A geometric local frame approach for flexible multibody systems

The notion of frame is ubiquitous in the kinematic description of flexible multibody models. In this work, a differential-geometric framework is selected to describe frame operations in a rigorous and systematic way. A frame transformation is thus seen as an element of the special Euclidean group $SE(3)$, which is represented by a four by four transformation matrix, and frame operations, such as spatial interpolation or time integration, rely on non-linear but analytical expressions in which translation and rotation contributions are inherently coupled. Based on this formalism, this thesis develops geometrically exact formulations of many classical components used in flexible multibody system modelling, which includes the formulation of a rigid body, several kinematic joints, a flexible beam, a flexible shell and a superelement.

As opposed to most popular techniques in the literature, a local frame representation of the equations of motion is adopted in this work. This means that the unknown kinematic variables such as the motion increments, the velocities and the accelerations, as well as the generalized forces are all expressed in a local frame attached to the body. After spatial semi-discretization, the equations of motion of a multibody system take the form of differential-algebraic equations on a Lie group which can be conveniently solved in a global parametrization-free approach using a Lie group integration scheme. This thesis presents numerous arguments to recommend this framework for the development of efficient codes for the numerical simulation of flexible multibody systems.

On the one hand, the proposed framework leads to novel and interesting theoretical aspects. For instance, it features a naturally singularity-free description of large rotations and it leads to inherently shear-locking free beam and shell finite elements.

On the other hand, the formulation leads to unprecedented computational properties. The geometric non-linearities are naturally filtered out of the equilibrium equations such that non-linearities are significantly reduced, as compared to classical formulations. In particular, the iteration matrix, which is used in implicit integration schemes, is insensitive to overall large amplitude motions and is only affected by local relative transformations, such as deformations in flexible elements and relative motions in kinematic joints. This property can be exploited to strongly reduce computational costs, as compared to classical formulations.