Abstract
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.
Highlights
The finite element method (FEM) [1, 2] is widely used to study the stress-strain state (SSS) of complex-shaped elastic composite bodies
The dimension of discrete models can be effectively reduced by means of the multigrid finite elements (MgFE) [11,12,13,14,15,16,17,18], which serve as a basis for the multigrid finite element method (MFEM) [13,14,15,16,17], based on the FEM
It is noteworthy that the proposed MgFE of the first and second types can be used in analyzing the three-dimensional SSS of corrugated plates, panels, and floors [19, 20]
Summary
The finite element method (FEM) [1, 2] is widely used to study the stress-strain state (SSS) of complex-shaped elastic composite bodies. Calculating composite bodies on the basis of the FEM in the formulation of a threedimensional problem of the elasticity theory [10] with account for their structure can be reduced to constructing high-dimension discrete models, of the order of 109 1012. The base functions of the coarse grids of the 2gFE of these shells are determined in the form of Lagrange polynomials and using the power polynomials P(x, y, z) of the first, second, and third orders [12], written in the local. It is possible to use arbitrarily small basic partitions Rd , allowing one to arbitrarily exactly account for the complex shape and inhomogeneous (microinhomogeneous) structure of the 2gFE Vd and its complex fixing and loading, as well as to arbitrarily exactly describe a three-dimensional SSS in the Vd 2gFE domain (with no increase in the dimension of the Vd 2gFE, which is noteworthy). Some colour figures will degrade or suffer loss of information when converted to black and white, and this should be taken into account when preparing them
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