#52 Department of Mathematical Modeling and Optimal Design

Head of Department

Corresponding Member of the National Academy of Sciences of Ukraine

Yurii G. Stoyan

E-mail: stoyan@ipmach.kharkov.ua

Deputy Head of Department

Doctor of Technical Sciences, Principal Scientist

Mykola I. Gil

full-time personnel:

Tatiana I. Sheiko – Doctor of Technical Sciences, Principal Scientist, Professor
Tetiana Ye. Romanova – Doctor of Technical Sciences, Principal Scientist, Professor
Oleksandr V. Pankratov – Doctor of Technical Sciences, Principal Scientist
Sergii M. Sklepus – Doctor of Technical Sciences, Principal Scientist
Volodymyr M. Patsuk – Candidate of Technical Sciences, Senior Scientific Researcher
Andrii M. Chugay – Candidate of Technical Sciences, Senior Scientific Researcher
Georgii M. Yaskov – Candidate of Technical Sciences, Senior Scientific Researcher
Roman O. Uvarov – Candidate of Technical Sciences, Scientific Researcher
Mykola I. Bychkov – Leading Engineer


Department of Mathematical Modeling and Optimal Design was established in 1972. In 2016 the department was united with the department of applied mathematics and computing methods.

The scientific direction of the simulation of the placement of geometric objects was formed in the mid-1960s in the academic school of the academician of the National Academy of Sciences of Ukraine V.L. Rvachev. The first scientific developments and publications that appeared in 1965-1966 were devoted to analytical description of contours of geometric figures using R-functions.

One of the branches of application of R-functions is the task of studying fields of different physical nature whose mathematical models are boundary value problems for equations with partial derivatives under certain boundary and initial conditions.

R-functions allowed to solve the inverse problem of analytic geometry and also to develop a unified approach to the construction of coordinate sequences for the main variational and projection methods for different types of boundary conditions and for geometric objects of virtually arbitrary shape. The bundles of functions were constructed – the structure of solutions, which exactly satisfy the given boundary conditions, which stimulated the further development of variational and projection methods in mathematical modeling of deformation, temperature, hydrodynamic, electromagnetic and other fields of different physical nature. In recent years, the theory of R-functions has been further developed both in terms of constructing equations of geometric objects and in the mathematical modeling of various physical and mechanical fields.

R-function in 3D. In recent years much attention has been paid to the construction of equations of machine-building components in 3D. The model stored in the computer memory allows the researcher to manipulate the received spatial images using interactive 3D computer graphics software.

The equations of the surface of the body of the car, the steady bush, the stepped shaft with two-toothed pulley, a rotary valve, a revolver cylinder, a screw with a shaped head, a slot and a locking surface, a scissor lift, an oil filter bracket, and a hexagonal cassette with 91 Fuel Elements were constructed. Nowadays three-dimensional physical objects are quite promising to use 3D-printers and R-functions in the task of information about geometric objects for the implementation of 3D printing. Based on the theory of R-functions, mathematical and computer model turbine blades, chevron bearings, oil filter brackets, car suspension bushings, Fuel Element shells with polyzonal and chevron ribbing, ship hulls, car body, building structures and others were created. Many of them are implemented on a 3D printer.

Mathematical modeling of physical fields with screw type of symmetry. The problem of constructing mathematical models of fields having a screw type of symmetry arises in many application areas. Twisted Pipe is a simple and convenient tool for providing flow rotational motion; in heat engineering, numerous applications of coils are known. The use of a twist of the flow has great prospects in vortex MHD generators, for regulating the propulsion of rocket motors, in chambers of nuclear power plants, with the maintenance of plasma with current in equilibrium, in the chemical, oil, gas and other industries.

Creep and damage to bodies of complex form from materials with characteristics depending on the type of load. There is a wide class of initially isotropic materials, the characteristics of which, when creep, depend on the type of load. These are, first of all, light alloys, superalloys, some types of structural steels, powder materials, composite materials of various structures, plastics, polymers, ceramics. These materials are used in aviation, space, heat and power, chemical industry for the manufacture of elements of constructions of aircraft, missiles, aircraft, jet engine parts, chemical equipment, pipelines, and others. Thus, for example, aluminum alloys are widely used in subsonic aircraft. In supersonic aircraft and missile technology, titanium alloys are used. They are of low weight, high strength, heat resistance, corrosion resistance, the minimum thermal expansion coefficient of all metals. From the titanum, elements of the roofing of planes, rockets, engines, power elements of wings, fire barriers, hydrosystems, thin-walled pipeline systems, and others are made.

