Modeling of Parachute Dynamics with Front Tracking Method

Abstract

We use the front tracking method on a spring system to model the dynamic evolution of parachute canopies. The canopy surface of a parachute is represented by a triangulated surface mesh with preset equilibrium length on each side of the simplices. The stretching and wrinkling of the canopy and its supporting string chords (risers) are modeled by the spring system. The spring constants of the canopy and the risers are chosen based on the analysis of Young's modulus for fabric surface and string chord. The mass-spring system is a nonlinear ODE system. Added by the numerical and computational analysis, we show that the spring system has an upper bound of the eigen frequency. We analyzed the system by considering three spring models and we proved in one case that all eigenvalues are imaginary and there exists an upper bound for eigen-frequency. Based on this analysis, we analyzed the numerical accuracy and stability of the nonlinear spring mass system for fabric surface and its tangential and normal motion. We used the fourth order Runge-Kutta method to solve the ODE system and showed that the time step is linearly dependent on the mesh size for the system. And also high order method helps to control amplification of system. Damping is added to dissipate the excessive spring internal energy. The current model does not have radial reinforcement cables and has not taken into account the canopy porosity. This mechanical structure is coupled with the incompressible Navier-Stokes solver through the "Impulse method" which computes the velocity of the point mass by superposition of momentum. We analyzed the numerical stability of the spring system and used this computational module to simulate the flow pattern around a static parachute canopy and the dynamic evolution during the parachute inflation process. The numerical solutions have been compared with the available experimental data and there are good agreements in the terminal descent velocity and breathing frequency of the parachute.