Method of determining a posteriori error estimation in calculations of composite shells using multigrid finite elements

A numerical method of determining a posteriori error estimates of solutions created for composite cylindrical shells using multigrid finite elements (MFE) has been proposed. The suggested method is based on ZZ method proposed by O. C. Zienkiewicz and J. Z. Zhu for both energy norm and $L_2$ norm of solution errors estimates. In contrast to ZZ method, the suggested method uses MFE that takes into account complex shapes, heterogeneous and micro-heterogeneous body structures and forms small dimension discrete models for creating «precise» solutions. To give examples there was carried out analysis of error estimates for displacements and stresses in calculation of stress-strain state (SSS) of three-layer cylindrical shells with and without cutouts under local loading. It has been stated that analysis of SSS using MFE causes converging sequences of approximate solutions in norm $L_2$. Calculations that use mean square error for stresses in each finite element of the shell show that MFE allow to use arbitrarily small regular discretization grids all over the shell area without the necessity to tighten the grid in local areas for calculating SSS. This leads to simple algorithms of calculating SSS with the help of MFE and ensures considerable saving of computer resources. In the given examples the use of MFE decreases the dimension of the system of MFE algebraic equations and reduces computer memory volume by $1 500$ and $8\cdot 10^4$ times respectively, compared to the finite elements base model that doesn't use MFE.