A rational finite element method based on the absolute nodal coordinate formulation

The NURBS method is widely used for representing the geometry of engineering and physics systems in computer aided design (CAD) software. However, no finite element formulations currently used in computer aided analysis (CAA) can produce complex geometry identical to NURBS representations. Most contemporary finite element methods, in fact, are at best only crudely capable of representing the complex shapes that can be represented using NURBS. However, finite elements based on the nonlinear absolute nodal coordinate formulation (ANCF) are known to utilize similar shape functions to those employed in Bezier and B-spline representations of geometry. In order to facilitate the integration of computer aided design and analysis (ICADA), this investigation proposes the creation of a new finite element formulation based on the ANCF that utilizes rational polynomials for shape functions. This will lead to the new rational absolute nodal coordinate formulation (RANCF). To demonstrate the feasibility of developing RANCF finite elements, it is shown that rational cubic Bezier curve elements can be converted through a simple coordinate transformation into a rational ANCF form, in which the coordinates are the nodal positions at the end points of the element and the gradient vectors at those nodal points. Furthermore, it is shown that because a cubic NURBS curve can be converted to a series of rational cubic Bezier elements that such a NURBS curve can be converted into a series of these RANCF elements. This investigation only proposes the kinematic description of this new RANCF element. The full development of the nonlinear dynamic equations of motion as well as the development of fully parameterized RANCF finite elements will be explored in future investigations.