Mixed variational principle in elasticity theory and canonical systems of equations

The paper proposes a modification of the mixed variational principle from which stationarity conditions are derived in the form of a mixed system of equations resolved for the first derivatives of the displacement and stress components acting in a plane perpendicular to one of the coordinate axes. The variational principle allows decreasing the dimension of the problem of elasticity thus reducing the system of equations to a canonical form. The modified mixed principle helps immediately obtain a canonical system of equations for various applied theories. This possibility is demonstrated with the example of the Timoshenko theory of plates __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 55–62, May 2007.     

Inertial motion of a body in an ideal fluid from the state at rest

The motion of a body in an ideal incompressible fluid flow without vortices in the absence of external forces is considered. It is demonstrated that the body can move inertially from the state at rest if its shape satisfies certain conditions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 214–219, July–August, 2008.     

On the canonical equations of Kirchhoff-Love theory of shells

The paper outlines a procedure to derive the canonical system of equations of the classical theory of thin shells using Reissner’s variational principle and partial variational principles. The Hamiltonian form of the Reissner functional is obtained using Lagrange multipliers to include the kinematical conditions that follow from the Kirchhoff-Love hypotheses. It is shown that the canonical system of equations can be represented in three different forms: one conventional form (five equilibrium equations) and two forms that are equivalent to it. This can be proved by reducing them to the same system of three equations. For problems with separable active and passive variables, partial variational principles are formulated __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 99–107, October 2007.     

Hierarchial structure theory for basic equations of fluid mechanics (BEFM) and the simplified Navier-Stokes equations (SNSE)

A hierarchial structure for the basic equations of fluid mechanics (BEFM) is found through the analysis of scales of length and time that proves a measure of the rate of change of the quantities describing the motion of the fluid as well as an estimation of the order of magnitude of various terms included in BEFM. The hierarchial structure theory shows that if (1) the characteristic Reynolds numbersRe is larger than unity and (2) the length scale in one coordinate direction is larger than that in other coordinate directions. BEFM can be classified into some levels according to the estimation of the order of magnitude of various terms included in BEFM. The hierarchial structure of BEFM has two branches: one is from BLE- to BEFM inner hierarchy, the other is from EE- to BEFM outer hierarchy, where BLE and EE are abbreviations of the boundary-layer equations and of Euler equations, respectively. The relationship between the two branches of the hierarchial structure, the characteristics, subcharacteristics and mathematical properties of the hierarchial equations are studied. A comparison between the present hierarchial equations and the Simplified Navier-Stokes equations (SNSE) appeared in literatures is also made. BLE-, EE-and Inner-outer-matched (IOM) equations hierarchies are the most important and useful three levels for solving viscous flow-fields approximately.     

Restrictions on the possible flows of scalar retarded functional differential equations in neighborhoods of singularities

The paper addresses the occurrence of possible restrictions on the flows defined by scalar retarded functional differential equations (FDEs), locally around certain simple singularities, compared with the possible flows of ordinary differential equations (ODEs) with the same singularities. It is found that for the Hopf and the Bogdanov-Takens singularities, there are no restrictions on the local flows defined by scalar FDEs, even when the nonlinearities depend on just one delayed value of the solutions. On the other hand, for the singularity associated with a zero and a conjugated pair of pure imaginary numbers as simple eigenvalues, it is shown that there occur restrictions on the flows defined by scalar FDEs with nonlinearities involving just one delay, as well as two delays satisfying a certain resonance condition. These restrictions are of geometric significance, since they amount to the impossibility of observing the homoclinic orbits that occur in arbitrarily small neighborhoods of the singularity for ODEs. Versal unfolfings for the considered singularities by FDEs and the possible restrictions on the associated flows are also studied.     

