Method of R-functions in problems on the nonsteady vibration of plates

Rvachev, V. L.; Kurpa, L. V.; Шевченко, А. Ф.

doi:10.1007/bf01529963

Cited by 15 publications

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“…The system of equations (4), supplemented with the appropriate boundary conditions, as well as a linear problem of free oscillations of multi-layered shallow shells, can be solved using RFM method [3,4] for almost any form of the shell's plan and various types of boundary conditions. Substituting the expression (3) for the unknown functions in the system (1) and applying the BubnovGalerkin procedure, we obtain the nonlinear ordinary differential equation…”

Section: The Solution Methodsmentioning

confidence: 99%

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Research of Nonlinear Vibrations of Laminated Shallow Shells with Cutouts by R-functions Method

Kurpa¹,

Timchenko²,

Osetrov³

2016

Bulletin NTU "KhPI": JDSM

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У цій роботі запропоновано ефективний метод дослідження геометрично нелінійних власних коливань тонкостінних елемен-тів конструкцій, які можуть моделюватися багатошаровими пологими оболонками зі складною формою в плані. Запропоно-ваний метод базується на сумісному використанні теорії R-функцій, варіаційних методів і процедури Бубнова-Галеркина. Це дозволяє звести спочатку нелінійну систему рівнянь руху до задачі Коші. Математична постановка задачі виконана у рамках уточненої теорії першого порядку. Створено відповідне програмне забезпечення в системі POLE-RL для поліноміальних результатів і з використанням C++ програми для сплайнів. Розв'язані нові задачі лінійних і нелінійних коливань багатошаро-вих пологих оболонок з вирізами. Для підтвердження надійності отриманих результатів проведено їх порівняння з отрима-ними за допомогою сплайн-апроксимації та з відомими із літератури. Досліджений ефект граничних умов на вирізі.Ключові слова: теорія R-функций, теорія Тимошенко, багатошарові пологі оболонки, нелінійні коливання.В настоящей работе предложен эффективный метод исследования геометрически нелинейных собственных колебаний тон-костенных элементов конструкций, которые могут моделироваться многослойными пологими оболочками со сложной фор-мой в плане. Предложенный метод базируется на совместном использовании теории R-функций, вариационных методов и процедуры Бубнова-Галеркина. Это позволяет свести изначально нелинейную систему уравнений движения к задаче Коши. Математическая постановка задачи выполнена в рамках уточненной теории первого порядка. Создано соответствующее про-граммное обеспечение в системе POLE-RL для полиномиальных результатов и используя C++ программы для сплайнов. Решены новые задачи линейных и нелинейных колебаний многослойных пологих оболочек с вырезами. Для подтверждения надежности получаемых результатов проведено их сравнение с полученными посредством сплайн-аппроксимации и извест-ными в литературе. Исследован эффект граничных условий на вырезе. Ключевые слова: теория R-функций, теория Тимошенко, многослойные пологие оболочки, нелинейные колебания.In present work an effective method to research geometrically nonlinear free vibrations of elements of thin-walled constructions that can be modeled as laminated shallow shells with complex planform is applied. The proposed method is based on joint use of Rfunctions theory, variational methods and Bubnov-Galerkin procedure. It allows reducing an initial nonlinear system of motion equations of a shallow shell to the Cauchy problem. The mathematical formulation of the problem is performed in a framework of the refined first-order theory. The appropriate software is created within POLE-RL program system for polynomial results and using C++ programs for splines. New problems of linear and nonlinear vibrations of laminated shallow shells with cutouts are solved. To confirm reliability of the obtained results their comparison with the ones obtained using spline-approximation and known in literature is provided. Effect of boundary condition on cutout is stud...

