A relative configuration vector method for solving geometrically exact beam problems

The geometrically exact beam problems are often solved based on total Lagrangian method and updated Lagrangian method. However, these methods are prone to singularities and poor convergence when dealing with large deformations. The relative configuration vector method is proposed to obtain a singularity-free solution of the static equilibrium equations even for models with large deformation. First, the kinematic description of geometrically exact beams based on Lie group SE(3) is presented, along with an SE(3)-interpolation for the beam elements. The advantages of SE(3)-interpolation, such as objectivity and shear-locking-free behavior are then discussed. Next, an interpolation scheme for the variation of the average convective strain is derived, which allows for the computation of strain operators and static equilibrium equations without introducing complex operators. Subsequently, a detailed comparison is made among the three computing methods in terms of singularities, equivalence conditions for multivariable substitution, and the tangent stiffness matrix. Theoretically, the relative configuration vector method can inherently avoid the logarithm map and thereby eliminates the associated singularities. Finally, numerical examples are taken to show that the proposed relative configuration vector method avoids singularities and enhances convergence behavior under large deformations compared with the total and updated Lagrangian methods.