Features of the linear stage of development of 3D wave packets in a plane Poiseuille flow

In the framework of the asymptotic theory of free interaction, the linear stage of the development of 3D wave packets in a plane Poiseuille flow is studied. Numerical results show the presence of “ripples” in the lateral direction in the first phase of the linear stage. The disturbances propagate within a certain angle.     

On the problem of output feedback control synthesis under uncertainty and finite-time observers

We consider the problem of controlling a linear system of ordinary differential equations with a linear observable output. The system contains uncertain items (disturbances), for which we know only “hard” pointwise constraints. The problem of synthesizing a control that brings the trajectories of the system into a given target set in finite time is solved under weakened conditions without assuming that the control and the disturbance are of the same type. To this end, we suggest an approach that amounts to constructing an information set and a weakly invariant set with subsequent “aiming” of the first set at the second. Both stages are carried out in a finite-dimensional space, which permits one to use an efficient algorithm for solving the synthesis problem approximately on the basis of the ellipsoidal calculus technique. The results are illustrated by an example in which the control of a linear oscillation system is constructed.     

超音速尖锥边界层中扰动演化特征的数值研究

采用高精度紧致格式, 对超音速尖锥边界层中二维扰动的空间演化, 进行了直接数值模拟．结果表明,虽然尖锥边界层流动存在一定的锥面法向速度,但小扰动的幅值及相位的演化都与由平行流假设得到的线性理论结果吻合．还研究了有限幅值扰动的演化,给出了其演化规律．并在扰动幅值增长到一定值时,发现了小激波．     

Stabilization for the vibrations modeled by the ‘standard linear model’ of viscoelasticity

We study the stabilization of vibrations of a flexible structure modeled by the ‘standard linear model’ of viscoelasticity in a bounded domain in ℝ ⁿ with a smooth boundary. We prove that amplitude of the vibrations remains bounded in the sense of a suitable norm in a space $ \mathbb{X} $ \mathbb{X} , defined explicitly in (22) subject to a restriction on the uncertain disturbing forces on $ \mathbb{X} $ \mathbb{X} . We also estimate the total energy of the system over time interval [0, T] for any T > 0, with a tolerance level of the disturbances. Finally, when the input disturbances are insignificant, uniform exponential stabilization is obtained and an explicit form for the energy decay rate is derived. These results are achieved by a direct method under undamped mixed boundary conditions.     

Adaptive robust control with guaranteed outcome

We consider the adaptive control problem for a linear discrete system in the presence of adaptive and multiplicative deterministic disturbances. Prior information about disturbances is so poor that the system becomes unidentifiable. We show that stability of the closed-loop system and a uniform (in the parameters) bound of the performance criterion can be achieved by a simple projection algorithm. Translated from Algoritmy Upravleniya i Identifikatsii, pp. 141–155, 1997.     

Subsonic and supersonic disturbances in the generation of surface waves in a liquid layer adjacent to a high-speed gas-stream

In an effort to shed further light upon the nature of “supersonic” disturbances as distinct from that of ‘subsonic’ disturbances in parallel compressible flows, this paper makes an investigation of the stability characteristics of the surface waves generated in a liquid layer adjacent to a high-speed gas-stream. It turns out that the nature of the surface waves generated in the liquid layer depends markedly upon the type of disturbances present in the high-speed gas-stream. For the case of ‘subsonic’ disturbances it is shown that the energy transfer from the gas stream to the surface waves is contributed predominantly by the Fourier component of the normal gas-pressure force-field in phase with the slope of the wavy surface. For the case of ‘supersonic’ disturbances, this energy transfer is shown to be predominantly due to the component of the pressure-field in phase with the surface-wave displacement and is related to the presence of travelling periodic waves in the gas-stream—this energy transfer is shown to promote always the growth of the surface waves.     

Fast computation of optimal disturbances for duct flows with a given accuracy

This work is devoted to the numerical analysis of small flow disturbances, i.e. velocity and pressure deviations from the steady state, in ducts of constant cross sections. The main emphasis is put on the disturbances causing the most kinetic energy density growth, the so-called optimal disturbances, whose knowledge is important in laminar-turbulent transition and robust flow control investigations. Numerically, this amounts to computing the maximum amplification of the 2-norm of a matrix exponential exp{tS} for a square matrix S at t ≥ 0. To speed up the computations, we propose a new algorithm based on low-rank approximations of the matrix exponential and prove that it computes the desired amplification with a given accuracy. We discuss its implementation and demonstrate its efficiency by means of numerical experiments with a duct of square cross section.     

