Innovations
in aviation and astronautics-2010 Scientific
practical Conference of young scientists and students of the Moscow Aviation
Institute (April 26-30, 2010, Section
«Applied mathematics and physics» A.A.Puntus Moscow
Aviation Institute ( The scientific practical conference of young
scientists, graduate students and students of the Moscow Aviation Institute,
“Innovations in aviation and astronautics On November 26, 2009, by the order of the Ministry of Education and
Science of the As a result
of the conference on April 26-30, the authors of the best reports are
recommended to participate in the second International Scientific and Practical
Conference “Scientific-technical creativity of youth is a way to a
society based on knowledge.” This conference
will be held on June 29 - July 2, So, this
publication contains the theses of reports of students and young specialists
presented in the section "Applied Mathematics and Physics" at the
conference “Innovations in aviation and astronautics Member of the
Conference Organizing Committee, the Chair of
the Section “Applied mathematics and physics” Professor
A.A.Puntus A.I.Emelkin, V.I.Polezhaev, A.A.Puntus. (Moscow Aviation Institute, Numerical
modeling and research of structure of the current, local and average heat
exchange at free convection in inclined layers. Rapid development of numerical
methods and constant increase in capacity of computers give more and more ample
opportunities for numerical research of problems of mechanics of a liquid and
gas. The great value has computing experiment. Computing experiment is
intermediate between natural experiment and analytical research. Researches
of last years are concentrated to the decision of convection problems at
various orientation of a layer with various complicating factors, for example,
internal sources of heat, and also radiation, vibrations, a magnetic field,
porous and abnormal environments, change of an angle of slope in time etc. But,
in spite of the fact that last decade on a first line works on experimental and
theoretical research of spatial convection effects in inclined layers are put
forward, complexity of structure of a current, possibility of management and
requirement of new technologies force to come back to insufficiently studied
question on two-dimensional effects of convection interactions in inclined
layers. The most general mathematical
model for the description of currents of a continuous viscous liquid are the equations
of Navier-Stokes, which in itself are difficult enough and differ nonlinearity,
non-stationary, multiscale and presence of a great number of defining
parameters, strongly depend on initial and boundary conditions. In this work modes of
interaction of convection mechanisms, caused by loss of stability of mechanical
balance (having warmed up from below, lift-lowering current) and absence of
balance (having warmed up sideways, cell structure) are studied. In intermediate cases which
are observed at change of an angle of slope, occurs convection interaction
between these two mechanisms, leading to essential reorganization of structure
of a current, local and average
characteristics of heat exchange. Development of convection research includes
studying of following questions: definition of critical conditions of loss of
stability, the analysis of structure stationary convection currents and its
influence on characteristics of heat-weight transfer. To studying of a critical
angle of such reorganization depending on parameters of lengthening, also this
work (including trivial, but from it not less interesting case of square area)
is devoted. All data is received during
computing experiment, i.e. a method of studying of devices or physical
processes by means of mathematical modeling. He assumes, what after
construction of mathematical model its numerical research allowing is carried
out to simulate behavior of investigated object in various conditions or in
different variants. Calculations were spent with the help of researching
program COMGA (Cоnvection in Micro
Gravity and Applications), developed at Institute of Problems of Mechanics of
the Russian Academy of Sciences still in the late eighties, whence follows that
the problem of creation of new programs is actual. A.O.Simonenko. (Moscow Aviation Institute, The
decision of the nonlinear equations by means of the modified The interval and interval
uncertainty are initial concepts of the interval analysis. Interval uncertainty
is a condition of incomplete knowledge of size for which the accessory to some
interval, that is the closed numerical interval is known only. The mathematical
discipline studying problems with interval неопределённостями and methods of
their decision, is called as the interval analysis. In the present work the
problem of search of roots of the nonlinear equation on the set initial
interval is considered. One of methods of the decision
of the given problem is The work purpose is creation
of the algorithm, allowing to find all roots of the equation on any set
interval (including on an interval containing zero or infinity). As a basis the algorithm of an
interval method of Further it has been generated
two algorithms of the modified interval method of The software realizing the
above described algorithms in which on base examples it is possible to check up
method work is created. As a base example the nonlinear equation in which left
part there is a polynomial which degree is set by the user is taken. Also
entrance parameters are polynomial factors, an initial interval and accuracy of
the decision. In the program the tree of step-by-step
results, the schedule of function is under construction and the decision is
deduced by the specified method that allows to consider a decision course
visually and most full. Efficiency of the developed software has been checked
up on set of examples. A.S.Kozhevnikov. (Moscow Aviation Institute, The
application of spectral method analysis for systems with a random period of
quantinization to models of asset price dynamics. At present the stochastic processes
with jumps have become more popular than diffusion processes for modeling
fluctuations of the market to manage risks and to rate the fair value of an
option. In particular, when the process of price dynamics includes Brownian
motion the asset price can be changed by a small value in a short period of
time, while real prices are moving jumps (have big changes in small time
intervals). Financial models with jumps
divided into two categories. The first category includes jump-diffusion models.
