Geometrically accurate floating frame of reference finite elements for the small deformation problem

Abstract

In the current implementations of the finite element floating frame of reference (FFR) formulation, conventional beam, plate, and shell elements that employ rotations as nodal coordinates are used. These elements cannot describe accurately curved geometries and are not related to the more geometrically accurate B-spline and non-uniform rational B-splines (NURBS) representations by a linear mapping. This paper proposes a new geometrically accurate FFR implementation based on the consistent rotation-based formulation (CRBF) obtained from the kinematic description of the absolute nodal coordinate formulation (ANCF). The ANCF position vector gradients are used effectively to separate the initial geometry from the displacement coordinates. A crucial step in the development of the new approach is proving that the new FFR element displacement fields have a proper set of rigid body modes. Specifically, the new FFR approach has the following important features: complex curved geometries can be accurately captured eliminating the tedious work required for converting solid models to analysis meshes; the same number of nodal coordinates as conventional elements is used; no distinction is made between plate and shell geometries; no changes need to be made in the main solvers since only preprocessing changes are required; and a local linear problem can still be created in order to systematically eliminate insignificant deformation modes. Furthermore, while the new elements employ infinitesimal rotations as nodal coordinates, the solution algorithm can be designed to ensure the continuity of the gradients at the nodal points. The use of the new FFR implementation will contribute to a successful integration of computer-aided design and analysis (I-CAD-A).