are discussed. In addition, a method to minimize the phase noise of oscillators by numerical optimization is outlined and applied to the design of a low phase noise oscillator. Computer simulations are compared with measurement results of the fabricated oscillators. © 1996 John Wiley & Sons, Inc.     

Efficient spectral collocation and continuation methods for symmetry-breaking solutions of rotating Bose–Einstein condensates

We describe an efficient spectral collocation method (SCM) for symmetry-breaking solutions of rotating Bose–Einstein condensates (BECs) which is governed by the Gross–Pitaevskii equation (GPE). The Lagrange interpolants using the Legendre–Gauss–Lobatto points are used as the basis functions for the trial function space. Some formulas for the derivatives of the basis functions are given so that the GPE can be efficiently computed. The SCMs are incorporated in the context of a predictor–corrector continuation algorithm for tracing primary and secondary solution branches of the GPE. Symmetry-breaking solutions are numerically presented for both rotating BECs, BECs in optical lattices, and two-component BECs in optical lattices. Our numerical results show that the numerical algorithm we propose in this paper outperforms the classical orthogonal Legendre polynomials.     

A Numerical Method for Simulation of Microflows by Solving Directly Kinetic Equations with WENO Schemes

A numerical method for simulation of transitional-regime gas flows in microdevices is presented. The method is based on solving relaxation-type kinetic equations using high-order shock capturing weighted essentially non-oscillatory (WENO) schemes in the coordinate space and the discrete ordinate techniques in the velocity space. In contrast to the direct simulation Monte Carlo (DSMC) method, this approach is not subject to statistical scattering and is equally efficient when simulating both steady and unsteady flows. The presented numerical method is used to simulate some classical problems of rarefied gas dynamics as well as some microflows of practical interest, namely shock wave propagation in a microchannel and steady and unsteady flows in a supersonic micronozzle. Computational results are compared with Navier–Stokes and DSMC solutions.     

Comparison of Eulerian Vlasov solvers

Vlasov methods, which instead of following the particle trajectories, solve directly the Vlasov equation on a grid of phase space have proven to be an efficient alternative to the Particle-In-Cell method for some specific problems. Such methods are useful, in particular, to obtain high precision in regions where the distribution function is small.Gridded Vlasov methods have the advantage of being completely free of numerical noise, however the discrete formulations contain some other numerical artifacts, like diffusion or dissipation. We shall compare in this paper different types of methods for solving the Vlasov equation on a grid in phase space: the semi-Lagrangian method, the finite volume method, the spectral method, and a method based on a finite difference scheme, conserving exactly several invariants of the system. Moreover, for each of those classes of methods, we shall first compare different interpolation or reconstruction procedures. Then we shall investigate the cost in memory as well as in CPU time which is a very important issue because of the size of the problem defined on the phase space.     

Kalman filter and joint tracking and classification based on belief functions in the TBM framework

The paper develops an approach to joint tracking and classification based on belief functions as understood in the transferable belief model (TBM). The TBM model is identical to the classical model except all probability functions are replaced by belief functions, which are more flexible for representing uncertainty. It is felt that the tracking phase is well handled by the classical Kalman filter but that the classification phase deserves amelioration. For the tracking phase, we derive a minimal set of assumptions needed in the TBM approach in order to recover the classical relations. For the classification phase, we distinguish between the observed target behaviors and the underlying target classes which are usually not in one-to-one correspondence. We feel the results obtained with the TBM approach are more reasonable than those obtained with the corresponding Bayesian classifiers.     

具有自学机制和退火选择的教学优化算法

为了克服教学优化(TLBO)算法容易早熟,解精度低的弱点,提出一种具有教师自学和学生选择学习的改进教学优化算法。在每次迭代过程中教师个体首先通过反向学习(OBL),实现教师的自我提高,加强优秀个体周围邻域的搜索,引导算法向包含全局最优的解空间逼近,保证算法具有较好的平衡和探索能力。学生个体通过随机执行反向学习进行自学习,同时亦向教师个体进行学习,计算两种学习方法后的状态相对教师个体的突跳概率,并以此概率为基础进行轮盘赌产生子个体。通过在多个标准测试函数上的实验仿真并与相关的算法对比,结果表明所提出的改进算法具有更高的收敛速度和收敛精度。     

Time-dependent generalized pseudospectral method for accurate treatment of multiphoton processes of diatomic molecules in intense laser fields

We present a numerical method based on the generalized pseudospectral discretization in prolate spheroidal coordinates and split-operator time propagation. The method has been applied for the study of multiphoton processes of H⁺₂ and N₂ diatomic molecules in intense laser fields. It proves very accurate while using only moderate computer resources.     

