Formulation of 3D finite elements using curvilinear coordinates

The objective of this paper is to gain insight into the implementation of 3D finite elements in curvilinear coordinates using the fundamental equations of 3D elasticity and the Principle of Virtual Displacements. After reviewing the mathematical model of the geometry, we present the formulation of hexahedral finite elements that represent a generalization of classical shell finite elements, in which also the thickness direction can be curvilinear. These finite elements are equivalent to the widely-used continuum mechanics based solid finite elements, but the adoption of local curvilinear reference system will allow in future works to apply mixed methods, such as the Mixed Interpolation of Tensorial Components (MITC), to contrast the locking phenomenon in the three spatial directions. For the assessment of the present elements, we consider different examples of curved geometries: cylindrical, conical, toroidal and spherical. A free-vibration analysis of curved 3D components is performed and the results, in terms of natural frequencies, are compared with the convergence solutions computed with the commercial software MSC Patran/Nastran.