=Paper=
{{Paper
|id=Vol-2744/short31
|storemode=property
|title=Modeling Geometric Varieties with Given Differential Characteristics and Its Application (short paper)
|pdfUrl=https://ceur-ws.org/Vol-2744/short31.pdf
|volume=Vol-2744
|authors=Evgeniy Konopatskiy,Andrey Bezditnyi,Oksana Shevchuk
}}
==Modeling Geometric Varieties with Given Differential Characteristics and Its Application (short paper)==
https://ceur-ws.org/Vol-2744/short31.pdf
Modeling Geometric Varieties with Given Differential
Characteristics and Its Application1
[0000-0003-4798-7458] [0000-0003-0528-9731]
Evgeniy Konopatskiy1 and Andrey Bezditnyi2 and
[0000-0002-9224-0671]
Oksana Shevchuk1
1 Donbas National Academy of Civil Engineering and Architecture, Ukraine
2 Sevastopol branch of ยซPlekhanov Russian University of Economicsยป, Russian Federation
e.v.konopatskiy@mail.ru, pereverten_1985@mail.ru,
o.a.shevchuk@donnasa.ru
Abstract. The article describes an approach to modeling geometric manifolds
with specified differential properties, which is based on the use of geometric in-
terpolants of a multidimensional space. A geometric interpolant is understood as
a geometric object of a multidimensional space passing through predetermined
points in advance, the coordinates of which correspond to the initial experimental
and statistical information. Principles for determining geometric interpolants and
an example of an analytical description of a 3-parameter geometric interpolant
belonging to a 4-dimensional space in the form of a geometric scheme and a
computational algorithm based on a sequence of point equations are given. The
main direction of practical use of geometric interpolants is geometric modeling
of multifactor processes and phenomena, but they can also be an effective tool
for multivariate approximation. Based on this, the article presents a general ap-
proach to modeling geometric manifolds with given differential properties and
its application in the form of a numerical solution of differential equations by
approximating it using geometric interpolants of a multidimensional space. To
implement an approach to modeling geometric manifolds with specified differ-
ential properties, it is proposed to use a computational algorithm consisting of 10
points. The advantages of using geometric interpolants for the numerical solution
of differential equations are highlighted.
Keywords: Geometric Varieties, Differential Characteristics, Geometrical In-
terpolant.
1 Introduction
Differential characteristics of geometric manifolds have important theoretical and ap-
plied significance. The need to study them in parallel with the development of mathe-
matical analysis led to the emergence of differential geometry as a separate branch of
mathematics.
Copyright ยฉ 2020 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
2 E. Konopatskiy, A. Bezditnyi, O. Shevchuk
In differential geometry, the geometric meaning of differentiation and integration
operations is considered. So, the geometric meaning of the first derivative of the func-
tion is reduced to determining the tangent to the curve, and the second derivative โ to
determining the curvature of the curve. The study of differential characteristics often
led to the creation of new geometric objects based on their prototypes. For example, an
evolute, that has found wide application in engineering practice, is a geometric place of
points (a curve) that are the centers of curvature of the original curve. Another example
of using differential characteristics of curved lines is the work [1-3], in which the author
proposed the use of functional curves for geometric modeling of physical processes in
the form of a set of functionally interconnected lines of trajectory, speed and accelera-
tion. This approach is valid for defining other geometric manifolds: compartments of
surfaces, hypersurfaces, etc., which leads to differentiation of the function of many var-
iables. For example, the point equation of the torso surface as a geometric place of
tangents to its return edge is nothing more than a differential equation of a function of
two variables. If we consider the analytical description of geometric manifolds obtained
by differential-geometric transformations, then in their structure they are nothing more
than a certain differential equation and are its geometric equivalent. Then, between dif-
ferential equations and geometric manifolds with given differential characteristics,
there is a mutually inverse logical relationship. That is, geometrical manifolds with
given differential characteristics can serve as a particular solution of differential equa-
tions in the same way as differential equations are used to analytically describe geo-
metric manifolds with given differential characteristics.
Differential equations can have both exact and approximate (numerical) solutions.
