The Method of Local Approximations in the Construction of High-Precision Small-dimensional MgFE for the Calculation of Composite Bodies

Abstract

The method of multigrid finite elements is effectively used to analyze the stress state in composite bodies (CB). When constructing a multigrid finite element (MgFE), briefly a standard MgFE, using known procedures, a fine grid, and large ones nested in a fine one are used. The fine grid is generated by partitioning, which takes into account the heterogeneous structure of the MgFE, large grids are used to reduce its dimension. For a standard MgFE, it is characteristic that every large grid and the corresponding approximations of displacements are determined throughout its entire area. This leads to an increase in the dimension of the standard MgFE when constructing high-order approximations on large grids, which are used to increase its order of accuracy. Standard high-precision MgFE, i.e. of high order of accuracy, have a large dimension, which makes their application difficult.
 In this paper, a method of local approximations (MLA) for constructing high-precision small-dimensional MgFE (short — small-sized MgFE) is proposed. Such MgFE are used to calculate elastic CB and are designed on the basis of standard. The main idea of the MLA is that local approximations of high-order displacements are determined on large grids in the central part of the region of a small-sized MgFE, and in the vicinity of the boundary of the region — of a small order, which allows using various local approximations to vary the dimension and order of accuracy of a small-sized MgFE. Two approaches to the construction of small-sized MgFE are shown, in the case of their complex shape, forming finite elements are used. Calculations show that small-sized MgFE generate stresses in the CB, the errors of which are 15÷50 smaller than the errors of similar stresses corresponding to standard MgFE, i.e. small-sized MgFE are more effective than standard ones. The use of smallsized MgFE in calculations makes it possible to determine stresses with a small error for large CB partitions.