Title: Closure Models for Multimaterial Cells in Arbitrary Lagrangian-Eulerian Hydrocodes Speaker: Mikhail Shashkov, Los Alamos National Laboratory Date/Time: Monday, April 23, 2007, 9:00 am – 10:00 am Location: CSRI Building, Room 90 (Sandia NM) Brief Abstract: High-speed multimaterial flows with strong shear deformations occur in many problems of interest. Due to the nature of shock wave propagation in complex materials, the Arbitrary Lagrangian-Eulerian (ALE) Methods are currently the only proven technology to solve such problems. In ALE methods, the mesh does not move with the fluid, and so it is unavoidable that mixed cells containing two or more materials will appear. Multimaterial cells are introduced in ALE methods to represent material interfaces that undergo high deformation. The main difficulties in this case are how to accurately determine the thermodynamic states of the individual material components and the nodal forces that such a zone generates, despite the lack of information about the velocity distribution within multimaterial cells. A separate set of material properties is normally maintained for all the materials in each multimaterial cell along with the volume fractions that define the fraction of the cell’s volume occupied by each material. The volume fractions also can be used to reconstruct material interfaces inside mixed cell. There are several different types of subcell models, or closures, are in routine use in hydrocodes (see, e.g., [6]). The simplest of these assumes that volumetric strain is equal (equal compressibility) in all the materials within multimaterial cell and so the volume fractions are unchanged during the Lagrangian phase. The main deficiency of this approach is clearly illustrated by considering a cell containing a mixture of gas and metal. The gas clearly more compressible then the metal and so should take most of the volume change, but the simple equal strain model forces metal to undergo the same volumetric strain as the gas. Different class of methods is based on assumption of pressure equilibrium (PE), or on introducing some pressure relaxation (PR) mechanism (see, e.g., [9, 10, 8, 7, 4, 2]). The pressure at a material interface should be continuous; however, the pressure within a computational cell represents an average pressure integrated over the cell volume. This means that there is no physics requirement for absolute pressure equilibrium within a multimaterial cell. In fact for the entire computational cell to come to pressure equilibrium a shock wave would have to cross the cell many times, while CFL stability condition restricts a shock wave from crossing any cell in a single time step. However, pressure continuity at material interfaces does suggest that the pressure within a multi-material cell should move towards pressure equilibrium, rather than to diverge from it. This can be achieved by introducing mechanism like viscosity in the model (see, e.g., [10, 2]), where authors require equilibrium of sum of pressure and artificial viscosity. In addition to PE or PR this class of method includes conservation of volume and some form of conservation of total internal energy, which is still not enough equations to close the model. One of the possible closures is to assume thermal equilibrium (TE), which is questionable, as it implies the need for unrealistically large thermal conductivity within each multimaterial cell. There are several other possibilities, e.g., one can use the assumption of equality of change in entropy for each material. Such a model is proposed in [4] and has the important property that it leads to a hyperbolic system of equations that satisfy an entropy inequality under CFL-like restrictions. Methods in this class also differ with respect to how the equations are approximated, from fully implicit, [2, 4], to fully explicit, [10]. It is important to note that in this class of method we assume no knowledge of the actual configuration of materials in the cell. Recently, new class of closure models that attempts to emulate the behavior of separate Lagrangian subcells has been developed, [1, 3, 5]. In this class of methods one estimates the velocity normal to the interface between materials and then estimates the change in the volume for each material. This changes can be corrected according to some heuristic rules. For example, one can require that if entire cell is expanding (contracting) then all materials in this cell have to expand (contract), [1]. In ideal situation the position and orientation of the material interface may be known (e.g., from interface reconstruction). Each material has distinct set of material properties. Internal energy is updated separately for each material from its own p dV equation. Common pressure for mixed cell, which is used in momentum equation, is computed using principles of conservation of total energy. One also can introduce some internal energy exchange between materials inside mixed cells, which allows more freedom in definition of common pressure, [3, 5]. In my presentation I will describe different closure models and present numerical comparison of different models in Lagrangian calculations with mixed cells. This work is result of numerous fruitful discussions with A. Abgrall, D. Bailey, A. Barlow, Yu. Bondarenko, D. Burton, B. Despres, J. Grove, A. Harrison, B. Kashiwa, M. Kucharik, R, Liska, R. Loubere, R. Lowrie, P.-H. Maire, R. Schmitt, P. Vachal, B. Wendroff, Yu. Yanilkin. This work was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396 and the DOE Office of Science Advanced Scientific Computing Research (ASCR) Program in Applied Mathematics Research. CSRI POC: Randy Summers, (505) 844-6296; Allen Robinson, (505) 844-6614; and Bill Rider (505) 844-1572 |