Abstract. We consider a nonlinear, moving boundary, fluid-structure interaction problem between a time dependent incompressible, viscous fluid flow, and an elastic structure composed of a cylindrical shell supported by a mesh of elastic rods. The fluid flow is modeled by the time-dependent Navier- Stokes equations in a three-dimensional cylindrical domain, while the lateral wall of the cylinder is modeled by the two-dimensional linearly elastic Koiter shell equations coupled to a one-dimensional system of conservation laws defined on a graph domain, describing a mesh of curved rods. The mesh supported shell allows displacements in all three spatial directions. Two-way coupling based on kinematic and dynamic coupling conditions is assumed between the fluid and composite structure, and between the mesh of curved rods and Koiter shell. Problems of this type arise in many ap- plications, including blood flow through arteries treated with vascular prostheses called stents. We prove the existence of a weak solution to this nonlinear, moving boundary problem by using the time discretization via Lie operator splitting method combined with an Arbitrary Lagrangian-Eulerian approach, and a non-trivial extension of the Aubin-Lions-Simon compactness result to problems on moving domains.
Solving Cosserat Rod Models via Collocation and the Magnus Expansion
Choosing a kinematic model for a continuum robot typically involves making a tradeoff between accuracy and computational complexity. One common modeling approach is to use the Cosserat rod equations, which have been shown to be accurate for many types of continuum robots. This approach, however, still presents significant computational cost, particularly when many Cosserat rods are coupled via kinematic constraints. In this work, we propose a numerical method that combines orthogonal collocation on the local rod curvature and forward integration of the Cosserat rod kinematic equations via the Magnus expansion, allowing the equilibrium shape to be written as a product of matrix exponentials. We provide a bound on the maximum step size to guarantee convergence of the Magnus expansion for the case of Cosserat rods, compare in simulation against other approaches, and demonstrate the tradeoffs between speed and accuracy for the fourth and sixth order Magnus expansions as well as for different numbers of collocation points. Our results show that the proposed method can find accurate solutions to the Cosserat rod equations and can potentially be competitive in computation speed.
- Award ID(s):
- 1734461
- Publication Date:
- NSF-PAR ID:
- 10281088
- Journal Name:
- 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)
- Page Range or eLocation-ID:
- 8653 to 8660
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Publisher's Version of Record
- https://doi.org/10.1109/IROS45743.2020.9340827
