A generalization of the spherical linear interpolation (or slerp) for the finite rotations to the case of more than two control variables on SO(3) is introduced to design an objective FE-formulation for the non-linear space Cosserat rod model. The interpolation uses the De Casteljau’s algorithm. In this way, the same interpolation degree can be used for the placement of the centroid curve and for the finite rotation of the cross-section. A recursive formula is obtained for the interpolation of the rotations. Similar recursive formulas are derived for the spin and the curvature vector, leading to a generalization of the B´ezier basis functions on the manifold SO(3). The rod formulation so obtained is invariant under a rigid rotation (objective), in the sense that the patch-test with respect to the rigid body motion is satisfied. Furthermore, an optimal path-independence is achieved as verified by several numerical investigations.

An objective FE-formulation for Cosserat rods based on the spherical Bézier interpolation

Leopoldo Greco
Investigation
;
Alessandro Cammarata
Investigation
;
Domenico Castello
Investigation
;
Massimo Cuomo
Investigation
2024-01-01

Abstract

A generalization of the spherical linear interpolation (or slerp) for the finite rotations to the case of more than two control variables on SO(3) is introduced to design an objective FE-formulation for the non-linear space Cosserat rod model. The interpolation uses the De Casteljau’s algorithm. In this way, the same interpolation degree can be used for the placement of the centroid curve and for the finite rotation of the cross-section. A recursive formula is obtained for the interpolation of the rotations. Similar recursive formulas are derived for the spin and the curvature vector, leading to a generalization of the B´ezier basis functions on the manifold SO(3). The rod formulation so obtained is invariant under a rigid rotation (objective), in the sense that the patch-test with respect to the rigid body motion is satisfied. Furthermore, an optimal path-independence is achieved as verified by several numerical investigations.
2024
Cosserat rod, Spherical linear interpolation, Spherical B´ezier interpolation, Invariant formulation, Path-independence
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11769/609009
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? ND
social impact