Aviation
and Astronautics-2009 8-th International Conference ( Section "Applied and Mathematical methods" A.A.Puntus Moscow
aviation institute (State Technical University), Russia On On The international conference
"Aviation and astronautics" is a large-scale event in which participate
annually more than 70 space-oriented organizations among which about 30 various
universities from more than 20 regions of At this conference, in the
section "Applied and mathematical methods", interesting reports were prepared
by both students and graduate students from МАI, as well as by representatives
of other organizations and high schools. Theses of some reports that have made
a big impression by the results are given in this edition. M.R.Akchurin, A.A.Egorova ( Mathematical
models for estimation methods to motivate staff. In the report the conception of personnel motivation and
its importance for modern enterprises, the classification of personnel
motivation methods and short analyses of possible problems appearing during
this methods implementation process have been reviewed. Due to high cost of implementation of the most motivation
methods, there is obvious necessary to estimate the effect of their using
before. The efficient and reliable result of the research will decrease the
enterprise exigencies. The most problem is that the criteria of the methods
valuation are weak-formalizing and, in some cases, mediately
interacting. In addition, two groups of the criteria should be marked out. One
of them is the criteria which estimated the personnel motivation methods from
an employee's point of view and another one is considered point of view of an
employer. The review of existed methods of weak-formalizing
estimation measures and specific of their usage in personnel management systems
have been done. A mathematic model of the results valuation of
personnel motivation methods usage, based on the hierarchy system and the
system of balanced characteristics is suggested. An actuality of the system
implementation by software tools and its integration into automated personnel
management systems is substantiated. Also, in the report are some suggestions how to
collect initial information for the described mathematical model for the
certain enterprise in the most effective way. M.A.Lebedev, A.I.Emelkin, A.O.Voronin ( Parametrical
researches of structure of the stream, local and average heat exchange at free
convection in inclined layers. In development of theoretical
bases of hydrodynamics and its enclosures the important place occupies studying
of elementary processes convection heat exchange on the basis of 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. The problem about free
convection in inclined layers is interest for estimation of their
heat-insulating properties. In such problems, depending on incline angle,
operates two mechanisms of thermal gravitational convection: A)
balance is possible, but is unstable (a vertical
slice, lifting-dipping stream) B)
balance is impossible (a horizontal layer, tesseral structure) In intermediate cases occurs
convection interaction between these two mechanisms, leading to essential
reorganization of a stream structure, local and average characteristics of heat
exchange. The critical angle of such reorganization depends on many parameters,
in particular, on number of Prandtl, number of Rayleigh, lengthening parameters
etc. In work on numerical modelling
of convection problems on the basis of the equations of Navier-Stokes modes of
convection mechanisms interaction, caused by loss of stability of mechanical
balance, influence of a hysteresis and change of stream structure are studied
at dynamic (dependent on time) layer turn. In work calculations continuing a
cycle of researches are executed at change of values of a critical angle, and
also effects of a hysteresis depending on physical properties (number of
Prandtl, a parity of the parties, thermal boundary conditions). 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 incline angle 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
stream structure, 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. I.F.Dmitrakov
( The analysis of efficiency of metaheuristic methods for the global extremum
search of multivariable functions. The problems of search of
unconditional and conditional global extremum of
multivariable functions are considered. For their decision it is offered to use
metaheuristic optimization methods. Detailed
algorithms of application of the Simulated Annealing method and the
Differential Evolution method are generated. 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 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 software environment of
visualization of optimizing algorithms work process, where two 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. O.V.Zoteeva, S.N.Khonina (Samara state
aerospace university named by academician S. P. Korolev,
The
modelling of astigmatic converters of laser modes for capture and manipulation
of nanoparticles The goal of this work is the
modeling of astigmatic converters for laser Gaussian modes. The theme attracts
especial attention because of unique facilities, detected by using the annular Laguerre-Gaussian modes for the manipulation of micro
objects, capture and control of single atoms and for the acceleration of
nanoparticles. The object of research play the important role in the aircraft
building, because using of nanotechnologies in aircraft building will allow us
to get advantages in the sphere of strength of aircrafts. It's worth to say, that using
of nanotechnologies permits to solve the problem of the ice formation on
constructions, to raise the safety of flying, and to reduce the fuel
consumption. Using of some Hermit-Gaussian
mode's specific properties and also properties of Fourier and Fresnel transformations allow to apply several methods to
get Laguerre-Gaussian modes from available
Hermit-Gaussian modes. Annular modes can be obtained
owing to the Hermit-Gaussian mode's properties. It is reached by using of
linear combinations of ones with defined coefficients. But it is much more easy and
efficient to design laser astigmatic converters, those allow using only one
Hermit-Gaussian mode instead of tens of these modes, required in linear
combinations. At the same time the necessary result will be reached by
corresponding modifying of the optical-scheme configuration. In the course of the science
work it was found out that astigmatic converters make it possible to carry out
the effective converting of linear distributions into vortical ones. So, we can come to the
following conclusions: - using of the multilevel
diffractions optical instruments allows to produce every mode superposition,
but there is complication to produce multilevel relief's, and also in case of
binary relief's the efficiency loss is watched; - astigmatic converters enable
to get vortical distributions from the binary distributions, and at the same
time the efficiency is near to 100%, but there are some restrictions for
parameters of the optical scheme and the laser beam. Thus every method has both
advantages and disadvantages. And in every case it becomes to choose more
optimal way to get required results. A.S.Kozhevnikov, K.A.Rybakov ( Application of the
spectral methods for the stochastic system analysis in problems of financial
mathematics. Heston model. At present two types of models are applied in the financial industry.
