Implementation of AMR in Multi-Material ALE Hydrocodes

Robert W. Anderson^1

^1 Lawrence Livermore National Laboratory 7000 East Ave., Livermore 94513, CA

The transformation of an existing multi-material hydrocode to allow dynamicadaptive mesh refinement (AMR) is a complex undertaking requiringunderstanding of both algorithms as well as implementation and designissues. I will discuss a recent project of this type involving amulti-material hydrodynamics model including multi-material zones withinterface reconstruction, material strength, sliding surfaces, and reactiveflow in an ALE formulation. The solved and unsolved problems associatedwith performing AMR in this context will be surveyed.

ALE formulation with mixed elements

C. Aymard1, J. Flament1, J.Ph. Perlat1

1CEA, BP12, 91680 Bruyères­le­Châtel, France

Arbitrary  Lagrangian  Eulerian   (ALE)   formulation  with  multi­material   elements  has  been implemented   in   a   Lagrangian   hydrocode   to   improve   the   robustness   of   the   Lagrangian simulation, especially on thin meshes.In a same simulation, we mix Lagrangian blocks (single material) and Ale blocks (multi­material). Disjoint blocks can interact at the boundaries through sliding surfaces.In an Ale Block, the basic computational cycle consists in a Lagrangian step followed by a rezone one.The Lagrangian step uses the classical Wilkins second order scheme. The assumption of equal material   volumetric   strain   rate   which   governs   the   average   values   computation   in   mixed elements is improved by an iterative pressure relaxation algorithm.The rezone step is splitted in two phases : the mesh smoothing phase in which a new grid is defined and a remapping phase in which the material quantities are interpolated on the new grid.The mesh smoothing phase uses specific mesh smoothing schemes for boundary nodes and classical equipotential methods for interior nodes.The remapping is based on the reconstruction of unstructured lagrangian mesh with variable connectivities   for   each  material   in   the  Ale  block.  The  position  of   the   interface  between materials in mixed elements is computed by Young’s method. The material  quantities are mapped by an intersection mesh method on the new regular grid.

Interface Reconstruction and

Sub-Zone Physics Models

D. Bailey and G. ZimmermanLawrence Livermore National Laboratory

Conference/Workshop

Numerical methods for multi-material fluid flowsCzech Technical University in Prague

September 10 - 14, 2007

Abstract

We present our recent work on interface reconstruction in a logi-

cally structured Lagrangian CFD code that now incorporates the MoF

system developed by Shashkov’s group at LANS. We also discuss the

models used to update and re-map the state variables in the mixed

cells.

∗This work was performed under the auspices of the U. S. Department of Energy by

the University of California Lawrence Livermore National Laboratory under Contract W–

7405–Eng–48.

Compatible Finite Element Multi­Material ALE HydroAndrew J Barlow

Design Physics Department, AWE, Aldermaston, Berkshire, RG7 4PRE­mail: andy.barlow@awe.co.uk

The main ideas of compatible Lagrangian hydro were originally developed in the form of a Finite Volume scheme by Caramana, Shashkov and Burton et al at LANL. The compatible approach is based around two key ideas; a stronger Lagrangian assumption, where corner masses are  treated as Lagrangian objects as well  as   the elements and the enforcement  of consistency   between   the   solution   of   the   momentum   and   internal   energy   equations.   This provides a means of improving total energy conservation and allows greater flexibility in the types of force that can be allowed in a zone. This potentially offers significant benefits in terms   of   improved   accuracy   and   robustness   over   traditional   staggered   grid   hydrocode schemes which employ a PdV based internal energy update. 

A   new   compatible   finite   element   Lagrangian   hydro   method   has   been   developed   and implemented in CORVUS, AWE’s 2D Arbitrary Lagrangian Eulerian (ALE) code. The new finite element method was developed in preference to the published finite volume schemes for a  number  of   reasons:   to   see   if   the   fundamental  principles  of  compatible  hydro  could  be translated across to other numerical methods in use in hydrocodes, to facilitate a more direct comparison  of   the  performance  of   the   compatible  hydro   scheme with   the  existing   finite element scheme in CORVUS and enabled rapid progress to be made as the existing physics in the code could be used immediately.

The key changes required to transform the finite element scheme used for the Lagrangian step in CORVUS to make the scheme into a compatible hydro scheme are; redefinition of the real and area weighted nodal masses and the replacement of the PdV internal energy update with a compatible work update expressed in terms of   the corner forces applied in the momentum step and the distance moved by the nodes during the timestep. Once this was established edge artificial viscosities and subzonal pressures were introduce via the introduction of subzonal finite elements with additional nodes this created being treated as non­dynamic points.  

The  new  finite   element   scheme  provides   total   energy   conservation   to   round  off   for   the Lagrangian step without slide lines. The edge artificial viscosities and sub­zonal pressures that have been introduced through the framework of the compatible hydro scheme provide further improvements in terms of accuracy and robustness for Lagrangian calculations. The energy conservation and symmetry of the slide and void closure algorithms have also been improved by making use of the ideas of compatible hydro. 

In order to apply this compatible hydro scheme as the Lagrangian step of a multi­material ALE code a number of problems have had to be overcome. These include how to; calculate 

the   work  done  on   individual   material   component  within  multi­material   zones  where   the volume fraction may vary during the Lagrangian step, advect momentum given the new nodal mass definitions and advect the corner masses required by the compatible hydro scheme. 

The talk will discuss the details of the compatible finite element Lagrangian scheme, and the extensions required to apply the scheme as the Lagrangian step of a multi­material Arbitrary Lagrangian Eulerian code. This will include recent work on local mesh movement algorithms which attempt to maximise the benefits of the compatible Lagrangian hydro scheme. Test problems and real applications will be presented to demonstrate the benefits and performance of the new method for hydrocode and radiation hydrodynamics applications.

Improved numerical modelling of surface tension effects via a novel discretization of the Continuum Surface Force model

C.A.Batha, R. J. R. Williams, D. L. YoungsAWE plc

http://awe.co.uk

Surface tension plays an important part in the dynamics of many interfacial and free surface flows, and is thus an important phenomenon in many industrial and engineering applications. Surface   tension   effects   classically   appear   in   the   fluid   equations   as   jump conditions   at   fluid interfaces where fluid properties vary discontinuously. The CSF model of Brackbill et al, [1], reformulates the discontinuous jump conditions, due to surface tension, at fluid interfaces by a smoothly varying volume force acting over the fluid interface. The method is extensively used, and  has  been  extended   to  model   compressible   flows,  but   is   known  to  generate   un­physical "spurious   currents"   at   fluid   interfaces   due   to   an   imbalance   in   surface   tension   forces   and associated pressure gradients, due to discretization errors in the static equation

∇p = σκnδs

By maintaining consistency with the discrete form of the jump condition at a steady interface, a novel numerical technique is presented in which the only potential source of "spurious currents" lies in curvature estimation errors.

