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 threedimensional 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.


Introduction
The finite element method (FEM) [1,2] is widely used to study the stress-strain state (SSS) of complex-shaped elastic composite bodies. In order to simplify calculations, the deformation of bodies of a certain type (for example, beams, plates, shells) is described using engineering theories based on hypotheses [3][4][5][6][7][8][9]. However, engineering solutions do not always meet modern requirements. 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 12 9 10 10  . For such discrete models, it is difficult to use computational software such as ANSYS, NASTRAN, etc. [2]. The dimension of discrete models can be effectively reduced by means of the MgFE [11][12][13][14][15][16][17][18], which also serve as a basis for the multigrid finite element method (MFEM) [13][14][15][16][17], based on the FEM algorithms. The main advantages and features distinguishing the MFEM from the FEM are as follows.
1. In the MFEM (without increasing the dimension of MgFE), one can use arbitrarily small basic partitions of bodies, i.e., MgFE, which allows one to arbitrarily accurately account for their complex shape and inhomogeneous structure, as well as the complex nature of fixing and loading bodies. In the FEM, it is impossible to use arbitrarily small basic partitions because the PC resources are limited, which means that the MFEM is more effective than the FEM.
2. Application the MFEM (based on the basic models of bodies) requires a significantly smaller amount of RAM ( 6 3 10 10  times) and time compared to the FEM used for basic models, i.e., the MFEM is more economical than FEM.
3. The MFEM uses homogeneous and composite MgFE constructed using nested grids, which expands the field of application of this method. The FEM uses single-grid homogeneous finite elements (FE). It is noteworthy that boundary-value problems can always be solved using the MFEM instead of the FEM because MgFE can always be used instead of single-grid FE. As mgFE are constructed using m nested grids instead of one (m>1), the MFEM can be regarded as the generalization of the FEM, i.e., the FEM is a special case of the MFEM. Hence, if the FEM-based calculations of bodies are carried out using MgFE, then this case is essentially the implementation of the MFEM.
In this paper, the two types of complex-shaped three-dimensional composite MgFE are designed using the generating FEM. According to this method, the MgFE domain is obtained using the specified displacement of a given complex-shaped plane single-grid FE (below referred to as a generating FE) in a three-dimensional space. The MgFE domain of the first type is obtained by rotating the generating FE around a specified axis at a specified (small) angle, and the MgFE domain of the second type is obtained by the parallel displacement of the generating FE along a specified straight line at a given distance. The nodes of the generating FE are the nodes of a coarse grid of the MgFE, and the nodes of any cross section of the coarse grid of the MgFE are the nodes of the generating FE. This approach simplifies the construction of approximating displacement functions on the coarse grids of the complex-shaped MgFE, where the basis functions of the generating FE are used, and Lagrange polynomials are applied along the direction of the generating FE. The MgFE of the first type are used to calculate the composite shells of revolution, and the MgFE of the second type are used to calculate the composite cylindrical shells of revolution (with a variable curvature radius) and complex-shaped plates and beams. It is assumed that there are ideal bonds between the components of the inhomogeneous structure of the MgFE. The calculation of composite shells of double curvature using the MgFE of the first type is described in [16], and the application of the MgFE of the first and second types for determining the power elements of standard designs is demonstrated in [18].

Multigrid FE of the first type. Complex-shaped composite shells of revolution
The fundamental principles of constructing the MgFE of the first type, applied to analyze a three-dimensional SSS of composite shells of revolution, are considered on the example of a complex-shaped shell two-grid FE (2gFE), Fig. 1 H  . It is noteworthy that some nodes of the coarse grids of the MgFE can generally fail to match the nodes of the fine grids. We construct the e V 1gFE by using the equations of the three-dimensional problem of the elasticity theory [10], written in the local Cartesian coordinate system of the e V FE [12]. Thus, a three-dimensional SSS is implemented in the 2gFE. Radii         The calculations of the elastic composite round cylindrical shells and panels, carried out using MgFE, are described in [12]. 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 ) , , ( z y x P of the first, second, and third orders [12], written in the local Cartesian coordinate systems. The MgFE of the shells of revolution are verified using the known numerical method [2,12]. You are free to use colour illustrations for the online version of the proceedings but any print version will be printed in black and white unless special arrangements have been made with the conference organiser. Please check whether or not this is the case. If the print version will be black and white only, you should check your figure captions carefully and remove any reference to colour in the illustration and text. In addition, 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.

