Mixed Finite Elements for Geometrically Exact Beams using Discontinuous Rotations and Discrete Curvature

TL;DR

This paper introduces a novel mixed finite-element formulation for geometrically exact beams that allows for discontinuous rotations and provides a consistent treatment of discrete curvature, leading to efficient and accurate simulations.

Contribution

It presents a new mixed finite-element approach with discontinuous rotations and discrete curvature for geometrically exact beams, improving simplicity and efficiency.

Findings

Achieves optimal convergence rates and accuracy across various beam slenderness and approximation orders.

The lowest-order element avoids rotation interpolation by using element-constant rotations.

Demonstrates the method's effectiveness through several benchmark tests.

Abstract

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.