In engineering practice, there are often differential equations whose solution cannot be
expressed in elementary mathematical functions. Therefore, numerical methods based
on multidimensional interpolation and approximation are used for their particular solu-
tion. An example is the iso-geometric method for solving partial differential equations
[4-5], which is based on the concept that the same set of functions is used both for an
analytical description of an approximating geometric object and for approximating a
solution. In the works of Sophus Li [6-7], a detailed study of point groups of plane
transformations is proposed, whereas differential equations are found not as the main
object of research, but as an auxiliary apparatus. Another approach based on modeling
geometric manifolds with pre-defined properties is proposed in [8-10]. The main con-
cept of the proposed approach to the numerical solution of differential equations [8-10]
is to use geometric interpolants of a multidimensional space [11] to model geometric
manifolds with specified differential characteristics.
2 Principles for determining geometric interpolants in a
multidimensional space
In General, a geometric interpolant is a geometric object of a multidimensional affine
space that passes through pre-defined points whose coordinates correspond to the orig-
inal experimental and statistical information [11]. The process of determining geomet-
ric interpolants consists of two principal stages: building a tree of a geometric model
Modeling Geometric Varieties with Given Differential Characteristics and Its Application 3
and its analytical description using point equations [12] and computational algorithms
based on them. The tree of a geometric model is a geometric scheme for the graphical
construction of a geometric interpolant based on the method of a moving simplex [12]
using as formative elements algebraic curves passing through pre-defined points ob-
tained on the basis of Bernstein polynomials [13].
1 the principle. From simple to complex. Any, even the most complex geometric
interpolant, is represented by a set of simpler geometric interpolants, united with each
other by a generating line. For example, a geometric interpolant of 2-dimensional space,
which is a section of the response surface, can be represented as a set of 1-dimensional
geometric interpolants (reference lines passing through pre-defined points), connected
to each other using a generating line. The more we set the current parameters that de-
termine the geometric interpolant, the more complicated will be the structure of the tree
of the geometric model and, accordingly, the computational algorithm for constructing
it. However, it still consists of simpler interconnected elements.
2 the principle. Belonging of one geometric object to another. The geometric the-
ory of multidimensional interpolation [11] is based on the simplest principle of belong-
ing of one geometric object to another accepted in descriptive geometry, according to
which a straight line belongs to a plane if two points of this line belong to a plane. In
turn, a point belongs to a plane if it belongs to a straight line lying in that plane. Con-
sidering a straight line as a special case of a curve, and a plane as a special case of a
surface, we obtain the statement, that is necessary for defining a 2-parameter geometric
interpolant in the form of a compartment of a surface, that passes through predeter-
mined points and belongs to 3-dimensional space.
Statement 1. In order for the source points to belong to a 2-parameter geometric
interpolant, you must organize them as reference lines (in descriptive geometry, the
term guide lines is used), and then combine them using a generating line that passes
through the current points of the reference line family.
Then all points, that belong to the reference lines will belong to the desired 2-param-
eter interpolant. It should be noted that the formative element of both the reference and
forming lines of the geometric interpolant are arcs of algebraic curves passing through
pre-set points. Only when forming reference lines, the curves pass through the source
points (whose coordinates correspond to the original experimental and statistical infor-
mation), and when forming the generating line-through the current points of reference
lines of the geometric interpolant.
Generalizing the proposed approach to defining geometric interpolants on a 4-di-
mensional space, we obtain a statement akin to statement 1.
Statement 2. In order for the source points to belong to a 3-parameter geometric
interpolant, you must organize the source points as 2-parameter geometric interpolants
(a family of reference surfaces), united in a hypersurface by a forming line passing
through the current points of the family of reference surfaces.
Similarly, you can define an n-parametric geometric interpolant, that belongs to an
(n+1)-dimensional space.
As an example, consider the tree of a geometric model of a 3-parameter geometric
interpolant (Fig. 1).
4 E. Konopatskiy, A. Bezditnyi, O. Shevchuk
Fig. 1. Tree of the geometric model of the 3-parametric geometric interpolant
An analytical description of a 3-parametric geometric interpolant in the form of a 3-
parametric hypersurface is represented by the following sequence of point equations:
๐
๐๐๐ = โ ๐๐๐๐ ๐๐๐๐ (๐ข)
๐=1
...............................