There are models of investment banks and hedge funds. The hypothesis of the
arbitration absence is used and fair value of an option is rated in models of
investment banks. Analysts try to create effective and more applicable models
to gain more profit. All variety of hedge funds strategies are based on
hypothesis about the predictability of some variable. Therefore now the
financiers turn to models of hedge funds more often. Such models are models of
local, undetermined, and stochastic volatility. We consider the stochastic volatility model, known as the Heston model, which generalizes the Black-Scholes model. It is used to estimate asset and option
prices. This model allows us to find distribution function close to real for
the asset price and also includes the correlation between the asset price and
its volatility. The problem of finding asset price probability density function,
expected asset price, and its dispersion as well as the option price is studied
in this work. To solve this problem (to find probability density function for asset
price) we have developed the algorithm based on the spectral form of
mathematical description. Simulated results for different variants of the asset price, volatility,
and value of the option behaviors are analyzed. D.E.Pivovarov, V.I.Polezhaev,
A.A.Puntus ( The
application of Bessel's functions for solution the Poisson equation in cylinder
with piecewise continuous boundary conditions. In this work it's considered
the Fourier's method for solution of Let's consider scalar function
of some physical quantity characterizing physical process into cylinder of
certain height and radius. Naturally it's comfortable to describe such process
with help of cylindrically coordinate system. Assume the constant zero value of
our function on the sidewall and consequently such problem is axisymmetric. That's mean the value of function is
independent of angle of section. The value of function has finite quantity
because of the problem is considered in limited spatial region. Finally we must set the
boundary conditions on bases of cylinder. We can set any type of boundary
conditions. Our function also depends of time, so we need in initial
conditions. So, the problem is solved with specified conditions. Solving D.Vasilevskiy, B.Preobrazhenskiy,
G.Spirin ( Non-stationary thermal conductivity of dispersed
materials. Calculation of the conductivity
of the dispersed materials on the base of principle of generalized conductivity
is proceeding in the framework of stationary thermal problem. On the other
hand, it is obvious, that the consideration of the non-stationary temperature
field within the unit cell is possible also in the terms of non-stationary
thermal conductivity, the asymptotic approximation of which, in the case of
great time periods, give us the true thermal conductivity. Resembling situation takes
place for the non-stationary thermal experiment, in particular for the methods
of measurements, using boundary condition of the second order. In this case,
temperature field propagates from the surface of the sample. The initial stage
of heating is sensitive to the structural irregularities of the sample and can
not be the information source of the true thermal conductivity of the sample.