扩散方程单内点精细积分法与差分法比较研究

一维扩散方程初值问题可以用全域或子域精细积分求解。子域积分可以采用不同数量的内点，单内点是其最简单的情况。当单内点精细积分中的传递函数即指数函数用其泰勒展开式的一阶近似来替代时，精细积分转化为差分方程。本文研究了这一对应关系。各种常见差分格式均找到了对应的单点精细积分格式，并在单点精细积分一般公式中得到了统一表达形式     

Conjugate shear flows of a weakly stratified fluid

This paper studies the problem of pairs of horizontal shear flows of weakly stratified fluids with identical mass, momentum, and energy fluxes. The initial problem is reduced to a system of two scalar equations for the main- and perturbed-flow parameters by using bifurcation methods. The existence conditions for nontrivial branches of conjugate flows close to the main flow are investigated. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 79–88, March–April, 2009.     

Oscillations of a cylinder piercing a linearly stratified fluid layer

The plane problem of the small steady-state oscillations of a horizontal cylinder arbitrarily located in a three-layer fluid whose upper and lower layers are homogeneous and whose middle layer is linearly stratified is considered in the linear formulation using the Boussinesq approximation. The fluid is assumed to be ideal and incompressible. The method of mass sources distributed along the body contour is used in the internal wave generation regime and an integral equation for the fluid pressure is derived in the non-wave regime. The hydrodynamic load acting on the body is calculated as a function of the oscillation frequency of the cylinder and its location. The results are compared with experimental data.     

Three-dimensional motion of a viscous incompressible fluid in a narrow tube

An approximate solution of the problem of unsteady motion of a viscous incompressible fluid in a long narrow deformable tube at low Reynolds numbers is obtained. Pressure oscillations and tube deformation are shown to be related by an integrodifferential equation. The solution obtained extends the Poiseuille solution in elliptic tubes to the case of comparatively arbitrary small deformations in terms of the tube length and angle. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 28–32, July–August, 2009.     

非对称的结构正则方程的分块求解策略及程序实现

对于结构与介质耦合分析及非关联的弹塑性分析，常导致非对称的结构正则方程。直接求解非对称的结构正则方程，由于内存有限，会限制一些超大型结构的求解，本文针对非对称结构正则方程，详述了分块求解的策略，并在ＪＩＧＦＥＸ程序系统中成功实现，极大提高了该系统的解题规模及应用范围。     

Numerical analysis of funnel formation during unsteady fluid outflow from a rotating cylindrical vessel

The main features of the unsteady outflow of a fluid from a cylindrical vessel rotating together with it at a constant angular velocity through a central drain hole in the bottom are studied. The software package STAR-CD tested on the results of experiments with water in the absence of rotation is used. Certain important features of the phenomenon under consideration related with the formation of vortex funnels in the fluid are established. The effect of the main control parameters of the problem, namely, angular velocity, viscosity, initial depth, etc., is analyzed.     

Phase velocity spectrum of internal waves in a weakly-stratified two-layer fluid

The problem of steady-state internal waves in a weakly stratified two-layer fluid with a density that is constant in the lower layer and depends exponentially on the depth in the upper layer is considered. The spectral properties of the equations of small perturbations of a homogeneous piecewise-constant flow are described. A nonlinear ordinary differential equation describing solitary waves and smooth bores on the layer interface is obtained using the Boussinesq expansion in a small parameter.     

Some existence theorems of common and coincidence solutions for a class of systems of functional equations arising in dynamic programming

Some existence theorems of common and coincidence solutions for a class of moregeneral systems of functional equations arising in dynamic programming are shown.Theresults presented in this paper not only contain the corresponding results of[6,7]as specialcases,but also give an existence theorem of solutions for a class of functional equationssuggested by Wang recently.     

Approximate symmetries of creeping flow equations of a second grade fluid

Creeping flow equations of a second grade fluid are considered. Two current approximate symmetry methods and a modified new one are applied to the equations of motion. Approximate symmetries obtained by different methods and the exact symmetries are contrasted. Approximate solutions corresponding to the approximate symmetries are derived for each method. Symmetries and solutions are compared and advantages and disadvantages of each method are discussed in detail.     

Continuous solutions of nonlinear functional difference equations and their properties

We establish conditions for the existence of continuous bounded solutions of systems of nonlinear functional difference equations and study their properties.     

