A geometrically exact discrete elastic rod model based on improved discrete curvature

Abstract

This paper presents a discrete, geometrically exact elastic rod model based on improved discrete curvature. The model is an extension of finite difference type discretization of Kirchhoff–Love rod from Discrete Differential Geometry, with 4 unknowns, including translational displacements of the axis and the axial cross-sectional rotation. The orthogonality of the rod axis and cross-section is guaranteed by parallel transport in pseudotime. A newly defined unified discrete curvature with five weighting function schemes is presented. Three of them are proposed by reducing the max nonlinear term of the Taylor expansion of the rotation vector. Theory and numerical tests show that these three methods can increase the accuracy of the model compared to the traditional method. Scheme 5 performs the best out of the recommended schemes. A transformation of freedoms to the translational displacements and end rotations is introduced to the Discrete Differential Geometry model. By using this technique, rigid connections between rod segments, rotational boundaries and external ending moments can be realized conveniently and naturally. Important properties such as convergence, objectivity and insensitivity to the slenderness ratio are also verified and investigated with several suitable numerical examples. Furthermore, results show the use of discrete curvature has better performance for the stiff bar system with flexible connections than the continuous beam.