CUBATURE FORMULA FOR AN OCTAHEDRON OF THE SEVENTH ALGEBRAIC ORDER OF ACCURACY
DOI:
https://doi.org/10.32782/KNTU2618-0340/2020.3.2-2.18Ключові слова:
quadratic octahedron, cubature formula, algebraic order of accuracy, finite element, stiffness matrixАнотація
When solving the problems of mathematical physics by the finite element method for volume regions using lattices of a tetrahedral-octahedral structure, there is the problem of choosing a specific basis for the octahedron and the formula for numerical integration over this polyhedron. The numerical solution of the problem is the solution of a system of linear algebraic equations with coefficients that are elements of the stiffness and mass matrices. The accuracy of the solution of the boundary problem depends on the accuracy of the cubature formulas for the octahedron.
When the computational domain is discretized by the linear octahedron and tetrahedron, the problem of numerical integration over the octahedron region is partially solved. Cubature formulas are constructed for calculating the local stiffness matrix for an octahedron with piecewise linear, trigonometric and second-order polynomial bases. The cubature formula for calculating the elements of the local mass matrix is constructed for an octahedron with a trigonometric basis. Cubature formulas for an octahedron with trigonometric and second-order polynomial bases are exact for a trigonometric partial form and third-order algebraic polynomials, respectively, and contain a minimal number of interpolation nodes.
In this paper, a cubature formula for a quadratic octahedron with a fourth-order polynomial basis is constructed. This formula is exact for seventh-order algebraic polynomials and has two different sets of node coordinates and weight coefficients. An estimate of the remainder term of the cubature formula for integrand functions of the class is obtained. Theoretical results were verified by calculating the elements of the local stiffness matrix for a fourth-order polynomial system of basis functions of a quadratic octahedron. Based on the calculation results, the cubature formula optimal in accuracy is determined. The weighting coefficients of the formula are positive; one of the four groups of interpolation nodes does not belong to the region of the octahedron.
This cubature formula can be used to solve the boundary problems of mathematical physics for volume regions that are discretized by the lattice of the tetrahedral-octahedral structure.
Посилання
Grosso, R., & Greiner, G. (1998). Hierarchical Meshes for Volume Data. Computer Graphics International: International Conference (Germany, Hannover, June 22–24, 1998). Washington: IEEE Computer Society Press, pp. 761–771.
Motaylo, A. P. (2014). O chislennom reshenii statsionarnoy zadachi teploprovodnosti metodom konechnyih elementov na reshetke tetraedralno-oktaedralnoy strukturyi. Nauchnyie vedomosti BelGU. Matematika. Fizika. 25(196), 119–127.
Motailo, A. P., & Bilousova T. P. (2019), Pobudova kubaturnoi formuly dlia oktaedra. Proceedings of the Suchasni enerhetychni ustanovky na transporti, tekhnolohii ta obladnannia dlia yikh obsluhovuvannia: 10-ta mizhnarodna naukovo-praktychna konferentsiia (Kherson, September 12−13, 2019). Kherson: KDMA, pp. 277–280.
Motailo, A. P., & Aleksenko V. L. (2019). Kubaturna formula po oktaedru dlia tryhonometrychnoho polinoma okremoho vydu. Proceedings of the Perspektyvni napriamky suchasnoi elektroniky, informatsiinykh i kompiuternykh system:
IV vseukrainska naukovo-praktychna konferentsiia (Dnipro, November 27−29, 2019). Dnipro: DNU, pp. 58−60.
Motailo, A. P. (2019). Pobudova harmonichnoho bazysu kvadratychnoho oktaedra. Proceedings of the Suchasni tekhnolohii promyslovoho kompleksu: V Mizhnarodnа naukovo-praktychnа konferentsiiа (Kherson, September 10−15, 2019). Kherson: KNTU, pp. 178−180.
Myisovskih, I. P. (1964). O postroenii kubaturnyih formul dlya prosteyshih oblastey. Zhurnal vyichislitelnoy matematiki i matematicheskoy fiziki. 4, 1, 3–14.
Kryilov, V. I. (1967). Priblizhennoe vyichislenie integralov. Moskva: Nauka.
Popov, A. S. (2015). Kubaturnyie formulyi na sfere, invariantnyie otnositelno gruppyi vrascheniy diedra s inversiey D4h. Sibirskie elektronnyie matematicheskie izvestiya. 12, 457–464. DOI: 10.17377/semi.2015.12.039
Kalitkin, N. N. (2011). Chislennyie metodyi: ucheb. posobie. SPb: BHV-Peterburg.