๐
๐๐ = โ ๐๐๐ ๐๐๐ (๐ฃ) ,
๐=1
...............................
๐
๐ = โ ๐๐ ๐๐ (๐ค)
[ ๐=1
where M โ the current point of the hypersurface compartment that passes through the
pre-set points in the number of mา nา l ;
M i โ current point of the generator i a section of the surface that passes
through pre-defined points, which is a reference for building a hypersurface;
M ij โ current point j the reference arc of a curve that passes through pre-
set points;
M ijk โ the starting points through which the desired hypersurface compart-
ment should pass. The coordinates of these points correspond to the original
experimental and statistical data;
pijk (u ) โ functions from the parameter that define the type of reference lines;
qij (v ) โ functions from the parameter v , defining the type of forming lines
compartments surfaces;
ri (w) โ functions from the parameter w , defining the form of the hypersur-
face compartment forming line; l โ number of source points in each reference
line;
n โ the number of control lines to build the reference surfaces;
m โ number of reference surfaces for building a hypersurface;
i โ serial number of the reference surface;
Modeling Geometric Varieties with Given Differential Characteristics and Its Application 5
j โ sequential number of the reference line;
k โ the sequential number of the starting point in each reference line;
u , v and w โ current parameters that change from 0 to 1.
To move from point equations to parametric ones, you need to perform a coordinate
calculation, which is the analytical equivalent of projecting a 3-parameter hypersurface
on the axis of the global coordinate system.
The main direction of practical use of the geometric theory of multidimensional in-
terpolation is geometric modeling of multi-factor processes and phenomena [14]. In
addition, it can be effectively used in geometric computer modeling as a forming tool
for modeling geometric shapes and bodies passing through pre-set points. However,
another effective application of the geometric theory of multidimensional interpolation
is possible. In particular, in [8-10, 15-16], 3 variants of using geometric interpolants for
multidimensional approximation are proposed and studied:
โ by selecting nodal points from the original set of points [15].
โ by minimizing the sum of lengths between nodes and source points [16-17].
โ by setting the required differential characteristics at the nodal points [8-10].
The second variant [16-17] is presented as a result of generalization of the least
squares method to a multidimensional space using its geometric interpretation. In [8-
10], an additional field of application of the third variant of multidimensional approxi-
mation is proposed: numerical solution of differential equations using geometric inter-
polants, which is based on the modeling of geometric manifolds with specified differ-
ential characteristics.
3 General approach to modeling geometric manifolds with
specified differential characteristics
The main idea of the proposed approach to modeling geometric manifolds with given
differential properties is as follows: in multidimensional nodes interpolation network
of points the condition of the original differential equation is satisfied, which provides
the required differential properties of the simulated geometric object. Multidimensional
interpolation is performed automatically between nodes in the point network by using
geometric interpolants. This is the numerical solution of the original differential equa-
tion.
To implement an approach to modeling geometric manifolds with specified differ-
ential characteristics, we suggest using the following algorithm:
1. We should to form a network of points of the necessary dimension and density
depending on original differential equation. We recommend using a regular point
network, but if necessary, you can also use an irregular point network.
2. Structure the generated network of points to build a tree of the geometric model.
3. Use formative arcs of algebraic curves passing through the front set points, for the
analytical description of the geometric interpolant. In this case, we form a compu-
tational algorithm equivalent to the tree of the geometric model.
6 E. Konopatskiy, A. Bezditnyi, O. Shevchuk
4. Perform a coordinate calculation and, if using a regular network of points, go from
the system of parametric equations for the analytical description of the geometric
interpolant to its explicit equation. If an irregular network of points is used, addi-
tional differentiation of parametric equations by parameters is necessary, followed
by solving a homogeneous system of first-degree differential equations by the Kra-
mer method [18].
5. Enter the coordinates of the points corresponding to the initial and boundary con-
ditions of the original differential equation. Geometric representation of initial and
boundary conditions can be geometric objects: points, lines, surfaces, etc., that pass
through pre-defined points.
