Semi-implicit hybrid finite volume/finite element method for the GPR model of continuum mechanics

TL;DR

This paper addresses efficient simulation of incompressible and weakly compressible $GPR$ models for continuum mechanics, which unify elastoplastic solids and viscous fluids. It proposes a semi-implicit hybrid finite-volume/finite-element method on unstructured staggered grids that splits the system into a transport/non-conservative subsystem (solved explicitly) and a pressure subsystem (solved implicitly via finite elements), with stiff relaxation terms for the distortion $A$ and heat-flux $J$ treated implicitly. The scheme is designed to be asymptotic-preserving in the fluid relaxation limit toward $Navier ext{-}Stokes ext{-}Fourier$ and in the low Mach regime, enabling stable, efficient computations without prohibitive time steps. Extensive 2D and 3D benchmarks demonstrate second-order spatial accuracy, robustness across solid and fluid media, and good agreement with reference solutions, highlighting the method’s potential for complex fluid-structure interactions and multiphysics applications, with future extensions to ALE and higher-order strategies.

Abstract

We present a new hybrid semi-implicit finite volume / finite element numerical scheme for the solution of incompressible and weakly compressible media. From the continuum mechanics model proposed by Godunov, Peshkov and Romenski (GPR), we derive the incompressible GPR formulation as well as a weakly compressible GPR system. As for the original GPR model, the new formulations are able to describe different media, from elastoplastic solids to viscous fluids, depending on the values set for the model's relaxation parameters. Then, we propose a new numerical method for the solution of both models based on the splitting of the original systems into three subsystems: one containing the convective part and non-conservative products, a second subsystem for the source terms of the distortion tensor and heat flux equations and, finally, a pressure subsystem. In the first stage of the algorithm, the transport subsystem is solved by employing an explicit finite volume method, while the source terms are solved implicitly. Next, the pressure subsystem is implicitly discretised using finite elements. Within this methodology, unstructured grids are employed, with the pressure defined in the primal grid and the rest of the variables computed in the dual grid. To evaluate the performance of the proposed scheme, a numerical convergence analysis is carried out, which confirms the second order of accuracy in space. A wide range of benchmarks is reproduced for the incompressible and weakly compressible cases, considering both solid and fluid media. These results demonstrate the good behaviour and robustness of the proposed scheme in a variety of scenarios and conditions.

Figures (17)

• Figure 1: Sketch of the face-based unstructured grids in 2D. Left: $T_{k}$, $T_{l}$, $T_{m}$ are the triangles of the primal grid and $V_{j},\, j=1,\dots, 5$ are their vertices. Right: Interior (grey elements) and boundary (white elements) elements of the dual mesh. The boundary between the two interior dual cells $C_{{\mathfrak{i}}}$ and $C_{{\mathfrak{j}}}$, $\Gamma_{{\mathfrak{i}}{\mathfrak{j}}}$, is highlighted in red.
• Figure 2: Lid-driven cavity. Left: contour plot of the distortion component $A_{12}$. Right: 1D cut in $x-$ and $y-$directions of the velocity components $u_{2}$ and $u_{1}$ computed using the new hybrid FV/FE method for the incompressible GPR model (blue solid line - $u_{1}$ and dark grey solid line - $u_{2}$) and reference solutions reported in GGS82 (blue squares - $u_{1}$ and black circles - $u_{2}$).
• Figure 3: Shear motion. 1D cut in $x-$direction of the velocity component $u_{2}$ of the numerical solution obtained using the hybrid FV/FE method for the weakly compressible GPR model with the local ADER-ENO approach (blue squares). Reference solution computed with a TVD-FV scheme on a mesh of $1000$ cells (black solid line). From left top to right bottom: first Stokes with $\mu=10^{-2}$, first Stokes with $\mu=10^{-3}$, first Stokes with $\mu=10^{-4}$, shear solid.
• Figure 4: Double shear layer. Contour plots of the distortion field component $A_{12}$ obtained with the new semi-implicit hybrid FV/FE solver at times $t\in\left\lbrace 0.4,0.8,1.2,1.8\right\rbrace$ (from top to bottom). Numerical results were obtained with the incompressible GPR (left) and the weakly compressible GPR (right).
• Figure 5: Solid rotor. Contour plots of the velocity field component $u_{1}$ obtained with the hybrid FV/FE approach (left) and the HTC-FV scheme in HTCTotalAbgrall (right).