Approach to the Construction of Geometric Models of Multifactor Processes and Phenomena by the Method of Multidimensional Interpolation

Konopatskiy, E. V.; Architecture, Makeyevka

doi:10.17587/prin.10.77-86

Cited by 11 publications

(8 citation statements)

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“…A method for the numerical solution of differential equations using a geometric interpolant is proposed. Moreover, it can easily be generalized to multidimensional space and therefore can be used to solve differential equations with a large number of variables, by analogy with the geometric modeling [20][21] of multifactor processes and phenomena [3][4][5]. The proposed method is considered as an example of solving the inhomogeneous heat equation using a 16-point twoparameter interpolant.…”

Section: Resultsmentioning

confidence: 99%

“…A geometrical interpolant is a parameterized geometrical object passing through predetermined points, whose coordinates correspond to the initial experimentalstatistical information, or possessing the necessary, predetermined, properties. In accordance with the geometric theory of multidimensional interpolation [3][4][5], the geometric interpolant is formed by analytically describing the tree of the geometric model. So, for a one-dimensional geometric interpolant (1parameter interpolant) the tree of the geometric model is just one line ( Fig.…”

Section: A Bit About Geometric Interpolantmentioning

confidence: 99%

“…The need to determine such curves lies in the fact that when modeling multi-factor processes for each separate problem, it is necessary to solve systems of linear algebraic equations (SLAE) in determining the desired equation. To obtain a universal approach to modeling multifactor processes [3][4][5], it was necessary to obtain such equations of arcs of algebraic curves into which you can substitute any values of the points coordinates (both fixed and variable), and immediately obtain the desired result. For this, the SLAE solution process was laid down directly at the stage of curve modeling.…”

Section: Analytical Description Of Geometric Interpolantsmentioning

confidence: 99%

See 2 more Smart Citations

About one method of numeral decision of differential equalizations in partials using geometric interpolants

Konopatskiy

Voronova

Бездитный

et al. 2020

CPT2020 the 8th International Scientific Conference on Computing in Physics and Technology Proceedings

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The article presents a new vision of the process of approximating the solution of differential equations based on the construction of geometric objects of multidimensional space incident to nodal points, called geometric interpolants, which have pre-defined differential characteristics corresponding to the original differential equation. The incidence condition for a geometric interpolant to nodal points is provided by a special way of constructing a tree of a geometric model obtained on the basis of the moving simplex method and using special arcs of algebraic curves obtained on the basis of Bernstein polynomials. A fundamental computational algorithm for solving differential equations based on geometric interpolants of multidimensional space is developed. It includes the choice and analytical description of the geometric interpolant, its coordinate-wise calculation and differentiation, the substitution of the values of the parameters of the nodal points and the solution of the system of linear algebraic equations. The proposed method is used as an example of solving the inhomogeneous heat equation with a linear Laplacian, for approximation of which a 16-point 2-parameter interpolant is used. The accuracy of the approximation was estimated using scientific visualization by superimposing the obtained surface on the surface of the reference solution obtained on the basis of the variable separation method. As a result, an almost complete coincidence of the approximation solution with the reference one was established.

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Section: Resultsmentioning

confidence: 99%

Section: A Bit About Geometric Interpolantmentioning

confidence: 99%

Section: Analytical Description Of Geometric Interpolantsmentioning

confidence: 99%

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About one method of numeral decision of differential equalizations in partials using geometric interpolants

Konopatskiy

Voronova

Бездитный

et al. 2020

CPT2020 the 8th International Scientific Conference on Computing in Physics and Technology Proceedings

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show abstract

“…In geometric modeling of multifactor processes and phenomena [9][10][11], a curved line, as a one-parameter set, serves as an analytical description of one-factor processes and phenomena. It should be noted that a straight line is a special case of a curve of a line with zero curvature.…”

Section: Features Of Visualization Of Flat Linesmentioning

confidence: 99%

“…Initially, the BN-calculus was created as a special mathematical apparatus for engineering calculations related to modeling curves and surfaces of any shape in accordance with predetermined requirements and in the required parameterization. However, in the process of its development, in addition to shaping geometric objects, it found wide application in the field of modeling and optimization of multifactor processes and phenomena using multidimensional interpolation and approximation in their geometric interpretation [9][10][11]. The basic element of the BN-calculus is a point, and all geometric objects are defined as an organized set of points, for the analytical description of which the invariant properties of the parameter with respect to parallel projection are used.…”

Section: Introductionmentioning

confidence: 99%

Features visualization of geometric objects in the BN-calculus

Konopatskiy¹,

Бездитный²,

Kokareva³

et al. 2020

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The article discusses an approach to imaging of geometric objects based on point equations of their formation. Spot equations that are represented in symbolic form, are reduced to a system of parametric equations using coordinate-wise calculation. The number of equations of the system depends on the dimension of the space in which geometric object is considered. The basic idea is that one part of this system of equations is used for spatial imaging of the object, and the other part-for color imaging. Thus, the combined use of spatial and color imaging allows to reduce the dimension of the space, involved in spatial visualization, and it becomes possible to visualize the additional properties of the geometric object. A distinctive feature of the proposed approach is the use of a continuous non-linear color encoding information by using a continuous function to expand the imaging capabilities of geometric multidimensional space objects. The work provides 6 examples of the practical use of the proposed approach for the visualization of one-parameter and two-parameter geometric objects. At the same time, the possibilities of visualizing one-parameter objects by the example of arcs of algebraic curves belonging to spaces of different dimensions and two-parameter ones by the example of visualizing a portion of a topographic surface are investigated. The prospect of further research is to summarize the proposed approach for visualizing three-parameter geometric objects and bodies, as well as the reconstruction of three-dimensional geometric objects based on color images.

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The use of interpolation methods for modelling multifactor processes based on an experiment planning matrix

Konopatskiy

Seleznev

Бездитный

2022

J. Phys.: Conf. Ser.

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This article is dedicated to the development of experimental data processing tools based on the geometric theory of multidimensional interpolation. At the same time, a scientific problem was solved, which is dedicated to the analytical determination of models of multifactor processes while maintaining the existing approach to experiment planning This makes it possible not only to use interpolation methods for the mathematical description of new experiments, but also to use the experimental data obtained earlier to improve their mathematical interpretation. The paper provides two examples that confirm the effectiveness of the proposed approach to constructing models of multivariate processes using multivariate interpolation methods. The first of the above examples contains two geometric models of the stress-strain state of metal polyhedral bent struts, which are presented in the form of ruled response hypersurfaces passing through 8 predetermined points in the 4-dimensional space. The second one is dedicated to the construction of a 2-factor process process using a 2- parameter parabolic geometric interpolant with subsequent optimization by methods of mathematical analysis of a function of two variables. As a result, the optimization of the aerated concrete manufacturing process was carried out to achieve the maximum values of the ultimate strength in compression after heat and moisture treatment.

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Approach to the Construction of Geometric Models of Multifactor Processes and Phenomena by the Method of Multidimensional Interpolation

Cited by 11 publications

References 0 publications

About one method of numeral decision of differential equalizations in partials using geometric interpolants

About one method of numeral decision of differential equalizations in partials using geometric interpolants

Features visualization of geometric objects in the BN-calculus

The use of interpolation methods for modelling multifactor processes based on an experiment planning matrix

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