The price dynamics is defined by the diffusion process, which is experiencing
jumps at random times. The second category includes models with an infinite
number of jumps on a small time interval. There is no need to introduce
Brownian motion in models of the second category, because the dynamics of jumps
already rich enough to simulate a nontrivial behavior at small time interval. We consider Merton model and
Bates model in which the behavior of asset price is described a process
generated by a mixture of diffusion and discontinuous processes. It is assumed
that asset price shocks are independent and identically distributed and form a
Poisson flow of events with constant intensity, which equal the average number
of jumps per unit time. Such models can be considered as a special case of
systems with a random period of quantization. The problem of finding asset
price probability density function, expected asset price, and its dispersion as
well as the option price in models of Merton and Bates is studied in this work. To solve this problem we have
developed the algorithm based on the spectral form of mathematical description
(spectral method).The main advantage of this approach is the universality of
application and ease of implementation. The spectral method allows us to reduce
the Kolmogorov-Feller equation for probability density of asset price in Merton
and Bates models to linear algebraic equations. We offer to use the logarithm
of asset prices as the state of a system and to convert equations of the Bates
model so that the asset price (or its logarithm) and the variation will be
measured in the same scale. These offers allow to simplify the solution of the
problem by the spectral method and to get more exact results at small
truncation of spectral characteristics. Simulated results for
different variants of the behavior of asset prices are analyzed, and made
calculations to estimate the influence of the intensity of jumps and their
distribution. The calculation results for models of Merton and Bates are
compared. V.N.Panovskiy. (Moscow Aviation Institute, Formation
of modified interval arithmetic and its implementation as a complex of programs
of interval analysis. Modern computational
mathematics considers many practical problems, among which there are problems
of solving systems of equations, finding the minima and maxima of functions of
several variables and others. If for the decision of these questions we
apply the interval analysis, creation of so-called interval arithmetic will be
necessary. Interval arithmetic is an algebraic system, formalizing arithmetic
operations on intervals as single objects. After describing such a system,
we gain a possibility to use all main theorems, corollaries and algorithms of interval
analysis. We consider the following
problem: to create and describe interval arithmetic, which can be used to solve
various mathematical problems, and implement it as a complex of programs of
interval analysis. It is obvious, that except for
the definition of simple binary operations (addition, subtraction,
multiplication and division) it is necessary to identify all unary operations
(taking the logarithm, module, power, etc.). It is also important to take
into account two facts. First, while describing the new algebraic system
we should use the extended real arithmetic to avoid questions of uncertainty of
an operation. In [1] it is proposed to supply the set of real numbers with
two ambiguities (plus and minus infinity) and realize all the known operations
above this new set. Secondly, instead of the set of intervals we should
refer to their associations, i.e. multiinterval, as many of the functions of
the asymptotic behavior or even having a gap have a special domain, to describe
which multiintervals fit better. The existence of interval arithmetic of
this objects is described in [2]. In order to optimize the
execution of an operation over multiintervals we should implement the operation
of merging intervals, reducing the already existing intervals and combining
overlapping. The main result is a new
algebraic system which is built on the basis of already existing interval
arithmetic: multiinterval and arithmetic of William Kahan. There is a
program written on Visual C#. NET 3.5, which visually shows the operations
realized over multiintervals and solves the equations with several unknown
persons. 1. E.Hansen, G.William Walster. Global
optimization using interval analysis. Marcel Dekker, Inc, 2004. 2. S.P.Shariy.
