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5References
1. Khairullina O.B., Sidorov A.F., and Ushakova O.V. Variational methods of construction of optimal grids // Handbook
of Grid Generation. Thompson J.F., Soni B.K., and Weatherill N.P., eds. Boca Raton, London, New York, Washington, D.C.:
CRC Press, 1999. P. 36-1�36-25.
2. Anuchina N.N., Volkov V.I., Gordeychuk V.A., Es'kov N.S., Ilyutina O.S., and Kozyrev O.M. Numerical simulation of
3D multi-component vortex flows by MAH-3 code // Advances in Grid Generation. ed by Ushakova O.V. Novascience
Publishers. 2007.
3. Ushakova O.V. Criteria for hexahedral cell classification // Applied Numer. Math. 2018. V. 127, P. 18�39.
4. Anuchina A.I., Artyomova N. A., Gordeychuck V. A., and Ushakova O. V. A Technology for Grid Generation in
Volumes Bounded by the Surfaces of Revolutions // Numerical Geometry, Grid Generation and Scientific Computing,
V. A. Garanzha et al. (eds.). Lect. Notes in Computational Science and Engineering. 2019. V. 131, P. 281-292.
Smoothed particle hydrodynamics method for numerical solution of filtering problems
of three-phases fluid
V. V. Bashurov
FSUE �Russian Federal Nuclear Center � All-Russian Research Institute of Experimental Physics�, Sarov, Nizhny
Novgorod Region
Email: bashurov@mail.ru
DOI 10.24412/cl-35065-2021-1-00-99
This work is devoted to solving the problem of filtration of a mixture of water, gas and oil in a homogene-
ous porous medium. The basic equations of filtration theory [1] are converted into a special form for numerical
approximation by the smoothed particle method. A numerical difference scheme is constructed on the basis of
the smoothed particle hydrodynamics method [2]. An algorithm for setting the boundary conditions is pro-
posed and a number of isothermal one-dimensional and two-dimensional test numerical calculations of the
filtration process of a mixture of water, oil and gas are presented.
References
1. Parker J.C., Lenhard R., Kuppusami T. A parametric model for constitutive properties governing multiphase flow in
porous media. -Water Resources Research. 1987. V. 23, no. 4. p. 618�624.
2. Gingold R.A., Monaghan J.J. Smoothed particle hydrodynamics: theory and application to non-spherical stars.
Mon. Not. Roy. Astron. Soc. 1977. 375 p.
The least-squares collocation method and its applications to problems of continuum mechanics
V. A. Belyaev1
1Khristianovich Institute of Theoretical and Applied Mechanics SB RAS
Email: belyaevasily@mail.ru
DOI 10.24412/cl-35065-2021-1-00-03
The report is devoted to an application of the developed versions of the least-squares collocation (LSC)
method to solving continuum mechanic problems. The efficiency of their combination with various methods of
accelerating iterative processes is shown. Possibilities of the LSC method for solving boundary value problems
for differential equations of various orders in canonical and irregular domains, including those with singulari-
ties, are investigated [1]. Mathematical modelling and numerical simulation of composite beam bending, cal-
culation of thin plates bending, and numerical analysis of polymer fluid flows are carried out. Comparison with
the results of other authors shows the advantages of the LS� method, as well as satisfactory agreement with
experimental data in calculations.
The research was partly supported by the Ministry of Science and Higher Education of the Russian Federation within
the framework of the state assignment (project Nos. 121030500137-5 and ����-�19-119051590004-5).
References
1. Belyaev V. A. Solving a Poisson equation with singularities by the least-squares collocation method // Numer. Anal.
Appl. 2020. V. 13, N. 3, P. 207-218.
Numerical simulation of a stabilizing Poiseuille-type polymer fluid flow in the channel with elliptical
cross-section
A. M. Blokhin1,2, B. V. Semisalov1
1Novosibirsk State University
2Sobolev Institute of Mathematics SB RAS
Email: blokhin@math.nsc.ru, vibis@ngs.ru
DOI 10.24412/cl-35065-2021-1-00-04
Stabilization of the Poiseuille-type flows of an incompressible viscoelastic polymer fluid is studied using
non-linear rheological relations from [1]. Channels of elliptical and circular cross-sections are considered. In [2]
it was shown that the corresponding stationary formulation admits three different solutions. The process of
stabilization of the flow after the jump of pressure gradient in the channel was simulated using the algorithm
from [3]. The stabilized flow shows which of the three solutions of the stationary problem is implemented in
practice. Simulations in a wide range of values of the physical parameters enable us to discover the effect of
"switching" the limiting solution of the non-stationary problem from one solution of the stationary equations
to another. The scenario of this switch is discussed in detail.
The research has been done under the financial support of the Russian Science Foundation (project No. 20-11-
20036)
References
1. Altukhov Yu. A., Gusev A. S., Pyshnograi G. V. Introduction to the Mesoscopic Theory of Flowing Polymer Systems.
Barnaul: Altai State Pedagogical Academy Press, 2012 [in Russian].
2. Blokhin, A. M., Semisalov, B. V. Simulation of the Stationary Nonisothermal MHD Flows of Polymeric Fluids in
Channels with Interior Heating Elements // J. Appl. Ind. Math. 2020. V. 14. P. 222�241.
3. Semisalov B. V. Fast Nonlocal Algorithm for Solving Neumann�Dirichlet Boundary Value Problems with an Error
Control // Vychisl. Metody. Programmirovanie. 2016. V. 17, N. 4. P. 500�522 [in Russian].
The discontinuous shapeless particle method for quasi-linear transport
S. V. Bogomolov1, A. E. Kuvshinnikov2
1Lomonosov Moscow State University
2Keldysh Institute of Applied Mathematics RAS
Email: kuvsh90@yandex.ru
DOI 10.24412/cl-35065-2021-1-00-06
Simulation of gas dynamic problems deals with the appearance of discontinuities, more precisely, strong
gradients. The quality of computational methods is assessed primarily by their ability to convey this behavior
of a solution as adequately as possible. In our opinion, the discontinuous particle method [1-3] allows one to
cope with these difficulties better than alternative, traditionally more commonly used difference and finite
element methods. This is achieved because the particle method is based on the Lagrange approach, and this,
in turn, provides automatic mesh generation.
