  Mechanics of Solids A Journal of Russian Academy of Sciences   Founded
in January 1966
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A. F. Starikov, "Estimates of accuracy of approximate solutions of some combined modular elastic continuum systems with discrete interaction," Mech. Solids. 43 (5), 722725 (2008) 
Year 
2008 
Volume 
43 
Number 
5 
Pages 
722725 
DOI 
10.3103/S0025654408050051 
Title 
Estimates of accuracy of approximate solutions of some combined modular elastic continuum systems with discrete interaction 
Author(s) 
A. F. Starikov (Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Profsoyuznaya 65, V342, GSP7, Moscow, 117997, Russia) 
Abstract 
At present, it happens increasingly often that one needs to study complicated modular systems with discrete interaction between the units. If a system contains a distributed unit, then it is said to be combined. One method for studying such systems is to analyze their frequency models; in this case, it is sometimes required to calculate the Green functions of the distributed unit and study the block structures generated by a model of a system with superelements. A technique for studying such systems was developed in [1] and is called the factorized perturbation method; in foreign literature, this technique is called the Green function method (e.g., see [2]).
The main idea of such methods is first to construct an equivalent system in the characteristic space that arises in the discrete macrostructure of the original system as a result of the discrete interaction between the units of the latter and then to reduce the characteristic solution to the original solution by using a simple transfer relation. A typical characteristic of structural methods is their algorithmic universality, which is independent of the special form of the discrete microstructure and requires solving a characteristic matrix equation. To obtain approximate solutions by these methods, one should approximately calculate the Green functions of the distributed units and explicitly indicate their operating frequency ranges ( see [3, 4]). The concept of structural methods is to obtain a system solution by using a priori studied elements.
There are a variety of methods for studying combined modular systems. In view of this, the accuracy of the approximate solutions, which are also obtained by numerous methods, should be estimated from the algorithmic standpoint. At present, this is done at the level of practicalempirical considerations like doubling the number of grid points for the analysis of the distributed unit, taking into account additional vibration modes in the Green functions, etc.
In the present paper, we describe a structural method for solving elastic onedimensional distributed systems with discrete interaction and present an efficient scalar a priori estimate and a universal a posteriori estimate for the accuracy of an approximate solution, which can be used in many approximate methods, e.g., FEM [5]. 
Received 
09 July 2004 
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