Finite interval analysis. Publishing «XYZ», 2009. V.O.Kalas, P.S.Krasilnikov. (Moscow Aviation Institute, Equilibrium
position stability of the Sitnikov problem. Sitnikov problem is one of the
non-integrable problems in celestial mechanics that represents a rectilinear
motion in the restricted problem of three bodies. Two equal primary masses move
on two coplanar elliptic orbits, around
their barycentre while a third infinitesimal moves on a line perpendicular to
the motion plane of the first two masses and going throw the center of mass.
Position stability in the first approximation has been studied and it was shown
that equation in the first approximation is the linear differential equation of
second order with periodic coefficient (Hill equation). Also it was shown that
numerical investigation of stability using Lagrange series, which introduce the
eccentric anomaly as the function of the time, is invalid when the eccentricity
is “large” because any discarded term, which has more than first
power of the eccentricity, affects on the edge of the stability very much and
results are erroneous for all degrees of approximation. So that investigation
of stability using series expansion parameter is reliable when eccentricity is
sufficiently small. Stability
for eccentricity over the range zero to one was investigated by regularization
original equation set. Regularization was made because equal unity value of
eccentricity is a critical point of equation. The eccentricity-dependent
magnitude of the trace of monodromy matrix was plotted. By using certain
stability criterion (stability is the case if the magnitude of the trace of
monodromy matrix is strictly less two) it has drown a conclusion about
equilibrium position stability in the first approximation for most eccentricity
values. Instability is the case on discrete set of eccentricity values, which
satisfy the equality two for magnitude of monodromy matrix. Equal unity value
of the eccentricity is an accumulation point for this set. The first ten
eccentricity values of instability were calculated. Research of stability for
nonlinear approximation is needed serious consideration. G.S.Filippov,
V.S.Nikolaenko. (Moscow Aviation Institute, The mathematical
modeling radiation process of the complicated radiator radiation process by
Monte–Carlo method. In addition to the heat radiation of exhaust gas
stream influence, the indicatrix radiation which goes from airship to the conventional back half-sphere, is also influenced by
the radiation from the inner surface of the motor aggregate. We tale up the compound radiator that is a
system of geometric adjustable surfaces kind of cone, cylinder, hyperbola.
Inside arranged the radiant surfaces in the form of adjustable ellipsoid, cone, ring.
The inner parts are the source of radiation. Accordingly the heat
radiation of the heated components has the complicated form, taking
into account the difference of temperatures, reflection coefficients,
complicated geometry. Moreover, the radiation inside the surface can be
repeatedly reflected. To make the calculation of such a
complicated radiator, there are several methods, for example the method of
directing cosine. But the calculation algorithms in this case, can be rather
comprehensive, recursive. Realization of this algorithm on computer, even on a
very powerful one, takes too much time (hundreds hours). The “ To make the calculation a complicated mathematical
model of the radiation
surface was built. The method main point is that for the
microelement of the radiation
surface, direction of the ray, going from the centre of that
microelement, is given accidentally. It is considered, that all the radiation
energy of the microelement or the cell is spread in the mentioned accidental
direction. After that we examine the ray expansion, before it goes out of the
nozzle or crosses other surfaces. When the ray crosses other surface,
the coordinate of the cross point is calculated and the new accidental
direction of the reflected ray is defined (as the mirror reflection is
missing). The energy of the reflected ray falls depending on the surface
reflection coefficient. When the ray goes out of the nozzle we define the
coordinate of it’s cross point with the conventional half-sphere, parted in
areas. The radiant energy stream, falling on the mentioned half-sphere parts,
is kept and sum up, in case the rays fall in one and the same part. In such a
way the spatial
distribution of the radiation or radiation indicatrix is defined. The algorithm uses the probabilistic
approach, that approximates the simulated process of emitting and reflecting
the ray to the real physical process. This method is precise enough and saves
the machine time. The developed algorithm goes with any other complicated
radiator, and can be also improved for adding other factors’ investment. For the derived result analysis the
comparison with other methods’ calculation (directing cosine) was made.
