Dynamic materials with formation of clots: optimal mass transport in one spatial dimension without takeover

Abstract

The concept of dynamic materials (DM) was introduced in the earlier work [1,2,9] and developed there for the case of non-colliding characteristics. The present paper extends this concept toward the case when the characteristics collide. The extension is formulated for a linear continuity equation in 1D with a no takeover condition applied to the moving masses. This implies the formation of clots with the preservation of both mass and momentum, which makes the approach resemble the method of "sticky particles" known in pressureless gas dynamics. The extension introduces non-linearity into the transportation problem that was originally missing, thus placing it closer to a non-linear conservation law.

Propagation of dilatation and shear waves through a dynamic checkerboard material geometry in 1D space + time

“…The formation of a shock wave is just one example of such extension. An alternative (simpler) scenario has been introduced in [5] for the continuity equation in 1D space + time, where a "no take over" policy was adopted for traveling elementary masses. As a consequence, when such masses collide, they become concentrated into clots (the "δ-particles").…”

On Material Optimization in Continuum Dynamics

“…3; as to Problem 2, this one allows for various answers depending on the physics of the phenomena that develop after collisions of characteristics. One of such answers was given in [12].…”

“…With "no take over" policy, this may require concentration of elementary masses into delta-particles (clots), i.e., the intersection of their wave routes at some points down the road, which makes the solution irregular. A solution of this particular kind was given for the "at" problem in [12].…”

“…One of such scenarios, traditional in gas dynamics, requires formation of the shock waves, this time artificially maintained through properly arranged controls. Another scenario was suggested in [12] for a traffic control problem with a "no take over policy". This concept allows the formation of clots, i.e., concentration (clinging) of elementary masses into the "delta-particles" of finite mass arising due to the collision of characteristics.…”

“…With reference to [11], we investigate the problem (1)-(3), (12), under material property constraints…”