Analysis of a Time Implicit Scheme for the Oseen Model Driven by Nonlinear Slip Boundary Conditions

This work is concerned with the time discrete analysis of the Oseen system of equations driven by nonlinear slip boundary conditions of friction type. We study the existence of solutions of the time discrete model and derive several a priori estimates needed to recover the solution of the continuous problem by means of weak compactness. Moreover, for the difference between the exact and approximate solutions, we obtain the rate of convergence of order one with respect to the time step without imposing extra regularity on the weak solution.     

One-dimensional statistical models

The properties of one-dimensional statistical systems are studied. A consistent comparison of the values of the binary correlation function obtained from the configuration integral and from Bogolyubov chain equations in various approximations is presented. The results obtained here are discussed briefly.The existing methods for studying the behavior of real statistical systems are usually based on perturbation theory, and the presence of small parameters characterizing the proximity of the system to an ideal system is assumed. Strongly interacting systems do not permit the isolation of small parameters; therefore, there are no effective methods for studying them at present. In this connection, it appears to be of interest to turn to one-dimensional systems which enable the investigation to be advanced much further and, in particular, permit a consideration of the case of strong interaction. Comparison of the exact results with approximate results obtained by methods for decomposing chains of recurrence equations for correlation functions [1] may be regarded as a qualitative criterion of the accuracy of the latter.Configuration integrals for one-dimensional statistical systems were first obtained in reference [2]. Papers have recently appeared in which one-dimensional models are studied by methods of the theory of stochastic processes [3–6].     

集中力作用下的两相饱和介质位移场Green函数

以复模两相饱和介质Biot动力学方程为基础,根据该方程D'Alembert解的Fourier变换所属的Homholtz方程特性,由Biot方程解的相容性条件及δ函数性质较好地解决了快、慢纵波位势的耦合问题.较为简便地得到了两相饱和介质在集中力作用下低频(ω＜ωc)时的频域和时域的Green函数.     

On higher order integration of angular velocities using quaternions

In the present paper the integration of angular velocities is studied. Both exact and approximate results are expressed in terms of rotational quaternions. Analytical solution is found using the theory of analytic differential systems. This exact solution serves as a suitable basis for derivation of various numerical methods. Approximative approaches based on Taylor series and several maps from pure to unit quaternions are presented. A special care is taken in describing the higher order approximations. The computational performance and comparison of numerical methods is demonstrated by examples.     

Calculation of viscous compressible gas flow past a corner

We consider the problem of calculating the parameters for supersonic viscous compressible gas flow past a corner (angle greater than ). The complete system of Navier-Stokes equations for the viscous compressible gas is solved in the small vicinity Q₁. (characteristic dimensionl~1/R) of the corner point. The conditions for smooth matching of the solution of the Navier-Stokes equations and the solution of the ideal gas or boundary layer equations are specified on the boundary of Q₁. All these solutions are a priori unknown, and the conditions for smooth matching reduce to certain differential equations on the boundary of Q₁. Here account is taken of the interaction of the flows near the wall surface and in the so-called outer region [1].We note that no a priori assumptions are made in Q₁ concerning the qualitative behavior of the solution, in contrast with other studies on viscous flow past a corner (for example, [2–4]).The Navier-Stokes system in Q₁ is solved numerically, using the difference scheme suggested in [5]. This scheme permits obtaining the steady-state solution by the asymptotic method for large Reynolds numbers R, and also has an approximation accuracy adequate to account for the effects of low viscosity and thermal conductivity.     

Theoretical and experimental investigation of the frequency characteristics of the negative corona discharge in a hot turbulent air jet

The negative corona discharge in a hot turbulent air jet is investigated experimentally and theoretically, within the framework of a discrete dischargemodel making it possible to determine its frequency characteristics. When using the discrete dischargemodel, in contrast to describing the discharge as a continuous electric charge motion, it is assumed that, like in the experiment, the charge is transferred within the discharge space by separate portions. This was noted in an old monograph [1]. Some results on this subject were presented in [2]. In [3], a discrete model of the corona discharge was developed and realized numerically for a system of spherical electrodes in the presence of a hydrodynamic source at the center of the system. In the present study, certain results earlier obtained are generalized. An approximate, using similarity and dimensional methods, model is proposed for the negative corona discharge in a hot turbulent air jet and the frequency discharge characteristics under these conditions are investigated experimentally.     

