DOI: https://doi.org/10.20535/1810-0546.2015.2.91569

### Projection-Grid Method of Elasticity Problems Solution in Flight Dynamics

Олександр Сергійович Цибенко, Олександр Станіславович Конюхов

#### Abstract

Background. The development of efficient projection-grid method for solving initial-boundary value problems of the elastic dynamics of the aircraft.

Objective. Theoretical study of numerical methods for solving the elastic dynamics of aircraft in order to create a generalized method of numerical integration of initial-boundary value problems for discrete-continuum.

Methods. As a generalized mathematical description of the initial-boundary value problem by using the operator formulation of the first order main part. Approximate solution of initial value problems of elastic dynamics of aircraft represented as a linear form on the class of admissible functions of non-degenerate projective basis. Algebraization of the spatial variables is realized because of orthogonalization residuals of equations and boundary conditions for the system of functions defining non-degenerate weight basis. The greatest effect is achieved by computing the matching item in the projection and a weight basis in conjunction with the "weak" formulation of the Galerkin method in the form of the finite element method. The general form of the finite difference method is used for algebraization of unknown functions on a temporary argument. For solving systems of nonlinear algebraic equations on time layers, Newton's method and its modifications were applied.

Results. A general approach to solving the problems of the elastic dynamics of aircraft using the procedure of algebraization based on projection-grid schemes of the method of weighted residuals. A posteriori estimates for the accuracy, convergence and stability of numerical solutions of the elastic dynamics of aircraft were presented.

Conclusions. The developed technique of algebraization tasks in elastic dynamics of aircraft can be widely used in the simulation of the dynamics of liquid carrier rockets in different parts of the flight.

#### Keywords

Dynamics of aircraft; Initial-boundary value problem; Galerkin method; Finite element method; Finite-difference schemes; Accuracy; Convergence; Stability

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