Constructing Steklov-type cubature formulas for a finite element in the shape of a bipyramid

Authors

DOI:

https://doi.org/10.15587/1729-4061.2021.238024

Keywords:

bipyramid, octahedron, stiffness matrix, cubature formula, interpolation nodes, weight coefficients

Abstract

This paper reports the construction of cubature formulas for a finite element in the form of a bipyramid, which have a second algebraic order of accuracy. The proposed formulas explicitly take into consideration the parameter of bipyramid deformation, which is important when using irregular grids. The cubature formulas were constructed by applying two schemes for the location of interpolation nodes along the polyhedron axes: symmetrical and asymmetrical. The intervals of change in the elongation (compression) parameter of a bipyramid semi-axis have been determined, within which interpolation nodes of the constructed formulas belong to the integration region, while the weight coefficients are positive, which warrants the stability of calculations based on these cubature formulas. If the deformation parameter of the bipyramid is equal to unity, then both cubature formulas hold for the octahedron and have a third algebraic order of accuracy.

The resulting formulas make it possible to find elements of the local stiffness matrix on a finite element in the form of a bipyramid. When calculating with a finite number of digits, a rounding error occurs, which has the same order for each of the two cubature formulas.

The intervals of change in the elongation (compression) parameter of the bipyramid semi-axis have been determined, which meet the requirements, which are employed in the ANSYS software package, for deviations in the volume of the bipyramid from the volume of the octahedron.

Among the constructed cubature formulas for a bipyramid, the optimal formula in terms of the accuracy of calculations has been chosen, derived from applying a symmetrical scheme of the arrangement of nodes relative to the center of the bipyramid. This formula is invariant in relation to any affinity transformations of the local bipyramid coordinate system. The constructed cubature formulas could be included in libraries of methods for approximate integration used by those software suites that implement the finite element method.

Author Biographies

Anzhelika Motailo, Kherson State Maritime Academy

PhD

Department of Natural Sciences

Galina Tuluchenko, National Technical University "Kharkiv Polytechnic Institute"

Doctor of Technical Sciences

Department of Higher Mathematics

References

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Published

2021-08-30

How to Cite

Motailo, A., & Tuluchenko, G. (2021). Constructing Steklov-type cubature formulas for a finite element in the shape of a bipyramid . Eastern-European Journal of Enterprise Technologies, 4(4(112), 40–46. https://doi.org/10.15587/1729-4061.2021.238024

Issue

Vol. 4 No. 4(112) (2021): Mathematics and Cybernetics - applied aspects

Section

Mathematics and Cybernetics - applied aspects

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Copyright (c) 2021 Anzhelika Motailo, Galina Tuluchenko

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