There was also made a comparison with the results, got by the ANSYS program.
The analysis confirmed that the correct method in whole, and algorithm in
particular were chosen. Besides, the calculation of the
radiation of gas flow
by going out of the complicated radiator surface was made. At
that we made the calculation of the radiation on the collecting sphere in
different radiation conditions. We made the calculation of the
radiation of gas flow by
going out of the complicated radiator surface, on conditions that
a protective screen made of special sprayed material is
put around it. We also made the calculation of this method effectiveness and
practicability. D.V.Metlitskaya, A.V.Panteleev. (Moscow Aviation Institute, Creation of a complex of software «Genetic
algorithms of conditional optimization with binary and real coding». This work presents
the genetic algorithms (GA) which are representatives of evolutionary methods
of search. Genetic algorithms are based on modeling of processes of natural
evolution. One of the most popular areas of the application of genetic
algorithms is optimization of multiple parameter functions. As well as other
methods of evolutionary calculations, genetic algorithms do not guarantee
detection of the global optimum, but successfully work, when it is required to
find "good enough" result for comprehensible time. Genetic algorithms
can be applied, when the information on character and properties of
investigated function almost completely is absent. There are two
groups of genetic algorithms: genetic algorithms with binary coding [1-3] and
genetic algorithms with real coding [3-5]. The first group uses binary alphabet
for coding points on the set of possible solutions. The second group has
resulted from refusal of idea of coding; the solution in a chromosome is
represented in the form of a set of real numbers. The objective
function defined on set of possible solutions is considered in this work. It is
required to find a global conditional maximum of function on this set. On the
basis of the genetic algorithms was generated the program of search of a global
conditional maximum. A working environment is Microsoft Visual Studio 2005,
programming language is C #. The program works in a mode of dialogue with the
user. The user interface includes the basic window with forms for data input
and display of received results, the push-button panel for management of a
course of computing process (it is possible to receive the solution at once or
on steps). Program work begins with input of the initial data: type of
optimized function, set of value of variables, and also from input of
parameters of algorithm: population characteristics (the size, a maximum
quantity of chromosomes (individuals), length of bit lines of genes), type of
operators of selection, crossing, a mutation, and quantity of elite individuals
(if elite strategy is applied). During work with the program it is possible to
receive a population graphic representation, and also the schedule of change of
the greatest value of objective function at transition from one population to
another. Also it is possible to keep a report of the program and to keep the
received results in memory of the computer. On test examples
(Rosenbrock saddle and sphere function) have been considered efficiency of
algorithm. The analysis of the received results shows that the generated
algorithms allow finding the comprehensible solution when function has
difficult structure, and the decision of a problem of search of an optimum of
sphere function does not cause difficulties. 1. J.N.Holland.
Adaptation in Natural and Artificial Systems. 2.
D.Goldberg. Genetic Algorithms in Search, Optimization and Machine learning. Addison-Wesley,
1989. 5. Z.Michalewicz.
Genetic algorithms, Numerical optimization and constraints// Proceedings of the
6th International conference on genetic algorithms, 151–158, 1995. D.E.Pivovarov,
V.I.Polezhaev, A.A.Puntus. (Moscow Aviation Institute, Computational
solution of third system of Navier–Stokes equations in Boussinesq
approximation for convectional heat transfer problem. Last time straight
computational solution of Navier–Stokes equations is considered powerful
and reliable tool for investigation of turbulence flows. The computational
results of numerous examples correspond with experimental results. It impel us
to apply this method for simulation convection processes of heat exchange. There are a lot of algorithms
of computational solutions today. All these ones differ in discretization
scheme, accuracy and stability. Finite-differences scheme of N.Nikitin is
interested in use of curvilinear coordinate system. In addition it uses
semi-implicit third order accurate Runge-Kutta method, local error estimation
and time-step control. This paper describes above
mentioned algorithm in application to convection heat exchange problem,
realization it on the PC and check its applicability on the model problems such
as Davis test, lifting-movable flow, Rayleigh-Benar problem, convection in
inclined layers. Tecplot presents graphical realization of computational data.
