L. N. Yasnitsky – Dr.Sc. (Eng.), Professor, Professor of Perm State University, National Research University “Higher School of Economics”

E-mail: yasn@psu.ru

S. L. Gladky – Ph.D. (Phys.-Math.), Head of Analytics Development Group, LLC “VIPAKS” (Perm)

E-mail: lrndlrnd@mail.ru

I. I. Nikitenko – Senior Design Engineer, LLC “CROC Region” (Moscow)

E-mail: ignat.nikitenko@gmail.com

Currently, the vast majority of boundary value problems arising in engineering practice are solved with the use of generic packages that implement numerical methods: ANSYS, COSMOS, LS-DYNA, NASTRAN, PATRAN, FEMAP, BEAST, etc. These packages allow engineers to obtain the numerical solution of boundary value problems of almost any complexity. But there is a serious problem, which is that to estimate the errors of such solutions for complex engineering problems, as a rule, is not possible.

The fact is that with the grinding of finite element grids, the conditionality of the matrix of the system of resolving algebraic equations always deteriorates. This means that approximate numerical solutions with mesh refinement ultimately tend not to the sought solutions of boundary value problems at all. Realizing this, the developers of engineering structures of responsible purpose are forced to recheck the results of such computer simulation by conducting expensive and long-term field experiments.

This article is devoted to the creation of a high-precision analytical method for solving boundary value problems, based on the idea of using a genetic algorithm to optimize the selection of basic functions, in the form of the sum of which the solution of the boundary value problem is sought.

This article shows that the use of genetic optimization algorithm instead of gradient algorithm can significantly reduce the error of solving boundary value problems. This effect is achieved due to the fact that, firstly, the genetic algorithm, unlike the gradient algorithm, is not subject to “stuck” in local minima. Secondly, the genetic algorithm allows to solve the optimization problem in a complex way. The genetic algorithm not only finds the optimal type of basis functions, but also optimizes their number, as well as the type and number of constituents retained in the basis expansions.

In this article it is shown that the development and application of artificial intelligence methods, in particular, the proposed genetic algorithm, allows you to develop computer programs that implement high-precision analytical methods for solving boundary value problems, which are not inferior in complexity to the widely used in engineering practice programs that implement numerical methods. Joint application of analytical methods for solving boundary value problems and methods of artificial intelligence, can improve the accuracy and reliability of the results of computer simulation, which is necessary in the calculation of modern engineering structures responsible purpose.

