ICNPAA 2010: Mathematical Problems in
Engineering, Aerospace and Sciences

To prevent aircraft loss-of-control incidents, knowledge of the safe maneuvering envelope is of extreme importance. The safe maneuvering envelope can be determined by means of reachability analysis from a known safe set, for example the set of all trim states. Implicit surfaces provide a convenient way to describe sets because of their ability to inherently handle merging and pinching of sets. Level set methods add dynamics to implicit surfaces, and the evolution can be tracked by the viscosity solution of a Hamilton-Jacobi partial differential equation.

Unfortunately, uniform grid based solvers for level set methods suffers greatly from the curse of dimensionality. Additionally, for Eulerian methods the time step is limited by the Courant-Friedrich-Lewy condition. This combination makes application to aircraft dynamics too computationally demanding. In this paper an alternative method is used based on a Semi-Lagrangian solution method on adaptive kd-tree grids. This method allows a larger timestep, and adapts the grid as required during the evolution of the implicit surface.

The method is applied to several examples in two- and three dimensions. The results are compared with an Eulerian solution method. Based on the results conclusions are drawn and future research directions are formulated.