Mathematical Modelling of Cable Entanglement in Deployable Space Structures

Background
The trend in deployable space structures is to use flexible material that have a high packing ratio. Thin cords and films belong to this category of materials. While membranes have been used frequently, cords and cables are more delicate to implement as they can become entangled in protruding structures on the spacecraft during the deployment - end of mission! A recent example is the failure of the zero-gravity experiment of a Japanese antenna due to a cable getting stuck. The reason for the failure might have been residual stresses in the cable from the spools; the effect of these small stresses can be neglected in a gravity environment, but not in a space environment.

Modelling tool
The commercial software ABAQUS and LS-DYNA can model contact between line element, i.e. beams and trusses, but the algorithms can sometimes be difficult to tune with unrealistic bounding between elements. Line elements with no-compression material properties can accurately model cable behaviour in term of stiffness, but when modelling dynamic entanglement it is unclear how many line elements that are required to simulate the actual behaviour. An alternative approach is to discretise the cable by solid elements. This can be done in the software discussed above, but also in FEniCS.

FEniCS is a free software (open source) project which automates the discretization of Partial Differential Equations (PDE) by the adaptive Finite Element Method (FEM). Ko is a component in FEniCS which describes equations for elasto-visco-plasto solid mechanics in an updated Lagrange (current configuration) formulation. Ko models contact by a mass-spring model including friction, which has been demonstrated as robust in computer animation settings. Both FEniCS and Ko aim for full generality and efficiency through state-of-the-art algorithms.

FEniCS allows specification of arbitrary PDE and finite elements, including time stepping. The PDE is described in a form language which
closely resembles mathematical notation on paper. The FEniCS programming interface can be accessed through C++ or Python.

Aims and scope

Project 1:
- Verify the Ko solid and contact model against established benchmarks.
- Construct a new contact model in a PDE formulation and verify it.

Project 2:
- Analyse the differences in dynamic behaviour between line element and solid element discretisations in the software ABAQUS or LS-DYNA.
- Analyse the effects of cross-sectional shape of the cable on the entanglement probability in a deployable space structures.
- If possible, verify the results against experiments.