•High order schemes for a unified first order hyperbolic formulation of continuum mechanics.•The mathematical model applies simultaneously to fluid mechanics and solid mechanics.•Viscous fluids are ...treated in the frame of hyper-elasticity as generalized visco-plastic solids.•Formal asymptotic analysis reveals the connection with the Navier–Stokes equations.•The distortion tensor A in the model appears to be well-suited for flow visualization.
This paper is concerned with the numerical solution of the unified first order hyperbolic formulation of continuum mechanics recently proposed by Peshkov and Romenski 110, further denoted as HPR model. In that framework, the viscous stresses are computed from the so-called distortion tensor A, which is one of the primary state variables in the proposed first order system. A very important key feature of the HPR model is its ability to describe at the same time the behavior of inviscid and viscous compressible Newtonian and non-Newtonian fluids with heat conduction, as well as the behavior of elastic and visco-plastic solids. Actually, the model treats viscous and inviscid fluids as generalized visco-plastic solids. This is achieved via a stiff source term that accounts for strain relaxation in the evolution equations of A. Also heat conduction is included via a first order hyperbolic system for the thermal impulse, from which the heat flux is computed. The governing PDE system is hyperbolic and fully consistent with the first and the second principle of thermodynamics. It is also fundamentally different from first order Maxwell–Cattaneo-type relaxation models based on extended irreversible thermodynamics. The HPR model represents therefore a novel and unified description of continuum mechanics, which applies at the same time to fluid mechanics and solid mechanics. In this paper, the direct connection between the HPR model and the classical hyperbolic–parabolic Navier–Stokes–Fourier theory is established for the first time via a formal asymptotic analysis in the stiff relaxation limit.
From a numerical point of view, the governing partial differential equations are very challenging, since they form a large nonlinear hyperbolic PDE system that includes stiff source terms and non-conservative products. We apply the successful family of one-step ADER–WENO finite volume (FV) and ADER discontinuous Galerkin (DG) finite element schemes to the HPR model in the stiff relaxation limit, and compare the numerical results with exact or numerical reference solutions obtained for the Euler and Navier–Stokes equations. Numerical convergence results are also provided. To show the universality of the HPR model, the paper is rounded-off with an application to wave propagation in elastic solids, for which one only needs to switch off the strain relaxation source term in the governing PDE system.
We provide various examples showing that for the purpose of flow visualization, the distortion tensor A seems to be particularly useful.
In this paper we develop a new well-balanced discontinuous Galerkin (DG) finite element scheme with subcell finite volume (FV) limiter for the numerical solution of the Einstein–Euler equations of ...general relativity based on a first order hyperbolic reformulation of the Z4 formalism. The first order Z4 system, which is composed of 59 equations, is analyzed and proven to be strongly hyperbolic for a general metric. The well-balancing is achieved for arbitrary but a priori known equilibria by subtracting a discrete version of the equilibrium solution from the discretized time-dependent PDE system. Special care has also been taken in the design of the numerical viscosity so that the well-balancing property is achieved. As for the treatment of low density matter, e.g. when simulating massive compact objects like neutron stars surrounded by vacuum, we have introduced a new filter in the conversion from the conserved to the primitive variables, preventing superluminal velocities when the density drops below a certain threshold, and being potentially also very useful for the numerical investigation of highly rarefied relativistic astrophysical flows.
Thanks to these improvements, all standard tests of numerical relativity are successfully reproduced, reaching three achievements: (i) we are able to obtain stable long term simulations of stationary black holes, including Kerr black holes with extreme spin, which after an initial perturbation return perfectly back to the equilibrium solution up to machine precision; (ii) a (standard) TOV star under perturbation is evolved in pure vacuum (ρ=p=0) up to t=1000 with no need to introduce any artificial atmosphere around the star; and, (iii) we solve the head on collision of two punctures black holes, that was previously considered un–tractable within the Z4 formalism.
Due to the above features, we consider that our new algorithm can be particularly beneficial for the numerical study of quasi normal modes of oscillations, both of black holes and of neutron stars.
•Strongly hyperbolic first order Z4 formulation of the Einstein-Euler equations.•New and simple well-balanced discontinuous Galerkin schemes for numerical general relativity.•Robust conversion from conservative to primitive variables also in the presence of vacuum.•Stable long-time simulations of black holes and TOV stars in two and three space dimensions.•Head-on collision of two puncture black holes in three space dimensions using the Z4 formulation.
•First order hyperbolic reformulation of the Navier-Stokes-Korteweg system.•Combination of augmented Lagrangian approach with the Godunov-Peshkov-Romenski model.•Restoration of hyperbolicity for ...non-convex equations of state via augmented Lagrangian approach.•Thermodynamically compatible GLM curl cleaning.•ADER DG schemes with a posteriori subcell finite volume limiting.
