Pierre-Henri Maire

A Cell-Centered Lagrangian Scheme for Two-Dimensional Compressible Flow Problems

We present a new Lagrangian cell-centered scheme for two-dimensional compressible flows. The primary variables in this new scheme are cell-centered, i.e., density, momentum, and total energy are defined by their mean values in the cells. The vertex velocities and the numerical fluxes through the cell interfaces are not computed independently, contrary to standard approaches, but are evaluated in a consistent manner due to an original solver located at the nodes. The main new feature of the algorithm is the introduction of four pressures on each edge, two for each node on each side of the edge. This extra degree of freedom allows us to construct a nodal solver which fulfills two properties. First, the conservation of momentum and total energy is ensured. Second, a semidiscrete entropy inequality is provided. In the case of a one-dimensional flow, the solver reduces to the classical Godunov acoustic solver: it can be considered as its two-dimensional generalization. Many numerical tests are presented. They are representative test cases for compressible flows and demonstrate the robustness and the accuracy of this new solver.

A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes

We present a high-order cell-centered Lagrangian scheme for solving the two-dimensional gas dynamics equations on unstructured meshes. A node-based discretization of the numerical fluxes for the physical conservation laws allows to derive a scheme that is compatible with the geometric conservation law (GCL). Fluxes are computed using a nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The first-order scheme is conservative for momentum and total energy, and satisfies a local entropy inequality in its semi-discrete form. The two-dimensional high-order extension is constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess this new scheme. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new scheme.

A two-dimensional unstructured cell-centered multi-material ALE scheme using VOF interface reconstruction

We present a new cell-centered multi-material arbitrary Lagrangian-Eulerian (ALE) scheme to solve the compressible gas dynamics equations on two-dimensional unstructured grid. Our ALE method is of the explicit time-marching Lagrange plus remap type. Namely, it involves the following three phases: a Lagrangian phase wherein the flow is advanced using a cell-centered scheme; a rezone phase in which the nodes of the computational grid are moved to more optimal positions; a cell-centered remap phase which consists of interpolating conservatively the Lagrangian solution onto the rezoned grid. The multi-material modeling utilizes either concentration equations for miscible fluids or the Volume Of Fluid (VOF) capability with interface reconstruction for immiscible fluids. The main original feature of this ALE scheme lies in the introduction of a new mesh relaxation procedure which keeps the rezoned grid as close as possible to the Lagrangian one. In this formalism, the rezoned grid is defined as a convex combination between the Lagrangian grid and the grid resulting from condition number smoothing. This convex combination is constructed through the use of a scalar parameter which is a scalar function of the invariants of the Cauchy-Green tensor over the Lagrangian phase. Regarding the cell-centered remap phase, we employ two classical methods based on a partition of the rezoned cell in terms of its overlap with the Lagrangian cells. The first one is a simplified swept face-based method whereas the second one is a cell-intersection-based method. Our multi-material ALE methodology is assessed through several demanding two-dimensional tests. The corresponding numerical results provide a clear evidence of the robustness and the accuracy of this new scheme.

ReALE: A reconnection-based arbitrary-Lagrangian-Eulerian method

Abstract We present a new reconnection-based arbitrary-Lagrangian–Eulerian (ALE) method. The main elements in a standard ALE simulation are an explicit Lagrangian phase in which the solution and grid are updated, a rezoning phase in which a new grid is defined, and a remapping phase in which the Lagrangian solution is transferred (conservatively interpolated) onto the new grid. In standard ALE methods the new mesh from the rezone phase is obtained by moving grid nodes without changing connectivity of the mesh. Such rezone strategy has its limitation due to the fixed topology of the mesh. In our new method we allow connectivity of the mesh to change in rezone phase, which leads to general polygonal mesh and allows to follow Lagrangian features of the mesh much better than for standard ALE methods. Rezone strategy with reconnection is based on using Voronoi tessellation. We demonstrate performance of our new method on series of numerical examples and show it superiority in comparison with standard ALE methods without reconnection.

A cell-centered diffusion scheme on two-dimensional unstructured meshes

We propose a new cell-centered diffusion scheme on unstructured meshes. The main feature of this scheme lies in the introduction of two normal fluxes and two temperatures on each edge. A local variational formulation written for each corner cell provides the discretization of the normal fluxes. This discretization yields a linear relation between the normal fluxes and the temperatures defined on the two edges impinging on a node. The continuity of the normal fluxes written for each edge around a node leads to a linear system. Its resolution allows to eliminate locally the edge temperatures as function of the mean temperature in each cell. In this way, we obtain a small symmetric positive definite matrix located at each node. Finally, by summing all the nodal contributions one obtains a linear system satisfied by the cell-centered unknowns. This system is characterized by a symmetric positive definite matrix. We show numerical results for various test cases which exhibit the good behavior of this new scheme. It preserves the linear solutions on a triangular mesh. It reduces to a classical five-point scheme on rectangular grids. For non orthogonal quadrangular grids we obtain an accuracy which is almost second order on smooth meshes.

