An arbitrary Lagrangian-Eulerian semi-implicit hybrid method for continuum mechanics with GLM cleaning

TL;DR

The paper tackles the numerical solution of the thermodynamically compatible GPR model for fluids and solids by introducing a semi-implicit ALE hybrid FV/FE method. It decouples fast pressure waves from bulk motion via operator splitting and employs GLM curl cleaning to control curl involutions in distortion A and thermal impulse J, preserving energy consistency. The method uses unstructured staggered grids, space-time FV transport, and P1 FE pressure solves with Picard iterations, achieving robust all-Mach-number performance validated by extensive 2D/3D benchmarks. Results indicate strong accuracy and stability across fluid and solid regimes, with potential extensions toward exact involution preservation and higher-order schemes for broader practical impact.

Abstract

This paper proposes a semi-implicit arbitrary Lagrangian-Eulerian (ALE) method for the solution of the unified Godunov-Peshkov-Romenski (GPR) model of continuum mechanics. To handle the curl free involutions arising in the solid limit of the model, the original system is augmented by adopting a thermodynamically compatible generalized Lagrangian multiplier (GLM) approach. Next, an operator splitting strategy decouples the computation of fast pressure waves from the bulk velocity of the medium yielding a transport subsystem, containing convective terms and non-conservative products, and a Poisson-type subsystem, for the pressure. A second splitting yields an ODE subsystem comprising only the potentially stiff source terms, responsible for the relaxation of the model between its fluid and solid limits. The mesh motion can be driven by two sources: the local fluid velocity and a prescribed boundary displacement. For the spatial discretization, we employ unstructured staggered grids, with the pressure defined on the primal mesh and all remaining variables on the dual grid. The transport subsystem is advanced via an explicit finite volume method, in which integration over closed space-time control volumes ensures verification of the geometric conservation law (GCL). On the other hand, implicit continuous finite elements are used for the discretization of the pressure subsystem and an implicit DIRK scheme is employed to solve the ODE subsystem. Consequently, the proposed approach is well suited to address all Mach number flows. A comprehensive set of benchmarks is employed to assess the accuracy and robustness of the proposed methodology in tackling both fluid and solid mechanics problems.

Figures (28)

• Figure 1: Construction of two interior dual elements $C_{{\mathfrak{i}}}$ and $C_{{\mathfrak{j}}}$ from primal elements $T_{k}$, $T_{l}$, $T_{m}$.
• Figure 2: Construction of a space-time control volume $\widetilde{C}_{{\mathfrak{i}}}$ generated by the deformation of the dual cell $C_{{\mathfrak{i}}}$ between times $t^{n}$ and $t^{n+1}$. $C_{{\mathfrak{i}}}^{n}$ and $C_{{\mathfrak{i}}}^{n+1}$ are shadowed in green while the space time boundary is indicated with dashed lines. The black triangles at time $t^{n}$ plane correspond to the primal elements generating the dual cell $C_{{\mathfrak{i}}}^{n}$.
• Figure 3: Map of a space-time face $\widetilde{\Gamma}_{{\mathfrak{i}}{\mathfrak{j}}}$ to the reference element in the local coordinate system $\chi-\tau$. The face $\widetilde{\Gamma}_{{\mathfrak{i}}{\mathfrak{j}}}$ (shadowed in gray) shared by the space-time control volumes $\widetilde{C}_{{\mathfrak{i}}}$ and $\widetilde{C}_{{\mathfrak{j}}}$ is determined by vertex $\widetilde{\mathbf{X}}_1$, $\widetilde{\mathbf{X}}_2$, $\widetilde{\mathbf{X}}_3$ and $\widetilde{\mathbf{X}}_4$.
• Figure 4: RP1 Sod. 1D cuts of the solution obtained using the hybrid FV/FE method for the compressible GPR model with the fully Eulerian code (blue squares) and the ALE scheme (green dashed line) compared against the exact solution for the Navier-Stokes equations (black line) and the solution of the GPR model obtained using the HTC FV scheme in HTCA2 (red dashed line). From left-top to right-bottom: density, horizontal velocity component, distortion field component $A_{11}$, and pressure.
• Figure 5: RP2 smooth double rarefaction. 1D cuts of the solution obtained using the hybrid FV/FE method for the compressible GPR model (blue squares) compared against the exact solution (black line). From left to right: density, horizontal velocity component and pressure.

Theorems & Definitions (2)

• Proposition 2.1
• proof