It is represented parametric investigations, maximal characteristics of heat
exchange, comparison derived data with data of other authors and exposed third
features. I.F.Dmitrakov. (Moscow Aviation Institute, The
application of metaheuristic methods for the global extremum search of
functions. The problems of search of
unconditional and conditional global extremum of multivariable functions by
means of metaheuristic optimization methods are considered. Detailed algorithms of
application of the Simulated Annealing
method, the adaptive Simulated
Annealing method and the Differential Evolution method are generated. The Differential Evolution
method is based on the analysis of evolutionary processes. The Differential
Evolution algorithms feature is use of differences between individuals
(argument of objective function). It is realized by the linear operator, named
"differentiation". The Simulated Annealing method is based on the analysis of
process of freezing of liquids or recrystallization of metals in annealing
process. Algorithms of Simulated Annealing with some probability suppose
transition in a condition with higher value of objective function in the course
of solution search, for that the point could leave a local minimum
neighborhood.
The software environment of
visualization of optimizing algorithms work process, where above-listed methods are added, is
developed. This software environment
allows to: -
easily operate with parameters of a tested method; -
analyze efficiency of its operation on test functions; -
analyze method convergence; -
"see" process of its operation; -
effectively form a policy of method parameters
choice that has crucial decisive
importance in metaheuristic
optimization methods; -
see computation intermediate outcome; -
efficiently make a comparison of the chosen method
with other global optimization methods for the following result analysis. The efficiency of the
developed software is checked up on various base examples. M.A.Zhmakin, A.O.Burmistrov. (Moscow Aviation Institute, Minimax
Identification of the kinematic model of the movement of aircraft in terms of a
priori uncertainty. In this paper we study the
problem of minimax estimation of motion parameters of an aircraft (LA) with
inaccurate given probability characteristics of observation errors. Movement
measurements of an aircraft are made by measuring system, consisting of diverse
radio instrumentation [1]. A major
disadvantage of the optimal methods of estimation is the assumption that the
probability characteristics of the model parameters of aircraft motion and
observation errors are given a priori [2]. In addition, it is usually assumed
that the model parameters relate to any particular type, which restrict the
sound application of methods of statistical estimation. This paper describes a
model of movement measuring aircraft, in which noises are partially unknown
probability characteristics. The type of the distribution vector of random
observation errors and element wise constraints imposed on the covariance matrix
of errors are considered to be given. The very covariance of measurement errors
is unknown. The paper presents formulation
and solution of the problem of minimax estimation with the use of some sufficient
conditions that allow the operator to reduce the problem of finding minimax
estimation to solve the dual problem, which is a standard problem of convex
programming. The meaning of the dual problem is to define a set of probability
characteristics of the parameters of the observation model, in which the
optimum for the selected criteria of quality assessment is the least accurate
(i.e., the distribution parameters and the noise model is the "worst"
for a given class of admissible distributions). It is shown, that the minimax
estimate the model parameters of motion coincides with the optimal in the mean
integrated squared criterion of assessment, calculated under the assumption
that the true covariance of errors is the same as the "worst". The algorithm of searching
"the worst" of the covariance matrix with the use of the software
package SeDuMi system MatLab, designed to solve special problems of convex
programming (e.g., SDP and SOCP), is shown in the report. Features of this
package helped to find the "worst" covariance matrix taking into
account the conditions of its symmetry and nonnegative definiteness. The report
provides a method of constructing the set of uncertainty for the covariance
matrix of observation errors in the form of multidimensional confidence region
with the given level of reliability. Obtained results allow to
reduce significantly the demands on the level of a priori information about the
characteristics of the observation model and to improve the reliability of the
results of estimation in terms of a priori statistical uncertainty. This
work was supported by RFBR (grant № 09-08-00369), as well as in
activities 1.1 FTP (state contract 30.09.2009 № 02.740.11.0471). 1.
