The Solution of the Shells Theory Problems by the Numerical-Analytical Boundary Elements Method

2019 ◽  
Vol 968 ◽  
pp. 460-467
Author(s):  
Yurii Krutii ◽  
Mykola Surianinov ◽  
Vitalii Chaban
Keyword(s):  
Differential Equation ◽  
Strain State ◽  
Order Differential Equation ◽  
Arbitrary Point ◽  
Boundary Elements ◽  
System Of Equations ◽  
Eighth Order ◽  
Boundary Elements Method ◽  
Long Cylindrical Shell ◽  
Analytical Expressions

The application of the numerical-analytical boundary elements method (NA BEM) to the calculation of shells is considered. The main problem here is due to the fact that most of the problems of statics, dynamics and stability of shells are reduced to solving an eighth-order differential equation. As a result, all analytical expressions of the NA BEM (fundamental functions, Green functions, external load vectors) turn out to be very cumbersome, and intermediate transformations are associated with eighth-order determinants. It is proposed along with the original differential equation to consider an equivalent system of equations for the unknown state vector of the shell. In this case, calculations of some analytical expressions related to high-order determinants can be avoided by using the Jacobi formula. As a result, the calculation of the determinant at an arbitrary point reduces to its calculation at the point , which leads to a significant simplification of all analytical expressions of the numerical-analytical boundary elements method. On the basis of the proposed approach, a solution is obtained of the problem of bending a long cylindrical shell under the action of an arbitrary load, the stress-strain state of which is described by an eighth-order differential equation. The results can be applied to other types of shells.

scholarly journals For the calculation of flat shells by the numerical analytical method of boundary elements

2021 ◽  
Vol 2 (92) ◽  
pp. 37
Author(s):  
Mykola Suryaninov ◽  
Oleksii Boiko
Keyword(s):  
Differential Equation ◽  
Shallow Shell ◽  
Arbitrary Point ◽  
Boundary Elements ◽  
Order Partial Differential Equation ◽  
Eighth Order ◽  
Shallow Shells ◽  
Direct Integration Method ◽  
Boundary Elements Method ◽  
Analytical Expressions

Abstract. The application of the numerical-analytical boundary elements method (NA BEM) to the calculation of shallow shells is considered. The method is based on the analytical construction of the fundamental system of solutions and the Green’s function for the differential equation of the problem under consideration. The theory of calculation of a shallow shell proposed by V. Z. Vlasov, which for the problem under consideration leads to an eighth-order partial differential equation. The problem of bending a shallow shell is two-dimensional, and in the numerical-analytical boundary elements method, the plate and shell are considered in the form of generalized one-dimensional modules, therefore, the Fourier separation method and the Kantorovich-Vlasov variational method were applied to this equation, which made it possible to obtain ordinary differential equations of the eighth order. It is noted that until recently, the main problem in the subsequent implementation of the algorithm of the numerical-analytical boundary element method was due to the fact that all analytical expressions of the method (fundamental functions, Green’s functions, vectors of external loads) are very cumbersome, and intermediate transformations are associated with determinants of the eighth order. It is proposed to use the direct integration method at the first stage, when, along with the original differential equation, an equivalent system of equations for the unknown shell state vector is considered. In this case, the calculations of some analytic expressions associated with determinants of higher orders can be avoided by using the Jacobi formula. As a result, the calculation of the determinant at an arbitrary point is reduced to its calculation at a zero value of the argument, which leads to a significant simplification of all intermediate transformations and analytical expressions of the numerical-analytical boundary elements method.

scholarly journals To the solution of the problem of bending of a cylindrical shell by the boundary elements method

2018 ◽  
Vol 230 ◽  
pp. 02032 ◽  
Author(s):  
Mykola Surianinov ◽  
Yurii Krutii
Keyword(s):  
Differential Equation ◽  
Cylindrical Shell ◽  
Boundary Elements ◽  
Order Partial Differential Equation ◽  
Eighth Order ◽  
Various Boundary Conditions ◽  
Analytical Construction ◽  
Load Vector ◽  
Plates And Shells ◽  
Boundary Elements Method

