scholarly journals Приближенные граничные условия для задачи нахождения оптических коэффициентов ультратонких металлических пленок в СВЧ и ТГЦ диапазонах

2020 ◽  
Vol 128 (9) ◽  
pp. 1327
Author(s):  
П.С. Глазунов ◽  
В.А. Вдовин ◽  
В.Г. Андреев
Keyword(s):  
Boundary Conditions ◽  
Maximum Error ◽  
Dielectric Substrate ◽  
Thickness Dependence ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Transmission Coefficients ◽  
Fabry Perot ◽  
Approximate Boundary Conditions ◽  
Analytical Expressions

Approximate boundary conditions for a problem of calculating the optical coefficients of a system composed of inhomogeneous ultrathin metallic film with an arbitrary thickness dependence of conductivity deposited on dielectric substrate are obtained. The derivation of the boundary conditions is based on the Picard method of successive approximations. Analytical expressions for the errors in calculating the optical coefficients with use of the proposed approximate boundary conditions are presented. It is shown that the error increases with the frequency and the film thickness increasing. The maximum error for films of 10 nm-thickness does not exceed 10.7% at 1 THz. As an example, the complex optical coefficients of a system similar to Fabry-Perot etalon and a metal film without a substrate with model thickness dependence of conductivity are calculated. The coincidence between the results of numerical simulation and calculations performed with approximate boundary conditions is shown. The possibility of direct calculating the average conductivity of a film from experimentally measured reflection and transmission coefficients is demonstrated.

scholarly journals Application of the successive approximation method to the analysis of bending plates

2018 ◽  
Vol 251 ◽  
pp. 04058
Author(s):  
Radek Gabbasov ◽  
Vladimir Filatov ◽  
Nikita Ryasny
Keyword(s):  
Boundary Conditions ◽  
Successive Approximation ◽  
Approximation Method ◽  
High Accuracy ◽  
Numerical Results ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Successive Approximation Method ◽  
Medium Thickness ◽  
Bending Plates

This work presents an algorithm for calculating the bending plates of medium thickness according to the Reissner’. To obtain numerical results, the method of successive approximations (MSA) is used. This method has high accuracy and fast convergence, which was confirmed by the solution of a range of tasks. Publication of the results of the calculation of plates of medium thickness with the boundary conditions revised here is supposed to be in the following articles.

scholarly journals INTEGRAL PARAMETERS OF CONCRETE DIAGRAMS FOR CALCULATIONS OF STRENGTH OF REINFORCED CONCRETE ELEMENTS USING THE DEFORMATION MODEL

2020 ◽  
Vol 16 (1) ◽  
pp. 25-37
Author(s):  
Vladimir Eryshev ◽  
Nickolay Karpenko ◽  
Artur Zhemchuyev
Keyword(s):  
Reinforced Concrete ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Deformation Model ◽  
Concrete Element ◽  
Normative Documents ◽  
Functional Relationships ◽  
Regulatory Documents ◽  
Base Points ◽  
Analytical Expressions

In accordance with the requirements of regulatory documents, restrictions are introduced on stress levels at the end of the falling branch of the diagrams at the maximum normalized strain values. We have developed mathematical models that establish a uniform sequence for calculating the unambiguous values of deformations at the base points of concrete diagrams, taking into account the accepted functional relationships and the rules for their use according to the tables of normative documents. It was shown that for equal values of deformations and stresses at base points, analytical expressions of diagram recommended by regulatory documents, even if it differs in structure, give identical outlines, diagram branches coincide. The correlation between the calculation models by Russian and foreign regulatory documents was established by comparing the values of the integral parameters of the diagrams and the ultimate forces obtained by calculating the reinforced concrete element according to the deformation model. As integral parameters of concrete deformation diagrams, it was recommended to use areas bounded by diagram branches and diagram completeness coefficients. Analytical modeling of integral parameters allowed us to exclude the procedure for numerically summing stresses along elementary strips in a section and solving nonlinear equations by the method of successive approximations when calculating the strength of an element.

