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The consistency of pressure‐gradient approximations used in multi‐dimensional shock hydrodynamics

This paper presents a study of the consistency properties of the pressure‐gradient approximation used in multi‐dimensional finite‐element shock hydrodynamics codes today. In specific, consideration is given to the so‐called ‘bent‐element blues’ problem associated with the pressure‐gradient approximation when using the Q1Q0 element. On arbitrary grids comprised of distorted elements, the piecewise‐constant representation of the pressure field leads to a low‐order pressure‐gradient approximation at the global (nodal) level. This results in spurious nodal forces that are not aligned with the pressure gradient. There are several side‐effects of this behavior that include (a) incorrectly exciting physical modes in problems that exhibit unstable behavior, e.g. Rayleigh–Taylor problems (both magnetic and hydrodynamic), (b) potentially seeding hourglass modes, and (c) exhibiting non‐stationary behavior for steady‐state problems. A series of commonly used pressure‐gradient approximations are reviewed and evaluated based on linear consistency—the ability of the approximation to annihilate constant terms and exactly reproduce a linear gradient. The deeper theoretical issues associated with the proper selection of function spaces for the finite‐element hydro formulation are not discussed here. There are two gradient approximations that use piecewise‐constant data and deliver a consistent pressure‐gradient approximation on arbitrary grids. The first is the well‐known least‐squares gradient construction, and the second is a corrected gradient approximation that imposes linear consistency at the (global) nodal level. At the time of this writing, the corrected gradient approximation appears to be the most viable candidate for resolving the consistency issues associated with the Q1Q0 element technology. Copyright © 2009 John Wiley & Sons, Ltd.