Mixed Finite Elements for Geometrically Exact Beams using Discontinuous Rotations and Discrete Curvature
This work provides a simpler and more efficient discretization framework for geometrically exact beam problems, which is relevant for computational mechanics researchers.
The paper proposes a mixed finite-element formulation for geometrically exact beams that introduces the moment vector as an additional independent field, enabling element-local discontinuous rotations and discrete curvature. The method achieves optimal convergence rates and accuracy regardless of beam slenderness, with the lowest-order element using only element-constant rotations.
We propose a novel mixed finite-element formulation for geometrically exact (Simo--Reissner) beams that introduces the moment vector as additional independent field. The specific mixed form allows for an element-local, discontinuous approximation of rotations, which is key to a simple and efficient discretization framework. The concept of discrete curvature provides a mathematically consistent treatment of rotation discontinuities. For linear constitutive laws, the mixed form is derived via a Legendre transform of the curvature-related strain energy. Objectivity is retained at the discrete level by interpolating relative rotations through a multiplicative split of the rotation field; path-independence is inherent to the total Lagrangian setting and verified numerically. Several benchmarks demonstrate optimal rates of convergence and accuracy, irrespective of the beam's slenderness and order of approximation. Notably, the lowest-order element entirely avoids rotation interpolation by employing element-constant rotations only.