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A Thermodynamically Consistent High-Order Framework for Staggered Lagrangian Hydrodynamics

Zhiyuan Sun, Jun Liu, Pei Wang

Abstract

We present a consistent high-order staggered Lagrangian hydrodynamics framework designed to reconcile an underlying disparity in existing curvilinear formulations: the mismatch between quadrature-based "strong" mass conservation and the discrete degrees of freedom (DOFs) of thermodynamic variables. By mathematically coupling the numerical quadrature rule with the density representation, our approach ensures rigorous point-wise consistency between density, internal energy, and pressure. This synchronization eliminates the ambiguity of equation-of-state (EOS) updates inherent in previous high-order staggered methods. To stabilize the discretization, we develop a high-order generalization of the subzonal pressure method by conceptually enriching the pressure field from the $Q^{m-1}$ to the $Q^m$ finite element space. We prove that evaluating this enriched field using a high-order quadrature rule naturally generates a restorative anti-hourglass force, which exactly recovers the classical $Q^1-P^0$ compatible hydrodynamics algorithm as a limiting case for $m=1$. Furthermore, we introduce a concise, algorithmic formulation of tensor artificial viscosity that streamlines implementation and significantly reduces computational overhead in high-order settings. The resulting framework yields strictly diagonal mass matrices for both momentum and energy equations, enabling highly efficient, fully explicit time integration without global linear solves. Extensive numerical benchmarks, including smooth convergence tests and complex shock-dominated flows, demonstrate that the proposed method achieves optimal high-order accuracy while maintaining superior geometric robustness.

A Thermodynamically Consistent High-Order Framework for Staggered Lagrangian Hydrodynamics

Abstract

We present a consistent high-order staggered Lagrangian hydrodynamics framework designed to reconcile an underlying disparity in existing curvilinear formulations: the mismatch between quadrature-based "strong" mass conservation and the discrete degrees of freedom (DOFs) of thermodynamic variables. By mathematically coupling the numerical quadrature rule with the density representation, our approach ensures rigorous point-wise consistency between density, internal energy, and pressure. This synchronization eliminates the ambiguity of equation-of-state (EOS) updates inherent in previous high-order staggered methods. To stabilize the discretization, we develop a high-order generalization of the subzonal pressure method by conceptually enriching the pressure field from the to the finite element space. We prove that evaluating this enriched field using a high-order quadrature rule naturally generates a restorative anti-hourglass force, which exactly recovers the classical compatible hydrodynamics algorithm as a limiting case for . Furthermore, we introduce a concise, algorithmic formulation of tensor artificial viscosity that streamlines implementation and significantly reduces computational overhead in high-order settings. The resulting framework yields strictly diagonal mass matrices for both momentum and energy equations, enabling highly efficient, fully explicit time integration without global linear solves. Extensive numerical benchmarks, including smooth convergence tests and complex shock-dominated flows, demonstrate that the proposed method achieves optimal high-order accuracy while maintaining superior geometric robustness.

Paper Structure

This paper contains 24 sections, 130 equations, 28 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Compressible Euler equations and discretization space
  3. Quadrature Rules and Nodal Arrangements
  4. Examples of $Q^m-Q^{m-1}$ Space Pairs
  5. The Gauss quadrature rule and the mass conservation law
  6. The relationship between the quadrature rule and the number of DOFs of the density variable
  7. The Issues with Exact Integration
  8. Momentum conservation law
  9. Calculation of the right-hand side
  10. Mass matrix discretization
  11. Nodal force assembly
  12. Internal energy balance equation
  13. Numerical stabilization: Hourglass control and artificial viscosity
  14. Hourglass distortion and rank deficiency analysis
  15. The high-order hourglass control algorithm
  16. ...and 9 more sections

Figures (28)

  • Figure 1: Reference element of the $Q^1-P^0$ FEM space pair, showing the DOFs of the kinematic variable (black circles) and the thermodynamic variable (red circle).
  • Figure 2: Reference element of the $Q^2-Q^1$ FEM space pair, showing the DOFs of the kinematic variables (black circles) and the thermodynamic variables (red circles).
  • Figure 3: Reference element of the $Q^3-Q^2$ FEM space pair, showing the DOFs of the kinematic variables (black circles) and the thermodynamic variables (red circles).
  • Figure 4: Reference element $(\xi, \eta)$ of the $Q^1$ element and its 2D Gauss quadrature points.
  • Figure 5: Reference element $(\xi, \eta)$ of the $Q^2$ element and its 2D Gauss quadrature points.
  • ...and 23 more figures