Convergence of the finite element method and the semi-analytical finite element method for prismatic bodies with variable physical and geometric parameters

Abstract

In this paper, a numerical study of the convergence of solutions obtained on the basis of the developed approach [1, 3, 4, 5] is carried out. A wide range of test problems for bodies with smoothly and abruptly varying physical and geometric characteristics in elastic and elastic-plastic formulation are considered. The approach developed within the framework of the semi-analytical method to study the stress-strain state of inhomogeneous curvilinear prismatic bodies, taking into account physical and geometric nonlinearity, requires substantiation of its effectiveness in relation to the traditional FEM and confirmation of the reliability of the results obtained on its basis.
 The main indicators that allow comparing the SAFEM and FEM include the rate of convergence of solutions with an increase in the number of unknowns and the amount of charges associated with solving linear and nonlinear equations. For the considered class of problems, the convergence is determined by such factors as the nature of the change along Z3’ of the geometric and mechanical parameters of the object. The uneven distribution of mechanical characteristics is associated with the presence of the initial heterogeneity of the material, the development of plastic deformations, and the dependence of material properties on temperature. The same factors also affect the convergence of the iterative process, since the conditionality of the SAFEM matrix depends on them. In order to determine the area of effective application of the SAFEM, a wide range of test cases are considered.
 In all cases, the semi-analytic finite element method is not inferior in approximation accuracy, and in some problems it is 1.5-2 times superior to the traditional method of scheduling elements. finite element method.

Research of the stressed and deformed state of metal stripin the broaching process

“…Keywords: finite element method, semi-analytical finite element method, stress-strain state, elastic and elastic-plastic deformation, shape change, strip broaching, metal embossing, flat and spatial production. В данійроботі, використовуючи методику, викладену в роботах [1,2,[4][5][6] було виконано на основі напіваналітичного методу скінчених елементів розв'язання просторової задачі пластичного формозмінення призматичного тіла і проведено аналіз процесу протягування смуги в процес обробки металу тисненням. Мета цьогодо слідження полягає у визначенні просторової картини напружено-деформованого стану смуги у процесі протяжки та зіставленні просторового розв'язання з плоским.…”

“…9 назв. In this work, using the methodology outlined in works [1,2,[4][5][6], the solution of the spatial problem of plastic deformation of a prismatic body was solved on the basis of the semi-analytical method of finite elements, and the analysis of the strip broaching process in the process of metal embossing was carried out. The purpose of this study is to determine the spatial picture of the stress-strain state of the strip during drawing and to compare the spatial resolution with the flat one.…”

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In papers [2,4,5] the solution relations and the algorithm of the method of block iterations of solving linear and nonlinear equations by the semivanalytic finite element method for curvilinear inhomogeneous prismatic bodies are realized. In paper [1], a numerical study of the convergence of solutions was performed, and a wide range of test problems for bodies with smoothly and abruptly changing physical and geometric characteristics in elastic and resilient-plastic formulation was considered. In paper [6], to confirm the reliability of the results obtained on the basis of the semi-analytical finite element method, the effectiveness of this approach for the calculation of curvilinear inhomogeneous prismatic objects is shown.

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“…[2,4,5] реалізовано розв'язувальні співвідношення та алгоритм методу блочних ітерацій розв'язання лінійних і нелінійних рівнянь напіваналітичним методом скінчених елементів для криволінійних неоднорідних призматичних тіл. У роботі [1] виконано чисельне дослідження збіжності розв'язання, розглянуто широке коло тестових задач для тіл з плавно і стрибкоподібно мінливими фізичними та геометричними характеристиками в пружній і пружно-пластичній постановці. В [6] для підтвердження достовірності одержуваних результатів на основі напіваналітичного методу скінчених елементів, показано ефективність застосування даного підходу для розрахунку криволінійних неоднорідних призматичних об'єктів.…”

Analysis of structures with arbitrary kinematic boundary conditions by the semi-analytical finite element method

“…One of the most common examples for estimating the convergence of numerical methods for calculating thinwalled objects is the problem of elastic equilibrium of a non-flat rectangular cylindrical panel, hinged on two curved sides and free on the other two, loaded with its own weight (Fig. 1) [6]. ( 0) ( )…”