1. Yasnitskij L.N., Danilevich T.V. Sovremennye problemy nauki. M.: BINOM. Laboratoriya znanij. 2011.
2. Yasnitskij L.N. Sovremennyj krizis prikladnoj matematiki i perspektivy ego preodoleniya. Vestnik Permskogo universiteta. Ser. Matematika. Mekhanika. Informatika. 2007. № 7. S. 192–197. [in Russian]
3. Tarkhov D., Vasilyev A. New neural network technique to the numerical solution of mathematical physics problems. I: Simple problems. Optical Memory and Neural Networks (Information Optics). 2005. V. 14. P. 59–72.
4. Tarkhov D., Vasilyev A. New neural network technique to the numerical solution of mathematical physics problems. II: Complicated and nonstandard problems. Optical Memory and Neural Networks (Information Optics). 2005. V. 14. P. 97–122.
5. Tarkhov D.A. Nejronnye seti kak sredstvo matematicheskogo modelirovaniya. Nejrokomp'yutery: razrabotka, primenenie. 2006. № 2. S. 3–48. [in Russian]
6. Gorbachenko V.I., Zhukov M.V. Podkhody i metody obucheniya setej radial'nykh bazisnykh funktsij dlya resheniya zadach matematicheskoj fiziki. Nejrokomp'yutery: razrabotka, primenenie. 2013. № 9. S. 12–18. [in Russian]
7. Gorbachenko V.I., Zhukov M.V. Reshenie kraevykh zadach matematicheskoj fiziki s pomoshch'yu setej radial'nykh bazisnykh funktsij. Zhurnal vychislitel'noj matematiki i matematicheskoj fiziki. 2017. T. 57. № 1. S. 133–143. [in Russian]
8. Kolbin I.S., Reviznikov D.L. Reshenie zadach matematicheskoj fiziki s ispol'zovaniem normalizovannykh radial'no-bazisnykh nejropodobnykh setej. Nejrokomp'yutery: razrabotka, primenenie. 2012. № 2. S. 12–19. [in Russian]
9. Yasnitskij L.N. Metod fiktivnykh kanonicheskikh oblastej v mekhanike sploshnykh sred. M.: Nauka. 1992.
10. Yasnitsky L.N. The possibilities of error estimation in the boundary element type methods. Boundary Elements Communications. 1994. V. 5. № 4. P. 181–182.
11. Gladkij S.L., Yasnitskij L.N. Ob otsenke pogreshnosti metoda fiktivnykh kanonicheskikh oblastej. Izvestiya Rossijskoj akademii nauk. Mekhanika tverdogo tela. 2002. № 6. S. 69–75. [in Russian]
12. Trefftz E. Ein Gegenstuck zum Ritzschen Verfahren. Verhandl des 2. Intern. Kongress fur tehnische Mechanik. Zurich. 1926. P. 12–17.
13. Yasnitskij L.N. Ob odnom sposobe resheniya zadach teorii garmonicheskikh funktsij i linejnoj teorii uprugosti. Prochnostnye i gidravlicheskie kharakteristiki mashin i konstruktsij. Perm': Izd-vo Permskogo politekhnicheskogo instituta. 1973. S. 78–83.
14. Yasnitskij L.N. Kompozitsiya raschetnoj oblasti v metode fiktivnykh kanonicheskikh oblastej. Izvestiya Rossijskoj akademii nauk. Mekhanika tverdogo tela. 1990. № 6. S. 168–172. [in Russian]
15. Gladkij S.L., Talantsev N.F., Yasnitskij L.N. Verifikatsiya chislennykh raschetov metodom fiktivnykh kanonicheskikh oblastej. Vestnik Permskogo universiteta. Ser. Matematika. Mekhanika. Informatika. 2006. № 4. S. 18–27. [in Russian]
16. Yasnitskij L.N. Superpozitsiya bazisnykh reshenij v metodakh tipa Treffttsa. Izvestiya Rossijskoj akademii nauk. Mekhanika tverdogo tela. 1989. № 2. S. 95–101. [in Russian]
17. Yasnitskij L.N. K raschetu napryazhennogo sostoyaniya ellipsoidal'noj obolochki postoyannoj i peremennoj tolshchiny na osnove reshenij teorii uprugosti dlya sfericheskikh oblastej. Prikladnaya mekhanika. 1989. T. 25. № 6. S. 111–114. [in Russian]
18. Kirko I.M., Terrovere V.R., Yasnitskij L.N. Novaya optimal'naya forma makhovichnogo nakopitelya energii. Doklady Akademii nauk. Tekhnicheskaya fizika. 1989. T. 307. № 6. S. 1373–1375. [in Russian]
19. Klimenko I.P., Yasnitskij L.N. K raschetu deformirovannogo sostoyaniya vtulki plunzhernoj pary metodom fiktivnykh kanonicheskikh oblastej. Izvestiya vuzov. Mashinostroenie. 1991. № 4–6. S. 32–34. [in Russian]
20. Tomilov V.A., Klimenko I.P., Yasnitskij L.N. Stabilizatsiya velichiny zazora plunzhernoj pary za schet uprugikh deformatsij plunzhera. Problemy mashinostroeniya i nadezhnosti mashin. 1994. № 4. S. 109–113. [in Russian]
21. Dobrynin G.F., Yasnitskij L.N. Prochnostnye raschety izolyatorov. Steklo i keramika. 1994. № 7. S. 40–43. [in Russian]
22. Gorchakov A.I., Semenova A.V., Syrovatskaya Yu.V., Shcherbakov Yu.V., Yasnitskij L.N. Vliyanie geometricheskikh parametrov mikrodugovogo oksidirovaniya na ravnomernost' pokrytij, formiruemykh na alyuminievykh splavakh. Fizika i khimiya obrabotki materialov. 2004. № 1. S. 43–47. [in Russian]
23. Ashmanov V.D., Oshchepkov V.A., Yasnitskij L.N. Novaya konstruktsiya mnogoslojnogo polotna shiny tsepnoj pily. Izvestiya vuzov. Mashinostroenie. 2004. № 6. S. 28–36. [in Russian]
24. Gladkij S.L., Stepanov N.A., Yasnitskij L.N. Intellektual'noe modelirovanie fizicheskikh problem. Moskva–Izhevsk: NITs «Regulyarnaya i khaoticheskaya dinamika». 2006.
25. Gusman S.Ya., Yasnitskij L.N. Obosnovanie vybora fiktivnykh kanonicheskikh oblastej. Vestnik Permskogo universiteta. Ser. Matematika. Informatika. Mekhanika. 1994. № 1. S. 55–65. [in Russian]
26. Polyanin A.D. Spravochnik po linejnym uravneniyam matematicheskoj fiziki. M.: FIZMATLIT. 2001.
27. Gladkij S.L., Yasnitskij L.N. Algoritmy optimizatsii bazisnykh razlozhenij v metode fiktivnykh kanonicheskikh oblastej. Vestnik PGTU. Ser. Dinamika i prochnost' mashin. 2001. № 3. S. 131–141. [in Russian]