Z. Lu1,2, L. Li1, L. Cao1, Ch. Hou3 1Chongqing Three Gorges University, Chongqing, 404000, P.R. China 2Tianjin University of Finance and Economics, Tianjin, 300222, P.R. China 3Guangdong University of Finance, Guangzhou, 511300, P.R. China
Keywords: априорные оценки ошибки, нелинейная задача оптимального управления, метод конечных объемов, вариационная дискретизация, a priori error estimates, nonlinear optimal control problem, finite volume method, variational discretization
In this paper, we study a priori error estimates for a finite volume element approximation of a nonlinear optimal control problem. The schemes use discretizations base on a finite volume method. For the variational inequality, we use a method of the variational discretization concept to obtain the control. Under some reasonable assumptions, we obtain some optimal order error estimates. The approximate order for the state, costate, and control variables is Oh2) or O(h2√|lnh|) in the sense of L2-norm or L∞-norm. A numerical experiment is presented to test the theoretical results. Finally, we give some conclusions and future works.
This paper presents an algorithm for solving boundary value problems of the elasticity theory, suitable to solve contact problems and those whose scope of deformation contains thin layers of a medium. The solution is represented as a linear combination of subsidiary solutions and fundamental solutions to the Lame equations. Singular points of fundamental solutions of the Lame equations are located as an external layer of the deformation around the perimeter. Coefficients of the linear combination are determined by minimizing deviations of a linear combination from the boundary conditions. To minimize deviations, the conjugate gradient method is applied. Examples of calculations for mixed boundary conditions are presented.
In the polar coordinates, a discrete analog of the conjugate-operator model of the heat conduction problem preserves the structure of the original model. The difference scheme converges with the second order of accuracy for the cases of discontinuous parameters of the medium in the Fourier law and irregular grids. An efficient algorithm for solving the discrete conjugate-operator model in the case when the thermal conductivity tensor is a single operator.
I.A. Shalimova, K.K. Sabelfeld
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, 630090, Russia
Keywords: полиномиальный хаос, метод стохастических коллокаций, стационарное уравнение Дарси, метод Монте-Карло, разложение Кархунена-Лоэва, polynomial chaos, probabilistic collocation method, Darcy equation, Monte Carlo method, Karhunen-Loeve expansion
This paper deals with the solution of a boundary value problem for the Darcy equation with a random hydraulic conductivity field. We use an approach based on the polynomial chaos expansion in the probability space of input data. We use the probabilistic collocation method to calculate the coefficients of the polynomial chaos expansion. A computational complexity of this algorithm is defined by the order of a polynomial chaos expansion and the number of terms in the Karhunen-Loève expansion. We calculate different Eulerian and Lagrangian statistical characteristics of the flow by the Monte Carlo and probabilistic collocation methods. Our calculations show a significant advantage of the probabilistic collocation method in comparison with the conventional direct Monte Carlo algorithm.
Based on a collocation technique, we introduce a unifying approach for deriving a family of multi-point numerical integrators with trigonometric coefficients for the numerical solution of periodic initial value problems. A practical 3-point numerical integrator is presented, whose coefficients are generalizations of classical linear multistep methods such that the coefficients are functions of an estimate of the angular frequency ω . The collocation technique yields a continuous method, from which the main and complementary methods are recovered and expressed as a block matrix finite difference formula which integrates a second order differential equation over non-overlapping intervals without predictors. Some properties of the numerical integrator are investigated and presented. Numerical examples are given to illustrate the accuracy of the method.
Dynamics
of a disperse phase in a swirling two-phase flow behind a sudden tube expansion
is simulated with the aid of Eulerian and full Lagrangian descriptions. The
carrier phase is described by three-dimensional Reynolds averaged Navier–Stokes
equations with consideration of inverse influence of particles on the transport
processes in gas. The velocity profiles calculated using these two
approaches are practically the same. It is shown that the main difference
between the Eulerian and Lagrangian approaches is presented by the
concentration profile of the dispersed phase. The Eulerian approach
underpredicts the value of particle concentration as compared with the
Lagrangian approach (the difference reaches 15-20 %).
The dispersed phase concentration predicted by the Lagrangian approach agrees
with the measurement data somewhat better than the data obtained through the
Eulerian approach.
The
Particle Image Velocimetry (PIV) technique and laser Doppler anemometer (LDA)
were used to measure the components of tangential and axial velocities of
gas and particles in a vortex chamber with a fluidized bed, particle layer
dynamics was estimated qualitatively, and the flow in the vortex chamber with a
centrifugal fluidized bed of solid particles was simulated numerically. It is
shown that with the growth of gas velocity in the swirler slots, the rotation
velocity of bed grows almost linearly, and with an increasing bed mass, the
rotation velocity decreases. Data on distributions of the volume fraction of
particles and gas flow velocity inside the bed were obtained by numerical
calculation.
Using planar optical methods based on laser-induced fluorescence and
particle image velocimetry instantaneous velocity fields and passive tracer
concentration are measured simultaneously in a model of GT-combustor at
realistic flow rates. Spatial distributions of velocity pulsations and passive
tracer concentration pulsations are measured at air flow rate about 0.4 kg/s.
Correlations of velocity and concentration pulsations are measured. The most
intense turbulent mass flux in the region of swirling flow mixing layer was
observed. The contribution of advective and turbulent components in the
transfer of a passive tracer in the axial direction was estimated.
The numerical simulation of
the laminar viscous flow past a cylinder performing rotary oscillations around
its axis is carried out. The Navier–Stokes equations are solved by finite
volume method using the program package OpenFOAM. The values of the amplitude
and frequency of forced oscillations are found, at which the maximum reduction of the drag coefficient of the
cylinder is achieved.
The plane nonlinear
initial boundary value problem about the separated flow past a plate set in
motion at a constant velocity from the state of rest has been considered.
Results of a numerical experiment which have allowed us to trace in detail the
vortex-wake formation process behind a vertical plate are reported. It is shown
that, after the beginning of the plate
motion, several stable vortical structures, including a Karman street, form in
succession behind the plate. It is found that, on the emergence of
the Karman street, there occurs a sharp and substantial growth of vortex-wake
intensity and hydrodynamic drag force with a pulsating time behavior. A
conclusion about the origination, in this regime, of self-sustained oscillations
of the liquid in the vicinity of the plate is drawn.
