Finite point set method: A mesh-free approach to model airbag inflation

摘要

The updated regulatory requirements (FMVSS 208), and the necessity of a better understanding of early stages of airbag inflation have led airbag designers to the challenge of predicting the airbag fluid flow and deployment kinematics. This multi-physics application brings Finite Element and Finite Volume simulation approaches to their limits in terms of accuracy and computational efficiency. Some inherent difficulties of these classical methods, like the necessity to discretize a complex fluid domain, have motivated the development of an alternative mesh-free mathematical model for solving the gas dynamics equations. An overview on the theoretical bases for this model is presented: the Euler equations are solved on a Lagrangian set of points. Derivatives have to be computed on a set of discrete function values. For this purpose, a smooth interpolation of the discrete function values is constructed using polynomial functions, best fitted to the discrete values using a moving least square method. The most simple finite point set scheme, using central differences for time integration, is not stable. Stabilization is obtained using an upwind scheme. The convergence of the scheme in terms of spatial discretization refinement is shown on a simple example. In order to maintain an even distribution, points are generated or removed automatically during the simulation. The weak coupling of this mesh-free model with the membrane Finite Element model used for the airbag membrane is done in a classical way: the fluid mesh-free code provides pressures to the structural explicit Finite Element code, which computes forces at nodes, accelerations, velocities and displacements considered as a new boundary condition for the fluid domain. A specific algorithm has been developed in order to separate points belonging to different airbag folds, and achieve a realistic propagation of pressure waves. Correlation of the simulation against theoretical and experimental benchmarks are presented: shock tubes, convergent-divergent channel, airbag, inflator tank tests, flat airbag against pendulum, folded airbags. Some improvements aiming at accelerating the convergence are presented. Further industrialization of the method is discussed, with, among other possible improvements, parallel programming.