THE CONSTRUCTING OF BIPYRAMID’S BASIS

A. P. Motailo, A. N. Khomchenko, G. Ya. Tuluchenko

Abstract


The bipyramid for the first time is considered as a 6-knots finite element (FE) in the article. For the construction of her biquadratic base
two different approaches are used: matrix method and internal condensation method for the bipyramid’s base as a 7-knots Lagrange FE. The first approach allows to investigate the fundamentally possible amount of bases for FE, and second approach does not give such possibility, but it is more economical. It is shown that after satisfaction of the traditional requirements to the basic functions in FEM in the biquadratic basic functions of bipyramid as a 6-knots FE, which are built by means of the named before approaches, there are a different amount of the indefinite coefficients. These coefficients are used future for the giving of the special properties to the basic functions, which adapt them to the solving of a boundary problems with Laplace’s equation. The value of trace of stiffness matrix is chosen as a criterion for the prognostic evaluation of approximation properties of FE in bipyramid’s form. The minimization of the trace value of the stiffness matrix results in the construction of the same biquadratic base of the bipyramid at the both approaches. On the basis of the got base the borders of the possible deformations of the bipyramid’s geometrical form are analysed. It is first wellproven in theory, that FE exists, at using of that as a cell of finite-element mesh, the best accuracy is arrived at the deviation of FE geometrical form from a regular polyhedron, in this case from an octahedron. The critical value of the aspect ratio which provides the achievement of a minimum of trace of stiffness matrix for the bipyramid of the studied geometrical form is found. A calculable experiment the results of that confirm the theoretical prognosis for properties of bipyramid as FE is conducted. The found
regularities allow to suppose the expediency of application of bases with higher-order for FE in form bipyramid.

Keywords


finite element method, bipyramid, trace of stiffness matrix, tetrahedral-octahedral mesh.

References


Алгоритм построения трехмерной адаптированной сетки для задач аэродинамики, решаемых методом конечных элементов / [Ю. А. Крашаница, А. В. Бахир, В. А. Тараненко, Ю. С. Мащенко] // Открытые информационные и компьютерные интегрированные технологии. – 2014. – № 66. – С. 105–110. 2. Greiner G. Hierarchical tetrahedral-octahedral subdivision for volume visualization / G. Greiner, R. Grosso // The Visual Computer. – 2000. – І. 16. – Р. 357–369. 3. de Bruijn H. Numerical Method for 3D Ideal Flow [Electronic resource] / Han de Bruijn – Access mode: http://hdebruijn.soo.dto.tudelft.nl/jaar2010/octaeder.pdf. 4. Мотайло А. П. Базисы шестиузлового октаэдра [Электронный ресурс] / А. П. Мотайло. – Материалы международной научно-практической конференции «Перспективные научные исследования – 2011». Серия: Математика: Прикладная математика (17–25 февраля 2011 г.). – София, Болгария. – Режим доступа: http://www.rusnauka.com/Page_ru.htm. 5. Сегерлинд Л. Применение метода конечных элементов / Л. Сегерлинд. – М. : Мир, 1979. – 392 с. 6. Шопов П. Й. Метод конденсации для задач механики несжимаемых флюидов / П.Й. Шопов // Сердика : Българско математическо списание. – 1984. – Т. 10. – С. 198–205. 7. Зенкевич О. Метод конечных элементов в технике / О. Зенкевич. – М. : Мир, 1975. – 541 с. 8. Секулович М. Метод конечных элементов / М. Секулович. – М. : Стройиздат, 1993. – 664 с. 9. Checking the Skewness // ANSYS Icepak 12.1: User’s Guide [Electronic resource]. – Access data: http://orange.engr.ucdavis.edu/ Documentation12.1/121/ICEPAK/iceug.pdf 10. ANSYS Fluent. [Electronic resource]. – Access data: https://www.sharcnet.ca/Software/Fluent6/html/udf/node1.htm 11. Несис Е. И. Методы математической физики / Е. И. Несис. – М. : Просвещение, 1977. – 199 с. 12. Мотайло А. П. О численном решении стационарной задачи теплопроводности методом конечных элементов на решетке тетраэдрально-октаэдральной структуры / А. П. Мотайло // Научные ведомости БелГУ. Математика. Физика. – 2014. – №25(196), Вып. 37. – Белгород: «НИУ БелГУ», 2014. – С. 119–127.


GOST Style Citations






DOI: https://doi.org/10.15588/1607-3274-2016-4-4



Copyright (c) 2017 A. P. Motailo, A. N. Khomchenko, G. Ya. Tuluchenko

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Address of the journal editorial office:
Editorial office of the journal «Radio Electronics, Computer Science, Control»,
Zaporizhzhya National Technical University, 
Zhukovskiy street, 64, Zaporizhzhya, 69063, Ukraine. 
Telephone: +38-061-769-82-96 – the Editing and Publishing Department.
E-mail: rvv@zntu.edu.ua

The reference to the journal is obligatory in the cases of complete or partial use of its materials.