Multigrid FE for calculating complex-shaped composite beams
The main provisions of the construction procedure for the MgFE of the second type used to analyze the three-dimensional SSS of composite beams are considered on the example of the p V 2gFE (Fig. 3 where p u ,

Multigrid FE for calculating complex-shaped composite cylindrical shells
The construction procedure of the 2gFE of the second type, used to analyze the threedimensional SSS of cylindrical shells of an inhomogeneous structure and complex shape, is considered on the example of the   (Fig. 5), which has a hole and the section of which is hatched in the figure. The generatrix (straight line) of the median surface of the a e V 2gFE is parallel to the Oy axis (Fig. 5). The a e V 2gFE domain is obtained by the parallel displacement of the generating complex-shaped 1gFE a V (Fig. 6) (Fig. 6), of the form (7)

Multigrid FE for calculating complex-shaped composite plates
We consider the construction procedure for the 2gFE of the second type in order to analyze the three-dimensional SSS of plates with an inhomogeneous structure on the example of the complex-shaped composite laminated 2gFE b g V (Fig. 7), where Oxyz is the Cartesian coordinate system. The characteristic dimensions B and H of the b g V 2gFE significantly exceed the dimension of h , with h being the thickness of the 2gFE. The b g V 2gFE domain is obtained by the parallel displacement of the generating 1gFE g V (Fig. 8) along the Oy axis at the given distance h . The coarse grid of the b g V 2gFE has 27 nodes marked by points in Fig. 7. The basic partition of the b g V 2gFE consists of cube-shaped homogeneous FE of the first order (rectangular parallelepiped [2]), in which a three-dimensional SSS is implemented. It is noteworthy that the basic partitions of the 2gFE can be arbitrarily small, i.e., can arbitrarily exactly account for the inhomogeneous structure and complex shape of the 2gFE. The base function   shaped as a rectangular parallelepiped, constructed with the help of power polynomials [2], and Lagrange polynomials [12]. 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] (corrugation can be shaped as a trapezium, rectangle, triangle, a part of a circular arc, etc.). Three-grid FE (3gFE) of the second type are designed using the 2gFE of the second type with the help of procedures similar to those in Secs. 2, 3.2, and 3.3.

Multigrid FE for calculating curvilinear composite beams
We consider a curvilinear beam 1 l (frame beam). The a L V 2gFE domain (Fig. 9), where the 2gFE approximates the beam 1 l , is obtained by the parallel displacement of the generating 1gFE L V (Fig. 10) along the Oy axis at the distance d , which is the beam width. The transverse section of the beam is d h  , where h is its height (thickness), corresponding to arc ds (Fig. 9). The thickness of the transverse beam can be variable. The coarse grid of the a L V 2gFE has 24 nodes marked by points (Fig.  9).   (Fig. 10), 12 ,..., 1  i , corresponding to a polynomial of the form (7), written in the local Cartesian coordinate system Oxz (Fig. 10), and ) (y L j denotes the Lagrange polynomials of the first order.

Conclusion
In this paper, three-dimensional composite and homogeneous MgFE of two types of complex shapes are considered, which are designed with the use of forming FE. The procedures for the construction of type 1 and type 2 MgFE are described, which are used for the calculation of composite (homogeneous) shells of rotation, cylindrical shells (with a variable radius of curvature), plates and beams of complex shape. The main advantages of the proposed MgFE are that they take into account the inhomogeneous, microinhomogeneous structure and complex shape of bodies, describe three-dimensional behavior in composite (homogeneous) bodies, form discrete models of small dimension and generate approximate solutions with a small error.