The information source of the above mentioned parameter is only that
non-stationary temperature field, for which the diffusion length into the
sample will be much greater than the characteristic dimension of its
heterogeneity. In this connection, by using
the experimental investigation, based on the method of the dispersed materials,
it is necessary to know the criterion of "quasihomogeneity"
of the sample. The given criterion can result quantitatively from the
consideration of the non-stationary temperature field for the system: the unit
cell and the semi-bounded medium, adjacent to it, thermal conductivity of which
equals the effective thermal conductivity of the sample. The result of the numerical
computation of the temperature field for the two diverse plates is presented in
this work. The comparison of the results were also carried out. The criterion
of "quasihomogeneity" was determined with help of the
received information. O.N.Tretiyakova, N.N.Svetushkov ( On the modelling of heat transfer in
two-dimensional heterogeneous domains for layered composite materials with
desired physical properties. The
problem of mathematical modeling of heat transfer in heterogeneous media is
relevant in connection with the variety of existing and developing new
materials, such as: microporous glass and ceramics,
composite materials, dispersed mixture of conductive and nonconductive
materials, etc. To calculate the temperature fields in such environments,
depending on the conditions of heat is required to solve the boundary problem
of thermal conductivity and the radiative-conductive heat transfer problem, if necessary,
taking into account radiative transfer. In the article [1] with finite difference
method was solved one-dimensional non-stationary radiative-conductive
heat transfer problem in the emitting, absorbing and anisotropically
scattering medium with variable depth of the material thermophysical
properties. Article [2] describes how can be solved boundary problems of
thermal conductivity to determine the heat transfer in the unit cell. We suggested an iterative process, which allows you to
choose the effective coefficient of thermal conductivity of a heterogeneous
environment suitable for calculating the
thermal characteristics of layered systems with layers parallel or orthogonal
to the flow of heat [1, 2]. Svetushkov N.N. [3] used method, named algorithm of "integral decomposition",
which allows describing the heat transfer problem by using the coupled system
of integral equations for heat flow. The solution of algebraic equations obtained as
a result of sampling conducted by the iterative method which converges rapidly
for two-dimensional problems, and allows us to estimate the error of
calculations on the current residual. On the basis of the algorithm are established
software environment for modeling heat transfer in a geometrically complex
heterogeneous environments. This approach allows simulating heat transfer in
layered structures, in which the coefficient of thermal conductivity varies
within a single layer on an arbitrary law, thus makes it possible to simulate
more complex heterogeneous structure. References 1. Tretiyakova O.N. Analysis
of Thermo-stressed state of plates of technical glass Author's abstract of diss. on Ph.D., M: 1983. 2. Nenarokov N. J. Mathematical modeling of heat transfer
processes in the study of thermophysical properties
of substances and materials under irregular treatment. Author's
abstract of diss. on Ph.D., M: 2000. 3. Svetushkov N.N. Simulation
of nonstationary thermal processes in heterogeneous
environments. Author's abstract of diss. on Ph.D.,
M: 2009. O.N.Tretiyakova, N.N.Svetushkov ( On numerical
modelling of heat transfer processes under laser radiation emitting and
absorbing medium. Development
of technological processes of laser heat treatment and thermo-cracking require
precise selection and optimization of conditions of laser irradiation. Selection of the experimental regimes by laser
irradiation is a rather difficult task, because the processes occur in small
spatial regions and time intervals. Therefore, the task of modeling such processes
is of considerable practical interest. Mathematical modeling of heat transfer depends
on many factors, such as the speed of the beam's profile, power, pulsed or
continuous wave laser modes, etc. An important advantage of mathematical
modeling is the ability to conduct a comprehensive study of the processes of
laser interaction with matter - a series of numerical experiments on
determination of temperature fields with the controlled accuracy. The main problem of numerical modeling of laser
heating is the presence of large temperature gradients in small spatial regions
near the source of radiation. Numerical instability is particularly pronounced
in two-dimensional case, when the finite-difference numerical solution methods
do not allow getting physically adequate temperature distribution, and want to
use additional methods to ensure sustainable behavior of the solution. Well-known computational packages (eg, ANSYS) is also given, in this conditions, oscillating
solutions. In
a more general setting, for example, by laser cracking of quartz glass,
sapphire and other materials necessary to consider the distribution of laser
intensity on the depth of the material, as well as to solve the problem on its
own sources of radiation. To correctly solve the problem in emitting,
absorbing and scattering media should be a joint decision of the heat equation
and the equation of radiative transfer, i.e. coupled problem of radiative-conductive heat
transfer rocketry. In solving problems RCT should be considered the temperature dependence
of optical and thermal properties of surfaces and materials in general. Accounting terms of
radiative heat transfer can significantly change the
pattern of distribution of temperature fields, especially at high temperatures. To solve the
problem using an algorithm which allows describing the
propagation of heat through the coupled system of integral equations for heat
flow. The solution of algebraic equations obtained as a result of sampling
conducted by the iterative method which converges rapidly for two-dimensional
problems, and allows us to estimate the error of calculations on the current
residual. On the basis of the algorithm are established
software environment for simulating heat transfer in a geometrically complex
heterogeneous environments, which allows to simulate the processes of heat
transfer during laser irradiation of a large set of materials with a wide range
of thermophysical properties in various technological
processes, such as: laser alloying, laser hardening, laser welding, laser thermo-cracking. There are calculation examples. K.A.Ivanov, A.I.Fedyushkin,
A.A.Puntus ( Study of influence of vibration on boundary layer and
convective heat transfer in applied problems of hydrodynamics. In this research work in the expected conditions of weightlessness, that
is, in approximation to possible research in space, a series of
multiple-parameter numerical calculations on the simulation of convective heat
and mass transfer under the vibration effect on the melt in the case of crystal
growth by the Czochralski method are produced. Cases
of vibration on the melt both in terrestrial conditions and in conditions of
weightlessness are also considered. A mathematical model for the problem is
based on the solution of two-dimensional unsteady Navier-Stokes equations for
incompressible fluid in the Oberbeck--Boussinesq approximation. The associated effects of
thermo-gravitation and forced convection, the effects of diffusion of heat and
momentum, and also the interaction of these mechanisms of flow are taken into
account. The numeral simulation is performed by the method of control volume,
implemented in the hydrodynamic complex software FLUENT. The vibrations were
set using technology Dynamic Mesh. To speed up the computation time, parallel
computing on multiple processors was used. The existence and nature of the
quasi-stationary averaged vibration flow (AVF) under vibration effects on the
melt for a wide range of defining parameters is shown. The influence of
thermal, dynamical and geometrical parameters on the AVF is shown in zero
gravity and normal gravity. For solving the Blasius problem of flow of a
viscous incompressible fluid around the heated vibrating plate in a rectangular
computational region, Navier-Stokes equations are used with given boundary
conditions. The solving of the problem was performed using a hydrodynamic
program FLUENT at different nonuniform grids with
concentration to the surface of the plate. The grid rebuild was carried out at
the top of the computational region, where the cell size is maximal, using the
Layering method (layered exceptions and additions of grid cells). To analyze
the influence of grids, three grids were built: coarse, medium and detailed.