Zhdanyuk B.F. Fundamentals of statistical processing of trajectory
measurements. M.: Soviet radio, 1978. 2. Malyshev
V.V., Krasilshchikov M.N., Karlov V.I. Optimization of observation and control
aircraft. M.A.Lebedev, V.I Polezhaev, A.A.Puntus. (Moscow Aviation Institute, The
hysteresis of characteristics current and heat an exchange in angle of
inclination. In a problem about convection
in inclined layers, depending on an angle of inclined, operates two mechanisms
of thermal gravitational convection: a balance is possible, depending on a
difference of temperatures between the top and bottom borders it can be steady
or not steady, and the balance is impossible. The Critical angle where
many-cells structure of current, which is characteristic for Benarovs cells
mechanism, changes with one cell, which is characteristic for vertical slice
with lateral heating, so we can’t define it unequivocally, because of
appearing of gusterisis. In this work, which continues
researches was considered a flat convection substance movement in a horizontal
layer, which was warmed up from below. At first, the angle of inclination
changes from 0 degrees to 180, and then upside-down from 180 to 0 degrees with
a step of The hysteresis is a characteristic
effect convection, when a current and heat exchange in one of directions does
not coincide with heat exchange in the opposite direction. This effect is known
and in other sciences and has practical application in the physicist,
chemistry, biology, but in convection heat exchange, despites on significant
effects, it wasn’t investigated regularly. The detailed research on the
basis of model with periodic boundary conditions is given in [4]. The work shows that both modes
lead to increasing of heating a stream in comparison with molecular,
corresponding to a plane-parallel current. The cellular mode provides much more
intensive heat over carrying, than vortical. It with higher speed of a
cross-section current in the cellular mode. As a result transitive character of
this phenomenon gusterisis type which leads close in relation to initial,
structure of convection and to the Nusselt number, which is different from
opposite rotation from the initial is accurately observed. So for air at a room
temperature this difference has made approximately 15 %, and for transformer
oil at temperature 100°С more than 20 %. Researches are executed by
means of the computer laboratory which description is given in [3]. 1.
D.Е.Pivovarov,
V.I.Polezhaev. The structures of the flow and feature of heat exchange at free
convection in inclined layers (The 17 School-seminar of young scientists and
specialists under the direction of the academician А. I. Leontyev
“The problem of gasdynamics and thermalmass exchange in aerospace
technologies”, Zhukovskij, May 25-29, 2009). 2.
V.I.Polezhaev, S.А.Nikitin.
Feature of heat exchange at freely-convective interactions in closed volumes.
Technical and technological annexes (The 17 School-seminar of young scientists and
specialists under the direction of the academician А. I. Leontyev
“The problem of gasdynamics and thermalmass exchange in aerospace
technologies”, Zhukovskij, May 25-29, 2009). O.V.Ryazantseva. (Moscow Aviation Institute, Global
optimization by continuous tight-fisted (greedy) randomized adaptive search
procedure. Global optimization problems
abound in many fields, including materials science, biology, chemistry, and
genetics, military science, electrical engineering, robotics, and
transportation science. Now at the decision of
problems of designing of space systems and space-rocket complexes metaheuristic
methods of global optimization are widely enough applied. They allow to receive
close enough to true result for comprehensible time and without considerable
simplifications of mathematical statements of problems. An example of such
method is the greedy randomized adaptive search procedure (GRASP) for global
optimization. The problem is: search of a
global extremum of functions of many variables on set of admissible decisions
where admissible values of corresponding variables are segments. The method is uses a
multi-start local search procedure, where each GRASP iteration consists of two
phases, a construction phase and a local search phase. The first phase (construction
phase) combines greediness and randomization to produce a diverse set of
good-quality starting solutions from which to start local search. Then the
received on the second phase points take as initial for the first phase and
procedure repeats. The best solution over all iterations is kept as the final
solution. On a
construction phase a line search in the directions parallel to co-ordinate axes
is made. After the line search a restricted candidate list (RCL) is formed.
