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A cell-centered implicit-explicit Lagrangian scheme for a unified model of nonlinear continuum mechanics on unstructured meshes
Numerical Analysis · Submitted 13 Jul 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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. 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 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.92 | 0.93 | 0.21 | asserted 1.00 |
| s1 | The scheme provably respects the stiff relaxation limits of the continuous model at the fully discrete level, thus it is asymptotic preserving. | result 0.57 | 0.87 | 0.21 | asserted 1.00 |
| s2 | 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. | method 0.94 | 0.89 | 0.05 | asserted 1.00 |
| s3 | Second-order 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. | method 0.99 | 0.88 | 0.06 | asserted 1.00 |
| s4 | 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. | method 0.93 | 0.92 | 0.05 | asserted 1.00 |
| s5 | 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. | result 0.88 | 0.95 | 0.69 | asserted 1.00 |
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Original 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. 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 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.
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1 arXiv paper cite this in 5 sentences
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Uses its data or values 1 paper
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The initial velocity field in this cross section is [equation] where $s = α(x + L/2)$ and $A, ω, a_1, a_2, α$ are constants with values (parameters retrieved from [ref] ): [equation] We use the constitutive relation [ref]-[ref] for with [equation] Complete elasticity is assumed and therefore $τ= ∞$.
Unified description of fluids and solids in Smoothed Particle Hydrodynamics · Beryllium plate · 0.83
Compares with it 1 paper
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Firstly, we plot the $y$ coordinate of the central point, which we then compare to data from the finite volume simulation [ref] .
Unified description of fluids and solids in Smoothed Particle Hydrodynamics · Beryllium plate · 0.97
Cites it as background 1 paper
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Previously, the unified model of continuum mechanics was discretized using various mesh-based techniques including Godunov-type finite volume methods and Discontinuous Galerkin methods [ref] , Arbitrary Lagrangian Eulerian methods [ref] , a finite volume methods in the Updated Lagrangian formulation with a high-order IMEX time integrator [ref] , semi-implicit staggered finite volume method [ref] for low-Mach problems, thermodynamically compatible finite volume scheme [ref] .
Unified description of fluids and solids in Smoothed Particle Hydrodynamics · Introduction · 0.98
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