Non linear side fraction functions of volume fractions, [3],  are used to determine normal vectors of second order accuracy for interface reconstruction within a compressible volume of fluid formulation. Curvature estimates are then naturally determined using a discrete divergence operator.  Accuracy  of   curvature   estimation  via   the  described  method,   relative   to   the  height function   approach,   [2],   is   highlighted   via   a   simple   linear   mode   Rayleigh­Taylor   instability problem. The method is then used to present results on surface tension effects in a variety of fluid mixing problems.

References

[1] J.U.Brackbill, D.B.Kothe, E.G.Puckett,  A continuum method for modelling surface tension, J.Comp.Phys, 100, 335­354 ,1992

[2] M. M. Francois, S. J. Cummins et al, A balanced force algorithm for continuous and sharp  interfacial surface tension models within a volume tracking framework, J.Comp.Phys,  213, 141­173, 2006

[3] D.L.Youngs,  Time   dependent   Multi­Material   flow   with   large   fluid   distortion,  Numerical methods for fluid dynamics: Proceedings of a first conference 1982, 273­285, 1982

Markus Berndt (speaker)Mathematical Modeling and Analysis Group, T-7

Mark A. KenamondX-3, Los Alamos National Laboratory

both from Los Alamos National LaboratoryLos Alamos, NM 87544, USA

Title:

A preconditioned condition number based mesh relaxer for2D dendritic/AMR meshes with with very bad aspect ratios.

Abstract:

In many applications, it is convenient to use a mesh that hashanging nodes. These are vertices whose coordinates are determined as an average of two neighbor vertices. Such constrained vertices can occur in two different situations: in a regularly adaptively refined mesh at the interface between refined and unrefined cells, as well as in meshes where quads are refined only by halving them(dendritic meshes). Such refined meshes present a challenge to meshsmoothing algorithms. We present an algorithm that is based on acondition number minimization approach and that can handle thesedifferent types of mesh refinement. Additionally, very bad aspectratio cells severely limit the efficiency of such a minimizationbased approach. We address this issue by preconditioning our conditionnumber based mesh smoother with a smart Laplacian smoother that takesinto account the principal directions of the set of edges that areconnected to each vertex. By using this approach we greatly accelerate the convergence of our condition number smoother.

A Pure Eulerian Scheme for Multimaterial Fluid FlowsJean­Philippe Braeunig^1^1 CEA DIF BP12, Bruyeres­le­Chatel 91680 France

This method named FVCF­NIP is designed to compute multimaterial fluid flows, compressibleand non­miscible. Each fluid behaviour is modeled using the compressibleeulerian model. We focus on the interface capturing between the fluids, thatprevents diffusion of eulerian quantities between fluids through theinterface. Moreover, it allows the free sliding of fluids on each others atthe interface. The method is locally conservative on each eulerianquantity.The Finite Volume scheme FVCF by Ghidaglia, Kumbaro and Le Coq (2001) isused on orthogonal fix meshes in 2D/3D. In a mixed cell, i.e. a cellcontaining two or more fluids, the interface is described by a piece of line thatseparate fluids. Thus fluids are pure on both side of the interface. Theeulerian quantities evolution is obtained, as in pure cells, by integrationof eulerian quantities and fluxes on the cell, taking into account theinterface motion and position within the cell.In the talk, the method algorithm will be described as well as associatednumerical studies. Finally, some numerical results wil be shown.

Progress toward an Improved Staggered­Grid Hydrodynamics Method

D. E. Burton1, M.J. Shashkov2

1X­3, MS F644, Los Alamos National Laboratory, Los Alamos, NM, USA2T­7, MS B284, Los Alamos National Laboratory, Los Alamos, NM, USA

The Lagrangian formulation of   the equations  of  hydrodynamics  has  a  very old and venerable history   going   back   over   60   years   as   a   practical   tool   for   large­scale   numerical   simulations. Problems associated with mesh tangling have been largely addressed through adaptivity  in  the forms   of   Arbitrary   Lagrange   Euler   (ALE),   free­Lagrange   reconnection,   and   more   recently Adaptive Mesh Refinement (AMR).  This has led to the development of formulations extended for unstructured polyhedral cells.

Most   Lagrange   formulations   have   employed   a   spatial   discretization   in   which   the   evolution equations for stress and velocity are solved on staggered control volumes arranged such that the logical center of each lies on the boundary of the other.   This overlap avoids the interpolation to obtain   boundary   fluxes   characteristic   of   cell­centered   schemes.     For   uniform   grids,   this formulation is second­order away from discontinuities and first­order near them.

The   basic   numerics   have   evolved   from   simple   finite   difference   approximations   to   multi­dimensional,   fully   conservative,   compatible,   finite­volume   formulations   that   mimic   the fundamental hydrodynamics equations.    In recent years, significant progress has been made in addressing major historical issues associated with energy conservation, hourglass instabilities, and shock­induced oscillations.

In   spite  of   successes,   the   staggered   formulation  has   flaws   that  bear   investigation.  Monotonic viscosity formulations have not been adapted to unstructured grids. The stress divergence operator is only first order for non­uniform grids, in effect transferring the burden for accuracy from the algorithm to grid generation tools.  The conventional nodal definition of kinetic energy cannot be conserved simultaneously with momentum during advection, and alternatives may conflict with energy compatibility requirements. Volumes calculated from coordinates and from velocity fluxes are not identical, leading to either energy or entropy error.  

The use of a higher­order differencing scheme was key to resolving hourglass problems.  This suggests that similar techniques might be helpful in improving some of the aforementioned flaws. This paper will address work in this area.

Numerial Calculations for Hydrodynamics Based on

Multi-dimensional Riemann Solvers

William W. Dai1, Paul R. Woodward

2, B. Kevin Edgar

2

1Los Alamos National Laboratory

2University of Minnesota

A Numerical scheme for multi-dimensional hydrodynamics will be reported based on an approximate multi-dimensional Riemann solver at grid points. The scheme is one of the first attempts to use multi-dimensional Riemann solvers in numerical simulations for hydrodynamics. The scheme is truly multi-dimensional. It is second order accurate in both space and time. It satisfies conservation laws for mass, momentum, and total energy exactly.

The set of the two-dimensional (2D) Euler equations may be written as

ρdU

dt=

∂Fx

∂x+

∂Fy

∂y.

Here, U , contains conserved quantities of

mass, mmentum, and energy, and Fx and Fy

are fluxes in the x- and y- directions. A 2D Riemann problem is the equation above with a set of constant states for each region surrounding a point, for example, four constant states in the four quadrants in a structured mesh. If the equation is integrated over a cell and one time step, 0 < t < ∆t , the following equation will be obtained.

U = U0

+∆t

∆m{ Fx∫ dy + Fydx}∫ .

