Multisymplectic Galerkin Lie group variational integrators for geometrically exact beam dynamics based on unit dual quaternion interpolation — no shear locking

Abstract

We present a Galerkin multisymplectic Lie group variational integrator. It is suitable for dynamical systems defined on a two dimensional space–time and the integrator allows arbitrary convergence orders independently for both dimensions. As an example we use geometrically exact beam dynamics where a slender structure is modelled as a centre line with a cross section at every point. The Lie group in question is the special Euclidean group in three-dimensional space, SE(3), which we parametrize using unit dual quaternions. This allows a very simple and efficient interpolation method to be used, which additionally prevents shear locking present in more naive discretizations of geometrically exact beams.