基于方向向量模拟技术的结构系统可靠性评价

对结构系统可靠性分析中的方向向量模拟方法进行了研究,指出一般方向向量模拟方法的不足,给出一个精度较高的结构失效概率近似积分公式,对示性函数I[g(r)]的几何意义进行了阐述,提出了一种生成服从n(n≥2）维单位球面上均匀分布的随机方向向量样本而不依赖于分布的方法,该方法简单易行,从教学上给出了严格证明。数值算例表明了该方法的有效性和广泛的适用性。     

Modelling of massive particulates for breakwater engineering using coupled FEMDEM and CFD

The seaward slope of many breakwaters consists of thousands of interlocking units of rock or concrete comprising a massive granular system of large elements each weighing tens of tonnes. The dumped quarry materials in the core are protected by progressively coarser particulates. The outer armour layer of freely placed units is intended to both dissipate wave energy and remain structurally stable as strong flows are drawn in and out of the particulate core. Design guidance on the mass and shape of these units is based on empirical equations derived from scaled physical model tests. The main failure mode for armour layers exposed to severe storms is hydraulic instability where the armour units of concrete or rock are subjected to uplift and drag forces which can in turn lead to rocking, displacement and collisions sufficient to cause breakage of units. Recently invented armour unit designs making up such granular layered system owe much of their success to the desirable emergent properties of interlock and porosity and how these combine with individual unit structural strength and inertial mass. Fundamental understanding of the forces governing such wave-structure interaction remains poor. We use discrete element and combined finite-discrete element methods to model the granular solid skeleton of randomly packed units coupled to a CFD code which resolves the wave dynamics through an interface tracking technique. The CFD code exploits several methods including a compressive advection scheme, node movement, and general mesh optimization. We provide the engineering context and report progress towards the numerical modelling of instability in these massive granular systems.     

Two-dimensional thermal elasticity problem for a body weakened by a system of thermally insulated cracks

The thermally elastic state of a body in two dimensions with cracks has been investigated in a number of articles (see the survey in [1]). However, in the majority of cases problems have been investigated in which temperature stresses in a body are weakened by a single crack. The existing solutions of problems on the interaction between thermally insulated cracks in an elastic body have been confined to simple cases either with collinear [2, 3] or with arched cracks [4, 5]. Below the two-dimensional thermoelastic problem for an infinite body with arbitrarily positioned straight-lined thermally insulated cracks is studied by reducing it to a system of singular integral equations. An approximate solution is found for large distances between cracks. An exact solution is obtained in the case of a periodic system of collinear cracks.     

Elliptic averaging methods using the sum of Jacobian elliptic delta and zeta functions as the generating solution

The present paper describes an improved version of the elliptic averaging method that provides a highly accurate periodic solution of a non-linear system based on the single-degree-of-freedom Duffing oscillator with a snap-through spring. In the proposed method, the sum of the Jacobian elliptic delta and zeta functions is used as the generating solution of the averaging method. The proposed method can be used to obtain the non-odd-order solution, which includes both even- and odd-order harmonic components. The stability analysis for the approximate solution obtained by the present method is also discussed. The stability of the solution is determined from the characteristic multiplier based on Floquet’s theorem. The proposed method is applied to a fundamental oscillator in a non-linear system. The numerical results demonstrate that the proposed method is very effective for analyzing the periodic solution of half-swing mode for systems based on Duffing oscillators with a snap-through spring.     

Control and synchronization of chaos systems using time-delay estimation and supervising switching control

In this paper, we present a new technique, developed using time-delay estimation (TDE) and supervising switching control (SSC), for the control and synchronization of chaos systems. The proposed technique consists of three units: a time-delay estimation unit that cancels system dynamics, a pole placement control unit that shapes error dynamics, and an SSC unit that is activated when the system dynamics are rapidly changing. We prove the stability of the closed-loop system using the Lyapunov analysis method. To verify the control and synchronization performance of the proposed technique (TDE-SSC), we compare it with TDC using numerical simulation. Our results indicate that the proposed scheme is an easily understood, numerically efficient, robust, and accurate solution for the control and synchronization of chaos systems.     

