CUBATURE FORMULA FOR AN OCTAHEDRON OF THE SEVENTH ALGEBRAIC ORDER OF ACCURACY
DOI:
https://doi.org/10.32782/KNTU2618-0340/2020.3.2-2.18Keywords:
quadratic octahedron, cubature formula, algebraic order of accuracy, finite element, stiffness matrixAbstract
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.
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