Three dimensional numerical simulations of long-span bridge aerodynamics, using block-iterative coupling and DES

The design of long-span bridges often depends on wind tunnel testing of sectional or full aeroelastic models. Some progress has been made to find a computational alternative to replace these physical tests. In this paper, an innovative computational fluid dynamics (CFD) method is presented, where the fluid-structure interaction (FSI) is solved through a self-developed code combined with an ANSYS-CFX solver. Then an improved CFD method based on block-iterative coupling is also proposed. This method can be readily used for two dimensional (2D) and three dimensional (3D) structure modelling. Detached-Eddy simulation for 3D viscous turbulent incompressible flow is applied to the 3D numerical analysis of bridge deck sections. Firstly, 2D numerical simulations of a thin airfoil demonstrate the accuracy of the present CFD method. Secondly, numerical simulations of a U-shape beam with both 2D and 3D modelling are conducted. The comparisons of aerodynamic force coefficients thus obtained with wind tunnel test results well meet the prediction that 3D CFD simulations are more accurate than 2D CFD simulations. Thirdly, 2D and 3D CFD simulations are performed for two generic bridge deck sections to produce their aerodynamic force coefficients and flutter derivatives. The computed values agree well with the available computational and wind tunnel test results. Once again, this demonstrates the accuracy of the proposed 3D CFD simulations. Finally, the 3D based wake flow vision is captured, which shows another advantage of 3D CFD simulations. All the simulation results demonstrate that the proposed 3D CFD method has good accuracy and significant benefits for aerodynamic analysis and computational FSI studies of long-span bridges and other slender structures.     

非最小相位系统的基函数型自适应迭代学习控制

针对重复运行的未知非最小相位系统的轨迹跟踪问题, 结合时域稳定逆特点, 提出了一种新的基函数型自适应迭代学习控制(Basis function based adaptive iterative learning control, BFAILC)算法. 该算法在迭代控制过程中应用自适应迭代学习辨识算法估计基函数模型, 采用伪逆型学习律逼近系统的稳定逆, 保证了迭代学习控制的收敛性和鲁棒性. 以傅里叶基函数为例, 通过在非最小相位系统上的控制仿真, 验证了算法的有效性.     

Lattice Boltzmann Model Based on the Rebuilding-Divergency Method for the Laplace Equation and the Poisson Equation

In this paper, a new lattice Boltzmann model based on the rebuilding-divergency method for the Poisson equation is proposed. In order to translate the Poisson equation into a conservation law equation, the source term and diffusion term are changed into divergence forms. By using the Chapman-Enskog expansion and the multi-scale time expansion, a series of partial differential equations in different time scales and several higher-order moments of equilibrium distribution functions are obtained. Thus, by rebuilding the divergence of the source and diffusion terms, the Laplace equation and the Poisson equation with the second accuracy of the truncation errors are recovered. In the numerical examples, we compare the numerical results of this scheme with those obtained by other classical method for the Green-Taylor vortex flow, numerical results agree well with the classical ones.     

厚板动力响应分析的一种辛算法

将厚板动力分析从Lagrange体系改换为Hamilton体系.根据古典阴阳互补和现代对偶互补的基本思想,首次建立了线性阻尼情形下厚板动力学的相空间非传统Hamilton变分原理.这种变分原理不仅能反映这种动力学初值—边值问题的全部特征,而且它的欧拉方程具有辛结构的特征.基于该变分原理,提出一种称之为辛空间有限元—时间子域法的辛算法.这种新方法是由空间域采用有限元法与时间子域采用La-grange插值多项式插值的时间子域法相结合而成.文中用这种辛算法分析了四种支承条件下厚板的动力响应问题.算例的计算结果表明,这种新方法的稳定性、收敛性、计算精度和效率都明显高于国际上常用的W ilson-θ法和Newm ark-β法.     

A Dynamic Multilevel Model for the Simulation of the Small Structures in Homogeneous Isotropic Turbulence

In turbulence simulations, the small scales of motion, even if they carry only a very small percentage of the whole kinetic energy, must be taken into account in order to accurately reproduce the statistical properties of the flows. This induces strong computational restrictions. In an attempt to understand and model the nonlinear interaction between the small and large scales, a dynamic multilevel procedure is proposed and applied to homogeneous turbulence. As in large eddy simulation, filtering operators are used to separate the different scales of the velocity field. In classical models (Smagorinsky), only the large scale equation is resolved. A different approach is proposed here. Indeed, by analyzing the nonlinear interaction term in the large scale equation, we show that they locally have a very small contribution to the whole dynamic of the flow. We then propose to treat them less accurately. Specific treatments for these terms are achieved by a space and time adaptative procedure; the cut-off value (filter width) which defines the scale separation varies as time evolves. Simulations at Re in the range of 60 to 150 have been performed until statistical steady states are reached, i.e. over long time period. Comparisons with direct simulations (DNS) show that this numerical modeling provides an efficient resolution of the nonlinear interaction term. The multilevel algorithm is shown to be stable; the corresponding simulated flows reach a statistically steady state very close to the DNS ones. The shape of the energy spectrum functions as well as the characteristic statistical properties of the velocity and its derivatives are accurately recovered.     

Goal-oriented error estimation and adaptivity for the finite element method

In this paper, we study a new approach in a posteriori error estimation, in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm. These so-called quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs. We present here the theory with respect to a class of elliptic boundary-value problems, and in particular, show how to obtain accurate estimates as well as upper and lower bounds on the error. We also study the new concept of goal-oriented adaptivity, which embodies mesh adaptation procedures designed to control error in specific quantities. Numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates.