Ideal loading history for an arbitrary sequence of shocks and ramps

Ramp-loading has become widely used in high energy density (HED) experiments to explore higher compressions and lower temperatures than attainable with a shock, as is relevant to the interiors of planets and compact stellar objects. For a limited drive duration, such as a laser pulse, the maximum amount of material experiences the ramp load if all the characteristics converge to form a shock simultaneously. We previously developed methods to perform this calculation deterministically and in a self-similar way, given states along the desired isentrope. However, that approach did not readily generalize to more complicated scenarios commonly encountered in HED experiments, such as a shock followed by a ramp, or a ramp propagating through a layered assembly. A more elegant method is to propagate characteristics backward in time from the point where they meet. The solution of the single-material ramp case is trivial, and the method applies to more general situations including arbitrary combinations or shocks and ramps, propagation of a ramp through a stack, and materials exhibiting more complicated properties such as strength.