The moment‐guided Monte Carlo method

In this work we propose a new approach for the numerical simulation of kinetic equations through Monte Carlo schemes. We introduce a new technique that permits to reduce the variance of particle methods through a matching with a set of suitable macroscopic moment equations. In order to guarantee that the moment equations provide the correct solutions, they are coupled to the kinetic equation through a nonequilibrium term. The basic idea, on which the method relies, consists in guiding the particle positions and velocities through moment equations so that the concurrent solution of the moment and kinetic models furnishes the same macroscopic quantities. Copyright © 2010 John Wiley & Sons, Ltd.     

A two-body continuous-discrete filter

In the theory of classical mechanics, the two-body central forcing problem is formulated as a system of the coupled nonlinear second-order deterministic differential equations. The uncertainty introduced by the small, unmodeled stochastic acceleration is not assumed in the particle dynamics. The small, unmodeled stochastic acceleration produces an additional random force on a particle. Estimation algorithms for a two-body dynamics, without introducing the stochastic perturbation, may cause inaccurate estimation of a particle trajectory. Specifically, this paper examines the effect of the stochastic acceleration on the motion of the orbiting particle, and subsequently, the stochastic estimation algorithm is developed by deriving the evolutions of conditional means and conditional variances for estimating the states of the particle-earth system. The theory of the nonlinear filter of this paper is developed using the Kolmogorov forward equation “between the observations" and a functional difference equation for the conditional probability density “at the observation." The effectiveness of the nonlinear filter is examined on the basis of its ability to preserve perturbation effect felt by the orbiting particle and the signal-to-noise ratio. The Kolmogorov forward equation, however, is not appropriate for the numerical simulations, since it is the equation for the evolution of “the conditional probability density." Instead of the Kolmogorov equation, one derives the evolutions for the moments of the state vector, which in our case consists of positions and velocities of the orbiting body. Even these equations are not appropriate for the numerical implementations, since they are not closed in the sense that computing the evolution of a given moment involves the knowledge of higher order moments. Hence, we consider the approximations to these moment evolution equations. This paper makes a connection between classical mechanics, statistical mechanics and the theory of the nonlinear stochastic filtering. The results of this paper will be of use to astrophysicists, engineers and applied mathematicians, who are interested in applications of the nonlinear filtering theory to the problems of celestial and satellite mechanics. Simulation results are introduced to demonstrate the usefulness of an analytic theory developed, in this paper.     

Boltzmann equation and moment equations in curvilinear coordinates

Various forms of writing the Boltzmann equation in an arbitrary orthogonal curvilinear coordinate system are discussed. The derivation is presented of a general transport equation and moment equations containing moments of the distribution function no higher than the fourth. For a gas of Maxwellian molecules it is shown that the system of moment equations for flows which differ little from equilibrium flows transforms into the system of hydrodynamic equations. The resulting equations may be useful in solving problems on motions of a rarefied gas by the moment methods. The results are valid for both the Boltzmann equation and model kinetic equations.The author wishes to thank A. A. Nikol'skii for discussions and helpful comments.     

Bulk equations and Knudsen layers for the regularized 13 moment equations

The order of magnitude method offers an alternative to the Chapman-Enskog and Grad methods to derive macroscopic transport equations for rarefied gas flows. This method yields the regularized 13 moment equations (R13) and a generalization of Grad’s 13 moment equations for non-Maxwellian molecules. Both sets of equations are presented and discussed. Solutions of these systems of equations are considered for steady state Couette flow. The order of magnitude method is used to further reduce the generalized Grad equations to the non-linear bulk equations, which are of second order in the Knudsen number. Knudsen layers result from the linearized R13 equations, which are of the third order. Superpositions of bulk solutions and Knudsen layers show good agreement with DSMC calculations for Knudsen numbers up to 0.5.      

