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Journal of Applied Mathematics and Mechanics 77 (2013) 445–451

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Journal of Applied Mathematics and Mechanics

journa l homepage: www.e lsev ier .com/ locate / jappmathmech

multiplicative method of solving boundary value problems of shellheory�

u.I. Vinogradovoscow, Russia

r t i c l e i n f o

rticle history:eceived 10 October 2012

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The boundary conditions are transferred to an arbitrarily chosen point by multiplication of matrices(multiplicatively). The transfer matrices of the boundary conditions are an analytic solution of a systemof first-order linear ordinary differential equations in canonical form of the mechanics of the deformationof shells in the form of values of Cauchy–Krylov functions. At an arbitrarily chosen point, the boundaryconditions are combined in a system of algebraic equations in matrix form, columns of the unknownquantities of which are parameters of the required values of the problem. The effectiveness of the method– the simplicity with which it can be realized on a computer, the low costs of computer time and theRAM – is based on the multiplicative transfer of the boundary conditions into matrix form. The class ofproblems is limited by the possibilities of the Fourier method of separation of the variables in partialdifferential equations.

© 2013 Elsevier Ltd. All rights reserved.

. Formulation of the problem

Suppose a system of linear ordinary differential equations is reduced to canonical matrix form and is represented as follows:

(1.1)

where y(x) is the transposed column of the required quantities, A(x) =∥∥ai,j

∥∥n

iis a matrix, the elements of which are the coefficients of

he system of ordinary differential equations and f(x) is a column of parameters of the right-hand sides of the system of ordinary differentialquations.

It is required to determine the solution in matrix form of ordinary differential equation (1.1) for the specified boundary conditions

(1.2)

ere

is the number of boundary conditions for x = 0, n is the order of the ordinary differential equations (1.1), Ht(0) and Hk(l) are rectangularatrices, the non-zero elements of which in the form of unities are entered for the choice of the elements of the columns of the required

uantities on the boundaries x = 0 and x = l:

n which known boundary conditions are imposed, and rl(0) = ‖r1, r2. . ., rs‖Tx=0 and rk(I) = ‖r1, r2, . . ., rn−s‖T

x=l are columns, the non-zerolements of which are specified values of the physical quantities or their parameters on the boundaries.

� Prikl. Mat. Mekh., Vol. 77, No. 4, pp. 620–628, 2013.E-mail address: vino.yuri@rambler.ru

021-8928/$ – see front matter © 2013 Elsevier Ltd. All rights reserved.ttp://dx.doi.org/10.1016/j.jappmathmech.2013.11.013

446 Yu.I. Vinogradov / Journal of Applied Mathematics and Mechanics 77 (2013) 445–451

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Fig. 1.

The solution of the boundary value problem is obtained by transferring boundary conditions (1.2) to an arbitrarily chosen point ofhe boundary interval1 using the solution2 of canonical matrix ordinary differential equation (1.1) and by solving a system of algebraicquations for the boundary conditions, combined at this point. The values of the required quantities, characterizing, for example, thetrength and stiffness of the shell, are then determined.

. The algorithm for transferring the boundary conditions when calculating the values of the Cauchy–Krylov function inirections from an arbitrarily chosen point of the boundary interval

The general solution of ordinary differential equation (1.1) has the form

(2.1)

f

(2.2)

hen

nd the general solution of the uniform ordinary differential equation has the form

(2.3)

particular solution of the ordinary differential equation is found using the formula2

(2.4)

For condition (2.2) the quantities Kxk−1xk= K and Tk = T are the same for all the equal intervals �xk = xk−1 − xk = �x(k = 1, 2, . . ., n), and the

articular solution y*(xi, xi – 1) in the fundamental interval takes the form

(2.5)

ere the superscript on the matrix K denotes raising to a power.We will abbreviate the writing of solution (2.1):

(2.6)

The values of the Cauchy–Krylov functions of the matrix Kxi−1xiof the homogeneous ordinary differential equation and the particular

olution of the inhomogeneous ordinary differential equation y∗xi−1i

are calculated in the direction from the arbitrarily chosen point xi tohe left boundary, i.e., from the point xi to the point xi – 1 (see the figure, where we show the directions in which the solutions of ordinaryifferential equation (1.1) are calculated). Fig. 1.

