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).

Geometrically accurate infinitesimal-rotation spatial finite elements

“…If the assumption of infinitesimal nodal rotations is used, one obtains the ANCF/FFR element formulation. 10 Using this assumption, the matrix A is approximated as A ¼ I þh, andh is the skewsymmetric matrix associated with the vector h ¼ 1 2 3 Â Ã T when the Euler angle sequence…”

“…, where e k o , e k d , and B k are defined, respectively, using equation 3as Using these definitions and the ANCF kinematic description, one has 10 u x, t…”

“…Therefore, the elements of J o can be used to define new geometric coefficients, allowing for accurately capturing the initial element shape. 10 Equation 5differs significantly from the displacement field used for conventional infinitesimal-rotation finite elements because this equation can be used for both straight and curved elements to capture accurately initial curvatures using the new geometric coefficients. Using equation 5, there is no need to distinguish between plate and shell elements, which are treated differently in the FE literature.…”

“…New spatial beam, plate, and solid (brick) elements are developed in terms of constant geometric coefficients obtained using the elements of the matrix of position gradients defined in the reference configuration. 10 The new spatial beam, plate, and solid element shape functions are obtained from the ANCF kinematic description. These elements can be used for developing reducedorder models for structural and MBS/FFR analysis.…”

“…12,13 Geometry and position vector gradients Three configurations are often used to describe the kinematics of continua with complex geometries. These are the straight configuration, the stress-free curved reference configuration, and the current deformed configuration described, respectively, by the parameters or coordinates 10 The position vector r of the material points can be written in the form r ¼ X þ u, where the vector u ¼…”