2011 SIAM Conference on Computational Science and Engineering Minisimposium on ”Advanced Discretization Techniques for Lagrangian Hydrodynamics” Tzanio Kolev and Mikhail Shashkov Feb 28–Mar 4, 2011 Reno, Nevada Abstract The discretization of the Euler equations of gas dynamics in a moving Lagrangian frame is at the heart of many large-scale multi-physics numerical algorithms. As part of the successful arbitrary Lagrangian-Eulerian (ALE) technique, the Lagrangian step in compressible shock hydrodynamics simulations still presents computational challenges, and much effort has been put into extending its accuracy and robustness. This minisymposium focuses on some recent efforts aiming to improve the symmetry preservation, shock resolution and energy conservation of Lagrangian methods. Highlighted topics include novel finite element and cell-centered discretization techniques, new forms of artificial viscosity and methods for axisymmetric problems.

Session 1 Exploration of a Cell-Centered Lagrangian Hydrodynamics Method D. Burton, S. Runnels, T. Carney and M. Shashkov We present a new cell-centered Lagrange hydro method, discuss some of its aspects that are still being explored, and demonstrate its performance on several test problems. The method is second-order in both space and time, enforcing conservation equations for volume, linear momentum, and total energy on the same control volume, while enforcing angular momentum on a dual control volume. The method employs a compatible decomposition of total energy that enables the computation of specific kinetic and internal energy at the cell’s center of mass. Trial values for stress and velocity at cell interfaces are determined using a multi-dimensional, two-shock Riemann-like solution with innovations that increase its effectiveness for skewed cells. These trial values are then combined to compute velocity and stress at the vertices that are then propagated back to the cell interfaces in a way that maintains geometric volume compatibility and angular momentum.

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High Order Finite Elements for Lagrangian Hydrodynamics, Part I: General Framework V. Dobrev, T. Ellis, T. Kolev and R. Rieben This talk presents a general Lagrangian framework for discretization of compressible shock hydrodynamics using high order finite elements. The novelty of our approach is in the use of high order polynomial spaces to define both the mapping and the reference basis functions. This leads to improved robustness and symmetry preservation properties, better representation of the mesh curvature that naturally develops with the material motion, significant reduction in mesh imprinting, and high-order convergence for smooth problems.

High Order Finite Elements for Lagrangian Hydrodynamics, Part II: Numerical Results V. Dobrev, T. Ellis, T. Kolev and R. Rieben This talk focuses on several practical considerations of the high order finite element discretization from Part I, including the generalization of traditional concepts such as corner forces and artificial viscosity. We consider an extensive set of test problems to examine shock wave propagation over unstructured/distorted meshes and symmetry preservation for radial flows in 2D, 3D and axisymmetric geometry. In each case we demonstrate robust performance of the high order FEM implementation in our research code BLAST.

Shock Hydrodynamics on Tetrahedral Meshes G. Scovazzi A new, variational multiscale stabilized formulation Lagrangian shock hydrodynamics is presented. To the author’s knowledge, it is the only hydrocode that can accurately compute highly unsteady shock hydrodynamics transients on triangular/tetrahedral meshes in two/three dimensions, as well as the more commonly used quadrilateral/hexahedral meshes. Piecewise linear, equal-order interpolation is adopted for velocities, displacements, and thermodynamic variables. This last aspect makes the current formulation insensitive to the typical pathologies affecting standard hydrocodes (namely hourglass on quadrilateral/hexahedral meshes, and artificial stiffness on triangular/tetrahedral meshes). Numerical tests for the unsteady Euler equations of gas dynamics are presented in two and three dimensions.

Session 2 A general formalism to derive cell-centered schemes for two-dimensional Lagrangian hydrodynamics on unstructured grids P.-H. Maire The aim of this work is to develop a general formalism to derive cell-centered schemes for 2D Lagrangian hydrodynamics on unstructured grids that meet the compatibility GCL requirement. The high-order extension of this general cell-centered scheme is constructed 2

using the two-dimensional extension of the Generalized Riemann Problem methodology in its acoustic version. Various numerical results on representative compressible fluid flows are presented to demonstrate the accuracy and the robustness of these schemes.

Staggered Lagrangian discretization based on cell-centered Riemann solver A bridge from staggered to cell-centered Lagrangian schemes R. Loubere, P.-H. Maire and P. Vachal In this presentation we provide a new formalism to bridge well-known staggered Lagrangian and newly developed cell-centered Lagrangian schemes. We will present this new approach that leads to formally second-order accurate scheme in space and time. A 3D implementation of this scheme will provide numerical results on classical test cases of hydrodynamics.

A high order cell centred Lagrangian Godunov scheme for shock hydrodynamics A. Barlow A new cell centred Lagrangian Godunov scheme for shock hydrodynamics is proposed. The new method uses a transient dual grid to define the motion of the vertices in a way that is consistent with the geometric conservation law. The extension of the scheme to second order accuracy in space is considered and an initial assessment of the performance of the method is made by comparison with results obtained with a compatible staggered grid scheme.

Issues in Designing a 2D Cylindrically Symmetric Conservative Lagrange Hydro Scheme D. Miller Work by Burton, Caramana, et al, developed practical energy conserving Lagrangian hydrodynamics schemes in the late 1990’s. The Wilkins area-weighted discretization, which preserves spherical symmetry in cylindrical geometry, can be made energy conserving, but the change requires Lagrangian corner masses, which in turn raise some issues about the definitions of corner volumes. We present one such scheme, some test results, and discuss the impact of our definitions on other packages in a multi-physics code.

Session 3 A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhedral Meshes K. Lipnikov and M. Shashkov We construct a new mimetic tensor artificial viscosity on general polygonal and polyhedral meshes. The tensor artificial viscosity is based on discretization of coordinate invariant operators, divergence of a tensor and gradient of a vector. We consider both 3

non-symmetric and symmetric forms of the tensor artificial viscosity. We demonstrate performance of the new viscosity for the Noh implosion, Sedov explosion and Saltzman piston problems on a set of various meshes.

Entropy Viscosity for Lagrangian Hydrodynamics J.-L. Guermond A new technique for approximating nonlinear conservation equations is described (entropy viscosity method). The novelty is that a nonlinear viscosity based on the local size of an entropy production is added to the numerical discretization at hand. The methodology initially introduced in Eulerian coordinates is adapted to Lagrangian Hydrodynamics. The methodology is numerically illustrated on standard benchmark problems.

A Corner and Dual Mesh ALE Remapping Algorithm for use with the Compatible Energy Lagrangian Discretization M. Owen and M. Shashkov The energy conserving Lagrangian hydrodynamic discretization introduced by Caramana et al. in the late 90’s has proven to be quite successful. However, the introduction of subzonal Lagrangian elements complicates ALE remapping algorithms for such methods, both in remapping the subzonal masses as well as properties on the dual mesh. We will discuss new ALE algorithms for computing such remapped properties which obey important goals such as consistency with the primary mesh remapping, conservation, and monotonicity.

Recent Advances in Lagrangian Hydro Methods M. Shashkov I will describe recent advances in Lagrangian hydro methods: the symmetry preservations, shock resolution and energy conservation; novel finite element and cell-centered discretization techniques, new forms of artificial viscosity and methods for axisymmetric problems.

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