About Modelling of Deformations of Plates

Of concern are the elliptic boundary problems of the fourth order which are underlying in mathematical models of deformations of plates on the elastic bases under the mixed boundary conditions of four theoretically possible types. Replacements of these problems in a variation form to their fictitious continuations are proposed. Solutions of the last problems by means of modifications of fictitious components methods are reduced to solutions of problems in a rectangular area. Optimum estimates of convergence of iterative processes at the continuous level are given. At simple sampling of fictitiously continued problems by method of final elements on parabolic completions, effective numerical modifications of fictitious components methods turn out to be suitable for practical realization on the computer. The received systems of linear algebraic equations can be optimally solved by means of iterative factorizations methods. As a result the proposed numerical methods are log-optimal or optimal by number of arithmetic operations necessary for achievement of set relative errors.

Keywords

deformations of plates; fictitious continuations.

References

1. Ushakov A.L. Modifikaziya metoda fiktivnykh komponent [Modification of a Method of Fictitious Components (Depp. in VINITI 11.11.1991, no. 4232-B1991)]. Chelyabinsk, Chelyabinsk State Technical University, 1991. 40 p.
2. Ushakov A.L. [Iterative Factorization on Fictitious Continuation for the Numerical Solution of the Elliptic Equation of the Fourth Order]. Bulletin of the South Ural State University. Series: Mathematics, Mechanics, Physics, 2014, vol. 6, no. 2, pp. 17-22. (in Russian)