Abstract

Most Lagrangian hydrocodes have been written within the framework of a staggered grid. These have proved extremely useful, but they share some defects such as mesh imprinting, failure to maintain symmetry, and some fail to conserve total energy. It is also possible to take a cell-centred approach, as is common with Eulerian hydrocodes, which makes full conservation straightforward. The fluxes across interfaces can be derived from Riemann solvers, which have proved very robust in the Eulerian context, and avoid the need for such measures as artificial viscosity and subzonal pressures. The outstanding issue seems to be the development of a good method for moving the mesh along with the flow. However, significant progress has recently been made in solving this problem. Most Lagrangian Godunov schemes either define the nodal velocities as averages of adjacent cell centred velocities or edge velocities (from the Riemann solver), or introduce a special nodal Riemann solver [1]. We propose here to derive the mesh motion by surrounding each cell vertex with a control volume to which the conservation laws are applied. We describe this as a dual grid. A first order version of this scheme was presented in [2]. This talk presents the extension of this first order scheme to second order and cylindrical geometry. An assessment is also made of the performance of the second order method in both plane and cylindrical geometry by comparison against results obtained with a staggered grid compatible finite element code [3]. Two different approaches are also considered for moving the nodes based on the dual grid approach, a method which reconstructs nodal velocities at the start of every time step and a second which carries the nodal velocities as an additional variable.

References

1. P.-H. Maire, A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured mesh, Journal of Computational Physics, 228 (7), 2391-2425(2009).

2. A. J. Barlow, P. L. Roe, A cell centred Lagrangian Godunov scheme for shock hydrodynamics, Comput. Fluids, 46, 133-136, (2011).

3. A. J. Barlow, ‘A compatible finite element multi-material ALE hydrodynamics algorithm.’, Int. J. Numer. Meth. Fluids 2008; 56:953-964.