Varying parameterization of an ellipsoidal thin shell with FEM-based implementation

This article describes an algorithm developed for the finite element analysis of the stress-strain state of a shell that takes the shape of a triaxial ellipsoid with varying parameterization of its mid-surface. A quadrangular fragment of the shell mid-surface with nodal unknowns in the form of displacements and their first derivatives along the curvilinear coordinates was used as the discretization element.
When approximating the displacements through the nodal values, two variants were considered. In the first variant, the known approximating functions were applied to each component of the displacement vector of the internal point of the finite element through the nodal values of the same component. In the second variant, the approximating expressions were used directly for the expression of the displacement vector of the internal point of the finite element through the vector unknowns of the nodal points. After the coordinate transformations, each component of the displacement vector of the internal point of the finite element was expressed through the nodal values of all components of the nodal unknowns. The approximating expressions of the required displacements of the internal point of the finite element also include the parameters of the curvilinear coordinate system used in the calculations.
The high efficiency of the developed algorithm was confirmed by the results of the numerical experiments.