The solution of the problem of the long cylindrical shell bending by a numerical and analytical boundary elements method is considered. The method is based on the analytical construction of a fundamental system of solutions and Green’s functions for the differential equation of the problem under consideration. This paper is devoted to the determination of these expressions. The semi-moment theory of the cylindrical shell calculation, proposed by V.Z. Vlasov, which for the problem under consideration leads to one eighth-order partial differential equation is used. The problem of the bending of a cylindrical shell is twodimensional, and in the numerical and analytical boundary elements method, plates and shells are considered as generalized one-dimensional modules, so the variational method of Kantorovich-Vlasov was applied to this equation to obtain an ordinary differential equation of the eighth order. Sixty-four expressions of all the fundamental functions of the problem are constructed, as well as an analytic expression for the Green’s function, which makes it possible to construct a load vector (without any restrictions on the nature of its application), and then proceed to the solution of boundary-value problems for the bending of long cylindrical shells under various boundary conditions.

scholarly journals INTERACTION OF A LONG PILE OF FINITE STIFFNESS WITH SURROUNDING SOIL AND FOUNDATION CAP

Vestnik MGSU ◽  
2015 ◽  
pp. 72-83
Author(s):  
Armen Zavenovich Ter-Martirosyan ◽  
Zaven Grigor’evich Ter-Martirosyan ◽  
Tuan Viet Trinh
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Carrying Capacity ◽  
Strain State ◽  
Order Differential Equation ◽  
Deformation Modulus ◽  
First Case ◽  
Equivalent Modulus ◽  
Underlying Soil ◽  
Surrounding Soil

The article presents the formulation and analytical solution to a quantification of stress strain state of a two-layer soil cylinder enclosing a long pile, interacting with the cap. The solution of the problem is considered for two cases: with and without account for the settlement of the heel and the underlying soil. In the first case, the article is offering equations for determining the stresses of pile’s body and the surrounding soil according to their hardness and the ratio of radiuses of the pile and the surrounding soil cylinder, as well as formulating for determining equivalent deformation modulus of the system “cap-pile-surrounding soil” (the system). Assessing the carrying capacity of the soil under pile’s heel is of great necessity. In the second case, the article is solving a second-order differential equation. We gave the formulas for determining the stresses of the pile at its top and heel, as well as the variation of stresses along the pile’s body. The article is also formulating for determining the settlement of the foundation cap and equivalent deformation modulus of the system. It is shown that, pushing the pile into underlying layer results in the reducing of equivalent modulus of the system.

scholarly journals Regime of small number of photons in the cavity for a singleemitter laser

2021 ◽  
Vol 2103 (1) ◽  
pp. 012158
Author(s):  
N V Larionov
Keyword(s):  
Differential Equation ◽  
Exact Solutions ◽  
Master Equation ◽  
Cavity Mode ◽  
Photon Number ◽  
System Of Equations ◽  
Equation Approach ◽  
Master Equation Approach ◽  
Single Emitter ◽  
Analytical Expressions

Abstract The model of a single-emitter laser generating in the regime of small number of photons in the cavity mode is theoretically investigated. Based on a system of equations for different moments of the field operators the analytical expressions for mean photon number and photon number variance are obtained. Using the master equation approach the differential equation for the phase-averaged quasi-probability Q is derived. For some limiting cases the exact solutions of this equation are found.

scholarly journals Stability of Atangana - Baleanu Fractional Order Differential Equation with Numerical Approximation

2021 ◽  
Vol 2070 (1) ◽  
pp. 012086
Author(s):  
A. George Maria Selvam ◽  
S. Britto Jacob
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Numerical Scheme ◽  
Numerical Approximation ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Banach Fixed Point Theorem ◽  
Stability Conditions ◽  
System Of Equations ◽  
Fractional Order Differential Equation

Abstract The field of Fractional calculus is more useful to understand the real-world phenomena. In this article, a nonlinear fractional order differential equation with Atangana-Baleanu operator is considered for analysis. Sufficient conditions under which a solution exists and uniqueness are presented using Banach fixed-point theorem method. The well-established Adams-Bashforth numerical scheme is used to solve the system of equations. Stability conditions are presented in details. To corroborate the analytical results, an example is given with numerical simulation. Mathematics Subject Classification [2010]: 26A33, 35B35, 65D25, 65L20.