We transfer the boundary conditions on the left boundary x = 0 at the point x = x1 as follows. Solution (2.6) for the interval (x0, x1), tohe left of the boundary, takes the form

(2.7)

Eliminating y(0) from condition (1.2), we obtain

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Yu.I. Vinogradov / Journal of Applied Mathematics and Mechanics 77 (2013) 445–451 447

The transfer of the boundary conditions to the point x2 is carried out in the same way as the transfer of the boundary conditions for x =to the point x1. Continuing the iterations, we obtain a formula for transferring the boundary conditions on the left boundary x = 0 to an

rbitrarily chosen point xi when calculating the values of the Cauchy–Krylov function in intervals in the direction of the left boundary

(2.8)

here

(2.9)

r

If ordinary differential equation (1.1) has constant coefficients, i.e., conditions (2.2) are satisfied and the lengths of the intervals (xi−1,i)(i = 1, 2, . . ., n) are equal, the matrices Kxi−1xi

of the values of the Cauchy–Krylov functions are the same. Iterational formulae (2.9) thenake the form

(2.10)

The superscript i is the exponent of the matrix, calculated in one of the intervals (xi–1, xi) (i = 1, 2,..., n).For the last interval (xn – 1, xn), lying to the right of the boundary, the solution of ordinary differential equation (1.1) is written in the

orm

Using this solution, we eliminate the columns y(l) from conditions (1.2) on the right boundary and, in this way, we transfer them to theoint xn – 1. Then

We similarly transfer the boundary conditions from the point xn – 1 to the point xn–2. Continuing the iteration, we obtain a formula forransferring the conditions on the right boundary to an arbitrarily chosen point xi when calculating the values of Cauchy–Krylov functionsn intervals in the direction of the right boundary. We have

(2.11)

here

(2.12)

For conditions (2.2) and equal lengths of the intervals, the matrices Kxi+2i+1 of values of the Cauchy–Krylov function are the same, and the

terational formulae (2.12) take the form

(2.13)

We combine boundary conditions (2.8) and (2.11), transferred to an arbitrarily chosen point xi, in a system of algebraic equations, thenknowns of which are the required quantities (in the case of a shell, characterizing its stress-strain state):

(2.14)

The solution of the boundary value problem for the points xi (the values of the argument) is completed by the determination of theolumn of the required quantities

(2.15)

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48 Yu.I. Vinogradov / Journal of Applied Mathematics and Mechanics 77 (2013) 445–451

. An algorithm for transferring the boundary conditions when calculating the values of the Cauchy–Krylov functions in theirection to an arbitrarily chosen point of the boundary interval

The general solution of ordinary differential equation (1.1) for calculating the values of the Cauchy–Krylov function in the assumedositive direction from xi – 1 to xi is written in the form of representation (2.6) with the replacement of the subscripts i ↔ i − 1, while thealculation of the solutions, shown in the figure, are carried out in the opposite directions. Proceeding in the same way as in Section 2, weransfer the boundary conditions at x = 0 to an arbitrarily chosen point xi:

(3.1)

he process is iterational and is carried out in accordance with the formulae

(3.2)

r

(3.3)

If ordinary differential equation (1.1) has constant coefficients and the lengths of the intervals are the same, iterational formulae (3.3)ake the form

(3.4)

he superscripts on the matrix K denote raising to the corresponding power.We transfer the boundary conditions on the right boundaryo an arbitrarily chosen point xi:

(3.5)

he process of transferring the boundary conditions is iterational and is carried out in accordance with the formulae

(3.6)

r

(3.7)

If ordinary differential equations (1.1) have constant coefficients and the lengths of the intervals are the same, iterational formulae (3.6)nd (3.7) take the form

(3.8)

he superscripts on the matrix K denote raising to the corresponding power.We combine boundary conditions (3.1) and (3.5), transferred to an arbitrarily chosen point xi, in system of algebraic equations (2.14),

he unknowns of which are the required quantities. The solution of the boundary value problem is completed in the same way.

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Yu.I. Vinogradov / Journal of Applied Mathematics and Mechanics 77 (2013) 445–451 449

. Orthonormalization of the boundary conditions

The differential equations of the mechanics of shell deformation belong to the “rigid” class. The solution of boundary value problemsased on them using a computer requires special procedures to ensure stability of the calculation. For the numerical integration of ordinaryifferential equations, Godunov, for example, proposed periodically carrying out procedures for the orthonormalization of the results ofhe integration.

Algorithms of the multiplicative method of transferring boundary conditions to an arbitrarily chosen point of the boundary interval,lso do not ensure stability of the calculation on a computer when solving boundary value problems. However, we further propose to userocedures of row orthonormalization of the boundary conditions, transferred to the point xi.