Then one direction in a random way gets out of the RCL. Found in result of
search in this direction the point undertakes as initial for the decision
continuation, the used direction is excluded from the list, corresponding value
of solution coordinate is fixed. The described sequence of actions proceeds
until all coordinates of a solution vector will be fixed. On a local search phase the
received point improves by search in the directions which quantity is defined
by dimension of a problem. Search in directions depends on a step which can
decrease in the course of search. The initial step gets out depending on the
set of search. This
stochastic search method is simple to implement, is widely applicable, and does
not make use of derivative information, thus making it a well-suited approach
for solving global optimization problems. Multi-start procedure, that is
repeated the algorithm of search for different initial random points, provides
additional accuracy because total result is the best of received on all starts. Program realization of the
presented method is executed by means of Builder C ++. Its functions: -
input parameters of a method; -
choice of criterion function from the
list; -
decision the problem and conclusion of
the detailed information, including step-by-step; -
construction
graphs: level lines with the found point of an extremum, values of criterion
function on starts; evolution of the best value of criterion function following
the results of the realized starts; -
viewing of a detailed protocol of the
decision; -
granting of the help information. The program possesses the
intuitively-clear interface and allows to show efficiency of application of
considered algorithm for the chosen functions by change of parameters of a
method. The program realization of
method can be useful in the course of training as demonstration of work of global
extremum search algorithm. It is also allows to receive set of the data
necessary for the analysis of its efficiency. T.E.Churkina. (Moscow Aviation Institute, On
satellite attitude motion stability at moon type resonance. The attitude planar motion of a satellite (solid body) in the central
Newtonian solar gravitational field in an elliptical orbit is considered. Moon
type motion is under consideration (i. e. the satellite in time of one
orbital revolution makes full turn in absolute space around the axis passing
its center of mass). Investigated is the nonlinear
problem on the motion stability at both plain and spatial perturbations. Curves of third and fourth
order resonances are built in the space of problem parameters in stability
regions of first approximation. For the parameters values corresponding
resonance curves Yu.O.Khan, A.I.Fedyushkin, A.A.Puntus. (Moscow Aviation Institute, Mathematical
model of calculation of thermal convection at normal and lowered gravitation. This work is devoted to
mastering the hydrodynamic set of programs Fluent and geometry and mesh
generation software Gambit through an example of solving a test problem of
convection in a square cavity heated from one side. The purpose of the work is
to construct a mathematical model for calculating thermal convection in
dimensionless variables for different Rayleigh numbers by using the complex of
programs Fluent, designed to solve problems of hydrodynamics in dimensional
variables. This raises the need for testing the results of numerical
calculation. The problem of thermal
convection of a viscous incompressible fluid in a square closed area with
thermally insulated horizontal walls and specified boundary temperatures on
vertical walls is examined. A mathematical model of this problem is
two-dimensional Navier-Stokes equations in dimensionless variables in the Boussinesq approximation. For speed, no-slip condition is
put at all boundaries, and for temperature, the following assumptions are made:
the lower and upper boundaries are heatproof (adiabatic), the right and left
boundaries are isothermal. As initial conditions, a stationary fluid with a
linear temperature distribution is given. This problem of natural
convection in a square area was solved in three natural variables: speed,
pressure, and temperature. To simulate this convection problem, the
dimensionless equations were used, including dimensionless parameters: Prandtl number, which characterizes the measure of
similarity of the temperature and velocity fields in the flow, as well as the
Grashof number. In addition, the system of Navier-Stokes-Boussinesq equations
was used, written in the following variables: current-temperature and
vortex-function (which does not include pressure). Calculations were made using
modern hydrodynamic program Fluent. Results of calculation showed
that the most accurate of all the solutions of this problem is the reference
solution obtained by extrapolation on the grid with zero step, by using the
finite-difference method on different grids. The calculation results also
showed that for given boundary and initial conditions with increasing Rayleigh
number, the maximal value of the absolute value of velocity is shifted more to
the boundaries of the area, and the isotherms tend to adopt a horizontal
position. As a result of the research using the program Gambit, a geometric
model and a non-uniform computational grid are reproduced, and further by using
the program Fluent, parametric calculations are conducted and comparison of the
obtained results with the “standard” solution is given for
different Rayleigh numbers, which showed good accuracy of calculations. |
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