Кубатурна Формула На Октаедрі

Мотайло, А. П.

doi:10.32839/2304-5809/2021-5-93-34

Cited by 1 publication

(2 citation statements)

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“…To find elements of the local stiffness matrix of an octahedron with quadratic functions, the procedures for integrating by analytical methods that are built into the Maple processor were applied. Formulas of numerical integration for FE in the form of an octahedron to the seventh algebraic order of accuracy were constructed in work [10]. These formulas reduce the time complexity of the FEM algorithm when using cells in the form of an octahedron but cannot be applied to FE of irregular geometric shape.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

“…Within a given interval, the resulting formulas have six interpolation nodes, which can be determined from (6) or (9). In addition, note that at p = 1, the systems of equations ( 5) and ( 8) have a common solution, which, when substituting into formula (7), is a cubature formula for the octahedron [10], which is accurate for the third-power algebraic polynomials.…”

Section: Determining the Conditions For Using Constructed Cubature Formulas In The Algorithmization Of Femmentioning

confidence: 99%

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

Мотайло

Tuluchenko

2021

EEJET

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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.

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Section: Literature Review and Problem Statementmentioning

confidence: 99%

Section: Determining the Conditions For Using Constructed Cubature Formulas In The Algorithmization Of Femmentioning

confidence: 99%