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8Section 1
Solving the hourglass instability problem using rare mesh variation-difference schemes
Abu Dawwas Yasser, D. T. Chekmarev
Lobachevsky State University of Nizhny Novgorod
Email: 4ekm@mm.unn.ru
DOI 10.24412/cl-35065-2021-1-00-01
The hourglass instability effect is characteristic of the Wilkins explicit difference scheme [1] or similar
schemes based on two-dimensional 4-node or three-dimensional 8-node finite elements with one integration
point in the element.
The hourglass effect is absent in schemes with cells in the form of simplexes (triangles in two-dimensional
case, tetrahedrons in three-dimensional case). But they have other disadvantages. In the paper [2], a rare
mesh scheme was proposed, in which elements in the form of a tetrahedron are located one at a time in the
centers of the cells of a hexahedral grid. This scheme showed the absence "hourglass" effect and other draw-
backs with high efficiency.
This approach was further developed for solving 2D and 3D problems.
References
1. Wilkins M. Calculation of Elastic-Plastic Flow // Methods in Computational Physics, V. 3, Fundamental Methods in
Hydrodynamics, Academic Press, pp. 211-263 (1964).
2. Spirin S. V., Chekmarev Dmitry T., Zhidkov A. V. �Solving the 3D Elasticity Problems by RareMesh FEM Scheme�,
Finite Difference Methods, Theory and Applications. V. 9045 of the series Lect. Notes in Comput Science. 379�384 (2015).
About grid generation in constructions bounded by the surfaces of revolution
N. A. Artyomova1, O. V. Ushakova1,2
1N. N. Krasovskii Institute of Mathematics and Mechanics UB RAS
2Ural Federal University, Ekaterinburg
Email: uov@imm.uran.ru
DOI 10.24412/cl-35065-2021-1-00-02
For constructions bounded by the surfaces of revolution, structured grid generation technique is present-
ed. Its technology has been elaborated within the variational approach [1] for constructing optimal grids. Grid
generation has been designed for numerical solution of the differential equations modeling the vortex pro-
cesses of multicomponent hydrodynamics [2]. The description of the technology for generation of grids has
appeared in [4]. The technology [4] has started to be developed by the elaboration of the grid generation algo-
rithms for the volume of revolution which has become the basic construction. The considered volume of revo-
lution is obtained by the rotation through 180� about the axis of a generatrix consisting of straight line seg-
ments, arcs of circles and ellipses. Then the deformed volumes of revolutions are considered along with the
generalizations of the volume of revolution which represent constructions obtained by the surfaces of revolu-
tion with parallel axes of rotation. The aim of the further development of the technology is to consider more
and more complicated constructions and elaborate the technology for them. In the presentation, the current
state of the development of the technology is given. Examples of generated grids are supplied.
References
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.