Stationary propagation of an exothermic-reaction front in a condensed medium

The problem of the propagation of an exothermic-reaction front in a condensed medium has been examined in a number of reports [1–8], in which various statements of the problem have been analyzed, the conditions of existence and uniqueness of a solution determined, and different approximate methods of determining the propagation velocity of the front suggested. In [1–6] the examination was conducted for a specific class of heat-release functions corresponding to the kinetics of exothermic transformation in homogeneous condensed systems. The propagation of the reaction front in heterogeneous sytems (condensed mixtures) was studied in [7, 8] for particular forms of the heat-release function corresponding to certain simplifying assumptions concerning the structure of the heterogeneous medium. In connection with the complexity of the structures of real condensed systems [9–11] and the diversity of the kinetic laws of interaction, in the present report the available results on a broader class of heat-release functions are generalized, and the possible approximate methods of determining the propagation velocity are analyzed.     

Modelling of massive particulates for breakwater engineering using coupled FEMDEM and CFD

The seaward slope of many breakwaters consists of thousands of interlocking units of rock or concrete comprising a massive granular system of large elements each weighing tens of tonnes.The dumped quarry materials in the core are protected by progressively coarser particulates.The outer armour layer of freely placed units is intended to both dissipate wave energy and remain structurally stable as strong flows are drawn in and out of the particulate core.Design guidance on the mass and shape of these units is based on empirical equations derived from sealed physical model tests.The main failure mode for armour layers exposed to severe storms is hydraulic instability where the armour units of concrete or rock are subjected to uplift and drag forces which can in turn lead to rocking,displacement and collisions sufficient to cause breakage of units.Recently invented armour unit designs making up such granular layered system owe much of their success to the desirable emergent properties of interlock and porosity and how these combine with individual unit structural strength and inertial mass. Fundamental understanding of the forces governing such wave-structure interaction remains poor.We use discrete element and combined finite-discrete element methods to model the granular solid skeleton of randomly packed units coupled to a CFD code which resolves the wave dynamics through an interface tracking technique.The CFD code exploits several methods including a compressive advection scheme, node movement, and general mesh optimization.We provide the engineering context and report progress towards the numerical modelling of instability in these massive granular systems.     

An indirect shooting method based on the POD/DEIM technique for distributed optimal control of the wave equation

This paper presents a fast numerical method, based on the indirect shooting method and Proper Orthogonal Decomposition (POD) technique, for solving distributed optimal control of the wave equation. To solve this problem, we consider the first‐order optimality conditions and then by using finite element spatial discretization and shooting strategy, the solution of the optimality conditions is reduced to the solution of a series of initial value problems (IVPs). Generally, these IVPs are high‐order and thus their solution is time‐consuming. To overcome this drawback, we present a POD indirect shooting method, which uses the POD technique to approximate IVPs with smaller ones and faster run times. Moreover, in the presence of the nonlinear term, to reduce the order of the nonlinear calculations, a discrete empirical interpolation method (DEIM) is applied and a POD/DEIM indirect shooting method is developed. We investigate the performance and accuracy of the proposed methods by means of 4 numerical experiments. We show that the presented POD and POD/DEIM indirect shooting methods dramatically reduce the CPU time compared to the full indirect shooting method, whereas there is no significant difference between the accuracy of the reduced and full indirect shooting methods.     

Regular wave integral approach to numerical simulation of radiation and diffraction of surface waves

A regular wave integral method is developed in the discretisation of a linear hydrodynamic problem on radiation and diffraction of surface waves by a floating or submerged body. The velocity potential of the problem is expressed as a solution of a body boundary integral equation involving the pulsating free surface Green function or pulsating free surface sources distributed on the body surface. With the use of a discretisation on the regular wave integral rather than discretisations on the singular wave integral of the Green function as in earlier investigations, the singular wave integral is approximated as an expansion of regular (or nonirregular) wave potentials. Influence coefficients between pulsating free surface source points are computed by the approximate expansion together with Hess–Smith panel integral formulas. Thus the velocity potential solution is evaluated by a boundary element algorithm. The numerical results produced from the proposed method agree well with semi-analytic solution results.     