Mathematical modeling of convective heat transfer in Fuel Element lattices. The calculation of the reactor at the design stage involves determining the main parameters of the active zone, temperature values, and others. The thermohydraulic calculation of the reactor core is fundamental in substantiating the safe operation of the NPP. Calculation of parameters of the coolant and temperatures of the fuel elements is carried out at all stages of designing and substantiating the safety of power plants. New constructive methods of the R-function method for mathematical and computer simulation of convective heat exchange in Fuel Element lattices have been developed, as well as the influence of the type of packing, the parameters of trimming of the shells of Fuel Elements on the distribution of velocity and temperature have been investigated.

The theory of R-functions is used both in Ukraine and abroad. In the United States at the University of Wisconsin, Madison led by prof. V. Shapiro created a new system SAGE, focused on solving boundary value problems of mathematical physics by the method of R-functions. In Germany at the University of Stuttgart, prof. K. Hollig uses RFM and spline approximation for calculating the problems of mechanics of a deformable solid. In Hungary, prof. A. Ivani with the help of RFM solves the problems of calculating electromagnetic fields. In Japan, at the universities of Tokyo and Aizy prof. Pashko, prof. Savchenko, prof. Wolves use RFM in blending tasks. In Russia, prof. V. Kravchenko uses RFM and atomic functions in the problems of radiophysics.

At the end of the 60s, the concept of dense placement function was defined, which allowed formalizing some tasks of placing geometric objects in relation to the cutting of industrial materials. The concept of the hodograph vector-function of dense placement (1970) determined the direction of scientific research in subsequent years. With the advent of F-functions (1980) it became possible to carry out an analytical description of the conditions for the placement of geometric 2D & 3D objects having arbitrary spatial form.

At present, constructive means of mathematical and computer modeling (phi-functions, quasi-phi-functions, pseudonormalized phi-functions, pseudonormalized quasi-phi-functions, gamma-functions) of placement (cutting, packing, layout, coating) taking into account technological constraints (allowable distances, changes in the orientation of objects, prohibition zones, features of the geometric and mechanical characteristics of the system) that appear when solving important scientific and applied problems in the priority fields of science and technology. Mathematical models of problems of optimal placement in the form of problems of mathematical programming were constructed, which allowed using local and global optimization methods for their solution using modern NLP-solvers and parallel computations.

Trainig of personnel

Over 30 Doctoral Dissertations and over 150 Candidate’s Theses have been prepared and defended.

Main scientific trends of research

  • Mathematical and computer simulation of optimization of geometric 2D & 3D objects placement of arbitrary spatial form
  • Development of local and global optimization methods focused on solving packaging, cutting, layout and coating tasks using modern NLP-solvers and parallel computing
  • Development of constructive means of the theory of R-functions for solving the inverse problem of analytic geometry
  • Development of the mathematical apparatus of the method of R-functions for the construction of beams of functions (decision structures) that exactly satisfy boundary conditions
  • Automation of the problem of geometric information and the process of solving boundary value problems of mathematical physics by the method of R-functions
  • Development of methods and algorithms of mathematical modeling of trajectories of high-speed processing of details on equipment with CNC spline functions


Modern directions of the theory of geometrical design:


  • Stoyan Yu., Pankratov A., Romanova T. (2017) Placement problems for irregular objects: mathematical modeling, optimization and applications/ Chapter in book Optimization Methods and Applications/ Editors: Butenko, Sergiy, Pardalos, Panos M, Shylo, Volodymyr (Eds.)/ Springer Optimization and Its Applications/ ISBN 978-3-319-68640-0, Series Volume 180, Springer International Publishing, eBook ISBN978-3-319-68640-0, DOI 10.1007/978-3-319-68640-0, P.612
  • Stoyan Yu., Romanova T., Pankratov A., Kovalenko A., Stetsyuk P. (2016) Modeling and Optimization of Balance Layout Problems. Chapter (pp. 177-208) in contributed book Space Engineering. Modeling and Optimization with Case Studies/ Springer Optimization and its Applications, Editors G. Fasano and J.Pintér, Springer Science + Business Media, New York, Vol. 114, XV, 487 p.
  • Stoyan Yu., Pankratov A., Romanova T., Chugay A. (2015) Optimized object packings using quasi-phi-functions. chapter (P. 265-291) in book Optimized Packings and Their Applications, Editors G. Fasano and J.Pintér/ Springer Optimization and its Applications,– Springer Science + Business Media, New York, Vol. 105. – 326 p
  • Stoyan Y., Stetsyuk P., Romanova T. (2014) Optimal Balanced Packing Using Phi-Function Technique. In S. Butenko, E. L. Pasiliao, and V. Shylo (Editors), Examining Robustness and Vulnerability of Networked Systems, pages 251–271. IOS Press. –309 p
  • Stoyan Yu., Romanova T. (2013) Mathematical Models of Placement Optimisation: Two- and Three-Dimensional Problems and Applications // Chapter in book “Modeling and Optimization in Space Engineering Springer Optimization and Its Applications”, Editors G.Fasano and J.Pintér, Springer, New York, Vol. 73, pp. 363–388.


