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Possibilities and prospects of application of artificial intelligence methods for solving boundary value problems of mathematical physics in engineering practice

DOI 10.18127/j19998554-201902-02

Keywords:

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.

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May 29, 2020

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