scholarly journals THE METHOD OF SUCCESSIVE APPROXIMATIONS IN THE MATHEMATICAL THE-ORY OF SHALLOW SHELLS OF ARBITRARY THICKNESS

World Science ◽  
2019 ◽  
Vol 1 (11(51)) ◽  
pp. 31-39
Author(s):  
Zelensky A. G.
Keyword(s):  
Boundary Conditions ◽  
High Efficiency ◽  
Boundary Value ◽  
Three Dimensional ◽  
Median Plane ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Shallow Shells ◽  
Common Solution ◽  
Arbitrary Thickness

The method of sequential approximations (MSA) in mathematical theory (MT) of transversal-isotropic shallow shells of arbitrary thickness is developed. MT takes into account all components of stress-strain state (SSS). SSS and boundary conditions are considered to be functions of three varia-bles. Three-dimensional problems are reduced to two- dimensional decompositions of all the compo-nents of the SSS into series in the transverse coordinate using Legendre polynomials and using the Reisner variational principle. The boundary conditions for stresses on the front surfaces of the shell are fulfilled precisely. Previous studies have shown the high efficiency of this MT. The boundary-value problem for a shallow shell is reduced to sequences of two boundary-value problems for the respective plates. One sequence describes symmetric deformation relative to the median plane, and the other sequence is skew symmetric. MSA makes it easier to find a common solution of differential equations (DE) for shallow shells. Highly accurate results for SSS are already in the first approxi-mation. MSA can be used when solving problems for shallow shells by other theories.

Stresses in a Stretched Slab Having a Spherical Cavity

1959 ◽  
Vol 26 (2) ◽  
pp. 235-240
Author(s):  
Chih-Bing Ling
Keyword(s):  
Boundary Conditions ◽  
Stress Function ◽  
Spherical Cavity ◽  
Analytic Solution ◽  
Linear Equations ◽  
Hankel Transform ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Biharmonic Functions ◽  
Infinite Slab

Abstract This paper presents an analytic solution for an infinite slab having a symmetrically located spherical cavity when it is stretched by an all-round tension. The required stress function is constructed by combining linearly two sets of periodic biharmonic functions and a biharmonic integral. The sets of biharmonic functions are derived from two fundamental functions specially built up for the purpose. The arbitrary functions involved in the biharmonic integral are first adjusted to satisfy the boundary conditions on the surfaces of the slab by applying the Hankel transform of zero order. Then the stress function is expanded in spherical co-ordinates and the boundary conditions on the surface of the cavity are satisfied by adjusting the coefficients of superposition attached to the sets of biharmonic functions. The resulting system of linear equations is solved by the method of successive approximations. The solution is finally illustrated by numerical examples for two radii of the cavity.

scholarly journals On the asymptotic behavior of the spectrum of a sixth-order differential operator, whose potential is the delta function

2020 ◽  
Vol 22 (3) ◽  
pp. 280-305
Author(s):  
Sergey I. Mitrokhin
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Delta Function ◽  
Discontinuous Coefficients ◽  
Sixth Order ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Order Differential Operator ◽  
Indicator Diagram ◽  
Dirac Delta

In this paper we propose a new method for studying differential operators with discontinuous coefficients.We consider a sequence of sixth-order differential operators with piecewise-smooth coefficients. The limit of the sequence of these operators’ potentials is the Dirac delta function. The boundary conditions are separated. To correctly determine solutions of differential equations with discontinuous coefficients at the points of discontinuity, “gluing” conditions are required. Asymptotic solutions were written out for large values of the spectral parameter, with the help of them the “gluing” conditions were studied and the boundary conditions were investigated. As a result, we derive an eigenvalues equation for the operator under study, which is an entire function. The indicator diagram of the eigenvalues equation, which is a regular hexagon, is investigated. In various sectors of the indicator diagram, the method of successive approximations has been used to find the eigenvalues asymptotics of the studied differential operators. The limit of the asymptotic of the spectrum determines the spectrum of the sixth-order operator, whose potential is the delta function.

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