In this paper we present a novel first order hyperbolic reformulation of the barotropic Navier-Stokes-Korteweg system. The new formulation is based on a combination of the first order hyperbolic Godunov-Peshkov-Romenski (GPR) model of continuum mechanics with an augmented Lagrangian approach that allows to rewrite nonlinear dispersive systems in first order hyperbolic form at the aid of new evolution variables. The governing equations for the new evolution variables introduced by the rewriting of the dispersive part are endowed with curl involutions that need to be taken care of. In this paper, we account for the curl involutions at the aid of a thermodynamically compatible generalized Lagrangian multiplier (GLM) approach, similar to the GLM divergence cleaning introduced by Munz et al. for the divergence constraint of the magnetic field in the Maxwell and MHD equations. A key feature of the new mathematical model presented in this paper is its ability to restore hyperbolicity even for non-convex equations of state, such as the van der Waals equation of state, thanks to the use of an augmented Lagrangian approach and the resulting inclusion of surface tension terms into the hyperbolic flux.
The governing PDE system proposed in this paper is solved at the aid of a high order ADER discontinuous Galerkin finite element scheme with a posteriori subcell finite volume limiter in order to deal with shock waves, discontinuities and steep gradients in the numerical solution. We propose an exact solution of our new mathematical model, show numerical convergence rates of up to sixth order of accuracy and show numerical results for several standard benchmark problems, including travelling wave solutions, stationary bubbles and Ostwald ripening in one and two space dimensions.
•Simple and general framework for constructing thermodynamically compatible schemes.•Compatibility is achieved via a simple scalar correction factor.•Thermodynamically compatible finite volume ...schemes.•Thermodynamically compatible DG schemes of type I and II.•A discrete cell entropy inequality is achieved by construction.•Nonlinear stability in the energy norm can be proven for all schemes.
We introduce a simple and general framework for the construction of thermodynamically compatible schemes for the numerical solution of overdetermined hyperbolic PDE systems that satisfy an extra conservation law. As a particular example in this paper, we consider the general Godunov-Peshkov-Romenski (GPR) model of continuum mechanics that describes the dynamics of nonlinear solids and viscous fluids in one single unified mathematical formalism.
A main peculiarity of the new algorithms presented in this manuscript is that the entropy inequality is solved as a primary evolution equation instead of the usual total energy conservation law, unlike in most traditional schemes for hyperbolic PDE. Instead, total energy conservation is obtained as a mere consequence of the proposed thermodynamically compatible discretization. The approach is based on the general framework introduced in Abgrall (2018) 1. In order to show the universality of the concept proposed in this paper, we apply our new formalism to the construction of three different numerical methods. First, we construct a thermodynamically compatible finite volume (FV) scheme on collocated Cartesian grids, where discrete thermodynamic compatibility is achieved via an edge/face-based correction that makes the numerical flux thermodynamically compatible. Second, we design a first type of high order accurate and thermodynamically compatible discontinuous Galerkin (DG) schemes that employs the same edge/face-based numerical fluxes that were already used inside the finite volume schemes. And third, we introduce a second type of thermodynamically compatible DG schemes, in which thermodynamic compatibility is achieved via an element-wise correction, instead of the edge/face-based corrections that were used within the compatible numerical fluxes of the former two methods. All methods proposed in this paper can be proven to be nonlinearly stable in the energy norm and they all satisfy a discrete entropy inequality by construction. We present numerical results obtained with the new thermodynamically compatible schemes in one and two space dimensions for a large set of benchmark problems, including inviscid and viscous fluids as well as solids. An interesting finding made in this paper is that, in numerical experiments, one can observe that for smooth isentropic flows the particular formulation of the new schemes in terms of entropy density, instead of total energy density, as primary state variable leads to approximately twice the convergence rate of high order DG schemes for the entropy density.
•Nonlinear large strain elasto-plasticity with crack formation and crack propagation.•Material failure is described via a scalar material damage parameter and reaction-type source terms.•Nonlinear ...hyperelasticity with diffuse interface representation of complex geometry.•Very large spectrum of different elasto-plastic material behaviors can be captured.•Rate-dependent, thermodynamically compatible model that is also able to describe material fatigue.
