On Constructing the External Contour of the Frankl Nozzle Using Quadratic Curvature

• Main Authors: Petro Stetsyuk, Oleksandr Tkachenko, Olga Gritsay
• Format: Article
• Language: English
• Published: V.M. Glushkov Institute of Cybernetics 2020-03-01
• Series: Кібернетика та комп'ютерні технології
• Subjects: nozzle contour; natural parameterization; curvature; nonsmooth optimization; r-algorithm
• Online Access: http://cctech.org.ua/13-vertikalnoe-menyu-en/95-abstract-20-1-3-arte

The aim of the article is to develop a method, the algorithm, and the 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. For the problem, where 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 the solutions of the original system and the scalable system, in which the coordinates of the points are multiplied by the same value, is proved. Based on the lemma, it is possible, using the obtained solution for a well-scalable system, to find easily the corresponding solution for a bad-scalable (singular) system. To find a solution to the system, we suggest using the modification of the r-algorithm [5] to solve the special problem of minimizing a nonsmooth function (the sum of the modules of the residuals of the system) while controlling constraints on unknown lengths 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 modification of the r-algorithm from a given number of starting points. The algorithm uses an analytical computing 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.