Strict anisotropic norm bounded real lemma in terms of matrix inequalities

This paper is aimed at extending the H _∞ Bounded Real Lemma to stochastic systems under random disturbances with imprecisely known probability distributions. The statistical uncertainty is measured in the terms of information theory using the mean anisotropy functional. The disturbance attenuation capabilities of the system are quantified by the anisotropic norm which is a stochastic counterpart of the H _∞ norm. A state-space sufficient criterion for the anisotropic norm of a linear discrete time invariant system to be bounded by a given threshold value is derived. The resulting Strict Anisotropic Norm Bounded Real Lemma involves an inequality on the determinant of a positive definite matrix and a linear matrix inequality. These convex constraints can be approximated by two linear matrix inequalities.     

Rates of convergence in the CLT for linear random fields

In this paper, we study sums of linear random fields defined on the lattice Z ² with values in a Hilbert space. The rate of convergence of distributions of such sums to the Gaussian law is discussed, and mild sufficient conditions to obtain an approximation of order n ^−p are presented. This can be considered as a complement of a recent result of [A.N. Nazarova, Logarithmic velocity of convergence in CLT for stochastic linear processes and fields in a Hilbert space, Fundam. Prikl. Mat., 8:1091–1098, 2002 (in Russian)], where the logarithmic rate of convergence was stated, and as a generalization of the result of [D. Bosq, Erratum and complements to Berry–Esseen inequality for linear processes in Hilbert spaces, Stat. Probab. Lett., 70:171–174, 2004] for linear processes.     

On the cutting stock problem under stochastic demand

This paper addresses the one-dimensional cutting stock problem when demand is a random variable. The problem is formulated as a two-stage stochastic nonlinear program with recourse. The first stage decision variables are the number of objects to be cut according to a cutting pattern. The second stage decision variables are the number of holding or backordering items due to the decisions made in the first stage. The problem’s objective is to minimize the total expected cost incurred in both stages, due to waste and holding or backordering penalties. A Simplex-based method with column generation is proposed for solving a linear relaxation of the resulting optimization problem. The proposed method is evaluated by using two well-known measures of uncertainty effects in stochastic programming: the value of stochastic solution—VSS—and the expected value of perfect information—EVPI. The optimal two-stage solution is shown to be more effective than the alternative wait-and-see and expected value approaches, even under small variations in the parameters of the problem.     

Oscillation and wandering of solutions to a second order differential equation

The Lyapunov’s oscillation and wandering characteristics of solutions to a second order linear equation are defined, namely, the mean frequency of a solution, of its derivative or their various linear combinations, the mean angular velocity of the vector composed of a solution and its derivative, also wandering indices derived from that velocity. Nearly all of the values introduced for any equation are proved to be the same: just all for the autonomic equation (moreover, they coincide with the absolute values of the imaginary parts of the roots of the characteristic polynomial), but even for the periodic one, generally speaking, not all.     

Influence of the gravity on interface shape during crystal growth of LICAF

A Galerkin finite element method, together with the boundary conformal mapping technique, is used to investigate the change of melWcrystal interface under low gravity during the growth of LEAF system. Results have shown that strong convection can cause a deeply concave interface toward the crystal, and significantly increase radial thermal gradients near the interface. The flow intensity and the change of the gravity have a linear relationship under low gravity (g _o u = 10⁻²-10⁻⁶). At smallMa number, the maximum acceleration for keeping a planar growth interface is g_max = 1 × 10⁻³ g under our given conditions. In addition, the growth velocity may have some influence on the growth interface shape even atpg gravity level, indicating that the growth velocity cannot be too fast even when convection is very weak.     

On the unbounded increase in vorticity and velocity circulation in stratified and homogenous fluid flows

We consider the Oberbeck-Boussinesq system without dissipation (ideal convection) in a horizontal layer and in a “barrel” with flat bottom and flat cover. It is shown that the velocity circulation along a fluid contour consisting of two fluid curves on the bottom and on the cover connected by two isothermic fluid curves can be calculated explicitly and is a linear function of time. The serre result stating that the azimuthal component of vorticity in rotationally symmetric ideal fluid flows between coaxial cylinders increases linearly is generalized to the case of stratified fluids. It is proved that all plane and axially symmetric isothermic flows in a layer or in a barrel are unstable with respect to nonisothermic perturbations and in the case of a homogenous fluid all axially symmetric flows between coaxial cylinders are unstable in the sense of Lyapunov with respect to perturbations of the azimuthal component of the velocity in any metric including the maximum of magnitude of vorticity. Translated fromMaternaticheskie Zametki, Vol. 68, No. 4, pp. 627–636, October, 2000.     

