A projection-based quaternion discretization of the geometrically exact beam model

Abstract

In the present work the geometrically exact beam model is formulated in terms of unit quaternions. A projection-based discretization approach is proposed which is based on a normalization of the quaternion approximation. The discretization relies on NURBS shape functions and, alternatively, on Lagrangian interpolation. The redundancy of the quaternions is resolved by applying the method of Lagrange multipliers. In a second step the Lagrange multipliers are eliminated circumventing the need to solve saddle point systems. The resulting finite elements retain the objectivity of the underlying beam formulation. Optimal rates of convergence are observed in representative numerical examples.