Later titles are Coarse, Medium and Fine denote respectively the coarse, medium
and detailed grids. All three grids showed almost the same result which
slightly differs from the theoretical value, which characterizes the practical
applicability of the data grid methods. Results of multi-parameter calculations showed that the vibration can be
a simple and effective control factor (compared, for example, weightlessness,
rotation or magnetic field), affecting the hydrodynamics, heat transfer and
kinetics of crystal growth. V.O.Kalas, P.S.Krasilnikov ( 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. Stability analysis was made by
the calculus of approximations. The eccentricity-dependent magnitude of the
trace of monodromy matrix is 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 drowned 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. Environment of this value demands
regularization for oscillation equation because this equation has a critical
point for this value of eccentricity. By regularization this
equation is solved and solution is plotted. Solution is the periodic function
with fast oscillation in the neighborhood of periodical points. For approach
unity of the eccentricity the time of oscillation in the neighborhood of
periodical points increases. Yu.O.Khan, A.I.Fedyushkin,
A.A.Puntus ( 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. V.M.Uskov (Central Aerohydrodynamic
Institute named after prof. N.E.Zhukovsky,
TsAGI, Bandwidth minimization of symmetrical sparse matrix by
means of local backward renumbering fronts. Today finite-element method
(FEM) is the widely-spread tool for solving boundary problems for partial
differential equations. FEM leads to the solution of systems of linear
equations with large symmetrical sparse matrices. Solution efficiency depends
on the matrix bandwidth that can be significantly minimized by means of
permutation of rows and columns. Cuthill-McKee (CM)
algorithm is usually used to find such permutation. The algorithm minimizes the
graph bandwidth corresponding to the matrix. The renumbering starts with one
vertex and comes through the graph as a front. The algorithm of bandwidth
minimization by means of local backward renumbering fronts has been proposed.
In the basis of the method is the suggestion that the graph bandwidth is less
when the quality of forward and backward fronts is approximately equal. The
concept front quality is introduced. It allows detecting the local front
irregularities. These irregularities can be eliminated by swapping vertex
numbers. The renumbering obtained by CM algorithm is taken as the initial one. The algorithm has been tested
for problems of finite-element analysis of large scale aviation structures and
for matrices from the collection of the T.E.Churkina (Moscow aviation
institute, state technical university, Moscow, Russia). On
Mercury attitude motion stability at 3-dimension perturbations. The attitude motion of Mercury in the central Newtonian solar
gravitational field is considered. Orientation of the planet in the orbital coordinate
system is specified by the Eulerian angles. The state of the system is defined
by It is known that the
motion of Mercury is resonant. It means that Mercury in time of two orbital revolutions makes
three full turns in absolute space around the axis passing its center of mass
(so called 3:2 resonance). Besides that it is customary
to model the motion of Mercury by plane periodical motions of a solid body in
the central Newtonian gravitational field. The second order differential
equation describing such type of motion is usually called the Beletskii equation. The boundary value problem
coincided with the described type of motion does not have an analytical
solution and is solved numerically with the help of computer. Investigated is the problem on
Mercury motion stability at both plain and spatial perturbations. Two inertial coefficients are
taken as parameters of the solving problem. Curves of 3rd and 4th order
resonances are built in the space of problem parameters in stability regions of
first approximation. For the parameters' values corresponding resonance curves The algorithm that allows
obtaining the coefficients of normal form of |
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