The method of fictitious discrete models in calculations bodies with an inhomogeneous regular structure

In calculations for the strength of elastic composite structures (plates, beams, shells), which are widely used in aviation and rocket and space technology, using the finite element method (FEM), it is important to know the error of the solution. To analyze the error of the solution, it is necessary to use a sequence of approximate solutions constructed according to the FEM using the grinding procedure for discrete basic models (BM), which take into account the inhomogeneous, micro-inhomogeneous structure of structures (bodies) within the micro-approach. Discrete models obtained by grinding BM have a high dimension, which makes it difficult for them to use FEM. In addition, there are BM of composite bodies (CB), for example, BM of bodies with a micro-inhomogeneous structure, which have such a high dimensionality that the implementation of FEM for such BM, due to the limited computer resources, is almost impossible. To solve these problems, it is proposed to use fictitious discrete models in the calculations of the CB according to the FEM.

In this paper, we propose a method of fictitious discrete models (MFDM) for calculating the strength of elastic bodies with an inhomogeneous, micro-inhomogeneous regular structure. MFDM is implemented using FEM with the use of adjusted strength conditions that take into account the error of approximate solutions. The method is based on the following statement. We believe that BM CB generates solutions that differ little from the exact ones. Due to the convergence of the FEM, such BM for CB always exist. The calculation of CB according to MFDM is reduced to the construction and calculation of the strength of fictitious discrete models (FM), the dimension of which is less than the dimension of the BM. FM reflects: the shape, characteristic dimensions, attachment, loading, and appearance of the heterogeneous structure of the CB, and the distribution of elastic modulus corresponding to the BM of the CB. The sequence consisting of the FM converges to the BM, i.e. the limiting FM coincides with the BM. The convergence of such a sequence ensures uniform convergence of the FM stresses to the corresponding BM stresses. The implementation of FEM for FM with the use of multigrid finite elements leads to a large saving of computer resources, which allows the use of MFDM for strength calculations of bodies with a micro-inhomogeneous regular structure. The calculation of the strength of CB according to MFDM requires 10³÷10⁶ less computer memory than a similar calculation using BM CB, and does not contain a procedure for grinding BM. The given example of calculating the strength of a beam with an inhomogeneous regular fiber structure according to the MFDM shows its high efficiency. The use of adjusted strength conditions allows us to use approximate solutions with a large error in the calculations of CB for strength, which leads to an increase in the efficiency of MFDM.