Elimination of artificial grid distortion and hourglass-type motions by means of Lagrangian subzonal masses and pressures

The bane of Lagrangian hydrodynamics calculations is the premature breakdown of grid topology that results in severe degradation of accuracy and run termination often long before the assumption of a Lagrangian zonal mass has ceased to be valid. At short spatial grid scales this is usually referred to by the terms "hourglass" mode or "keystone" motion associated, in particular, with underconstrained grids such as quadrilaterals and hexahedrons in two and three dimensions, respectively. At longer spatial lengths relative to the grid spacing there is what is referred to ubiquitously as "spurious vorticity," or the long-thin zone problem. In both cases the result is anomalous grid distortion and tangling that has nothing to do with the actual solution, as would be the case for turbulent flow. In this work we show how such motions can be eliminated by the proper use of subzonal Lagrangian masses, and associated densities and pressures. These subzonal pressures give rise to forces that resist these spurious motions. The pressure is no longer a constant in a zone; it now accurately reflects the density gradients that can occur within a zone due to its differential distortion. Subzonal Lagrangian masses can be choosen in more than one manner to obtain subzonal density and pressure variation. However, these masses arise in a natural way from the intersection of the Lagrangian contours, through which no mass flows, that are associated with both the Lagrangian zonal and nodal masses in a staggered spatial grid hydrodynamics formulation. This is an extension of the usual Lagrangian assumption that is often applied to only the zonal mass. We show that with a proper discretization of the subzonal forces resulting from subzonal pressures, hourglass motion and spurious vorticity can be eliminated for a very large range of problems. (C) 1998 Academic Press.