Nonlinear dynamic analysis of shear- and torsion-free rods using isogeometric discretization and outlier removal

TL;DR

This work develops a nonlinear, shear- and torsion-free Kirchhoff rod model discretized with isogeometric analysis, omitting the director as an independent field to reduce DOFs and yield $\mathbb{R}^3$-space solutions. It couples a robust hybrid midpoint-trapezoidal time integrator with an outlier removal strategy based on an extraction operator to suppress spurious high-frequency modes, and it assesses robustness through 2D/3D benchmarks and swinging-rod applications relevant to mooring lines. The results show that, while IGA can match standard Hermite-based discretizations in accuracy for static problems, it can exhibit reduced robustness in dynamics unless enhanced with outlier removal or smaller time steps; the configuration-dependent mass term also resists simple perturbative treatment. The combined methodology demonstrates a practical, energy- and momentum-preserving approach for nonlinear aero-hydro-elastic scenarios, with clear implications for offshore and cable-like structures and avenues for future theoretical and numerical refinements.

Abstract

In this paper, we present a discrete formulation of nonlinear shear- and torsion-free rods introduced by Gebhardt and Romero in [20] that uses isogeometric discretization and robust time integration. Omitting the director as an independent variable field, we reduce the number of degrees of freedom and obtain discrete solutions in multiple copies of the Euclidean space (R^3), which is larger than the corresponding multiple copies of the manifold (R^3 x S^2) obtained with standard Hermite finite elements. For implicit time integration, we choose the same integration scheme as Gebhardt and Romero in [20] that is a hybrid form of the midpoint and the trapezoidal rules. In addition, we apply a recently introduced approach for outlier removal by Hiemstra et al. [26] that reduces high-frequency content in the response without affecting the accuracy, ensuring robustness of our nonlinear discrete formulation. We illustrate the efficiency of our nonlinear discrete formulation for static and transient rods under different loading conditions, demonstrating good accuracy in space, time and the frequency domain. Our numerical example coincides with a relevant application case, the simulation of mooring lines.

Figures (27)

• Figure 1: Convergence of the relative error between the geometrically exact beam and the nonlinear rod model gebhardt_2021_beam, computed with different discretizations on a mesh of $40$ elements, obtained at the last load step.
• Figure 2: Deformed configurations of a clamped rod bent to a circle at different load steps, computed with quadratic $C^1$ B-splines ($p=2$) and a mesh of $40$ elements.
• Figure 3: Convergence of relative errors of the clamped rod bent to a circle computed with $C^{p-1}$ B-splines of different degrees $p$ (left column) and with $C^r$ B-splines, $1\,\leq\,r\,\leq\,p-1$ (right column). The reference rate shown in figures on the right column is the convergence rate of linear fourth-order problems tagliabue_error_2014
• Figure 4: Deformed configurations of a clamped rod subjected to an in-plane loading at different time steps, computed with different discretizations and a geometrically exact beam model.
• Figure 5: The energy of a clamped rod subjected to a vanishing in-plane loading, computed with different discretizations and a geometrically exact beam model.

Theorems & Definitions (7)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
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• Remark 7