A 3D GCL compatible cell-centered Lagrangian scheme for solving gas dynamics equations
Introduction
The Euler equations written under their Lagrangian form are usually employed for the simulation of compressible fluid flows with strong shock waves. Indeed, with such a formalism the mesh follows the fluid motion which leads to a natural refinement around the shock zones. The Lagrangian formalism enables the simulation of flows combining rarefaction and shock waves as well as flows with large changes of the computational domain such as those encountered in the domain of High Energy Density Physics (HEDP). Furthermore, the Lagrangian formalism enables a simple treatment of multi-material and free surface flows since the interface between different materials is naturally tracked.
Designing a scheme to solve the Lagrangian hydrodynamics implies to determine the nodes velocities in order to move the computational grid at each time step. Two main approaches are classically used. The staggered schemes consider the node velocity as being part of the primary variables. Such schemes compute the kinematic variables at the nodes and evaluate the thermodynamic variables at the center of the cells. The staggered schemes are the descendants of the seminal works of [40], [43] and have been considerably improved, refer to [10], [22], [30], [37]. In particular, advanced definitions of artificial viscosity, which enables the conversion of kinetic energy into internal energy through shock waves, have been proposed in [9], [12], [42] as well as the definition of subzonal forces [5], [8].
The scheme presented in this work belongs to the cell-centered finite volume schemes family developed from the work of Godunov [18]. In those schemes, all the variables are evaluated at the cell centers and the nodal velocity is determined using a nodal solver. This solver is based on approximate Riemann solver and is designed such that it respects the Geometric Conservation Law (GCL), conserves the momentum and total energy and satisfies the Second Law of thermodynamics. Multi-dimensional extensions of such schemes have been proposed in the works [1], [13], [26]. The second-order extension of such finite volume schemes can be performed using a MUSCL method (Monotonic Upstream-Centered Scheme for Conservation Laws) which consists in linearly reconstructing the pressure and velocity fields in order to improve the nodal velocity approximation and decrease the entropy production. This reconstruction step can lead to oscillations and needs a limiting procedure to enforce monotonicity. One can read [2], [3], [38] for the classical approach of the limiting step, and [7], [11], [19], [24], [27] for its application to cell-centered Finite Volume Lagrangian schemes.
Finite element schemes have also been developed to solve the Lagrangian hydrodynamics. All the variables are then defined at the nodes and the fields within the cells are interpolated. With such schemes, a high-order extension is quite natural by use of a high-order interpolation basis and curved geometry. In particular, one can refer to the works [14], [15], [34], [35].
The 3D extension of Finite Volume Lagrangian schemes were recently proposed in [4], [10], [11], [22], [28], [41] leading to two main difficulties: the GCL compatibility and the limiting of a reconstructed field in the 3D space.
The GCL compatibility is crucial and imposes the cell volume computed geometrically to coincide with the volume evaluated by the discretization of the volume conservation equation. The difficulty comes from the polyhedral cell and its non-planar faces. The two main solutions are to parametrize the cell [11], which means that a mapping is defined between the cell and a reference element, or to decompose the cell into tetrahedrons [6], [28]. However, the decomposition of a polyhedral cell is not unique in 3D and more complex than in the 2D framework.
In this paper, a systematic and symmetric decomposition of polyhedral cells is proposed. It relies on the decomposition of a non-planar face into planar triangular faces by adding a supplementary point on the face which is the barycenter of the face vertices. The kinematics of this supplementary point is solved by prescribing its velocity as being the barycenter of the vertices velocity. This amounts to assume a linear representation of the velocity field over the faces with respect to the space variable. Moreover, this decomposition enables to define a discrete divergence operator leading to the respect of the GCL.
