APPLICATION OF ISOGEOMETRIC APPROXIMATION BY CUBIC SPLINES IN THE CALCULATION OF THREE-DIMENSIONAL STRUCTURES BY THE VARIATIONAL METHOD
Authors:
1.
Kazan National Research Technological University
2.
KNITU
Type:
Article
Pages:
from 159
to 163
Status:
Published
Received:
01.03.2026
Accepted:
15.04.2026
Published:
05.05.2026
Language:
Russian
Keywords:
APPROXIMATION, FUNCTION APPROXIMATION, CUBIC SPLINES, ISOGEOMETRIC APPROXIMATIONS, VARIATIONAL CALCULATION METHOD, THICK PIPE
Abstract:
This paper presents the application of isogeometric approximation by cubic splines in strength calculations using the variational method based on finite-carrier functions, in particular when calculating the maximum stresses of a thick pipe under internal pressure. It is worth noting that the very technique of constructing an isogeometric approximation by cubic splines was presented by the authors in earlier works. The main properties and advantages of this approximation are: It does not create oscillations, does not require the introduction of additional shape smoothing parameters, and features a simple algorithm for constructing an approximating function. The main purpose of this study was to improve the method of calculating three-dimensional structures by using isogeometric approximation of curves by cubic splines over sets of specified points. In addition, the practical use of isogeometric approximation has revealed some specific properties that are important for its effective application. When calculating the thick pipe, the structure was divided into two final elements. Isogeometric approximation was used to describe the boundary lines in the form of circles of one finite element. Three points on the circle and the first two derivatives on the left and right sides of the circle were selected as the data set for approximating the boundary lines. Considering that these results are in good agreement with the known theoretical formulas for two verification problems, it can be stated that the variational method for calculating three-dimensional structures of complex shape, which uses isogeometric approximation by cubic splines, can be used to calculate the strength of three-dimensional structures and demonstrates the potential for solving a wide range of engineering problems.
This paper presents the application of isogeometric approximation by cubic splines in strength calculations using the variational method based on finite-carrier functions, in particular when calculating the maximum stresses of a thick pipe under internal pressure. It is worth noting that the very technique of constructing an isogeometric approximation by cubic splines was presented by the authors in earlier works. The main properties and advantages of this approximation are: It does not create oscillations, does not require the introduction of additional shape smoothing parameters, and features a simple algorithm for constructing an approximating function. The main purpose of this study was to improve the method of calculating three-dimensional structures by using isogeometric approximation of curves by cubic splines over sets of specified points. In addition, the practical use of isogeometric approximation has revealed some specific properties that are important for its effective application. When calculating the thick pipe, the structure was divided into two final elements. Isogeometric approximation was used to describe the boundary lines in the form of circles of one finite element. Three points on the circle and the first two derivatives on the left and right sides of the circle were selected as the data set for approximating the boundary lines. Considering that these results are in good agreement with the known theoretical formulas for two verification problems, it can be stated that the variational method for calculating three-dimensional structures of complex shape, which uses isogeometric approximation by cubic splines, can be used to calculate the strength of three-dimensional structures and demonstrates the potential for solving a wide range of engineering problems.
Keywords:
APPROXIMATION, FUNCTION APPROXIMATION, CUBIC SPLINES, ISOGEOMETRIC APPROXIMATIONS, VARIATIONAL CALCULATION METHOD, THICK PIPE
APPROXIMATION, FUNCTION APPROXIMATION, CUBIC SPLINES, ISOGEOMETRIC APPROXIMATIONS, VARIATIONAL CALCULATION METHOD, THICK PIPE