On the analysis of an endothermal/exothermal saturation problem

We consider an endo-or exo-thermal saturation problem that corresponds to a parabolic quasi-variational inequality. Applying regularity results and inequalities of Lewy-Stampacchia type, we prove the solvability of a modified problem (including the Steklov averaging and the mollification of the saturation velocity) for the nonlinear case and also of the exact problem for the linear case with a small coefficient in the temperature equation. Bibliography: 8 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 233, 1996, pp. 131–141.     

A limit problem in natural convection

We discuss a model limit problem which arises as a first step in the mathematical justification of our Boussinesq-type approximation [4], which takes into account dissipative heating in natural convection. We treat a simplified highly non linear system depending on a (perturbation) parameter ε. The main difficulty is that for ε ≠ 0 the velocity is not solenoidal. First we prove that our system has weak solutions for each fixed ε. Moreover, while the chosen perturbation parameter ε tends to zero we show, that we arrive at the usual incompressible case and the standard Boussinesq approximation.     

Functional Optimal Estimation Problems and Their Solution by Nonlinear Approximation Schemes

The design of state estimators for nonlinear dynamic systems affected by disturbances is addressed in a functional optimization framework. The estimator contains an innovation function that has to be chosen within a suitably defined class of functions in such a way to minimize a cost functional given by the worst-case ratio of the ℒ _p norms of the estimation error and the disturbances. Since this entails an infinite-dimensional optimization problem that under general hypotheses cannot be solved analytically, an approximate solution is sought by minimizing the cost functional over linear combinations of simple “basis functions,” represented by computational units with adjustable parameters. The selection of the parameters is made by solving a constrained nonlinear programming problem, where the constraints are given by pointwise conditions that ensure the well-definiteness of the functional and the existence of a solution. Penalty terms are introduced in the cost function to account for constraints imposed on points that result from sampling the sets to which the trajectories of the state and of the estimation error belong. To ensure an efficient covering of the sets, low-discrepancy sampling techniques are exploited that generate samples deterministically spread in a uniform way, without leaving regions of the space undersampled. Work supported by a PRIN grant from the Italian Ministry of University and Research (Project “New Techniques for the Identification and Adaptive Control of Industrial Systems”) and by the EU and the Regione Liguria trough the Regional Programs of Innovative Action of the European Regional Development Fund.     

Multigrid preconditioned Krylov subspace methods for three-dimensional numerical solutions of the incompressible Navier–Stokes equations

We consider numerical methods for the incompressible Reynolds averaged Navier–Stokes equations discretized by finite difference techniques on non-staggered grids in body-fitted coordinates. A segregated approach is used to solve the pressure–velocity coupling problem. Several iterative pressure linear solvers including Krylov subspace and multigrid methods and their combination have been developed to compare the efficiency of each method and to design a robust solver. Three-dimensional numerical experiments carried out on scalar and vector machines and performed on different fluid flow problems show that a combination of multigrid and Krylov subspace methods is a robust and efficient pressure solver. This revised version was published online in June 2006 with corrections to the Cover Date.     

On ranking of feasible solutions of a bottleneck linear programming problem

Summary An algorithm for the ranking of the feasible solutions of a bottleneck linear programming problem in ascending order of values of a concave bottleneck objective function is developed in this paper. The “best” feasible solution for a given value of the bottleneck objective is obtained at each stage. It is established that only the extreme points of a convex polytope need to be examined for the proposed ranking. Another algorithm, involving partitioning of the nodes, to rank the feasible solutions of the bottleneck linear programming problem is proposed, and numerical illustrations for both are provided.     

The study of three-dimensional gas flow in a nozzle using the completely implicit method

We study three-dimensional potential gas flow in a nozzle. The matrix of the system of linear equations obtained in approximating the equation for the velocity potential by symmetric differences has a strongly sparse form. This property is used for an approximate expansion of it as a product of triangular matrices. To guarantee stability in the supersonic region of flow additional terms of artificial viscosity type are introduced into the equation for the velocity potential. We study gas flows in axisymmetric and three-dimensional nozzles: elliptic, superelliptic, and nozzles with nonsymmetric subsonic part. Comparison of the results with the data of other authors has shown the high effectiveness of such an approach. Translated fromMetody Matematicheskogo Modelirovaniya, 1998, pp. 76–86.     

经过修正的层次流流动的流动稳定性问题(Ⅱ)——经过修正的平行剪切流的线性稳定性研究

本文根据文[1]给出的经过修正的层流流动的流动稳定性理论及平行剪切流中平均速度的一类修正剖面,研究了平行剪切流的线性稳定性性质,对于平面Couette流动和圆管Poiseuill流动,首次得到了二维扰动和轴对称扰动也能造成失稳的结果,并给出了这两种流动在某种定义下的中性曲线.