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936 Section 1
Section 1
METHODS OF COMPUTATIONAL ALGEBRA AND SOLVING MATHEMATICAL PHYSICS
EQUATIONS
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 theform 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 drawbacks
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 presented.
Its technology has been elaborated within the variational approach [1] for constructing optimal grids. Grid
generation has been designed for numerical solution ofthe differential equations modeling the vortex processes
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 algorithms
for the volume of revolution which has become the basic construction. The considered volume of revolution
is obtained by the rotation through 180� about the axis of a generatrix consisting of straight line segments,
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 revolution
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.