Gas bubble oscillations in a fluid in the presence of 2:1 resonance between the radial and deformation oscillation frequencies

Small nonlinear oscillations of an ellipsoidal bubble in a fluid in the presence of 2:1 frequency resonance between the radial and ellipsoidal modes are considered. The equations of motion are reduced to Hamiltonian form. The quadratic and cubic terms are taken into account in the expansion of the Hamiltonian. The Hamilton function is transformed to the normal form using the invariant normalization method in the first approximation. This makes it possible to construct an analogy between the system considered and the well-known problem of a pendulous spring. The radial and ellipsoidal bubble oscillation modes correspond to the vertical and horizontal coordinates of a material point, respectively. In the absence of resonance the solution of the nonlinear equations differs from the solution of the linear equations by only a small (quadratic in the amplitude) change in the oscillation frequency. In the resonance case the radial and ellipsoidal oscillation modes periodically change places and the energy of one mode is converted into that of the other. The interest in the system in resonance is associated with precisely this fact. The question of the dissipation effect in real media is considered. The decay rate depends significantly on the physical properties of the material and, in certain special cases, can be small enough for the energy transfer effect to manifest itself.     

Exact solutions of linearized equations of convection of a weakly compressible fluid

A mathematical model of fluid convection under microgravity conditions is considered. The equation of state is used in a form that allows considering the fluid as a weakly compressible medium. Based on the previously proposed mathematical model of convection of a weakly compressible fluid, unsteady convective motion in a vertical band, with a heat flux periodic in time set on the solid boundaries of this band, is considered. This model of convection allows one to study the problem with the boundary thermal model oscillating in an antiphase rather than in-phase mode, while the latter was required for the model of microconvection of an isothermally incompressible fluid. Exact solutions for velocity components and temperature are derived, and the trajectories of fluid particles are constructed. For comparison, the trajectories predicted by the classical Oberbeck-Boussinesq model of convection and by the microconvection model are presented.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 52–63, March–April, 2005.     

Flow of a Viscoplastic Fluid on a Rotating Disk

ＦＬＯＷＯＦＡＶＩＳＣＯＰＬＡＳＴＩＣＦＬＵＩＤＯＮＡＲＯＴＡＴＩＮＧＤＩＳＫＦａｎＣｈｕｎ（范椿）（ＩｎｓｔｉｕｉｅｏｆＭｅｃｈａｎｉｃｓ，ＡｃａｄｅｍｉａＳｉｎｉｃａ，Ｂｅｉｊｉｎｇ）（ＲｅｃｅｉｖｅｄＮｏｖ．２０，１９９２；Ｃｏｍｍｕｎｉｃａｔ...     

Acceleration of convergence by shifting the spectrum of implicit finite difference operators associated with the equations of gas dynamics

Eigensystem analysis techniques are applied to finite difference formulations of the Navier-Stokes equations in one dimension. Spectra of the resulting implicit difference operators are computed. The largest eigenvalues are calculated by using a combination of the Frechet derivative of the operators and Arnoldi's method. The accuracy of Arnoldi's method is tested by comparing the rate of convergence of the iterative method with the dominant eigenvalue of the original iteration matrix. On the basis of the pattern of eigenvalue distributions for various flow configurations, a shifting of the implicit operators in question is devised. The idea of shifting is based on the power method of linear algebra and is very simple to implement. This procedure has improved the rates of convergence of CFD codes (developed at NASA Ames Research Center) by 20%–50%. The sensitivity of the computed solution with respect to the shift is also studied. Finally, an adaptive shifting of the spectrum together with Wynn's acceleration algorithm are discussed. It turns out that the shifting process is a preconditioner for Wynn's method.     

Estimation of goal functional error arising from iterative solution of Euler equations

Estimation of the error arising in the cost (goal) functional due to stopping the iterative process is considered for a steady problem solved by temporal relaxation. The functional error is calculated using an iteration residual along with related adjoint parameters. Numerical tests demonstrate the applicability of this approach for the steady 2D Euler equations.