Here, U is a space-averaged value of U

over the cell at t = ∆t , U0 is its initial

value, ∆m is the mass in the cell, the

integral is count-clockwise along the perimeter of the cell, and the bar over the integral stands for the time-average during the time step. In our scheme, the time-averaged integral is approximately calculated through the time-averaged values obtained from an approximate multi-dimensional Riemann solver. To get the second order accuracy of the scheme, the states surrounding a grid point will not be the states of the cells, but are the states on domains of dependence.

The scheme has been tested in ALE calculations. The image below shows the pressure in an ALE calculation at t = 0.0002 for a 2D Riemann problem. The initial pressures on the four quadrants

are 106, 1.0, 10

6, 10 respectively. The

initial velocity is zero, and initial density is unity everywhere. No artificial viscosity has been used in the calculation.

Figure 1: Pressure of a two-dimensional Riemann problem.

Sources of Cartesian Mesh induced asymmetries based upon the Lagrangian + Remap Method

A.S. Dawes1

1Computational Physics Group, AWE plc, Aldermaston, Berkshire, RG7 4PR, UK

Partial Differential Equations (PDE’s), such as the Euler equations from fluid dynamics, can be solved analytically for simple idealized problems. However, for more general applications an approximate solution must be found by discretizing the PDE’s. For Computational Fluid Dynamics (CFD) there are a wide variety of methods in use, both in academia and at AWE. For example, finite differences, finite volume, finite element and the Arbitrary Lagrangian Eulerian (ALE) method to name but a few.

It is well known that the discretization of the PDE’s can produce inaccuracies. Experience has shown that simulating converging flow fields (such as Inertial Confinement Fusion or Noh’s Problem), where Cylindrical or Spherical is important, on an Orthogonal Cartesian mesh does not maintain radial symmetry. At AWE schemes are based upon the Lagrangian + Remap method. In this paper we will consider the sources of the numerical asymmetries and ways to eliminate them.

A 3D Finite Volume Lagrangian scheme

B. Despres1, Stephane Delpino1 and Emmanuel Labourasse1

1 CEA/DIF, 91 680 Bruyeres le Chatel, BP 12

In a recent work, a new Finite Volume Lagrangian scheme has been presented in 2D(see B. D. and C. Mazeran, ARMA, 2006). All unknowns are cell centered. Total energyis conserved. Cell centered schemes are attractive for ALE techniques.

We will present the 3D generalisation on arbitrary meshes. The construction is basedon some compatibility assumption which helps to have an algebraic presentation of thescheme. This algebraic construction encompasses all the desired geometrical properties,but in a more abstract framework.

We will show 3D tests cases which demonstrate the efficiency of this approach, evennear singular 3D points.

1

Interface Resolution in Multiphase FlowTimothy A. Dunn^1, David E. Stevens^1^1 Lawrence Livermore National Laboratory, 8000 East Ave, Livermore,CA 94550

The development of numerical methods to model the hydrodynamicinteraction of reactive materials with their surroundings will bepresented. Energetic materials often consist of a complexcompressible non-equilibrium mixture of gases and particles. Themultiphase character of this mixture must be taken into account whendeveloping models. However, the types of methods typically used tohandle the multiphase material are not necessarily the techniques bestsuited to accurately predict the response of the pure neighboringmaterial. Therefore, the algorithms employed must be able to resolvethe sub-scale interfaces embedded within the multiphase mixture aswell as its interface with the surrounding regions.

An Eulerian fluid-particle multiphase model is presented. This modelis based on the Discrete Equations Method (DEM) as presented inChinnayya et al. [J. Comput. Phys. 196 (2004) 490]. Modificationswere made to resolve the interface between the multiphase and purematerials. These modifications were integrated into the Riemannsolver to more accurately resolve the contact surface. A number oftechniques were attempted and will be presented along withdescriptions of the methods and comparisons of results.

Moment-of-Fluid Interface Reconstruction Method for Multi-Material Fluid Flows

Vadim Dyadechko^1, Mark Christon^1, Mikhail Shashkov^1 ^1 Los Alamos National Laboratory, Los Alamos, NM 87544, US

We present a new volume-conservative interface reconstruction method,offering several major advantages over the traditionalVolume-of-Fluid~(VoF) methods.The key feature of the new Moment-of-Fluid~(MoF) method is utilization of the cell-wise material centroids for the interface reconstruction.The location of the linear interface in each mixed cell is chosento preserve the volumes and provide the best possible approximation to the material centroids.The MoF construction of the linear interface in a mixed cell depends only on the moment data from within the cell and not on the data from its neighbors.Therefore, the MoF method is able to resolve interface detailsas small as the cell itself, which are 2-3 times smaller than conventional VoF methods can resolve.Also, the MoF interface reconstruction can be implemented as a cell-by-cell black-box routine, which is a great technological advantage over the VoF, especially in 3D.The technique proposed is 2nd-order accurate and is shown to be more accurate than similar VoF methods. Since the centroid of any Lagrangian parcel of incompressible fluidmoves very much like a Lagrangian particle, the cell-wise material centroids can be updated in hydro simulations with sufficient accuracy.

Molecular Dynamics Simulations of Dynamic Frictionand Mixing at Rapidly Moving Material Interfaces

Nicholas Epiphaniou1, Marco Kalweit1, Dimitris Drikakis1, Graham Ball21Aerospace Sciences, Fluid Mechanics and Computational Science Group, Cranfield

University, MK43 0AL, UK2AWE, Aldermaston, UK

Friction studies are important in applications to high-speed machining and ballisticpenetration modelling, two areas where it is important to understand the behaviour ofrapidly moving interfaces. Gaining insight into the velocity dependence of the effectivetangential force, and its time-evolution, under various external loads is also of particularinterest. Previous studies [1, 2, 3], have shown that for metals, a substantial velocityweakening occurs, i.e., a decrease in the friction stress with velocity, and this has been at-tributed to melting. Furthermore, experimental studies [4] have shown the developmentof characteristic micro structural changes during ductile metal sliding, which is distin-guished by a very highly strained plastic region near the interface and a nano-crystallineregion at the interface. The details of the phenomena that occur along and across theinterface between two materials cannot be modelled by continuum mechanics, but insteada microscopic analysis of these phenomena is required.

The present study concerns molecular dynamics (MD) simulations of dynamic frictionat Cu/Ag interface. MD simulations using the Embedded Atom Method (EAM) inter-atomic potentials have been performed for a box containing 1.3 · 106 atoms. Compressionforces of the order of 5.1GPa have been applied to Cu(010) and Ag(010) as well as slidingfriction velocities of up to 1Km/sec in the 〈100〉 crystallographic direction.

