Multi-scale Lagrangian shock hydrodynamics on Q1/P0 finite elements: Theoretical framework and two-dimensional computations

A new multi-scale, stabilized method for Q1/P0 finite element computations of Lagrangian shock hydrodynamics is presented. Instabilities (of hourglass type) are controlled by a stabilizing operator derived using the variational multi-scale analysis paradigm. The resulting stabilizing term takes the form of a pressure correction. With respect to broadly accepted hourglass control approaches, the novelty of the method resides in its residual-based character. The stabilizing residual has a definite physical significance, since it embeds a discrete form of the Clausius–Duhem inequality. Effectively, the proposed stabilization samples the production of entropy to counter numerical instabilities. The proposed technique is applied to materials with no shear strength (e.g., fluids), for which there exists a caloric equation of state, and extensions to the case of materials with shear strength (e.g., solids) are also envisioned. The stabilization operator is incorporated into a mid-point, predictor/multi-corrector time integration algorithm, which conserves mass, momentum and total energy. Encouraging numerical results in the context of compressible gas dynamics confirm the potential of the method.