Introduction.

Relevance of the problem:

The solution to the problem of weight reduction of structures is associated with their complication and the use of thin-walled elements. Even the simplest variant method of constructive optimization requires parametric studies on a computer using numerical methods for solving boundary value problems. The most famous among them are:

- finite-difference methods for constructing approximate solutions of differential equations using finite-difference approximations of derivatives;

- various modifications of the finite element method, the Bubnov-Galerkin method, the Rayleigh-Ritz method, which are based on approximations of solutions of differential equations by finite linear combinations of given functions: polynomials, trigonometric functions, etc .;

- methods for the numerical determination of the integrals of the ordinary differential equations Runge-Kutta, Volterra, Picard, etc.

The main success of the methods of finite differences and finite elements is that on their basis universal algorithms are constructed and packages of applied programs for calculating complex structures are created. The constructed computing facilities are able to detect the flow of forces in the structure and, therefore, the most strained elements of it. Nevertheless, they require unjustifiably high costs of the programmer's efforts and powerful computing tools when the task is to determine the stresses in the places of their concentration.

The most obvious effectiveness of the methods of numerical integration of ordinary differential equations consists in calculating individual parts of complex spatial structures and their individual thin-walled elements with the refinement of the stress-strain state in the places of its rapid change. Efficiency is determined by the small expenses of the programmer's work, by the small expenditure of computer time and the computer's operational memory.

Thus, increasing the effectiveness of known numerical methods, constructing their modifications and constructing new methods, is an urgent research task.

The proposed scientific novelty consists in the following:

1. S.K.Godunov’s method of orthogonal sweep has been improved,

2. A method of "transferring of boundary value conditions" (a direct version of the method) is proposed for solving boundary value problems with non-stiff ordinary differential equations,

3. A method of "additional boundary value conditions" is proposed for solving boundary value problems with non-stiff ordinary differential equations,

4. A "half of constants" method is proposed for solving boundary value problems with non-stiff ordinary differential equations,

5. A method of "transferring of boundary value conditions" (step-by-step version of the method) is proposed for solving boundary value problems with stiff ordinary differential equations,

6. The simplest method for solving boundary value problems with stiff ordinary differential equations without orthonormalization is proposed - the method of "conjugation of sections of the integration interval", which are expressed by matrix exponentials,

7. The simplest method for calculating the shells of composite and with frames is proposed.

Some of the works on which the methods are based are published jointly with Dr.Sc. Professor Yu.I.Vinogradov.

Contribution of Dr.Sc. Professor Yu.I. Vinogradov in these joint publications consisted either of 1) in the discussion of the results of verification calculations of those formulas and methods proposed by A.Yu. Vinogradov, or that 2) in addition to A.Yu.Vinogradov’s methods Yu.I.Vinogradov proposed statement that the Cauchy matrix can be computed not only in the form of matrix exponentials, but in addition there is the possibility of calculating them in the sense of Cauchy-Krylov functions, using for this purpose the analytical solutions of the systems of differential equations of the structural mechanics of plates and shells obtained by someone, that 3) Yu.I. Vinogradov proposed his own, different from A.Yu.Vinogradov’s formula, the formula for computing the vector of the particular solution of an inhomogeneous system of ordinary differential equations, which, however, looks much more complicated than the simple A.Yu.Vinogradov’s formula.

Also, in the co-authors of some articles, Yu.A.Gusev and Yu.I.Klyuyev. Their contribution to the publication material consisted in performing multivariate verification calculations in accordance with the formulas, algorithms and methods proposed by A.Yu. Vinogradov in his Ph.D. thesis. The Ph.D. thesis was defended in 1996.

In addition, we can say that on the basis of the material from A.Yu.Vinogradov’s Ph.D. thesis completed two more candidate's physical and mathematical dissertations under the direction of Yu.I.Vinogradov, whose material consists mainly of a multivariate application and verification by calculations of what was proposed by A.Yu.Vinogradov in his Ph.D. thesis, in application to various concrete problems of construction mechanics of thin-walled shells with the identification and analysis of properties of formulas, algorithms and methods from A.Yu.Vinogradov’s Ph.D. thesis.

Here are the data of these two dissertations:

Year: 2008 Petrov, Vitaliy Igorevich "Reduction of boundary value problems to initial problems and investigation of stress concentration in thin-walled constructions by the multiplicative method"

Scientific degree: Candidate of Physical and Mathematical Sciences

Specialty code of VAC: 05.13.18 Specialty: Mathematical modeling, numerical methods and program complexes.

 

Year: 2003 Gusev, Yuri Alekseevich "Multiplicative algorithms for the transfer of boundary conditions in problems of the mechanics of shell deformation"

Scientific degree: Candidate of Physical and Mathematical Sciences

Specialty code of VAC: 01.02.04 Specialty: Mechanics of a deformable solid.

In addition, in accordance with the modern capabilities of the Internet and in the presence of new theses in the open access it was revealed the use of materials from A.Yu.Vinogradov Ph.D. thesis with relevant references to A.Yu.Vinogradov’s relevant articles in the following candidate and doctoral dissertations of technical and physical and mathematical sciences: 

Year: 2005 Russian academy of sciences, Siberian Branch, Institute of Computational Technologies, Novosibirsk, Yurchenko Andrey Vasilevich “Numerical solution of boundary problems of elastic deformation of composite shells of rotation”

Scientific degree: Candidate of Physical and Mathematical Sciences

Specialty code of VAC: 05.13.18 - mathematical modeling, numerical methods and program complexes.

