[Paper Review] A cell-centered implicit-explicit Lagrangian scheme for a unified model of nonlinear continuum mechanics on unstructured meshes

Boscheri W., Chiocchetti S.+1 more|arXiv (Cornell University)|Jan 1, 2022

Computational Fluid Dynamics and Aerodynamics113 references21 citations

TL;DR

This paper presents a cell-centered implicit-explicit (IMEX) Lagrangian finite volume scheme for the Godunov-Peshkov-Romenski (GPR) unified model of nonlinear continuum mechanics on unstructured meshes. The scheme achieves second-order accuracy in space and time, satisfies the Geometrical Conservation Law (GCL), and is asymptotic preserving, accurately capturing fluid, solid, and transitional material behaviors—including viscous, heat-conducting, elastic, and elasto-plastic responses—across 2D and 3D test cases with high robustness and energy conservation.

ABSTRACT

A cell-centered implicit-explicit updated Lagrangian finite volume scheme on unstructured grids is proposed for a unified first-order hyperbolic formulation of continuum fluid and solid mechanics, namely the Godunov-Peshkov-Romenski (GPR) model. The scheme provably respects the stiff relaxation limits of the continuous model at the fully discrete level, thus it is asymptotic preserving. Furthermore, the GCL is satisfied by a compatible discretization that makes use of a nodal solver to compute vertex-based fluxes that are used both for the motion of the computational mesh as well as for the time evolution of the governing PDEs. Second order of accuracy in space is achieved using a TVD piecewise linear reconstruction, while an implicit-explicit (IMEX) Runge-Kutta time discretization allows the scheme to obtain higher accuracy also in time. Particular care is devoted to the design of a stiff ODE solver, based on approximate analytical solutions of the governing equations, that plays a crucial role when the visco-plastic limit of the model is approached. We demonstrate the accuracy and robustness of the scheme on a wide spectrum of material responses covered by the unified continuum model that includes inviscid hydrodynamics, viscous heat conducting fluids, elastic and elasto-plastic solids in multidimensional settings.

Motivation & Objective

• To develop a robust, second-order accurate Lagrangian finite volume scheme for the unified hyperbolic GPR model of continuum mechanics on unstructured meshes.
• To ensure the scheme is asymptotic preserving by accurately capturing the stiff relaxation limits of the GPR model, including Navier-Stokes-Fourier and elastic limits.
• To maintain geometric conservation (GCL) via a nodal solver that computes vertex-based fluxes for both mesh motion and PDE evolution.
• To enable accurate simulation of diverse material responses—ideal fluids, viscous fluids, elastic solids, and elasto-plastic materials—within a single unified framework.
• To achieve stability and accuracy in stiff regimes through an implicit treatment of relaxation source terms using a semi-analytical ODE solver.

Proposed method

• A cell-centered finite volume framework is used, with conserved variables (mass, momentum, energy) stored at cell centers and numerical fluxes computed via a nodal Riemann solver.
• The mesh motion is governed by a nodal velocity computed from one-dimensional Riemann problems across neighboring cells, ensuring GCL compliance via discrete Gauss theorem.
• Second-order spatial accuracy is achieved using a TVD piecewise linear reconstruction of conserved variables.
• An implicit-explicit (IMEX) Runge-Kutta time discretization is employed, treating stiff relaxation terms implicitly and flux terms explicitly.
• A custom semi-analytical ODE solver is developed to handle stiff relaxation terms in the GPR model, particularly near the Navier-Stokes-Fourier and hyperelastic limits.
• The scheme is applied to a reduced GPR model where only the symmetric part of the distortion tensor (metric tensor) is evolved, simplifying the system when rotational degrees of freedom are negligible.

Experimental results

Research questions

• RQ1 Can a cell-centered Lagrangian finite volume scheme be constructed that preserves the GCL and is asymptotic preserving for the unified GPR model on unstructured meshes?
• RQ2 How can second-order accuracy in space and time be achieved in a Lagrangian framework for a hyperbolic PDE system with stiff relaxation sources?
• RQ3 Can the same numerical scheme accurately simulate the full spectrum of material responses—from ideal fluids to elasto-plastic solids—within a single unified model?
• RQ4 What is the role of the semi-analytical ODE solver in stabilizing the scheme under stiff relaxation limits, and how does it ensure consistency with the Navier-Stokes-Fourier equations?
• RQ5 How does the scheme perform in terms of energy conservation and robustness across complex, multidimensional test cases involving large deformations and material transitions?

Key findings

• The LGPR scheme achieves second-order accuracy in both space and time, as confirmed by convergence studies on manufactured solutions and exact Riemann problems.
• The scheme is asymptotic preserving: it correctly captures the Navier-Stokes-Fourier limit for arbitrarily small relaxation times, with the numerical solution converging to the expected viscous, heat-conducting fluid behavior.
• The scheme preserves total energy to machine precision, as demonstrated in the twisting column test case with ω₀ = 100 and ω₀ = 200, where energy conservation is maintained over time.
• The method accurately simulates the transition from elastic to elasto-plastic behavior in the Kovalskii test, with numerical results closely matching analytical solutions for the plastic wave speed and stress evolution.
• The scheme successfully models complex 3D phenomena such as the twisting column, showing no spurious oscillations or nonphysical pressure distributions, even under strong compression and rotation.
• The scheme reduces to established methods in limiting cases: it recovers the hydrodynamics scheme of [83] in the ideal fluid limit and the hyperelasticity scheme of [27] in the elastic limit.