We are concerned with the numerical solution of a unified first order hyperbolic formulation of continuum mechanics that goes back to the work of Godunov, Peshkov and Romenski 65,68,97 (GPR model) and which is an extension of nonlinear hyperelasticity that is able to describe simultaneously nonlinear elasto-plastic solids at large strain, as well as viscous and ideal fluids. The proposed governing PDE system also contains the effect of heat conduction and can be shown to be symmetric and thermodynamically compatible, as it obeys the first and second law of thermodynamics. In this paper we extend the GPR model to the simulation of nonlinear dynamic rupture processes, which can be achieved by adding an additional scalar to the governing PDE system. This extra parameter describes the material damage and is governed by an advection-reaction equation, where the stiff and highly nonlinear reaction mechanisms depend on the ratio of the local equivalent stress to the yield stress of the material. The stiff reaction mechanisms are integrated in time via an efficient exponential time integrator. Due to the multiple spatial and temporal scales involved in the problem of crack generation and propagation, the model is solved on space–time adaptive Cartesian meshes using high order accurate discontinuous Galerkin finite element schemes endowed with an a posteriori subcell finite volume limiter. A key feature of our new model is the use of a twofold diffuse interface approach that allows the cracks to form anywhere and at any time, independently of the chosen computational grid, which is simply adaptive Cartesian (AMR). This is substantially different from many fracture modeling approaches that need to resolve discontinuities explicitly, such as for example dynamic shear rupture models used in computational seismology, where the geometry of the rupture fault needs to prescribed a priori. We furthermore make use of a scalar volume fraction function α that indicates whether a given spatial point is inside the solid (α=1) or outside (α=0), thus allowing the description of solids of arbitrarily complex shape. We show extensive numerical comparisons with experimental results for stress-strain diagrams of different real materials and for the generation and propagation of fracture in rocks and pyrex glass at low and high velocities. Overall, a very good agreement between numerical simulations and experiments is obtained. The proposed model is also naturally able to describe material fatigue.
We apply a hyperbolic cell-centered finite volume method to solve a steady diffusion equation on unstructured meshes. This method, originally proposed by Nishikawa using a node-centered finite volume ...method, reformulates the elliptic nature of viscous fluxes into a set of augmented equations that makes the entire system hyperbolic. We introduce an efficient and accurate solution strategy for the cell-centered finite volume method. To obtain high-order accuracy for both solution and gradient variables, we use a successive order solution reconstruction: constant, linear, and quadratic (k-exact) reconstruction with an efficient reconstruction stencil, a so-called wrapping stencil. By the virtue of the cell-centered scheme, the source term evaluation was greatly simplified regardless of the solution order. For uniform schemes, we obtain the same order of accuracy, i.e., first, second, and third orders, for both the solution and its gradient variables. For hybrid schemes, recycling the gradient variable information for solution variable reconstruction makes one order of additional accuracy, i.e., second, third, and fourth orders, possible for the solution variable with less computational work than needed for uniform schemes. In general, the hyperbolic method can be an effective solution technique for diffusion problems, but instability is also observed for the discontinuous diffusion coefficient cases, which brings necessity for further investigation about the monotonicity preserving hyperbolic diffusion method.
•High order cell-centered ADER-WENO ALE schemes for nonlinear hyperelasticity (GPR model).•Thermodynamically compatible symmetric hyperbolic formulation of continuum mechanics.•The mathematical model ...applies simultaneously to fluid mechanics and solid mechanics.•Fluids are treated in the frame of hyperelasticity as generalized visco-plastic solids.•Test cases for viscous heat conducting fluids and nonlinear elasto-plastic solids.
This paper is concerned with the numerical solution of the unified first order hyperbolic formulation of continuum mechanics proposed by Peshkov & Romenski Peshkov I, Romenski E. A hyperbolic model for viscous Newtonian flows. Continuum Mechanics and Thermodynamics 2016;28:85–104., which is based on the theory of nonlinear hyperelasticity of Godunov & Romenski 67Godunov S, Romenski E. Nonstationary equations of the nonlinear theory of elasticity in Euler coordinates. Journal of Applied Mechanics and Technical Physics 1972;13:868–885.Godunov S., Romenski E., Elements of Continuum Mechanics and Conservation Laws. Kluwer Academic/ Plenum Publishers; 2003., further denoted by GPR model. Notably, the governing PDE system is symmetric hyperbolic and fully consistent with the first and the second principle of thermodynamics. The nonlinear system of governing equations of the GPR model is overdetermined, large and includes stiff source terms as well as non-conservative products. In this paper we solve this model for the first time on moving unstructured meshes in multiple space dimensions by employing high order accurate one-step ADER-WENO finite volume schemes in the context of cell-centered direct Arbitrary-Lagrangian-Eulerian (ALE) algorithms.
The numerical method is based on a WENO polynomial reconstruction operator on moving unstructured meshes, a fully-discrete one-step ADER scheme that is able to deal with stiff sources Dumbser M., Enaux C., Toro E., Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. Journal of Computational Physics . 2008a;227:3971–4001., a nodal solver with relaxation to determine the mesh motion, and a path-conservative technique of Castro & Parés for the treatment of non-conservative products Parés C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM Journal on Numerical Analysis. 2006;44:300–321.Castro M, Gallardo J, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. applications to shallow-water systems. Mathematics of Computation 2006;75:1103–1134.. We present numerical results obtained by solving the GPR model with ADER-WENO-ALE schemes in the stiff relaxation limit, showing that fluids (Euler or Navier-Stokes limit), as well as purely elastic or elasto-plastic solids can be simulated in the framework of nonlinear hyperelasticity with the same system of governing PDE. The obtained results are in good agreement when compared to exact or numerical reference solutions available in the literature.
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