The prismatic finite element with the interpolation procedure vector form for the engineering structures’ strength calculations

The technique of forming a stiffness matrix of a volume prismatic finite element with a triangular base and with six nodes located at the vertices of the prism is presented. The discretization element is formed on the basis of the interpolation procedure vector form with consideration as the interpolation object of the displacement vector of an arbitrary point of the engineering structure. It is proposed to improve the compatibility of the prismatic discretization element at the boundaries of the docking of the bases by using Lagrange multipliers as additional nodal unknowns, which are introduced in additional nodes located in the middle of the prismatic discretization element bases’ sides. The presented vector form of interpolation allows one to obtain the correct finite element solutions in the problems of determining the stress-strain state of engineering structures using curvilinear coordinate systems.


Introduction
The modern development of the construction industry requires improving the methods of strength analysis of building structures and architectural objects. The fast-paced digitalization of the economy highlights the need for the development of numerical methods of calculation [1-5] based on the advanced computer technologies. Among all numerical methods, the finite element method (FEM) [6][7][8][9] should be considered as a priority, combining universality, the ability to fully algorithmize the computational process, and indifference to the types of external load and boundary conditions. Among the vast family of discretization elements developed to date, the researchers prefer volumetric finite elements. Therefore, the task of developing and improving three-dimensional discretization elements remains quite relevant. In addition, special attention should be paid to the fact that the geometric forms of structures and architectural forms are becoming more complicated, a large number of objects of this kind have a curved configuration. In this regard, the use of curved coordinate systems: cylindrical, spherical, toroidal, becomes most convenient in computational algorithms. However, the use of curvilinear coordinate systems leads to the problem of taking into account the displacement of the finite element as an absolutely rigid body. A correct solution to this problem becomes possible through the use of the vector form of the interpolation procedure, based on the displacement vector consideration itself as interpolation objects, rather than its individual components [10][11]. In the present work, the method for the formation of a stiffness matrix of a bulk prismatic finite element formed on the basis of the vector form of the interpolation procedure using Lagrange multipliers used to improve compatibility at the boundaries of the bases' junction of a sampling element of this type has been presented.

Geometric relationships
The radius vector of the reference surface of a building structure or architectural form with a curvilinear generatrix can be given by the following formula are the curved coordinates of the reference surface; are the Cartesian coordinates and their unit vectors.
As a result of differentiation (1) by 2 1 ,   we can obtain the expressions for the vectors of the local basis located in the plane that is in contact with the reference surface , ; where the comma indicates the differentiation in the corresponding coordinate. The unit vector normal to the reference surface can be obtained by the vector product of the vectors in (2) where       In the process of deformation of an engineering structure under the action of an applied load, the point Performing the scalar multiplication operation over (8) Deformations at a point h M 0 can be obtained on the continuum mechanics classical formula basis [12]  .

Prismatic discretization element with vector form of interpolation procedure
The sampling element is a prism with a triangular base and with six nodes located at the vertices of the prism. Columns of the sought values of the prismatic discretization element in the local  The standard interpolation procedure [6][7][8][9][10] in the prismatic discretization element corresponds to the following interpolation dependence where   T  is the form function column.
The use of formula (13) in algorithms for the strength calculation of engineering structures with curved outlines when using curved coordinate systems can lead to a very significant calculation error due to disregarding the displacements of the prismatic discretization element as an absolutely rigid body. The solution to this problem becomes possible using the vector form of the interpolation procedure. In this case, the interpolation dependence will have the following form where    ; ... ... ...
The interpolation expression (14), taking into account (15) Taking into account (6)    The compatibility of the proposed prismatic discretization element with the vector form of the interpolation procedure can be improved by using the Lagrange multipliers introduced as auxiliary unknowns in additional nodes located in the middle of the sides of the lower (1, 2, 3) and the upper (4, 5, 6) bases triangular prism [13][14]. It is proposed to ensure the equality of the derivatives of the normal components of the displacement vectors along the normals to the sides of the bases at additional nodes with the help of Lagrange correction factors is the normal to the base side of the prismatic discretization element in the additional node  ; The partial derivatives of the normal components of the displacement vectors along the normals to the sides of the bases at additional nodes can be represented by the sum , cos or taking into account (6) and (20) From (27), we can obtain the interpolation expressions for the partial derivatives of the first-order displacement vector along global curvilinear coordinates, for example, where   t is a polynomial whose terms are the partial derivatives of the first-order normal component along global curvilinear coordinates 1  or 2  .