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Jiun-Shyan Chen, "An oscillation limiting and flux conserving meshfree formulation for shock modelling"
High rate impulsive loading can generate hydrodynamic processes where shocks give rise to state and field variable discontinuities. The essential shock physics include the Rankine- Hugoniot (R-H) jump condition and the second law of thermodynamics for entropy production. Oscillatory instabilities in the form of Gibbs phenomenon occur at the jump when higher-order methods are used to approximate the discontinuous solution. In this work a stable (oscillation limiting) and flux conserved formulation under the reproducing kernel particle method framework is developed for shock modeling. The Rankine-Hugoniot condition is naturally satisfied and correct shock speed is obtained under the appropriate weak form. A Riemann-embedded flux divergence operator, formulated under the framework of stabilized conforming nodal integration, guarantees correct entropy production and flux conservation and thus produces the correct shock propagation, while a Godunov-type flux- corrected velocity limits shock oscillation. The Godunov velocity correction for oscillation limiting is constrained to the shock region using a detection algorithm based on the reproducing kernel spectral decomposition property, so that higher-order accuracy is maintained elsewhere in the domain.
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