Abstract
One of the main problems in modern mathematical modeling is to obtain high-precision solutions of boundary value problems. This study proposes a new approach that combines the methods of artificial intelligence and a classical analytical method. The use of the analytical method of fictitious canonic regions is proposed as the basis for obtaining reliable solutions of boundary value problems. The novelty of the approach is in the application of artificial intelligence methods, namely, genetic algorithms, to select the optimal location of fictitious canonic regions, ensuring maximum accuracy. A general genetic algorithm has been developed to solve the problem of determining the global minimum for the choice and location of fictitious canonic regions. For this genetic algorithm, several variants of the function of crossing individuals and mutations are proposed. The approach is applied to solve two test boundary value problems: the stationary heat conduction problem and the elasticity theory problem. The results of solving problems showed the effectiveness of the proposed approach. It took no more than a hundred generations to achieve high precision solutions in the work of the genetic algorithm. Moreover, the error in solving the stationary heat conduction problem was so insignificant that this solution can be considered as precise. Thus, the study showed that the proposed approach, combining the analytical method of fictitious canonic regions and the use of genetic optimization algorithms, allows solving complex boundary-value problems with high accuracy. This approach can be used in mathematical modeling of structures for responsible purposes, where the accuracy and reliability of the results is the main criterion for evaluating the solution. Further development of this approach will make it possible to solve with high accuracy of more complicated 3D problems, as well as problems of other types, for example, thermal elasticity, which are of great importance in the design of engineering structures.
Highlights
Modern methods of engineering structures design include math modeling as the obligatory stage
Math modeling of engineering structures leads to the solving of boundary value problems in the most cases: stress-strain state evaluation; temperature and electro-magnetic fields and so on
The existing computer application, based on the numerical methods, allows engineers, who are not experienced in the mathematical physics, to automatically get solutions of boundary value problems for complicated engineering structures
Summary
Modern methods of engineering structures design include math modeling as the obligatory stage. Math modeling of engineering structures leads to the solving of boundary value problems in the most cases: stress-strain state evaluation; temperature and electro-magnetic fields and so on. The main approaches of solving boundary value problems are numerical methods, namely, finite element and finite difference methods. The existing computer application, based on the numerical methods, allows engineers, who are not experienced in the mathematical physics, to automatically get solutions of boundary value problems for complicated engineering structures. This particular advantage is the reason of widely using the finite element and finite differences methods in the modern engineering practice. The numerical methods have some drawbacks which will be considered later in the article
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