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Section: The Solution Methodsmentioning

confidence: 99%

“…Therefore, development of methods solving such problems is an important task. In this paper we propose a method based on the theory of R-functions [3,4] and variational methods, which allows one to solve such problems.…”

Section: Introductionmentioning

confidence: 99%

Research of Nonlinear Vibrations of Laminated Shallow Shells with Cutouts by R-functions Method

Kurpa¹,

Timchenko²,

Osetrov³

2016

Bulletin NTU "KhPI": JDSM

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“…This problem was first investigated in the twodimensional case for linear polygons in [42,2]. Later, Shapiro extended the algorithm to handle curved polygons [46].…”

Section: Introductionmentioning

confidence: 99%

An evolutionary approach to the extraction of object construction trees from 3D point clouds

Fayolle

Pasko

2016

Computer-Aided Design

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In order to extract a construction tree from a finite set of points sampled on the surface of an object, we present an evolutionary algorithm that evolves set-theoretic expressions made of primitives fitted to the input point-set and modeling operations. To keep relatively simple trees, we use a penalty term in the objective function optimized by the evolutionary algorithm. We show with experiments successes but also limitations of this approach.

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“…If this formula has the property of completeness relative to the boundary-value problem (1)- (2), that is, it is possible to choose an undetermined component 9 that converts (3) into a solution of the problem with any prescribed precision (in some sense), it is called a general structure solution of the problem in question. -We recall first how the R-function method solves the problem of constructing the functions co and col that occur in formulas of the form (3). A complicated locus (also called a geometric object or a drawing) which the region f2, and hence also its boundary 0f~, may be, can be described by a logical formula connecting simple loci (primitives) after which transition to the usual equations (resp.…”

mentioning

confidence: 99%

“…Here r is a known function; B is an operator depending on the shape of the boundary Og2 and its parts OO i, i = 1, 2 ..... m, formed so that for any choice of an undetermined component ~ from a set M formula (3) satisfies the boundary conditions (2) exactly. If this formula has the property of completeness relative to the boundary-value problem (1)- (2), that is, it is possible to choose an undetermined component 9 that converts (3) into a solution of the problem with any prescribed precision (in some sense), it is called a general structure solution of the problem in question.…”

mentioning

confidence: 99%

TheR-function method in boundary-value problems with geometric and physical symmetry

Rvachev¹,

Шейко²,

Shapiro³

1999

J Math Sci

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The traditional mathematical models of fields of various physical nature are boundary-value problems for partial differential equations. The usual statement of a boundary-value problem has the following form:where ffZ is the region in which the solution u is being sought; OfZi, i = 1, 2 ..... m, is a covering of the boundary t?~ of the region f) (the parts c~f~ i are not necessarily different and may be equal to Of)); f and (Pi are known functions (perturbing the field), vector-valued functions, tensors, or possibly elements of some other type. In the Rfunction method developed in the present article geometric information is taken into account by functions co, co i, chosen as a rule from among the elementary functions, so that 0~2 and 0f~i are the sets of their zeros, and the solution u is sought in the pencilHere r is a known function; B is an operator depending on the shape of the boundary Og2 and its parts OO i, i = 1, 2 ..... m, formed so that for any choice of an undetermined component ~ from a set M formula (3) satisfies the boundary conditions (2) exactly. If this formula has the property of completeness relative to the boundary-value problem (1)-(2), that is, it is possible to choose an undetermined component 9 that converts (3) into a solution of the problem with any prescribed precision (in some sense), it is called a general structure solution of the problem in question. -We recall first how the R-function method solves the problem of constructing the functions co and col that occur in formulas of the form (3). A complicated locus (also called a geometric object or a drawing) which the region f2, and hence also its boundary 0f~, may be, can be described by a logical formula connecting simple loci (primitives) after which transition to the usual equations (resp. inequalities) of the form to(x, y)= 0 (resp. co(x, y)> 0) recognized in analytic geometry is accomplished using R-functions. The technique of constructing them will be shown through examples below.In the majority of cases the following R-functions are used to construct the functions to, co/:

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Method of R-functions in problems on the nonsteady vibration of plates

Cited by 15 publications

References 0 publications

Research of Nonlinear Vibrations of Laminated Shallow Shells with Cutouts by R-functions Method

Research of Nonlinear Vibrations of Laminated Shallow Shells with Cutouts by R-functions Method

An evolutionary approach to the extraction of object construction trees from 3D point clouds

TheR-function method in boundary-value problems with geometric and physical symmetry

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