RELIABILITY ASSESSMENT OF STRUCTURAL SYSTEMS BASED ON DIRECTIONAL VECTOR SIMULATION TECHNIQUE

The new improved directional vector simulation method for analyzing the reliability of structural systems failure probability is researched. This paper also points out the defects of general directional vector simulation, and gives rise to a new higher accuracy approximate integral formula of structural systems failure probability. A new geometric meaning of characteristic function is obtained. A new simple method of generating uniformly distributed random vector samples inn-dimensional unit hyper-spherical surface is put forward and strictly proved. This method is easy to put into practice. Numerical examples are given to show the applicability and effectiveness of the suggested approach to structural systems reliability problems. Supported by Chinese Postdoctor Fund([1998]6, 23).     

On radiative heat transfer in a plane layer of an absorbing medium

Recently there has arisen increased interest in the study of radiative heat transfer between geometrically simple systems, both as autonomous problems and as elements of more complex problems.Problems of this kind have been treated by many authors [1–111 who have considered gray, diffusely emitting and absorbing boundaries and gray nonscattering media. In most cases these investigations were restricted either to the derivation of approximate formulas for the net radiative flux, without an exact analysis of the temperature distribution in the layer [5–7], or to numerical computation [1–4], In the latter case, with the exception of [8], which contains a numerical analysis for the case of optical symmetry, no attempt was made to analyze the effect of the optical properties of the boundaries on the temperature field in the layer.These papers can be divided into two groups according to the method of analysis used. The first group includes papers based on the integral equations of radiative transfer, with the corresponding integral analytical methods [1, 2], Similar in nature are [3, 4] which use the slab method, applicable to electrical-analog computation, as well as a recent paper [8] based on probability methods.The second group of papers [5–7] is based on the so-called differential methods. Of particular interest is [7], which develops these methods to an advanced degree. In several papers the problem of radiative transfer is analyzed in conjunction with more complex problems (cf., e.g. [10, 11]).In the present work we shall attempt to carry out an approximate analytical study of problems connected with radiative heat transfer in a plane layer of an absorbing, emitting, nonscattering gray medium with temperature-independent optical properties. The layer is bounded by two parallel, diffusely emitting and diffusely reflecting, isothermal, gray planes.The paper presents the fundamental formulation of the problem, which consists in: (a) the determination of the net heat flux on the basis of given temperature distribution (direct formulation), and (b) the determination of the temperature distribution on the basis of given distribution of the net radiative heat source per unit volume and boundary temperatures (inverse formulation). The analysis is based on integral methods appropriate to the integral equations which represent the net total and hemispherical radiation flux densities [12].The author would like to thank S. S. Kutateladze for his interest in this work.     

The free convection of a conducting fluid in connected vertical channels

The convection of a conducting fluid in vertical channels in a magnetic field has been studied in a series of papers (see review [1]). Motion in isolated channels was considered in these papers. The present paper solves the problem of convection in a system of two connected plane vertical channels. A solution is obtained for the problem of steady-state motion when heating occurs from the side. Equilibrium stability is also investigated for a liquid which is heated from below (the magnetohydrodynamic generalization of the problem considered previously [2]).     

Approximate method of determining liquid oscillation frequencies in cavities of revolution

Variational methods [1–4], which require the use of computers, are widely used at present for determining the oscillation frequencies of liquid partially filling an arbitrary cavity of revolution. A technique is given in [5] for the approximate solution of this problem for a cavity which differs little from a cavity for which the solution is known. In the present paper we obtain an approximate first-order differential equation for the frequency squared, using the filling level as the independent variable. Calculations were made using this method for several cavities (sphere, cylinder with spherical base, cone with varying apex angles, torus). Comparison of these results with the results obtained experimentally by other theoretical methods shows that the proposed method is sufficiently accurate for engineering applications.The author wishes to thank I. V. Kolin for carrying out the calculations.     

Some aspects of calculating subsonic flow over wings of complex planform

The use of wings of complex planform is characteristic of the present stage of development of aviation; with discontinuities along the leading and trailing edges; with curved edges; with variable geometry (by pivoting the wing panels). This article considers some aspects of the calculation of the over-all and distributed aerodynamic characteristics of such wings for low and high subsonic velocities. The methods, based on the lifting surface scheme and the use of discrete vortical singularities, enable quite efficient and reliable digital computation of the flow about these wings at moderate angles of attack. For steady motion of the wing a further development of the method of [1] is obtained, for harmonic oscillations an extension of [2] is obtained, and for aperiodic motions of the wing and gust inputs a modification of the method of [3] is found.The author wishes to thank T. M. Muzychenko and N. G. Lavrenko for carrying out the calculations of the examples.