Lubrication equations are fourth order degenerate di usion equations of the form h t + r (f(h)r h) = 0, describing thin lms or liquid layers driven by surface tension. Recent studies of singularities in which h ! 0 at a point, describing rupture of the uid layer, show that such equations exhibit complex dynamics which can be di cult to simulate accurately. In particular, one must ensure that the numerical approximation of the interface does not show a false premature rupture. Generic nite di erence schemes have the potential to manifest such instabilities especially when under-resolved. We present new numerical methods, in one and two space dimensions, that preserve positivity of the solution, regardless of the spatial resolution, whenever the PDE has such a property. We also show that the schemes can preserve positivity even when the PDE itself is only known to be nonnegativity preserving. We prove that positivity preserving nite di erence schemes have unique positive solutions for all time. We prove stability and convergence of both positivity preserving and generic methods, in one and two space dimensions, to positive solutions of the PDE, showing that the generic methods also preserve positivity and have global solutions for su ciently ne meshes. We generalize the positivity preserving property to a nite element framework and show, via concrete examples, how this leads to the design of other positivity preserving schemes.
The paper deals with simulation of damage spread in special structures with waiting links." These structures are stable against dynamic impacts due to their morphology. They are able to transform partial damage" through a large region, thereby dissipating the energy of the impact. We s i m ulate various structures with waiting links and compare their characteristics with conventional designs.
Motivated by studies on a type of brain injury known as diffuse axonal injury, the dynamic azimuthal shearing of a mixture of a transversely isotropic viscoelastic material which is surrounded by a softer isotropic viscoelastic material is considered. It is demonstrated how the states of maximum strain and strain rate occur near the interface between the two materials. The dependency of these states of maximum strain and strain rate on the boundary conditions on the outer surface of the isotropic material is also discussed.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.