Modeling of technological process for aircraft structural components manufacturing based on the best parametrization and boundary value problem for nonlinear differential-algebraic equations

Аuthors

Budkina E. M.^*, Kuznetsov E. B.^**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: emb0909@rambler.ru
**e-mail: kuznetsov@mai.ru

Abstract

Among mathematical models describing various processes associated with manufacturing engineering, aerospace technology and aviation, there are models representing a system of ordinary differential equations and a system of nonlinear algebraic or transcendental equations, i.e. a system of differential-algebraic equations (DAE). Such problems often arise in applied mathematics and mechanics. Some hydrodynamic processes described by the DAE systems contain a small parameter (viscosity). Problems of this type describe the phenomenon of creep. The process of creep of the material in the first approximation can be modeled by DAE systems discussed in this paper.

The basic methods of solving boundary value problems for such systems are methods of collocation and shooting methods. With shooting method, a boundary value problem is reduced to some initial value problem. However, this method is applicable only in the case when the original problem is correct. We suggest to apply the best parameterization for regularization of this problem.

The paper considers the system of nonlinear differential-algebraic equations. It is shown that the best parameterization of the boundary value problem for a singularly perturbed differential-algebraic equations significantly improves the computational algorithm of the shooting method.

The numerical solution of the problem was obtained using the method of solution continuation with respect to parameter and the best parameterization. The boundary value problem is reduced to an initial value problem for differential-algebraic equations. We selected shooting method as a numerical solution.

By using these methods of solution all solutions of the boundary value problem were obtained regardless of the choice of initial values.

According to the results, following conclusions were made:

• the method of solution continuation with respect to parameter and the best parameterization can be used for solving singularly perturbed boundary value differential-algebraic problems;

• the method of solution continuation with respect to parameter and the best parameterization allows to find all solutions of the boundary value problem for nonlinear differential-algebraic equations.

Numerical studies of this work show that the parameterization of the boundary value problem for nonlinear differential-algebraic equations, proposed in this paper significantly improves the computational algorithm of the shooting method and allows to find all solutions of the boundary value problem. Thus, in this paper we propose a numerical method, which allows solve the applied problems related to technology of machine building, rocket and space technology, aviation.

Keywords:

creep of materials, differential-algebraic equations, shooting method, the best parameterization, the best parameter, solution continuation

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