  • Stoyan, Y., Pankratov, A., Romanova, T. (2016) Quasi-phi-functions and optimal packing of ellipses. J. Glob. Optim. 65 (2), 283–307.
  • Stetsyuk P., Romanova T., Scheithauer G. (2016) On the global minimum in a balanced circular packing problem// Optimisation Letters, Optim Lett 10:1347–1360 DOI 10.1007/s11590-015-0937-9.
  • Bennell JA, Scheithauer G, Stoyan Y, Romanova T, Pankratov A (2015). Optimal clustering of a pair of irregular objects. Journal of Global Optimization. 61(3):497-524.
  • Kovalenko, A., Romanova, T., Stetsyuk, P. (2015). Balance layout problem for 3D-objects: mathematical model and solution methods. Cybern. Syst. Anal. 51(4), 556-565 DOI 10.1007/s10559-015-9746-5.
  • Stoyan Yu., Pankratov A., Romanova T. (2015) Cutting and Packing problems for irregular objects with continuous rotations: mathematical modeling and nonlinear optimization. Journal of the Operational Research Society, Vol. 67, Issue 5, 786–800. DOI 10.1057/jors.2015.94
  • Kovalenko, A., Romanova, T., Stetsyuk, P. (2015) Balance layout problem for 3D-objects: mathematical model and solution methods. Cybern. Syst. Anal. 51(4), 556-565 DOI 10.1007/s10559-015-9746-5.
  • Bennell JA, Scheithauer G, Stoyan Y, Romanova T, Pankratov A (2015) Optimal clustering of a pair of irregular objects. Journal of Global Optimization. 61(3):497-524.
  • Stoyan Y. G. , Chugay A. M. (2014) Packing Different Cuboids with Rotations and Spheres into a Cuboid// Advances in Decision Sciences,
    ID 571743, https://www.hindawi.com/journals/ads/2014/571743/ref.
  • Stoyan Yu, Yaskov G. (2013) Packing congruent spheres into a multi-connected polyhedral domain.- Intl. Trans. in Op. Res. – 20(1). – P. 79-99. DOI: 10.1111/j.1475-3995.2012.00859.x.
  • G. Scheithauer, Y. Stoyan, T. Romanova, A. Krivulya (2011) Covering a polygonal region by rectangles/Comput. Optimiz. Appl., Springer, Netherlands, Vol. 48(3), 675-695.
  • Chernov N., Stoyan Yu, Romanova T. (2010) Mathematical model and efficient algorithms for object packing problem// Computational Geometry: Theory and Applications, Vol. 43:5, 535–553.
  • Stoyan Yu., Yaskov G. (2010) Packing identical spheres into a cylinder, International Transactions in Operational Research, Vol. 17, №1, 51–70.
  • Bennell J., Scheithauer G., Stoyan Yu., Romanova T. (2010) Tools of mathematical modelling of arbitrary object packing problems, J. Annals of Operations Research, Publisher Springer Netherlands: Vol. 179, № 1, 343-368.
  • Stoyan Y.G., Patsuk V.M. (2010) Covering a compact polygonal set by identical circles// Computational Optimization and Applications, 46, 75-92.
  • Scheithauer G., Stoyan Yu.G., Romanova T. (2005) Mathematical Modeling of Interactions of Primary Geometric 3D Objects// Cybernetics and Systems Analysis. Consultants Bureau, An Imprint of Springer Verlag New York LLC. ISSN: 1060-0396, Vol. 41, № 3, 332-342.
  • Stoyan Y., Gil N., Scheithayer G., Pankratov A., Magdalina I. (2005) Packing of convex polytopes into a parallelepiped // Optimization, Vol. 54. № 2, 215 – 235.
  • Stoyan Y., Gil M., Terno J., Romanova T., Scheithauer G. (2004) Phi- function for complex 2D objects// 4OR Quarterly Journal of the Belgian, French and Italian Operations Research Societies. Vol. 2, № 1, 69 – 84.
  • Stoyan Y., Gil N., Terno J., Romanova T., Scheithauer G. (2002) Phi-function for 2D primary objects// Studia Informatica, Paris. Vol. 2, № 1. – 1-32.
  • Stoyan Y., Gil M., Terno J., Romanova T., Scheithauer G. (2002) Construction of a Phi- function for two convex polytopes// Appliсationes Mathematicae. – Vol. 2. № 29. – 199-218.
  • Stoyan Yu. G. and Patsuk V. N. (2000) A method of optimal lattice packing of congruent oriented polygons in the plane// European Journal of Operational Research. Elsevier.- № 124.- 204-216.