视密度加权的时平均统一二阶矩两相湍流模型

在气粒两相湍流的双流体模型中,颗粒相的视（表观）密度是有脉动的,在时平均的统一二阶矩（USM）模型中出现了和颗粒数密度或视密度脉动有关的项和方程,使模型方程比较复杂。实际上,用LDV或PDPA测量的流体（用小颗粒代表）和颗粒速度都是颗粒数加权平均的结果。因此,在视密度加权平均基础上推导两相湍流模型更为合理。通过推导和封闭了视密度加权平均的统一二阶矩模型（MUSM）方程组,改进了两相速度脉动关联的封闭,并引入了颗粒遇到的气体脉动速度及其输运方程。MUSM模型可以减少所用方程数,节省计算量。视密度加权平均的统一二阶矩两相湍流模型是一种对原有时间平均的统一二阶矩模型和改进和发展。     

Linear and non-linear vibration and frequency response analyses of microcantilevers subjected to tip-sample interaction

Despite their simple structure and design, microcantilevers are receiving increased attention due to their unique sensing and actuation features in many MEMS and NEMS. Along this line, a non-linear distributed-parameters modeling of a microcantilever beam under the influence of a nanoparticle sample is studied in this paper. A long-range Van der Waals force model is utilized to describe the microcantilever-particle interaction along with an inextensibility condition for the microcantilever in order to derive the equations of motion in terms of only one generalized coordinate. Both of these considerations impose strong nonlinearities on the resultant integro-partial equations of motion. In order to provide an understanding of non-linear characteristics of combined microcantilever-particle system, a geometrical function is wisely chosen in such a way that natural frequency of the linear model exactly equates with that of non-linear model. It is shown that both approaches are reasonably comparable for the system considered here. Linear and non-linear equations of motion are then investigated extensively in both frequency and time domains. The simulation results demonstrate that the particle attraction region can be obtained through studying natural frequency of the system consisting of microcantilever and particle. The frequency analysis also proves that the influence of nonlinearities is amplified inside the particle attraction region through bending or shifting the frequency response curves. This is accompanied by sudden changes in the vibration amplitude estimated very closely by the non-linear model, while it cannot be predicted by the best linear model at all.     

SIMILARITY SOLUTIONS OF BOUNDARY LAYER EQUATIONS FOR A SPECIAL NON-NEWTONIAN FLUID IN A SPECIAL COORDINATE SYSTME

Two dimensional equations of steady motion for third order fluids are expressed in a special coordinate system generated by the potential flow corresponding to an inviscid fluid. For the inviscid flow around an arbitrary object, the streamlines are the phicoordinates and velocity potential lines are psi-coordinates which form an orthogonal curvilinear set of coordinates. The outcome, boundary layer equations, is then shown to be independent of the body shape immersed into the flow. As a first approximation, assumption that second grade terms are negligible compared to viscous and third grade terms. Second grade terms spoil scaling transformation which is only transformation leading to similarity solutions for third grade fluid. By ~sing Lie group methods, infinitesimal generators of boundary layer equations are calculated. The equations are transformed into an ordinary differential system. Numerical solutions of outcoming nonlinear differential equations are found by using combination of a Runge-Kutta algorithm and shooting technique.     

点涡动力系统的Lyapunov指数

平面上理想流体的三点涡系统是可积的Hamilton系统, 但其运动仍然相当复杂, 这给研究被动微粒在三点涡系统中运动带来了很大的困难. 着眼于点涡系统的被动微粒对初始小扰动的稳定性, 通过Oseledec定理定义被动微粒的Lyapunov指数, 给出了点涡系统中被动微粒稳定性的定量刻画. 同时, 由Hamilton系统的保体积性质得到的关于Lyapunov指数的简洁表达式, 避免了计算的繁琐. 利用这个定义, 点涡系的瞬时流场可以被划分成若干区域, 被动微粒的混沌运动只能在近涡的特定区域出现.      

Investigation of the Nonlinear Behavior of Beams*

ABSTRACT A system of equations that describes the nonlinear behavior of teams is presented. The differential equations are solved by using the Galerkin method, and a system of nonlinear algebric equations is obtained. A method to deal with discontinuities in structural properties and load distribution along a beam is presented. The derivation includes general expressions for gravity-type loads. Two methods of calculating the resultant moment along the beam are described. One method incorporates differentiation of the expressions for displacements, while the second method is based on integration of loads along the beam. The numerical model is used in order to investigate displacements and moment resultants along a cantilevered beam. Different problems that are associated with nonlinear behavior are presented and discussed. Good agreement is obtained between theoretical and experimental results.     