The aim of this work is to confirm the connection between velocity weakening andstructural transformation of nano-crystalline materials. The frictional force versus relativesliding velocity for the two interfaces reveals a linear region at low velocities and a highlylocalised plastic deformation region at high velocities with the frictional force decreasingwith velocity. The study also tries to shed light on the temperature dissipation in theproximity of the interface and its relationship with atomic diffusion. The temperaturedistribution across the interface of the two materials exceeded the melting point, especiallyat velocities greater than 500m/s. Mixing of the two materials was also observed at thesliding interface with the mixing layer width increasing when increasing the sliding speed.

References

[1] F.P. Bowden and P.A. Persson, Proc. Roy. Soc. 260A, 433 (1960)

[2] D.A.Rigney and J.E.Hammerberg, MRS Bulletin 23, 32 (1998)

[3] R.E.Winter, G.J.Ball and P.T.Keightley, J.Phys.D:Appl.Phys. 39, 5043 (2006)

[4] D.A.Rigney, M.G.S.Naylor, R.Divakar, and L.K.Ives, Mater. Sci. Eng. 81, 409 (1986).

1

A Numerical Algorithm for Transitioning from Sharp to Continuous Material Interface Representation

Marianne M. Francois1, Edward D. Dendy1, Robert B. Lowrie1

1Los Alamos National Laboratory, Los Alamos NM 87545, USA 

There   are   several   existing   approaches   to   model   material   interfaces   in   fluid   flow.   The interface can be captured (Eulerian approach) or tracked (purely Lagrangian approach or mixed Eulerian­Lagrangian approach). In this work we employ a purely Eulerian approach in which the different phases are represented by the volume fractions. The focus of this study is   the   transition   from a  sharp   representation  of  a  material   interface,   to   a  more  diffused representation whenever the interface curvature is unresolved. We consider a single velocity representation with averaged material  properties   in  mixed cells.  Within  this  context,  we devise an algorithm that combines the interface preserver capturing method (also known as “artificial   steepening”  or  “compressive   limiter”)  with  an  interface  reconstruction  method (volume   of   fluid:   VOF­PLIC).   The   VOF­PLIC   approach   reconstructs   a   linear   interface within each cell and this interface is used to compute accurate fluxes. This representation is considered “sharp” as it keeps the interface within a single cell as opposed to most capturing methods which diffuse the interface over a many cells.   In regions where the VOF­PLIC method fails,   (i.e.  unable  to  capture thin filament or  dispersed phase because of   lack of resolution) the method switches to the interface­preserver capturing method, which steepens the computed density  gradients   in  order   to  keep  the  mass  diffusion  to  a  minimum. The numerical algorithm for transitioning between volume tracking and interface capturing will be presented and examples will be shown on several test cases.

Interface Reconstruction Method in ALEComputation

Stephane Galera1, Jerome Breil1, Pierre-Henri Maire1

1 Centre Lasers Intenses et Applications, Universite Bordeaux I, CNRS, CEA351, cours de la Liberation, 33405 Talence, France

e-mail: galera@celia.u-bordeaux1.fr

In this paper we are interested in multimaterial flows simulations, where an interface existsbetween two immiscible fluids. In Lagrangian simulations, the treatment of interfaces isnaturally taken into account. When strong deformations occurs Arbitray LagrangianEulerian (ALE) methods are classically used to solve such problems. However, in thecontext of ALE, grid and interface move separatelly. Thus a special treatment is neededto take into account the interface. Futhermore, as mixed cells appear, we also need aclosure model. The goal of this work is the investigation of the coupling between interfacereconstruction methods and mixed cells models. A number of numerical methods exist forsolving the interface problems, and mixed cell closures. In this paper we first study thecoupling of two classical models: the Piecewise Linear Interface Construction – VolumeOf Fluid method (VOF PLIC) [3] coupled to a mixed cells modelling, in which we assumethat during the Lagrangian step of the ALE formulation, the volume fraction remainunchanged for each material in a mixed cell [2]. Our investigation will be illustrated bythe study of a Richtmyer-Meshkov instability problem [1].

References

[1] C. Mugler, L. Hallo, S. Gauthier and S. Aubert, Validation of an ALE Godunovalgorithm for solutions of the two-species Navier-Stokes equations AIAA paper, 96-2068.

[2] M. Shashkov. Closure models for multimaterial cells in Arbitrary Lagragian Eulerianhydrocodes, Proceedinds of ICFD 2007, Reading, UK., 2007.

[3] D. L. Youngs. Time dependent multimaterial flow wuth large fluid distortion. in K.W. Morton and M. J. Baines, Ed., Numerical Methods for Fluid Dynamics, 273–285,1982.

1

NEW METHODS FOR ORDER-INDEPENDENT MULTI-MATERIAL INTERFACE RECONSTRUCTION

R. Garimella^1, S. Schofield^1, M.~Francois^1, R.~Loubere^2

^1 Los Alamos National Laboratory, Los Alamos, NM 87545

^2 Universite Paul-Sabatier Toulouse, Toulouse, France

We present two new methods for volume-conservative order-independentinterface reconstruction in multi-material (more than 3 materials)flow simulations. This is different from the commonly used methodswhich, at best, carve out material regions from a cell sequentially,making the reconstruction material-order dependent. All the methods wepresent recover the approximate location of the material centroids incells from only volume fraction data. Then a weighted Voronoi diagramof these approximate centroids is constructed in each cell topartition the cell into material regions that match the input volumefractions exactly.

The first method we will present uses a particle attraction-repulsionmodel to compute approximate centroid locations in the cell. Thismethod can recover some features, such as filaments inside a cell,that traditional interface reconstruction methods cannot.

The second method we present computes the approximate centroid ofmaterials in the cell by performing a monotonic linear reconstructionof the ``volume fraction function''. This is followed as before by apower diagram subdivision into pure-material subcells. The methodgives very good results for regular grids and has been successfullyextended to general unstructured meshes.

In addition, we will present the results of our investigation intosmoothing techhniques for making these reconstructions second-orderaccurate. Finally, we will present our studies on the effects of thisreconstruction on advection procedures in multi-material flows.