Modern trends of the R-functions Theory:

  • Рвачев В.Л. Теория R-функций и некоторые ее приложения.—К., Наук.думка, 1982.—552 с.
  • Максименко-Шейко К.В. R-функции в математическом моделировании геометрических объектов и физических полей. – Харьков, ИПМаш НАН Украины, 2009. – 306 с.
  • Золочевский А.А. Нелинейная механика деформируемого твердого тела / А.А. Золочевский, А.Н. Склепус, С.Н. Склепус. – Харьков: «Бізнес Інвестор Групп», 2011.– 720 с.
  • Kurpa L., Rvachev V., Ventsel Е. The R-function method for the free vibration analysis of thin orthotropic plates of arbitrary shape // J. of Sound and Vibration. — 2003. — № 26. — Pp. 109-122.
  • Maksimenko-Sheiko K.V. Mathematical modeling of heat conduction processes for structural elements of nuclear power plants by the method of R-functions / M.Ye.Voronyanskaya, K.V.Maksimenko-Sheiko, T.I.Sheiko // Journal of Mathematical Sciences. — 2010. — Vol.170, No.6. — P.776-793.
  • Rvachev V.L., Sheiko T.I. R-functions in boundary value problems in mechanics // Applied Mechanics Reviews.—48, n.4.—1995.—Pp.151 – 188.
  • Rvachev V.L., Sheiko T.I., Shapiro V. Generalized Interpolation Lagrange-Hermite Formulas on Arbitrary Loci (Interlocation Operators of the R-functions Theory) // Journal of Mechanical Engineering.—1, n. 3-4.—1998.—Pp.150-166.
  • Rvachev V.L., Sheiko T.I., Shapiro V. The R-function method in boundary value problems with geometric and physical symmetry // Journal of Mаthematical Sciences. — 1999. — 97 (1). — Pp. 3888-3899.
  • Rvachev V.L., Sheiko T.I., Shapiro V., Tsukanov I. On Completeness of RFM Solution Structures // Computational Mech. — 2000. — 25. — Pp.305-316.
  • Rvachev V.L., Sheiko T.I., Shapiro V., Tsukanov I. Transfinite Interpolation over Implicitly Defined Sets // Computer Aided Geometric Design.— 2001.—N.18.— Pp.195-220.
  • Zolochevsky A. New model of unilateral creep damage / A. Zolochevsky, S. Sklepus, J. Betten // Journal of Theoretical and Applied Mechanics, Sofia.– 2001.– Vol.31.– № 1.– P. 64-70.
  • Sklepus S. M. Solution of the Axisymmetric Problem of Creep and Damage for a Piecewise Homogeneous Body with an Arbitrary Shape of a Meridional Section / S.М. Sklepus // Journal of Mathematical Sciences. – 2015. – Vol. 205, № 5. – P. 644-658.
  • Sklepus S. A Study of the Creep Damageability of Tubular Solid Oxide Fuel Cell / S. Sklepus, A. Zolochevsky // Strength of Materials – 2014. – Vol. 46, Issue 1 – P. 49-56.
  • Zolochevsky A. A Comparison between the 3D and the Kirchhoff-Love Solutions for Cylinders under Creep-Damage Conditions / A. Zolochevsky, S. Sklepus, A. Galishin, A. Kühhorn, M. Kober // Technische Mechanik. – 2014. – 34, 2 – P. 104-113.
  • Zolochevsky A. Analysis of creep deformation and creep damage in thin-walled branched shells from materials with different behavior in tension and compression / A. Zolochevsky, A. Galishin, S. Sklepus, G.S. Voyiajis // International Journal of Solids and Structures. – 2007. – 44. – Р. 5075-5100.
  • Рвачев В.Л., Шейко Т.И. Введение в теорию R-функций // Пробл. машиностроения. — 2001. — т.4, №1-2. — С. 46-58.
  • Рвачев В.Л., Шевченко А.Н. Проблемно-ориентированные языки и системы для инженерных расчетов.—К., Техніка, 1988.—197 с.
  • Рвачев В.Л., Шейко Т.И. Метод R-функций в задачах расчета полей для тел, физические характеристики которых имеют разрывы первого рода // Прикладная математика и механика. — 1984. — Т.48, №5. — С. 873-877.
  • Рвачев В.Л., Максименко-Шейко К.В. Математические модели движения несжимаемой вязкой жидкости по скрученным трубам // Математические методы и физико-механические поля.— 2003.—46, №2.—С.81-88.
  • Максименко-Шейко К.В., Шейко Т.И. R-функции в математическом моделировании геометрических объектов, обладающих симметрией // Кибернетика и системн. анализ. — 2008. — №6. — С. 75-83.
  • Максименко-Шейко К.В. R-функции в фрактальной геометрии. / К.В.Максименко-Шейко, А.В.Толок, Т.И.Шейко // Информационные технологии (г. Москва). — 2011. — №7. — С. 24-27.