非Gauss分布随机波浪力对海洋平台的非线性作用

为避免求解决定Maikov过程转移概率密度的Fokker—Planck方程，基于尺度分离的假设导出了一组描述非线性海洋平台受非Gauss分布随机波浪载荷作用所产生响应的矩量的常微分方程组。矩量方程清楚地反映出分别对应随机载荷和结构响应的两种不同统计特性的相互关系。由于矩量方程不依赖载荷的概率分布的具体细节，以它来模拟随机激励作用下的非线性系统将免于Monte Carlo方法所面临的正确模拟载荷概率分布的困难任务。将摄动法用于矩量方程可使线性化不再需要，这样就不会因为线性化而产生不可预料的误差。     

Piezoeffect in polar materials using moment theory

A method for describing the piezoelectric effect in a polar material is proposed based on the use of a composite particle model with seven degrees of freedom and a nonzero dipole moment. Based on micropolar theory, a system of equations is obtained which differs from the classical theory of piezoelectricity in the presence of additional terms. It is shown that under certain assumptions, the proposed system of equations becomes the classical system but the piezoelectric moduli depend strongly on the spontaneous polarization vector. It is shown that for anisotropic media with different symmetries, the structure of the third-rank piezoelectric tensors obtained using the proposed micropolar theory coincides with the structure of the tensors obtained using classical theory. For the media considered, dispersion relations are given and it is shown that in the proposed theory, unlike classical theory, the piezoelectric moduli are proportional to the spontaneous polarization.     

Iterative dipole moment method for calculating dielectrophoretic forces of particle-particle electric field interactions

Dielectrophoresis (DEP) is one of the most popular techniques for bio-particle manipulation in microfluidic systems. Traditional calculation of dielectrophoretic forces of single particle based on the approximation of equivalent dipole moment (EDM) cannot be directly applied on the dense particle interactions in an electrical field. The Maxwell stress tensor (MST) method is strictly accurate in the theory for dielectrophoretic forces of particle interaction, but the cumbersome and complicated numerical computation greatly limits its practical applications. A novel iterative dipole moment (IDM) method is presented in this work for calculating the dielectrophoretic forces of particle-particle interactions. The accuracy, convergence, and simplicity of the IDM are confirmed by a series of examples of two-particle interaction in a DC/AC electrical field. The results indicate that the IDM is able to calculate the DEP particle interaction forces in good agreement with the MST method. The IDM is a purely analytical operation and does not require complicated numerical computation for solving the differential equations of an electrical field while the particle is moving.     

Approximate kinetic equations in rarefied gas theory

The fundamental kinetic equation of gas theory, the Boltzniann equation, is a complex integrodiffcrential equation. The difficulties associated with its solution are the result not only of the large number of independent variables, seven in the general case, but also of the very complicated structure of the collision integral. However, for the mechanics of rarefied gases the primary interest lies not in the distribution function itself, which satisfies the Boltzmann equation, but rather in its first few moments, i.e., the averaged characteristics. This circumstance suggests the possibility of obtaining the averaged quantities by a simpler way than the direct method of direct solution of the Boltzmann equation with subsequent calculation of the integrals.It is well known that if a distribution function satisfies the Boltzmann equation, then its moments satisfy an infinite system of moment equations. Consequently, if we wish to obtain with satisfactory accuracy some number of first moments, then we must require that these moments satisfy the exact system of moment equations. However, this does not mean that to determine the moments of interest to us we must solve this system, particularly since the system of moment equations is not closed. The closure of the system by specifying the form of the distribution function (method of moments) can be considered only as a rough approximate method of solving problems. First, in this case it is not possible to satisfy all the equations and we must limit ourselves to certain of the equations; second, generally speaking, we do not know which equation the selected distribution function satisfies, and, consequently, we do not know to what degree it has the properties of the distribution function which satisfies the Boltzmann equation.A more reliable technique for solving the problems of rarefied gasdynamics is that based on the approximation of the Boltzmann equation, more precisely, the approximation of the collision integral. The idea of replacing the collision integral by a simpler expression is not new [1–4]. The kinetic equations obtained as a result of this replacement are usually termed model equations, since their derivation is usually based on physical arguments and not on the direct use of the properties of the Boltzmann collision integral. In this connection we do not know to what degree the solutions of the Boltzmann equation and the model equations are close, particularly since the latter do not yield the possibility of refining the solution. Exceptions are the kinetic model for the linearized Boltzmann equation [5] and the sequence of model equations of [6], constructed by a method which is to some degree analogous with that of [5].In the present paper we suggest for the simplification of the solution of rarefied gas mechanics problems a technique for constructing a sequence of approximate kinetic equations which is based on an approximation of the collision integral. For each approximate equation (i.e., equation with an approximate collision operator) the first few moment equations coincide with the exact moment equations. It is assumed that the accuracy of the approximate equation increases with increase of the number of exact moment equations. Concretely, the approximation for the collision integral consists of a suitable approximation of the reverse collision integral and the collision frequency. The reverse collision integral is represented in the form of the product of the collision frequency and a function which characterizes the molecular velocity distribution resulting from the collisions, where the latter is selected in the form of a locally Maxwellian function multiplied by a polynomial in terms of the components of the molecular proper velocities. The collision frequency is approximated by a suitable expression which depends on the problem conditions. For the majority of problems it may obviously be taken equal to the collision frequency calculated from the locally Maxwellian distribution function; if necessary the error resulting from the inexact calculation of the collision frequency may be reduced by iterations.To illustrate the method, we solve the simplest problem of rarefied gas theory-the problem on the relaxation of an initially homogeneous and isotropic distribution in an unbounded space to an equilibrium distribution.The author wishes to thank A. A. Nikol'skii for discussions of the study and V. A. Rykov for the numerical results presented for the exact solution.     

