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Some Observations on Artificial Bulk Viscosity in LS-DYNA: What Noh Knew in 1978

A review of the development background for artificial bulk viscosity is presented with a focus on the development implemented in LS-DYNA and similar to other explicit codes. The analytical solution and benchmark calculation results presented by Noh (1978) for a one dimension shock problem are used to indicate errors in the simulation results attributed by Noh to artificial bulk viscosity. Noh identifies and demonstrates three types of artificial bulk viscosity errors: (1) excess shock heating, (2) non-uniform meshing and (3) due to spherical geometries. The LSDYNA Multi-Material ALE solver is used to model Noh’s shock problem and the results are reported for plane strain, axisymmetric (cylindrical) and spherical geometries; the latter being of practical importance in modeling free air bursts. Several appendices are include that cover closely related topics: the role of bulk viscosity in time step stability calculations, solution of the Rankine–Hugoniot Jump Conditions for Noh’s plane strain case, and a comparison of four explicit solver results for the spherical geometry case. Perhaps the most lucid discussion of shock viscosity, a.k.a. artificial bulk viscosity, is Section 2.8 “The shock viscosity in one dimension” in the paper by Benson (1992), from which this manuscript borrows, in part. Calculations involving shocks have long used the numerical artifice of shock viscosity to obtain reasonable solutions in a numerical continuum mechanics framework for the discontinuities associated with shocks. The role of the shock viscosity is to spread out the shock thickness (discontinuity) over several elements. Thus the mesh refinement must be fine enough to allow for this approximation. Strong shocks in gases are dissipative due to both the gas heating caused by the large and rapid compression of the gas and viscosity among the gas molecules in the densified shock thickness. Most numerical shock viscosity algorithms lump these two dissipative mechanisms together and treat the resulting viscosity as a pressure term. Von Neumann and Richtmyer (1950) introduced a pressure viscosity term in their work with one dimensional shock propagation.