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ScienceDirect Procedia Engineering 165 (2016) 1705 – 1709

15th International scientific conference “Underground Urbanisation as a Prerequisite for Sustainable Development”

The use of iterative methods in solving tasks of structural mechanics Vladimir Kuroedov a, Luka Akimov b, Artem Frolov a,*, Aleksandr Zavylov d, Aleksey Savchenko a a

Peter the Great Saint-Petersburg Polytechnic University, 29 Polytechnicheskaya st., St.Petersburg, 195251, Russia b Politecnoco di Milano, 32 Piazza Leonardo da Vinci, Milano, 20133, Italian Republic

Abstract The article discusses some aspects of use of iterative methods for solving nonlinear tasks in structural mechanics. The example demonstrates the ability of the method of additional loads applied to various models of continuous media. Possibilities of the additional loads method in relation to different sections of structural mechanics by using few examples are also demonstrated. The advantages of the used method are discussed. © by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2016 2016Published The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the 15th International scientific conference “Underground Peer-review under scientific committee of the 15th International scientific conference “Underground Urbanisation as a Urbanisation as aresponsibility Prerequisite of fortheSustainable Development. Prerequisite for Sustainable Development Keywords: iterative method, method of successive approximations, nonlinear material, nonlinear tasks in structural mechanics, additional loads method, method of the variable parameters.

Introduction An iteration method, strictly speaking, could be called as a method of successive approximations. Special interest is paid to iterative method because it allows to find a solution with a predetermined accuracy. In addition, some small errors that are committed in the calculation process, can be corrected later, thus, additional feature of this

* Corresponding author. Tel.: +79110880935. E-mail address: [email protected]

1877-7058 © 2016 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the 15th International scientific conference “Underground Urbanisation as a Prerequisite for Sustainable Development

doi:10.1016/j.proeng.2016.11.912

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Vladimir Kuroedov et al. / Procedia Engineering 165 (2016) 1705 – 1709

method is that it is self-correcting. Also the method of successive approximations can be easily programmed, it is convenient to use in computer calculations [1-3]. Direct methods are an alternative to iterative methods. In comparison with usage of direct methods, the use of iterative methods reduce a time of calculation process. This method asks for less computer resources in the analysis of large models. Iterative methods are algorithmically easier and use matrix structure less. Iterative solvers provide approximate solution, converging from iteration to iteration. However, the convergence of conventional iterative methods is extremely slow (for example, the method of simple iteration in case of poorly conditioned matrix). Implicit iterative methods are more complex algorithmically. In these tasks a solution on new iteration is found in some direct methods, however, the advantage of implicit methods is significantly faster convergence [4,5]. Currently, the number of structural mechanics tasks (such as the physically nonlinear theory of elasticity, a plasticity, a strength considering the limiting behavior of material), that are solved within the physically nonlinear theory, increases. Number of such tasks increases in terms of solution of today's problems associated with construction, such as calculation of thin-walled members, micro-forming processes, analysis of structures made of different materials, etc. [6-9]. In this case the manifestation of physical nonlinearity could be caused by various factors, when the connection between tension V and deformation H is represented as:

V

D (H )H .

(1)

The solution of such tasks by finite element method is formally reduced to a system of nonlinear algebraic equations:

K (q)q

P.

(2)

Methods, that allow to reduce the solution of task Eq.2 to a sequence of linear tasks, are used in most studies devoted to this subject. The method of variable parameters is used in such tasks. In this method all coefficients of the stiffness matrix are recalculated on each step of the iterative process:

K j q j 1

P, j 1,2,..., N

(3)

The method of additional loads is also used in such tasks: (4) K j q j 1 P  R j , j 1,2,..., N Here K - a stiffness matrix, which corresponds to the initial approximation of q j ; R j - the vector of additional generalized forces, specified by nonlinear material properties. Both of these approaches could be combined with stepwise loading, when a predetermined load is applied in stages,