 

Year: 2003 Kochetov, Sergey Nikolaevich, Moscow "Methods and algorithms for determining the stress-strain state of thin-walled reinforced rotation structures from nonlinear elastic material"

Scientific degree: Candidate of Physical and Mathematical Sciences

Specialty code of VAC: 05.23.17 - construction mechanics.

Year: 1998 Chekanin, Alexander Vasilievich, Moscow "Development of the method of superelements applied to the problems of statics and dynamics of thin-walled spatial systems"

Scientific degree: Doctor of Technical Sciences

Specialty code of VAC: 05.23.17 - construction mechanics.

Year: 2005 Golushko, Sergey Kuzmich, Novosibirsk "Direct and inverse problems of the mechanics of elastic composite plates and shells of revolution"

Scientific degree: Doctor of Physics and Mathematics

Specialty code of VAC: 01.02.04 - mechanics of deformable solids.

Year: 2003 Gazizov, Khatib Sharifzyanovich, Ufa "Development of theory and methods for calculating the dynamics, stiffness and stability of composite shells of revolution" Scientific degree: Doctor of Technical Sciences

Specialty code of VAC: 01.02.04 - mechanics of deformable solids.

Year: 2001 Shlenov, Alexey Yurievich, Moscow "Dynamics of structurally inhomogeneous shell structures with allowance for the elastic-plastic properties of the material"

Scientific degree: Candidate of physical and mathematical sciences

Specialty code of VAC: 01.02.04 - mechanics of deformable of a solid.

Year: 1996 Rogov, Anatoly Alekseevich, Moscow "Dynamics of the pipeline after the break"

Scientific degree: Candidate of Physical and Mathematical Sciences

Specialty code of VAC: 01.02.04 - mechanics of deformable solids. 

A.Yu.Vinogradov’s articles were published in VAC magazines:

• Reports of the Academy of Sciences of the Russian Federation - 2 articles

• Mechanics of the Solid Body of the Russian Academy of Sciences - 2 articles

• Journal of Computational Mathematics and Mathematical Physics, Russian Academy of Sciences - 1 article

• Mathematical Modeling of the Russian Academy of Sciences - 2 articles

• Fundamental research - 1 article

• Modern problems of science and education - 1 article

• Modern high technology - 1 article.

In general, the formulas for solving boundary value problems for linear ordinary differential equations borrowed in this paper were taken from only 4 sources:

1. Abramov A.A. On the transfer of boundary conditions for systems of linear differential equations (variant of the sweep method) // Journal of Computational Mathematics and Mathematical Physics, 1961. - T.I. -N3. -C.542-545.

2. Berezin I.S., Zhidkov N.P. Methods of calculation. M.: Fizmatgiz, 1962.-T.2. - 635 p.

3. Gantmakher F.R. Theory of matrices. – М.: Science, 1988. - 548 p,

4. Godunov S.K. On the numerical solution of boundary value problems for systems of linear ordinary differential equations. Uspekhi Matematicheskikh Nauk, 1961. -T. 16, no. 3, (99). - p.171-174.

Chapter 1. Known formulas of the theory of matrices for ordinary differential equations.

All methods are given by the example of a system of differential equations of the cylindrical shell of a rocket - a system of ordinary differential equations of the 8th order (after the separation of partial derivatives by Fourier’s method).

The system of linear ordinary differential equations has the form:

,

where – the required vector-valued function of the dimension 8х1, – the derivative of the required vector-valued function of the dimension 8х1, – square matrix of coefficients of a differential equation of dimension 8х8, – vector-function of external action on the system of dimension 8х1.

Hereinafter, vectors are denoted by boldface instead of dashes over letters.

The boundary value conditions have the form:

where – the value of the required vector-function on the left-hand side x = 0 of dimension 8x1, – rectangular horizontal matrix of coefficients of boundary conditions of the left edge of dimension 4х8, – vector of external influences on the left edge of dimension 4х1,

– the value of the required vector-function on the right-hand side x = 1 of dimension 8x1, – rectangular horizontal matrix of coefficients of the boundary conditions of the right edge of dimension 4х8, – vector of external actions on the right edge of dimension 4х1.

In the case when the system of differential equations has a matrix with constant coefficients = const, the solution of Cauchy’s problem has the form [Gantmakher]:

,

where , where - is the unit matrix.

The matrix exponent can still be called Cauchy’s matrix and can be written as:

.

Then the solution of Cauchy’s problem can be written in the form:

,

where is the vector of a particular solution of an inhomogeneous system of differential equations.

From the theory of matrices [Gantmakher], the property of multiplication of matrix exponentials (Cauchy’s matrices) is known:

In the case when the system of differential equations has a matrix with variable coefficients , the solution of Cauchy’s problem is proposed, as is known, to be sought using the property of multiplication of Cauchy’s matrices. That is, the interval of integration is divided into small sections and in small parts Cauchy’s matrix approximately calculated by the formula for the constant matrix in the exponent. And then Cauchy’s matrices calculated on small sections are multiplied:

,

where Cauchy’s matrices are approximately computed by the formula:

, где .