Active control of an aircraft tail subject to harmonic excitation

Vibration of structures is often an undesirable phenomena and should be avoided or controlled. There are two techniques to control the vibration of a system, that is, active and passive control techniques. In this paper, a negative feedback velocity is applied to a dynamical system, which is represented by two coupled second order nonlinear differential equations having both quadratic and cubic nonlinearties. The system describes the vibration of an aircraft tail. The system is subjected to multi-external excitation forces. The method of multiple time scale perturbation is applied to solve the nonlinear differential equations and obtain approximate solutions up to third order of accuracy. The stability of the system is investigated applying frequency response equations. The effects of the different parameters are studied numerically. Various resonance cases are investigated. A comparison is made with the available published work. The English text was polished by Keren Wang.     

Direct numerical simulation of particle–fluid systems by combining time-driven hard-sphere model and lattice Boltzmann method

A coupled numerical method for the direct numerical simulation of particle–fluid systems is formulated and implemented, resolving an order of magnitude smaller than particle size. The particle motion is described by the time-driven hard-sphere model, while the hydrodynamic equations governing fluid flow are solved by the lattice Boltzmann method (LBM). Particle–fluid coupling is realized by an immersed boundary method (IBM), which considers the effect of boundary on surrounding fluid as a restoring force added to the governing equations of the fluid. The proposed scheme is validated in the classical flow-around-cylinder simulations, and preliminary application of this scheme to fluidization is reported, demonstrating it to be a promising computational strategy for better understanding complex behavior in particle–fluid systems.     

The uniqueness of the Einstein field equations in a four-dimensional space

The Euler-Lagrange equations corresponding to a Lagrange density which is a function of g _ij and its first two derivatives are investigated. In general these equations will be of fourth order in g _ij. Necessary and sufficient conditions for these Euler-Lagrange equations to be of second order are obtained and it is shown that in a four-dimensional space the Einstein field equations (with cosmological term) are the only permissible second order Euler-Lagrange equations. This result is false in a space of higher dimension. Furthermore, the only permissible third order equation in the four-dimensional case is exhibited.     

A finite volume scheme for two-phase compressible flows

In gas-particle two-phase flows, when the concentration of the disperesed phase is low, certain assumptions may be made which simplify considerably the equations one has to solve. The gas and particle flows are then linked only via the interaction terms. One may therefore uncouple the full system of equations into two subsystems: one for the gas phase, whose homogeneous part reduces to the Euler equations; and a second system for the particle motion, whose homogeneous part is a degenerate hyperbolic system. The equations governing the gas phase flow may be solved using a high-resolution scheme, while the equations describing the motion of the dispersed phase are treated by a donor-cell method using the solution of a particular Riemann problem. Coupling is then achieved via the right-hand-side terms. To illustrate the capabilities of the proposed method, results are presented for a case specially chosen from among the most difficult to handle, since it involves certain geometrical difficulties, the treatment of regions in which particles are absent and the capturing of particle fronts.     

Numerical calculation of nonstationary gharged-particle beams

Algorithms are considered for the solution of nonstationary electronics problems which reduce to calculation of electromagnetic fields and numerical integration of the equations of motion of charged particles. It is assumed that at each moment of time the potential distribution is described by the Poisson equation. Field calculation is performed by finite-difference methods. For simulation of the space charge a modified “large particle” method is described. The KSI-BÉSM compilation system is discussed as a means of automation of the problem-solving process. Examples of problem solutions are offered.