Generating finite element method in constructing complex-shaped multigrid finite elements

The calculations of three-dimensional composite bodies based on the finite element method with allowance for their structure and complex shape come down to constructing high-dimension discrete models. The dimension of discrete models can be effectively reduced by means of multigrid finite elements (MgFE). This paper proposes a generating finite element method for constructing two types of three-dimensional complex-shaped composite MgFE, which can be briefly described as follows. An MgFE domain of the first type is obtained by rotating a specified complex-shaped plane generating single-grid finite element (FE) around a specified axis at a given angle, and an MgFE domain of the second type is obtained by the parallel displacement of a generating FE in a specified direction at a given distance. This method allows designing MgFE with one characteristic dimension significantly larger (smaller) than the other two. The MgFE of the first type are applied to calculate composite shells of revolution and complex-shaped rings, and the MgFE of the second type are used to calculate composite cylindrical shells, complex-shaped plates and beams. The proposed MgFE are advantageous because they account for the inhomogeneous structure and complex shape of bodies and generate low-dimension discrete models and solutions with a small error.