High-order staggered Lagrangian hydrodynamics (I): framework of the discretization scheme

This paper presents a discretization framework for high-order staggered Lagrangian hydrodynamics, bridging two well-established algorithms algorithms: high-order curvilinear finite element Lagrangian hydrodynamics [Dobrev et al. 2012] and compatible hydrodynamics methods[Caramana et al.1998]. We emphasizes the critical relationship between the degrees of freedom (DOFs) associated with the density variable and the choice of numerical quadrature rules which is employed in the mass conservation law. The precise quadrature rule will lead the consistency issues arising from the density and internal energy variables being defined at different physical points. Our approach resolves this by unifying the quadrature rule for the specific discretization space, though this inevitably introduces hourglass distortion. Another key feature of the proposed framework is the strategic arrangement of DOFs for kinematic and thermodynamic variables, which enhances computational efficiency and leads to diagonal mass matrices in the momentum and energy equations. Finally, we present a smooth numerical example that validate the accuracy of the proposed high-order Lagrangian hydrodynamics framework.