On the approximation of class C(0) components of physical quantities in curved coordinate systems

Abstract

In numerical methods for calculating the strength of technospheric objects approximating expressions of the desired values in terms of their nodal values are widely used. The theory of approximating functions of scalar quantities is currently developed quite fully, but its direct use in curvilinear coordinate systems for approximating the components of displacement vectors and for the components of stress tensors can lead to significant inaccuracies with significant gradients of curvature and displacements of the calculated object as a rigid whole due to the lack of parameters in the approximating expressions of the curvilinear coordinate system used in the calculation. In this paper, in order to obtain approximating expressions for the individual components of the displacement vector of a finite element internal point in the form of a hexagon, the known approximating functions directly for the displacement vector through the displacement vectors of the nodal points are used. As a result of coordinate transformations, namely, using matrix expressions of the basis vectors of a finite element nodal points through the basis vectors of its internal point approximating expressions of each component of the displacement vector of the finite element internal point through all components of the displacement vectors of the finite element nodal points are obtained. In order to obtain approximating expressions for the components of the stress tensor of a finite element inner point a well-known approximating function is used directly to express the stress tensor of a finite element inner point through the stress tensors at its nodal points. Coordinate transformations consist in using matrix expressions of the dyad products of the node points basis vectors through the dyad products of the basis vector of the finite element inner point. As a result of the coordinate transformation, the approximating expression of each component of the stress tensor in the vicinity of the inner point of the finite element is determined through all the components of the stress tensors at the nodal points. The obtained approximating expressions for the components of the displacement vector and the components of the stress tensor using matrix expressions of the nodal points basis vectors through the basis vectors of the finite element internal point as well as through the matrix expressions of their dyad products allow taking into account the parameters of the curved coordinate system used. That leads to the solution of the well-known problem in the FEM which takes into account the displacement of the finite element as a solid.

Keywords

• displacement vector
• tensor of the second rank
• curvilinear coordinate system
• basis vectors
• approximating functions