Construction of high-precision low-dimensional MgFE using local approximations and generating FE

Composite structures (bodies), in particular, plates, beams, shells, are widely used in aviation and rocket and space technology. To analyze the stress state of elastic composite bodies (CB), the method of multigrid finite elements (MMFE) is effectively used, which is implemented on the basis of the Lagrange functional (in displacements). When constructing a multigrid finite element (MgFE), briefly a standard MgFE, using known procedures, a small base grid is used, which can be arbitrarily small, and large ones nested in a small one. The fine grid is generated by the partition of the MgFE, which takes into account its inhomogeneous, micro-inhomogeneous structure within the framework of the micro-passage. Large grids are used to reduce the dimension of the MgFE. The following is typical for a standard MgFE. Any large grid of a standard MgFE and corresponding approximations of displacements are determined on its entire region. This leads to an increase in the dimension of the standard MgFE with an increase in its order of accuracy, since in this case approximations of high-order displacements are determined on large grids. To reduce the error of solutions, high-precision MgFE are used, i. e., of a high order of accuracy, which have a large dimension. However, the use of high-precision MgFE is difficult, since they form discrete models of high-dimensional bodies. In this paper, we propose a method of local approximations (MLA) for constructing high-current MgFE of small dimension (short - small-sized MgFE), which are used to calculate elastic homogeneous and CB by MgFE. Two types of small-sized MgFE are considered. Small-sized MgFE of the 1st type are designed on the basis of standard ones with the use of local approximations of displacements, which are determined on the subdomains of standard MgFE, of the 2nd type - with the use of finite element generators (FE). The brief essence of the construction of small-sized MgFE of the 1st type is as follows. According to the MLA, we define a smaller Н grid on the V0 region of the standard MgFE than its base one. The V0 region is represented by the boundary and inner regions. The boundary (inner) regions have a common boundary, which does not degenerate into a point (do not have a common boundary), with the V0 region. On the boundary (inner) regions, we define large grids that are embedded in a small Н grid and generate local approximations of small (high) order displacements. On the V0 region, using local approximations of the displacements of the boundary and inner regions, we construct the MgFE. Then, using the condensation method, we express the movements of the internal nodes of the MgFE through the movements of the nodes lying on its boundary, i.e. on the boundary of the V0 region. As a result, we obtain a high-precision Vp MgFE of small dimension, i.e. a small-sized MgFE of the 1st type, the dimension of which is equal to the dimension of the standard one. It is important to note that with an increase in the order of accuracy of the Vp MgFE, its dimension does not change, i.e. it does not increase, and therefore it is called a highprecision MgFE of small dimension, i.e. small-sized. The procedure for constructing small-sized MgFE of the 1st type is described in detail. As is known, the calculation of the static strength of structures is reduced to determining the maximum equivalent stresses for them, the determination of which with a small error for CB is an urgent problem. Calculations show that small-sized MgFE of the 1st type generate maximum equivalent stresses in CB, the errors of which are 25  50 smaller than the errors of analogous stresses obtained using standard ones, on the basis of which small-sized, i.e. small-sized MgFE of the 1st type are more effective than standard ones. The use of small-sized MgFE of the 1st type in MMFE calculations makes it possible to determine the maximum equivalent stresses with a small error for large CB partitions. The construction of small-sized MgFE of the 2nd type is shown, which are designed on the basis of standard high-precision MgFE with the use of generating FE. A small-sized MgFE of the 2nd type has the same order of accuracy as the standard one, but its dimension is smaller than the dimension of the standard one. The advantage of small-sized MgFE of the 2nd type is that they give rise to discrete CB models of smaller dimension than standard ones.