Concerning the limiting of a reconstructed field in the 3D framework, the difficulty mainly appears in the case of a vectorial field. Most of the time, the vectors are limited component by component but this leads to symmetry preservation issue. A vectorial slope limiter has been proposed by Luttwak et al. [23] consisting in limiting all the extrapolated vectors lying outside a certain convex hull. However, let us mention that the construction of the convex hull is a rather complex task in 3D. A convenient approach, which enables to easily preserve the flow symmetries, is to limit the vectorial fields in a reference basis [22], [25] which utilizes the physical directions of the flow. For example, one can construct this basis upon the cell velocity. However, the classical limiting procedures are most of the time not sufficient to prevent overshoots and oscillations.
In the present work, a multi-dimensional minmod limiter is proposed. This new limiter enables to construct a cell gradient from nodal gradients and the minmod function. Combined with a classical Barth–Jespersen limiter, it provides a monotone solution with drastically reduced overshoots on strong shocks. Moreover, this limiter degenerates onto the classical minmod limiter in 1D.
The paper is structured as follows. The Lagrangian gas dynamics equations and the mesh notations are described in section 2. The section 3 proposes a systematic and symmetric geometric decomposition of the polyhedral cells and presents the resulting discrete divergence operator. The momentum and total energy conservation equations are discretized through section 4, which also presents the determination of the nodal fluxes and the detailed treatment of boundary conditions. The second order extension in space and time is then presented in section 5 with the details of the new multi-dimensional minmod limiter. Finally, the accuracy and robustness of the scheme is assessed in section 6 against several Cartesian test cases including strong shock waves.
Section snippets
Governing equations
Let be a domain of fluid moving with the fluid velocity and its boundary, then for a fluid of density ρ, velocity V, pressure P and total energy E, the gas dynamics equations under Lagrangian form write where n is the unit outward normal of . These equations correspond respectively to the mass conservation, the momentum conservation and the total energy conservation. This system is closed using an equation of
Discretization of the volume time rate of change
Using the previous face splitting, a polyhedral cell can be decomposed into tetrahedrons. Let be the tetrahedron such that its basis is the triangle and its remaining vertex is the space origin (refer to Fig. 2). The cell volume is then computed as where is the signed volume of the tetrahedron . Denoting by p, and the three points of the triangle , and by O the space origin, all the tetrahedrons can be represented by . This
Discretization of the Euler equations
In a Lagrangian scheme, the mass conservation equation is easily respected since it imposes the cell mass to be constant. Now, applying the previous discrete operators (7) and (8) to the semi-discrete momentum and total energy conservation equations writes where the subscript c indicates that the quantity has been mass averaged on the cell. For example, if φ is a physical variable denotes its mass
Second order in space
The second order extension in space of this Godunov-type scheme is performed using a MUSCL method and a piecewise linear representation of the pressure and velocity fields. Let us denote φ a scalar, the linear reconstruction of φ in cell writes where , and are respectively the extrapolated value at point x, the mean value and the gradient of variable φ in cell c. Let us recall that in the 3D space, the pressure gradient is a vector and the velocity
Numerical test cases
All the test cases presented in this section are characterized by the gamma gas law which considers a perfect gas ruled by the thermodynamics law where γ is the polytropic index of the gas. One has if the gas is monoatomic and if the gas is diatomic. The isentropic sound speed in the cell is defined as The test cases chosen present strong shock waves which lead to important mesh deformations and enable to prove the scheme robustness.
Conclusion
This paper proposes a symmetric and systematic geometric decomposition of the polyhedral cells using the faces barycenter. Using the assumption of a linear velocity field on the cell faces, this decomposition enables to define a discrete divergence operator and to respect the GCL on any unstructured polyhedral mesh. Moreover, a new multi-dimensional minmod limiter is constructed from nodal gradients and the minmod function. This method enables to drastically reduce overshoots around strong
Acknowledgements
This study has been carried out with financial support from the French State, managed by the French National Research Agency (ANR) in the frame of the “Investments for the future” Programme IdEx Bordeaux (ANR-10-IDEX-03-02), Cluster of excellence CPU.
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