Two­layer flows with free surface

S.L. Gavrilyuk 1

1Aix­Marseille University,  IUSTI, UMR CNRS 6595, 5 rue E. Fermi, 13453 Marseille Cedex 13. Also SMASH Project, INRIA

e­mail : sergey.gavrilyuk@polytech.univ­mrs.fr

We obtain a dispersive model for the description of large amplitude waves propagating in a two­layer system with free surface. The model is a ``two­layer'' generalization of the Green­Naghdi (GN) model. The novelty of the derived model in comparison with the work by Liska, Margolin and Wendroff (1995) is using the Lagrangian approach in the spirit of the work by Miles and Salmon (1985) done for the derivation of the GN model. The Lagrangian approach gives the background for application of general theoretical methods. In particular, this concerns the generalization of the notion of vortex  motions, which was proposed in our earlier paper (Gavrilyuk and Teshukov, 2001) for general class of Lagrangian models,  and which was developed here for a two­layer model. As in the case of the full problem, the present model captures the resonance between short waves and long waves. In this framework it is shown, by using numerical computations, the existence of homoclinic trajectories embedded into the continuous spectrum. They correspond to true solitary waves having the same velocities at infinity in each layer. Their study reduces to the analysis of a Hamiltonian system with two degrees of freedom. The traveling­wave solutions depend on three parameters: the density ratio, the depth ratio and the Froude  number based on the bottom layer. Two wave regimes, characterized by the elevation or depression of the interface between the layers are presented. A critical depth ratio separates these two regimes and it will be shown how it relates to a change of the structure of the potential for the Hamiltonian system. The analysis of the number and nature of critical points turned out to be decisive in this work. It was found that the number of critical points can be four or two, depending on the value of the Froude number (for fixed density and depth ratios).For sets of parameters corresponding to oceanic conditions we have perceived the existence of true solitary waves and their broadening whenever the speed wave increases towards a limit value. Finally, other sets of parameters are considered for which multi­humped solitons exist, highlighting the richness and complexity of the system considered.

An automatic ordering method for eulerian multi­materials schemes

Laurence Gozalo1

1CEA, BP12, 91680 Bruyères­le­Châtel, France

Eulerian   schemes   for   multi­materials   compressible   flows   have   proved   their   efficiency, especially when materials have to stand high deformations. One of the main issues remains in their  dealing  with   the  so­called  mixed cells,   that   is,   cells   in  which  several  materials  are present. In the Volume of Fluid (VOF) context, selected for conservative property, many high precision methods for interface reconstruction have been designed for two materials flows. However when it  comes to simulations with a greater number of materials,  a lot  of them appear unfitted. In our case, a Piecewise Linear Interface Calculation (PLIC) approach was chosen,   but   in   cells  where   three  or  more  materials   are   coexisting,   finding   their   relative positions with one another is not straightforward. The method we present here, related to Mosso and Clancy’s (1994), orders the materials in each mixed cell thanks to approximated centroids. We will give details of the method and some results on a few simple examples. 

On LES Modeling for Predictive Mixing

Fernando F. Grinstein 1

1Applied Physics Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA 

Accurate prediction of material mixing with quantifiable uncertainty is essential to achieving a   predictive   science   for   many   important   applications   in   engineering,   geophysics,   and astrophysics.  Typical applications  in realistic regimes and configurations exhibit  extreme flow complexity,  due to broad range of  length scales of physical processes and problem geometry,  and will  always require utilizing under­resolved computer  simulations.   In  this context, it is crucially important to have theory and computational evidence as to what type of flows and quantities can (or can not) be usefully predicted with insufficient resolution. Developing predictive numerical tools based on rational scientific principles for unresolved simulation of macroscale and microscale material mixing is very important in this context.It   is   not   feasible   to   compute   high   Reynolds­number   (Re)   turbulent   flows   by   directly resolving all scales of motion and material interfaces; instead, macroscale portions of the unsteady   turbulent   motion   are   computed   while   the   rest   of   the   flow   physics   including molecular  diffusion and other  microscale  physics   (e.g.,   combustion)   remains unresolved. One major approach in the turbulence community is large eddy simulation (LES) in which the large energy containing structures are resolved whereas the smaller, presumably more isotropic,  structures are filtered out  and their unresolved subgrid scale  (SGS) effects are modeled.   The   construction   of   SGS   models   is   pragmatic,   and   often   based   primarily   on empirical information. Adding to the physics­based difficulties in developing and validating SGS models, one is faced with simulations where contributions from numerical truncation terms can be as significant as those from SGS models in typical LES strategies. Extensive   recent   work   [1]   has   demonstrated   that   predictive   unresolved   simulations   of turbulent velocity fields are possible using any of the class of nonoscillatory finite­volume (NFV) numerical algorithms. This strategy is called implicit LES (ILES). This is a new area of research undergoing rapid evolution; scientific understanding and theory explaining the success   of   these   methods   have   been   proposed;   truncation   terms   associated   with   NFV methods implicitly provide SGS models capable of emulating the physical dynamics of the unresolved turbulent velocity fluctuations by themselves; the connection of these truncation terms   to   the   physical   theory   of   inviscid   dissipation   and   ultimately   to   irreversible thermodynamics   has   been   demonstrated.   The   extension   of   the   ILES   approach   to   the substantially more difficult problem of material mixing by an unresolved velocity field has not yet been investigated numerically, nor are there any theories as to when the methodology may be expected to be successful. Progress in addressing these issues with ILES in the cases of passive and shock­driven scalar mixing will be reported.

1. F.F. Grinstein, L.G. Margolin, and W.J. Rider 2007, Eds., Implicit Large Eddy Simulation: Computing Turbulent Flow Dynamics, Cambridge University Press.

Numerical Calculation Method for 2D Equation of Heat Conductivity for Multi­Component Environment in the EGAK Code

Guzhova A.R., Bondarenko Yu.A., Yanilkin Yu.V.Institute of Theoretical and Mathematical Physics,

Russian Federal Nuclear Center All­Russian Research Institute of Experimental Physics,  Sarov, Russia

In   this   paper   we   propose   two   approaches   to   the   solution   of   the   problem   of improvement of approximation accuracy of the equation of heat conductivity in the vicinity of multi­channel cells used in the EGAK code.

The first approach is based on the using of adaptive­embedded refined computational grids in the vicinity of the interfaces. The 1st­level refined grid is obtained by fragmentation of the primary grid cell ­ “mother cell” – into four fragments by the lines which connect the centers of its edges; the 2nd­level grid is obtained by the fragmentation of the 1st­level cells, etc. The features of the approximation of the equation of heat conductivity on the refined grid are discussed in the report; the calculation results for some test problems are presented.

The essence of the second approach is in the using of mixed cells of the specific model of multi­component heat conductivity, which does not imply the equity of the components, but is based on the fact that in the mixed cells heat exchange between the components takes place according to the same heat conductivity laws as those for the mean energy in regular heat conductivity. The main idea consists in the splitting of the heat conductivity process into tow scales; to separately take into account these two scales the splitting principle by physical processes is used. The “big scale” – heat exchange between the cells – is taken into account in a   regular   implicit  difference   scheme,  where   the  mean  parameters  of   the  mixed cells  are calculated with some sound method. After that heat exchange works on the “small scale” – heat exchange between the components inside the mixed cells; as the input data it uses the heat flows through the cell interfaces calculated at the first stage. At the second stage of the program these flows are divided between the components; the problem of distribution of the energy increment due to the heat conductivity between the components of the mixed cells is solved independently in each mixed cell. The results of the test and method­based calculations are given in the report. A considerably higher accuracy of the proposed method as compared to  the method that employs the supposition of  the components’  temperature equity in  the mixed cells, is shown.

