Construction of Multigrid Finite Elements to Calculate Shells, Plates and Beams Based on Generating Finite Elements
In practice, composite shells, plates and beams of complex shapes are widely used. Finite element calculations of three-dimensional composite bodies, taking into account their structure and complex shape are reduced to the construction of high dimensional discrete models. To reduce the discrete model dimension, multigrid finite elements (MgFE) are effectively used. When constructing m - grid finite element ( m gFE) m nested grids are used. The fine grid is generated by the base partitioning of m gFE, taking into account its heterogeneous structure and shape. On m - 1 coarse grids, the displacement functions used to reduce the dimension of the base partitioning are determined, which allows one to develop a small dimensional MgFE. The displacement functions and stress state in the MgFE described by the equations of the three-dimensional elasticity problems are represented in local rectangular coordinates. Characteristic properties of MgFE are as follows. When constructing MgFE (without increasing their dimension), arbitrarily small basic partitions can be used arbitrarily closely taking into account the complex inhomogeneous structure and shape of the MgFE, and arbitrarily closely describing the three-dimensional stress state in the MgFE. The present paper proposes a method of generating finite elements (FE) to construct complex-shaped three-dimensional composite multi-grid finite elements (MgFE) of two types. The principal of the generating FE method is as follows. The 1st type MgFE region is obtained by turning a given flat generating single-grid FE (complex shape) around a given axis by a given angle, the 2nd type MgFE region by a parallel moving the generated FE in a given direction to a specified distance. The 1st type MgFEs are used to calculate the composite rotational shells, the 2nd type MgFE are for composite cylindrical complex-shaped shells (with a variable radius of curvature), plates and beams. The advantages of the proposed MgFE are to take into account the complex heterogeneous and micro-heterogeneous body structure and shape, to give rise to small-dimensional discrete models and high accuracy solutions.