## 2020, issue 1, p. 23-31

*Received **09.02.2020;
Revised 18.02.2020; Accepted 10.03.2020*

*Published **31.03.2020;
First Online 26.04.2020*

*https://doi.org/10.34229/2707-451X.20.1.3*

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ON CONSTRUCtion of The EXTERNAL FRANKL Nozzle COntour Using QUADRATIC CURVATure

Petro
Stetsyuk ^{1 *} ,
Oleksandr Tkachenko ^{2}, Olga Gritsay ^{2}

^{1}* V.M. Glushkov Institute of Cybernetics,
Kyiv, Ukraine.*

^{2}* SE "Ivchenko-Progress", Zaporozhye,
Ukraine.*

^{*}* Correspondence: This email address is being protected from spambots. You need JavaScript enabled to view it.*

**The aim of the article** is to develop
a method, an algorithm, and appropriate software for constructing the external
contour of the Frankl nozzle in the supersonic part using S-shape curves.
The method is based on the problem of constructing a curve with the natural
parameterization. The curve passes through two given points with the given
inclination angles of the tangents and provides the given inclination angle of
the tangent at the point with the given abscissa [4]. To control the inflection
point of the S-shaped curve,
the inclination angle of the tangent at a point with the known abscissa is
used.

In the case, when the curvature is given by a
quadratic function,** the system** of five nonlinear equations **is
formulated**, among which three equations are integral. The system has five
unknown variables – three coefficients of the quadratic function, the total
length of the curve and the length of the curve to the point with a known
abscissa.

**The lemma** on the relation between
solutions of the original and the scalable systems, in which the coordinates of
the points are multiplied by the same value, **is proved**. Due to this
lemma, it becomes possible, using the obtained solution of the well-scalable
system, to find easily the corresponding solution of a bad-scalable (singular)
system.

To find a solution to the system, **we suggest**
to use
the modification
of the r-algorithm [5] solving special problem on minimization of the nonsmooth
function (the sum of the modules of the residuals of the system), under
controlling of the constraints on unknown lengths, in order to guarantee their
feasible values.

**The algorithm is implemented**** **using the
multistart method and the **ralgb5a** octave function [6]. It finds the best
local minimum of nonsmooth function by starting the modification of the
r-algorithm from a given number of starting points. The algorithm uses an
analytical computation of generalized gradients of the objective function and
the trapezoid rule to calculate the integrals.

**The computational experiment was carried
out****
**to
design the fragment of supersonic part in the external contour of a Frankl-type
nozzle. The efficiency of the algorithm, developed for constructing S-shape curves,
is shown.

**Keywords:** nozzle contour, natural
parameterization, curvature, nonsmooth optimization, r-algorithm

**Cite as: **Stetsyuk P., Tkachenko O.,
Gritsay O. On Construction of the External Frankl Nozzle Contour Using
Quadratic Curvature. *Cybernetics and Computer Technologies*. 2020. **1**.
23–31. (in Ukrainian) https://doi.org/10.34229/2707-451X.20.1.3

**References**

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2. Rashevskii P.K. Differential Geometry Course. 4 edition. M. Gostekhizdat, 1956. 420 p. (in Russian)

3. Borysenko V.,
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(in Ukrainian) http://nbuv.gov.ua/UJRN/gmtit_2016_1_6

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https://doi.org/10.15588/1727-0219-2018-1-7

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Modified *r*-algorithm to find the global minimum of polynomial functions.
*Cybernetics and Systems Analysis.* 1997. **33**(4) P. 482 – 497. https://doi.org/10.1007/BF02733104

6. Stetsyuk P.I. Computer program "Octave program ralgb5a: r(α)-algorithm with adaptive step". Svidotstvo pro rejestratsiju avtorskogo prava na tvir 8501. Ukraine. Ministerstvo ekonomichnogo rozvytku I torgivli. Derzhavnyi department intelektualnoji vlasnosti. Data reiestratsii 29.01.2019. (in Ukrainian)

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*ISSN 2707-451X (Online) *

*ISSN 2707-4501 (Print) *

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