Hierarchical Mixtures in an ICF Code

Alan K. HarrisonLos Alamos National Laboratory, MS T087, P. O. Box 1663, Los Alamos, NM   87545, USA

Flows   of   interest   in   ICF   problems   may   span   a   wide   range   of   length;   typically,   early­time hydrodynamic instabilities evolve eventually to fully­developed turbulence. To capture the essential features of such flows, it is promising to employ hybrid models that can describe instabilities by multifluid equations and turbulence by a turbulence model. However, in such a model only the fluid dynamics variables are treated differently for low­ and high­entropy flows. Other physics packages such as radiation  transport and thermonuclear burn have no way to distinguish  a priori  between poorly­mixed   "chunk"   mixtures   and   well­stirred   "atomic"   mixtures.   Consequently,   important phenomena that depend strongly on mixing structure cannot be modeled well, even with a structure­aware model in the hydrodynamics package. In order for all the physics packages to treat mixtures in a way appropriate to their structure, a representation of that structure must persist outside the hydro package. The representation must account not simply for chunk and atomically mixed cells, but for cells containing arbitrary combinations of both. In an ALE or Eulerian code, it must also be possible to represent unmixed material inside the same cell along with a mixture or combination of mixtures.

We have implemented such a description in an ALE hydro­based ICF code. Data structures describe a mixture hierarchy, for  instance, a chunk mixture  in which the "chunks" themselves are atomic mixtures. When ALE hydro is active, the division of a mesh cell by reconstructed interfaces is treated as the top level "mixture" in the hierarchy, with chunk and atomic mixtures as the second and third levels. Each element of any pure or mixture material in a mixed cell has its own thermodynamic and material properties, and multiple elements of the same material (e.g., bulk, chunk and atomic Be) may be present in the same cell. Mixing and ALE packages can create, maintain and trim the mixture hierarchy in each cell to correspond appropriately to the subgrid physics being modeled.This description of matter is coupled to our mix model, a hybrid model based on work by Cranfill. The   model   describes   turbulent   flows   by   a   turbulence   model,   including   an   energy   field,   and comparatively ordered mixing flows (hydrodynamic instabilities) by equations for drift velocities as well as a separate energy field. Since the latter flows are typically associated with coherent structures, we model them as producing chunk mixtures, while the turbulent flows create atomic mixtures and shred (atomically mix) preexisting chunks.This representation of mixtures is useful for several important reasons. First, it allows the code to model phenomena in which the same material may be present in two different conditions (e.g., bulk material and mixed material at a different temperature) in the same cell. Second, it enables us to explicitly track the evolution of material from one state to another within the same cell, to model processes such as chunk dissolution. Third, it provides a more complete description of a mixture, enabling other physics packages to model subgrid processes like transport and energy deposition more faithfully than would be possible based only on cell­averaged properties. Fourth, it makes it possible to model processes that depend critically on the characteristics of the mixture itself (rather than of its constituents).

Nodal Mesh Quality and ALE Computations forCompressible Fluids Flows

Philippe Hoch1

1 CEA/DAM, Ile de France, e-mail: philippe.hoch@cea.fr

We focus on numerical simulation of Lagrangian equations for 2D compressible fluidsflows. The mesh is formed by inhomogenous element (quadrangles and/or triangles) andnodes may have different degree (number of neighbors).In the Arbitrary Lagrangian-Eulerian framework, we present some extension of Escobaret al. algorithm for the mesh smoothing process. Here, we take into account explicitly the(arbitrary) mesh connectivity, moreover we extend the nodal quality notion (see multimat2005) which permits :

1. to control the region where singularity may appear (non convex element or the sinusof angles is too small, big variation of adjacent elements, etc..).

2. to obtain a generic tool to define non-linear mesh relaxation.

In a second step, we expose and show result for the “self-intersection” mapping for thedensity, speed and internal specific energy for the second order scheme using the approachof VanderHeyden W.B. and Kashiwa B.A.

Key Words:ALE, Mesh Quality, Conservative Projection, Positivity and maximum principle.

1

The Application of Multi-phase Flow Models in Simulations ofFluid-Structure Interaction

J. Knap and D.E. StevensLawrence Livemore National LaboratoryLivemore, CA 94550U.S.A.

In recent years, the issue of accurate prediction of thermo-mechanicalresponse of structures subjected to dynamical loading induced byfluids has gained renewed attention. Such loading scenarios, commonlyreferred to as fluid-structure interaction (FSI) phenomenon, have beeninvestigated in a wide range of applications, including: the effectsof blast waves on buildings, personnel protection, impulse failure ofmarine structures, and also, biomechanics of cells or arterial bloodflow. Often, simulations of FSI require development of large scalecomputer models that incorporate, however, only severely simplifiedconstitutive models for the thermo-mechanical response ofsolids. Moreover, frequently some of the most essential aspects ofFSI, such as structural failure due to fracture and fragmentation, areleft out of the model entirely. We apply the DEM multi-phase flow methodology to simulate FSI. Thefocus of this work is on the various aspects of failure in solids. Inparticular, we investigate the fracture and fragmentation ofstructures in response to blast waves. Verification and validationresults of our numerical predictions are also provided.

An Anti-Diffusive Method For Simulating InterfaceFlows with a Five-Equation Model

Samuel KOKH1, Frederic Lagoutiere2

1 DEN/DANS/DM2S/SFME/LETR,CEA Saclay, 91191 Gif sur Yvette CEDEX2 Laboratoire Jacques-Louis LIONS, Universite Paris VII

We present a work that deals with the simulation of compressible two-phase flowswith interfaces by means of a five-equation model. The interface position is described asa numerical transition zone of a color function z that takes the value 1 in the fluid 1 (resp.0 in fluid 0).

We are concerned here with the problem of controling the numerical diffusion of theinterface while keeping the algorithm free from any interface reconstruction process andalso preserving conservativity.

We follow the approach examined by Despres and Lagoutiere based on a detailed studyof a special anti-diffusive numerical scheme for the advection of characteristic functions.This analysis relies on tedious stability arguments which have already been transposedsuccessfully through a Lagrange-Remap strategy to another class of interface models fortwo-phase flows.

We consider here the so called “five-equation model with isobaric closure”. We supposeeach fluid k = 0, 1 to be equipped with an equation of state (ρk, εk) 7→ Pk, where ρk, εk

and Pk are respectively the density, the internal energy and the pressure of the fluid k.Both fluids have the same velocity u. We note y = zρ1/ρ the mass fraction of fluid 1,ρ = zρ1 + (1− z)ρ0 the density and ρε = zρ1ε1 + (1− z)ρ0ε0 the internal energy ρε of thematerial. Let ρe = ρε + ρu2/2 be the material total energy, then the system reads

(1)

∂tρV + ∂xF (ρV, z) = 0,

∂z + u∂xz = 0,P = P0 = P1,

ρV = (ρy, ρ, ρu, ρe)T ,F (ρV, z) = [ρyu, zρu, ρu2 + P, (ρe + P )u]T .