For brevity and generality, the boundary conditions can be written in the form

(4.1)

Orthonormalization is carried out in the following sequence:�11 =

√h1h1, where h1h1 is the scalar product of vectors (rows of the matrix H with dimension m);

orthonormalized rows (we will denote them by h̃i) are determined sequentially

tc.;we also change the elements of the column r on the right-hand sides of the boundary conditions

tc.As a result of the orthonormalization, the boundary conditions take the following matrix form

(4.2)

In general form, the iterational formulae for calculating the matrix H of the boundary conditions and the columns r of the right-handides, when the procedure of orthonormalization of the boundary conditions is carried out, are written as follows:

when the boundary conditions are transferred from the left boundary of the main interval to an arbitrarily chosen point x1

(4.3)

and when the boundary conditions are transferred from the right boundary of the main interval to the same point x1 in formulae (4.6),he indices are changed as follows: I → k, i − 1 → i + 1.

The proposed orthonormalization procedure when transferring the boundary conditions do not change the columns y(xi), the elementsf which are the required quantities of the boundary value problem.

. Monitoring of the calculation errors

We will rewrite the general solutions of ordinary differential equations (1) in the form

(5.1)

(5.2)

Solutions (5.1) and (5.2) for the same interval (xi – 1, xi) should be the same, and hence the solutions for the homogeneous ordinary

ifferential equation (1.1) should be the same, i.e.,

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50 Yu.I. Vinogradov / Journal of Applied Mathematics and Mechanics 77 (2013) 445–451

Hence, the absolute error of the calculations is determined by the difference of the matrix E from the unit matrix. It follows from thequality of the particular solutions in formulae (5.1) and (5.2) that the error of the calculation of the particular solution is given by thequation

. Comparison of the multiplicative method and Godunov’s method

Godunov’s method has long been used to calculate the strength of thin-walled structures, and computational systems have beenonstructed based on it. The important general features of Godunov’s method and of the multiplicative method is the transfer of theoundary conditions and orthonormalization to ensure stability of the calculation. These features connect them. It is natural to estimatend compare the effectiveness of these methods (the ease of use of the method on a computer, the reduction in the computer time andemory costs, and also the possibility of monitoring the error of the calculation).Godunov’s method is characterized by the following important features.

. The initial ordinary differential equations are written in the form of a system of first-order ordinary differential equations.

. The system of ordinary differential equations is solved by the numerical Runge–Kutta method of the fourth degree, taking the boundaryconditions into account: in the first interval an inhomogeneous ordinary differential equation with initial conditions on the givenboundary is solved. In the second and subsequent integration intervals, the initial conditions are the solutions of the correspondingordinary differential equations, calculated in the previous interval (step).

. Integration of the system of ordinary differential equations is accompanied by orthogonalization and normalization of the results of theintegration at the end of each interval of stable calculation. A conversion matrix is formed, and is stored after each orthonormalizationof the solution.

. The solutions of the initial ordinary differential equations at the end of the fundamental (boundary) interval form a system of algebraicequations, the right-hand side of which is the column of the condition specified on this boundary, by means of which the integrationconstants are determined.

. In the reverse run of the calculations, using conversion matrices, the unknown integration constants are determined at each step.

. If in the forward run of the calculations, the orthonormalized columns of the solutions are not stored at each step, the boundary valueproblem is solved once again but with certain integration constants. As a result the columns of the solution of the boundary valueproblem are determined.

he multiplicative method has the following important features.

. The initial ordinary differential equations are written in the form of a system of first-order ordinary differential equations.

. The solution of the matrix ordinary differential equation is determined by formulae.

. The boundary conditions in matrix form are transferred to an arbitrarily chosen point of the main (boundary) interval using a matrix ofvalues of the Cauchy–Krylov functions – the solution of the matrix ordinary differential equation. The algorithm of the transfer of theboundary conditions is realized solely by multiplication and addition of the matrices of the values of the Cauchy–Krylov functions.

. If necessary, to stabilize the calculation, line-by-line orthonormalization of the boundary conditions transferred in matrix form is carriedout without converting the column of the required values of the boundary value problem. The conversion matrices are not stored.

