A Conservative Three-Dimensional EulerianMethod for Coupled Solid–FluidShock Capturing

A new method is presented for the explicit Eulerian finite difference computation of shock capturing problems involving multiple resolved material phases in three dimensions. We solve separately for each phase the equations of fluid dynamics or solid mechanics, using as interface boundary conditions artificially extended representations of the individual phases. For fluids we use a new 3D spatially unsplit implementation of the piecewise parabolic (PPM) method of Colella and Woodward. For solids we use the 3D spatially unsplit Eulerian solid mechanics method of Miller and Colella. Vacuum and perfectly incompressible obstacles may also be employed as phases. A separate problem is the time evolution of material interfaces, which are represented by planar segments constructed with a volume-of-fluid method. The volume fractions are advanced in time using a second-order 3D spatially unsplit advection routine with a velocity field determined by solution of interface-normal two-phase Riemann problems. From the Riemann problem solutions we also determine crossinterface momentum and energy fluxes. The volume fractions in mixed cells may be arbitrarily small, which would ordinarily make the Courant–Friedrichs–Lewy time step stability limit arbitrarily small as well. We overcome this limitation using the mass-redistribution formalism to conservatively redistribute generalized mass in the neighborhood of the split cells.