The system (1) is discretized with the following Lagrange-Remap scheme

(2)

yi = yni , zi = zn

i

ρni (1/ρi − 1/ρn

i )− ∆t∆x

(un

i+1/2 − uni−1/2

)= 0

ρni (ui − un

i ) + ∆t∆x

(P n

i+1/2 − P ni−1/2

)= 0

ρni (ei − en

i ) + ∆t∆x

(P n

i+1/2uni+1/2 − P n

i−1/2uni−1/2

)= 0

(ρn+1

i Vn+1i − ρn

i Vi

)+ ∆t

∆x

(ρi+1/2Vi+1/2u

ni+1/2 − ρi−1/2Vi−1/2u

ni−1/2

)= 0

(zn+1i − zn

i ) + ∆t∆x

(zi+1/2uni+1/2 − zi−1/2u

ni−1/2)−

∆t∆x

zni (un

i+1/2 − uni−1/2) = 0

Our works shows that it is possible to transpose the lines of Despres and Lagoutiereto the system (1) and to the discretization (2). The final algorithm is conservative for thevariable ρV = (ρy, ρ, ρu, ρe)T . Numerical tests show that the scheme is anti-diffusive forboth variables y and z. We also verify that the method provide a good treatment of theRiemann Invariants (P, u) at the material interface.

1

Lock Method for the Equations of the Lagrangian Gas Dynamics in Mixed Cells, Based on the Equity of the Components’ Velocities.

Goncharov E.A., Kolobyanin V.Yu., Yanilkin Yu.V.Institute of Theoretical and Mathematical Physics,

Russian Federal Nuclear Center All­Russian Research Institute of Experimental Physics,Sarov, Russia

One of the most complicated problems of the Lagrangian­Eulerian methods (ALE) is the approximation of the equations of the Lagrangian gas dynamics for the case of multi­component environment because of the occurrence of the so­called mixed cells with two or more components. The mixed cells may occur in the calculations due to two reasons. First, when the interface moves along the Eulerian grid, and, second, if the problem has two zones of different materials mixing. The efficiency and the accuracy of both the Lagrangian gas dynamics individually, and the ALE method as a whole, where the Lagrangian gas dynamics is a constituent, depend on the solution of the specified problem. 

This paper proposes a novel calculation method for thermo­dynamic state of mixed cells (lock method), based on the leveling of the components’ mass velocities after passing of small perturbations through the heterogeneous mixture.

Test problems are used for the study of the precision of the results obtained under this method,   supplemented   with   the   algorithm   of   iterationless   leveling   of   the   components’ pressures.

Lagrangian Models and Remapping Algorithmsfor 2D Multimaterial ALE Methods

Milan Kucharik1, Richard Liska2, Mikhail Shashkov1, Pavel Vachal21 T-7, MS B284, Los Alamos National Laboratory, P.O. Box 1663,

Los Alamos, NM 87545, U.S.A.2 Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University

in Prague, Brehova 7, Praha 1, 115 19, Czech Republic

Most Arbitrary Lagrangian-Eulerian (ALE) methods for fluid dynamics consist ofthree stages: 1) Lagrangian solver updating the solution in the next time level; 2) meshrezoning technique providing smoothed computational mesh; and 3) remapping algorithminterpolating fluid quantities from the Lagrangian to the rezoned computational meshes.For relevant results of numerical simulations, multimaterial model is often suitable whichrequires generalization of all used methods to the case of multimaterial fluid flow. In thistalk, we will present 2D multimaterial versions of the Lagrangian and remapping stagesneeded for high-quality ALE method.

For the Lagrangian stage of the ALE algorithm, there exist plenty of models treatingthe multimatarial features of the fluid. Main differences among several of them (lyingmostly in the new volume fractions estimate) will be described. We will emphasize severalaspects of the multimaterial Lagrangian solver, such as incorporating of multimaterialartificial viscosity and tracking of pure material centroids. Also, basic comparison ofmentioned models will be presented.

Generally, the remapping stage interpolates the fluid quantities from the Lagrangiancomputational mesh to the smoothed one. In the multimaterial case, the remappingstage must also provide volume fractions of each material in the cells of the rezoned mesh.Moreover, status of cells can change during the rezoning/remapping process, which mustalso be detected by the remapping algorithm. We will present here such an algorithm forremapping of a complete set of fluid quantities and volume fractions.

We will present several numerical examples to demonstrate properties of the describednumerical methods. Effect of the particular interface reconstruction algorithm to the fi-nal solution will be presented. Finally, comparison of single/multi-material and pure La-grangian/ALE simulations for well known fluid dynamics testing problems will be shown.

Applications of ALE Method to Laser Plasma Studies

R. Liska1, M. Kucharık2, J. Limpouch1, P. Vachal11Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical

Engineering, Brehova 7, Praha 1, 115 19, Czech Republic2Los Alamos National Laboratory, T-7, MS B284, Los Alamos, NM 87545, U.S.A.

Laser plasma, created by the interaction of laser radiation with matter, is modelledas compressible fluid by Euler equations with heat conductivity and laser absorptionsource term. Simulated problems typically involve large scale corona expansion or targetcompression with moving boundaries. Lagrangian coordinates moving with the fluid aremuch more convenient for such problems than Eulerian static coordinates which are notsuited well for large scale changes of computational domain and for moving boundaries.For many problems, e.g. those with shear flow, however, the Lagrangian moving meshcan degenerate rather soon during the simulation. The mesh distortion problems can beavoided by using the Arbitrary Lagrangian Eulerian (ALE) method. The ALE method,either after several Lagrangian time steps or when the mesh becomes distorted, rezones themesh and remaps conservative quantities from the original Lagrangian mesh to the newrezoned (smoothed) mesh. After rezone and remap stage the Lagrangian computation cancontinue. We have developed 2D ALE code for laser plasma simulations using logicallyrectangular quadrilateral mesh in Cartesian or cylindrical coordinates.

On three particular laser plasma modelling problems, which we were unable to treatby pure Lagrangian simulation, we demonstrate the usefulness of the ALE method forlaser plasma simulations. The problems model particular physical experiments performedat the Prague Asterix Laser System (PALS) and include high velocity impact, double foiltarget and foam target.