. The boundary conditions, transferred to an arbitrarily chosen point of the main interval, are formed in a system of algebraic equationfor determining the column of the required values.

omparison of the multiplicative method and Godunov’s method reveals their basically different theoretical basis.The solution of linear ordinary differential equations – the mathematical model of the mechanics of shell deformation in the multiplica-

ive method, is determined, not by step-by-step numerical procedures, but in the form of formulae and ignoring the boundary conditions.he formulae enable one to calculate the solutions with a controlled error. The solution of the ordinary differential equations, ignoring theoundary conditions, involves combining the feature of the multiplicative method and the method of integration of systems of ordinaryifferential equations using a computer. However, the method of integration, and also the step-by-step Runge–Kutta method, solves theroblem with an accuracy to within the integration constants, which are determined when solving the system of algebraic equations, con-tructed using the specified boundary conditions. The proposed formulae for solving the system of ordinary differential equations from thehole manifold of systems of fundamental functions enable the Cauchy–Krylov functions (their values), which satisfy the arbitrary initial

onditions, to be determined (calculated). As a result of this property of the functions, the boundary value problem in the multiplicativeethod is reduced to solving a system of algebraic equations, not to determine the constants (the arbitrary integration constants), but to

etermine the required values of the problem.The computational advantages of the multiplicative method are the possibilities of monitoring the calculation errors. To ensure stability

f the calculation, both methods require the performance of orthogonalization and normalization operations. However, in the Godunovethod, the solutions of the systems of ordinary differential equations are subjected to this operation, which, after the boundary value

roblem has been solved, must be reestablished using the conversion matrices stored in the computer. In the multiplicative method, theolumn of the required quantities is stored unchanged until the system of algebraic equations is formed. Consequently, there is no need totore the conversion matrices in the computer.

The necessary number of intervals for which orthogonalization and normalization are required, exceeds severalfold (often by an orderf magnitude) in the multiplicative method the necessary length of the intervals in Godunov’s method.

When solving the problem using Godunov’s method, numerical integration of the system of ordinary differential equations is carriedut using a computer over the whole fundamental (boundary) interval. There is therefore a buildup of errors. When using the multiplicative

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Yu.I. Vinogradov / Journal of Applied Mathematics and Mechanics 77 (2013) 445–451 451

ethod, the solution in each interval of stable calculation, in which the basic interval is divided up, is determined independently and, ifecessary, with a controlled error.

The simplicity of the computational algorithm of the multiplicative method is the fact that its realization can be limited to the multi-lication and addition of matrices to construct the necessary system of algebraic equations.

In the second algorithm, where inversion of the matrix occurs, this operation is not carried out on the computer.The matrix of the Cauchy–Krylov functions is the matrix exponent eAx, the property of which enables the sign of matrix inversion to be

aken as its negative power: [eAx]−1 = e−Ax. Consequently, the matrix series of the calculation of the matrix exponents eAx and [eAx]

−1 = e−Ax

iffer solely in signs. The second matrix series is sign-variable. Hence, there is no calculation operation of matrix inversion in the secondlgorithm, as in the first.

Computational experiments confirm the theoretically validated effectiveness of the multiplicative method.

. Computational experiments

We solved the simplest problems of axisymmetric deformation of cylindrical, conical and spherical isotropic shells of constant thicknesssing the Vlasov mathematical model.3 The shells were fastened in cantilever style (for a cone and part of a sphere along the large crossection) and loaded uniformly along the edge with distributed ring forces. The need for orthonormalization did not arise when solving theroblems. For example, for a cylindrical shell with parameters R/h = 50 – 300 and l/R = 1 – 10, the calculation without orthonormalizationas stable.

Similar problems were solved when the shells were loaded at the edge with forces and moments, distributed over part of the arc ofcircle. We used the multiplicative method and Godunov’s method. When the relative thickness of the shells R/h was changed from 50

o 300, the critical length of the stable calculation when using the multiplicative method exceeded the corresponding value when usingodunov’s method by a factor from 11 to 5. The computer costs using the multiplicative method was one-two orders of magnitude less.

We solved problems for shells made of a composite material (the Grigorenko and Vasilenko mathematical models4). In all cases theultiplicative method turned out to be more effective. It can be recommended for calculating thin-walled structures when designing, for

xample, aircraft.

eferences

. Vinogradov AYu, Vinogradov YuI. A method of transferring the boundary conditions of Cauchy–Krylov functions for stiff linear ordinary differential equations. Dokl RossAkad Nauk 2000;373(4):474–6.

. Vinogradov YuI. A method of solving linear ordinary differential equations. Dokl Ross Akad Nauk 2006;409(1):15–8.

. Vlasov VZ. Collected Papers. Vol. 1. Moscow: Izd Akad Nauk SSSR; 1962.

. Grigorenko YaM, Vasilenko AT. Problems of the Statics of Anisotropic Inhomogeneous Shells. Moscow: Nauka; 1992.

Translated by R.C.G.