scholarly journals On the limit cycles for a class of eighth-order differential equations

2020 ◽  
Vol 6 (1) ◽  
pp. 53-61
Author(s):  
Chems Eddine Berrehail ◽  
Zineb Bouslah ◽  
Amar Makhlouf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Function ◽  
Periodic Solutions ◽  
Limit Cycles ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Rational Numbers ◽  
Eighth Order ◽  
Existence Of Periodic Solutions

AbstractIn this article, we provide sufficient conditions for the existence of periodic solutions of the eighth-order differential equation {x^{\left( 8 \right)}} - \left( {1 + {p^2} + {\lambda ^2} + {\mu ^2}} \right){x^{\left( 6 \right)}} + A\ddddot x + B\ddot x + {p^2}{\lambda ^2}{\mu ^2}x = \varepsilon F\left( {t,x,\dot x,\ddot x,\dddot x,\ddddot x,{x^{\left( 5 \right)}},{x^{\left( 6 \right)}}{x^{\left( 7 \right)}}} \right), where A = p2λ2 + p2µ2 + λ2µ2 + p2 + λ2 + µ2, B = p2 λ2 + p2µ2 + λ2µ2 + p2λ2µ2, with λ, µ and p are rational numbers different from −1, 0, 1, and p ≠ ±λ, p ≠±µ, λ ≠±µ, ɛ is sufficiently small and F is a nonlinear non-autonomous periodic function. Moreover we provide some applications.

Ion cyclotron electromagnetic wave structure in Elmo bumpy torus

1985 ◽  
Vol 34 (2) ◽  
pp. 191-211
Author(s):  
M. Cotsaftis ◽  
N. T. Gladd ◽  
N. A. Krall
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Wave Structure ◽  
Second Order ◽  
System Of Equations ◽  
Simple Mode ◽  
Ion Cyclotron ◽  
Plasma Radius ◽  
The Mean ◽  
Two Parameters

The problem of ion cyclotron wave structure in EBT has been analyzed using the smallness of the inverse aspect ratio ε = a/R0 and of the inverse cavity number ε' = 1/N. The procedure is to expand in these two parameters, reducing the complete toroidal problem to a system of equations to be solved in sequence. To second order in ε and ε', this system contains two ordinary differential equations of second order and one partial differential equation with periodic coefficients in a magnetically adapted system of co-ordinates. The smallness of the mean bumpiness parameter εB reduces the problem to a single second-order differential equation which, for a parabolic density profile, is a Whittaker equation. The EM wave structure corresponds to a simple mode with only a few wavelengths across the plasma radius, permitting multi-harmonic ion cyclotron heating with interesting efficiency, as observed in experiments.

Eigenvalue Problem for the Second Order Differential Equation with Nonlocal

2006 ◽  
Vol 11 (1) ◽  
pp. 13-32 ◽  
Author(s):  
B. Bandyrskii ◽  
I. Lazurchak ◽  
V. Makarov ◽  
M. Sapagovas
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Variable Coefficient ◽  
Order Differential Equation ◽  
Integral Condition ◽  
Second Order ◽  
Computational Results ◽  
Discrete Method ◽  
Complex Eigenvalues ◽  
Nonlocal Integral Condition

The paper deals with numerical methods for eigenvalue problem for the second order ordinary differential operator with variable coefficient subject to nonlocal integral condition. FD-method (functional-discrete method) is derived and analyzed for calculating of eigenvalues, particulary complex eigenvalues. The convergence of FD-method is proved. Finally numerical procedures are suggested and computational results are schown.

Difference Schemes for Nonlinear BVPS on the Semiaxis

2007 ◽  
Vol 7 (1) ◽  
pp. 25-47 ◽  
Author(s):  
I.P. Gavrilyuk ◽  
M. Hermann ◽  
M.V. Kutniv ◽  
V.L. Makarov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Adaptive Algorithm ◽  
Order Differential Equation ◽  
Constructive Method ◽  
Numerical Examples ◽  
Exact Difference ◽  
The Given

Abstract The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

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