A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry

The goal of this paper is to present high-order cell-centered schemes for solving the equations of Lagrangian gas dynamics written in cylindrical geometry. A node-based discretization of the numerical fluxes is obtained through the computation of the time rate of change of the cell volume. It allows to derive finite volume numerical schemes that are compatible with the geometric conservation law (GCL). Two discretizations of the momentum equations are proposed depending on the form of the discrete gradient operator. The first one corresponds to the control volume scheme while the second one corresponds to the so-called area-weighted scheme. Both formulations share the same discretization for the total energy equation. In both schemes, fluxes are computed using the same nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The control volume scheme is conservative for momentum, total energy and satisfies a local entropy inequality in its first-order semi-discrete form. However, it does not preserve spherical symmetry. On the other hand, the area-weighted scheme is conservative for total energy and preserves spherical symmetry for one-dimensional spherical flow on equi-angular polar grid. The two-dimensional high-order extensions of these two schemes are constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess these new schemes. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new schemes.

Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics

This work presents a multi-dimensional cell-centered unstructured finite volume scheme for the solution of multimaterial compressible fluid flows written in the Lagrangian formalism. This formulation is considered in the Arbitrary-Lagrangian-Eulerian (ALE) framework with the constraint that the mesh velocity and the fluid velocity coincide. The link between the vertex velocity and the fluid motion is obtained by a formulation of the momentum conservation on a class of multi-scale encased volumes around mesh vertices. The vertex velocity is derived with a nodal Riemann solver constructed in such a way that the mesh motion and the face fluxes are compatible. Finally, the resulting scheme conserves both momentum and total energy and, it satisfies a semi-discrete entropy inequality. The numerical results obtained for some classical 2D and 3D hydrodynamic test cases show the robustness and the accuracy of the proposed algorithm.

Two-step hybrid conservative remapping for multimaterial arbitrary Lagrangian-Eulerian methods

We present a new hybrid conservative remapping algorithm for multimaterial Arbitrary Lagrangian-Eulerian (ALE) methods. The hybrid remapping is performed in two steps. In the first step, only nodes of the grid that lie inside subdomains occupied by single materials are moved. At this stage, computationally cheap swept-region remapping is used. In the second step, nodes that are vertices of mixed cells (cells containing several materials) and vertices of some cells in a buffer zone around mixed cells are moved. At this stage, intersection-based remapping is used. The hybrid algorithm results in computational expense that lies between swept-region and intersection-based remapping We demonstrate the performance of our new method for both structured and unstructured polygonal grids in two dimensions, as well as for cell-centered and staggered discretizations.

A nominally second-order cell-centered Lagrangian scheme for simulating elastic-plastic flows on two-dimensional unstructured grids

In this paper, we describe a cell-centered Lagrangian scheme devoted to the numerical simulation of solid dynamics on two-dimensional unstructured grids in planar geometry. This numerical method, utilizes the classical elastic-perfectly plastic material model initially proposed by Wilkins [M.L. Wilkins, Calculation of elastic-plastic flow, Meth. Comput. Phys. (1964)]. In this model, the Cauchy stress tensor is decomposed into the sum of its deviatoric part and the thermodynamic pressure which is defined by means of an equation of state. Regarding the deviatoric stress, its time evolution is governed by a classical constitutive law for isotropic material. The plasticity model employs the von Mises yield criterion and is implemented by means of the radial return algorithm. The numerical scheme relies on a finite volume cell-centered method wherein numerical fluxes are expressed in terms of sub-cell force. The generic form of the sub-cell force is obtained by requiring the scheme to satisfy a semi-discrete dissipation inequality. Sub-cell force and nodal velocity to move the grid are computed consistently with cell volume variation by means of a node-centered solver, which results from total energy conservation. The nominally second-order extension is achieved by developing a two-dimensional extension in the Lagrangian framework of the Generalized Riemann Problem methodology, introduced by Ben-Artzi and Falcovitz [M. Ben-Artzi, J. Falcovitz, Generalized Riemann Problems in Computational Fluid Dynamics, Cambridge Monogr. Appl. Comput. Math. (2003)]. Finally, the robustness and the accuracy of the numerical scheme are assessed through the computation of several test cases.

A discontinuous Galerkin discretization for solving the two-dimensional gas dynamics equations written under total Lagrangian formulation on general unstructured grids

Based on the total Lagrangian kinematical description, a discontinuous Galerkin (DG) discretization of the gas dynamics equations is developed for two-dimensional fluid flows on general unstructured grids. Contrary to the updated Lagrangian formulation, which refers to the current moving configuration of the flow, the total Lagrangian formulation refers to the fixed reference configuration, which is usually the initial one. In this framework, the Lagrangian and Eulerian descriptions of the kinematical and the physical variables are related by means of the Piola transformation. Here, we describe a cell-centered high-order DG discretization of the physical conservation laws. The geometrical conservation law, which governs the time evolution of the deformation gradient, is solved by means of a finite element discretization. This approach allows to satisfy exactly the Piola compatibility condition. Regarding the DG approach, it relies on the use of a polynomial space approximation which is spanned by a Taylor basis. The main advantage in using this type of basis relies on its adaptability regardless the shape of the cell. The numerical fluxes at the cell interfaces are computed employing a node-based solver which can be viewed as an approximate Riemann solver. We present numerical results to illustrate the robustness and the accuracy up to third-order of our DG method. First, we show its ability to accurately capture geometrical features of a flow region employing curvilinear grids. Second, we demonstrate the dramatic improvement in symmetry preservation for radial flows.