6. Differentiate the resulting equations and substitute them into the original differen-
tial equation. Thus, we obtain an equation of an approximating geometric inter-
polant with pre-defined differential characteristics.
7. Substitute parameter values at the nodal points, thereby forming local system of
linear algebraic equations (SLAE).
8. In the case of using piecewise polynomial approximation, repeat the first 7 points
of the computational algorithm several times. With this, we accumulate local
SLAEs to form a global SLAE. In the case of using a continuous geometric inter-
polant, this point of the algorithm must be ignored and go to the next.
9. Solve the resulting SLAE and determine the necessary parameter values at the nodal
points of the geometric interpolant. Then insert the calculation result into the equa-
tion of the approximating geometric interpolant from the 6th point.
10. Analyze the result and check its validity. If the results are not accurate enough,
increase the number of nodal points in the geometric interpolant.
It should be noted that the order of the initial differential equation is directly related
to the order of the shape-forming curves passing through the predetermined points of
the geometric interpolant. Moreover, to ensure the correct result, it is necessary that
the order of the forming curves passing through the predetermined points is higher than
the order of the original differential equation.
4 Practical application of the proposed approach
The main practical application of the proposed approach to modeling geometric mani-
folds with specified differential characteristics is the numerical solution of differential
equations in computer-aided design and calculation systems by approximating their so-
lutions using geometric interpolants of a multidimensional space. At the same time, it
should be noted that the proposed approach is sufficiently universal and that it is pos-
sible to increase both the number of variables of the original differential equation and
its order. Moreover, if the solution can be represented with sufficient accuracy as a
smooth function, it is preferable to use continuous geometric interpolants. In other
cases, it is possible to use piecewise polynomial functions in the form of multidimen-
sional contours using algebraic curves of different smoothness order.
The proposed approach can be effectively used primarily for the numerical solution
of partial differential equations for which the solution is not expressed in elementary
Modeling Geometric Varieties with Given Differential Characteristics and Its Application 7
functions. In [9], an example of a numerical solution of an inhomogeneous heat equa-
tion using a 16-point geometric interpolant is given, and the results of the numerical
solution are compared with the reference solution obtained by the method of separating
variables. At the same time, further use of the resulting numerical solution in the form
of a polynomial dependence is more preferable for engineering calculations, in com-
parison with the equation obtained by the method of dividing variables.
It should be noted, that by covering a multidimensional network of source points
with a single geometric interpolant, a finite super element for approximating differen-
tial equations is formed, which has the following advantages:
1. Curved geometric objects of a multidimensional affine space are simultaneous car-
riers of several points, which significantly reduces the "piecewise" of the final func-
tion and the number of necessary calculations.
2. With regard to solving problems of strength and stability, the final super element can
be used on any network of points, while maintaining a curved component, which
ensures that geometric nonlinearity is taken into account when solving problems of
strength and stability. At the same time, the possibility of geometric modeling of
physical, structural, genetic and other non-linearity remains.
3. Due to the invariant properties of the parameter of point equations, there is no need
to use guide cosines.
4. Using the geometric theory of multidimensional interpolation, it is possible to gen-
eralize the approach to modeling geometric manifolds with given differential char-
acteristics both in the direction of increasing the number of variables and in the di-
rection of increasing the order of derivatives of the original differential equation.
Moreover, the number of variables is limited only by the dimension of the space, and
the order of derivatives is limited by the order of forming curves of the geometric
interpolant.
5 Conclusion
In conclusion, we would like to note the need for further research and development of
the proposed approach to modeling geometric manifolds with specified differential
characteristics, which can take its place in the number of innovative numerical methods
for solving differential equations. The prospect of further research is to conduct further
computational experiments to test the accuracy and speed of the proposed approach to
modeling geometric manifolds with specified differential characteristics and its practi-
cal application to the numerical solution of partial differential equations. It is assumed,
that the use of a geometric interpolant as an analog of a finite super element, will allow
us to obtain a numerical solution of partial differential equations and their systems of
the required accuracy in the shortest possible time, even taking into account the nonlin-
ear formulation of the original modeling problem.
8 E. Konopatskiy, A. Bezditnyi, O. Shevchuk
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