Implicit and Explicit Techniques for Space Elevator Simulation

The obvious way to simulate space elevator tether-climber dynamics is to discretize the tether using a mass-spring model, insert the climber into one of the springs, and simulate using an explicit solver. However, this approach either diverges or has unphysical disturbances when the climber passes nodes of the tether. The root of the problem is that as the climber approaches a node, the stiffness of the spring between the climber and the node goes to infinity, resulting in a mode with diverging frequency. Methods typically used to avoid this problem include: dropping and re-adding nodes as the climber passes, resampling the tether discretization, and attaching the climber to the spring it is on through a compliant spring. Instead, this presentation suggests keeping the simple discretization and changing the integration method such that a physically accurate result can be obtained. To achieve this, two problems must be solved. First, the integrator must avoid diverging due to the existence of a mode with diverging frequency. Second, a precise solution of the discretized problem actually has unphysical artifacts as the climber passes the node. Both problems can be addressed by using an implicit integrator with a time step tuned to exclude the highest frequency mode of the discretization. Surprisingly, excluding the highest frequency mode allows us to go from a precise but unphysical solution to the discretized problem to a less precise but more physically accurate solution. Unfortunately, using an implicit method implies solving large systems of equations, which is slow. Instead an Implicit-Explicit method can be used that only treats the forces on the climber implicitly, which limits the system of equations to the climber and its directly adjacent nodes.