1

Order-independent interface reconstruction viaPower Diagram in multi-material cells

Marianne M. Francois1, Rao V. Garimella2, Raphael Loubere3, Samuel P. Schofield1

1 CCS-2, Los Alamos National Laboratory MS-B296, Los Alamos, NM, 87545, USA2 Theoretical Division, T-7, Los Alamos National Laboratory MS-B284, Los Alamos,

NM, 87545, USA3 CNRS, Math.for Industry and Physics (MIP) UMR 5640 University Paul-Sabatier,

31062, Toulouse, France

The interface reconstruction in mixed fluid cells filled with more than two materialsis usually difficult without ad hoc assumption; as instance the onion skin model assumesan “order” in which materials must be processed by the algorithm, other methods mightassume the “shape” of the interface in a pre-defined list of shape. Such an order mightbe in some case easy to define but in the general case it is not obvious and of course theshape list is by nature limited. Moreover a bad interface approximation generaly leads tobad advection of the fluids, hence to bad approximation of the solution.Moreover, an erroneous interface reconstruction can lead to wrong materials advection,hence leads to inaccurate solution.

The purpose of our work is to develop an order-independent interface reconstructionfor mixed cells having more than two materials. A type of particle method is first usedto determine in mixed cells where each fluid/material is roughly located by using anattraction-repulsion particle system taking into account the particles in neighboring cells.The particles have the tendency to agglomerate and therefore define an approximate“centroid” for each fluid in the mixed cell. From the resulting particles agglomeration, weestimate a single approximate location point of each material. These location points arethen used as the power diagram generator points. Using these generators, we developeda Power Diagram method, a kind of weighted Voronoi diagram, to actually deduce thelocations of the interfaces between the fluids. The very interesting properties of such acoupled method are:

• it does not rely on an a priori order of material or fluids in mixed cells,

• it is independent of the dimentionality of the problem: “3D-easy”,

• it is independent of the number of material in mixed cell.

We will present the theory and several numerical examples showing the efficiency of sucha coupled method.

1

Radiative Shock Solutions

Robert B. LowrieLos Alamos National Laboratory, Computational Physics and Methods Group (CCS-2),

MS D413, Los Alamos, NM 87544 USA

Radiation hydrodynamics (RHD) is nonlinear and analytic solutions are rare. Thistalk will describe semi-analytic solutions of planar radiative shock waves for equilibriumand nonequilibrium diffusion radiation models. We will also compare these solutions withresults from a finite-volume simulation code. These are the first semi-analytic solutionswe know of for high-energy density, radiation hydrodynamics.

By “semi-analytic,” we mean that the solution requires the numerical integration ofnonlinear ordinary-differential equations (ODEs). The errors in this procedure are easy tocontrol, so that these solutions may be used for simulation code verification. Moreover, theODEs offer new insight into the shock structure and physics of these shocks. For example,previous work has assumed that the material temperature reaches its maximum on thedownstream side of the embedded hydrodynamic shock (Zel’dovich spike). We show thatin certain cases, the temperature may actually continue to increase after the hydrodynamicshock and reaches its maximum at a specific value of the local Mach number.

The semi-analytic solutions may be used to verify RHD simulation codes. Radiativeshocks are very demanding for a code to compute, because the extent of the radiationprecursor may be orders-of-magnitude larger than the relaxation region downstream ofthe Zel’dovich spike. Adaptive-mesh refinement (AMR) is necessary to resolve efficientlythe multiple length scales in the problem. We will quantitatively compare results fromthe RAGE AMR code with our semi-analytic solution. As an example of the utility ofAMR, using a 2:1 refinement ratio between mesh levels, RAGE requires 13 levels of meshrefinement in order to adequately resolve a certain Mach 5 radiative shock. The AMRmesh uses 967 mesh cells, as opposed to the 245,760 mesh cells required for the equivalentequally-spaced mesh.

1

Sliding and multifluid velocities in Staggerred Mesh MMALE codes  

Gabi Luttwak1

1Rafael , P.O.Box 2250,Haifa 31021,Israel 

In Multi­Material Arbitrary Lagrangian Eulerian (MMALE) codes the material boundaries may cut the mesh lines. The cells cut by the interface include several fluids and the interface position   is   resolved  by   a  multi­dimensional   interface   reconstruction   consistent   with   the volume of the fluids (VOF) in the neighbouring cells. The position of the interface serves to define  the material   fluxes preventing unphysical  mixing of  the materials.    At  a material interface the pressure and the normal component of velocity are continuous, but there may be a   jump   in   the   tangential   component   of   the   velocity.   Lagrangian   slide­line   and   sliding­impacting   surface   calculations   does   take   this   into   account,   however   most   Eulerian   and MMALE codes   traditionally   assume  a   common velocity   in  multi­material   cells.   Such  a procedure prevents sliding, or at least adds an effective numerical and mesh size dependant, thus  un­physical   friction.   In   a  hyperbolic  problem all   the   solution   field  depends  on   the motion at the boundaries thus in some cases this assumption can lead to large errors. This is not necessary. In one of the first published MMALE codes [1], we allowed distinct fluid nodal velocities,  while  enforcing a  common normal  to   interface component.  Walker and Anderson [2]    added   cell­centered multi­material  velocities  to  the code CTH. We have recently investigated the advantages of using a Staggered Mesh Godunov scheme [3] for ALE and MMALE calculations.  This  scheme was shown  to  capture  sharp shocks  while having a "natural" capability of damping the hourglass instability. In the current work, we add   multi­fluid nodal velocities to the code. This is done like in [1] which some changes to make the procedure quicker and to preserve full consistency between the vertex masses of the species and the masses of the amount of those materials in the neighbouring cells.[1]G. Luttwak, R.L.Rabie, "Multimaterial Arbitrary Lagrangian Eulerian code MMALE and its application to some problems of penetration and impact", LA­UR­85­2311,(1985)[2]J.D.Walker,C.E.Anderson,"Multimaterial   velocities   for   mixed   cells",p1773­1776,   High Pressure Science and Technology­1993,ed. Schmidt et al.,AIP, (1994)

[3] Gabi Luttwak, Joseph Falcovitz, "Staggered Mesh Godunov (SMG) Schemes for ALE 

Hydrodynamics", presented at the  Numerical methods for multi-material fluid flows held at Oxford, Sept.2005

A Cell-Centered Arbitrary Lagrangian-EulerianMethod

Pierre-Henri Maire1, Jerome Breil1, Stephane Galera1

1 UMR CELIA CEA–CNRS–Universite Bordeaux I, 33405 Talence, France

• Introduction

The purpose of this presentation is to describe an original and a complete ALE strategydevoted to the computation of Inertial Confinement Fusion (ICF) flows. The main ele-ments in an ALE simulation are an explicit Lagrangian step, a rezoning step in whichnodes of the Lagrangian grid are moved to improve geometric quality of the grid, and aremapping step in which the Lagrangian solution is reconstructed on the rezoned grid.We will describe each of these steps in the sequel.

• Lagrangian step

The Lagrangian step is based on a new second order cell-centered Lagrangian scheme. Theprimary variables in this scheme are specific volume, momentum and total energy. Thevertex velocities and the numerical fluxes through the cell interfaces are not computedindependently contrary to standard approaches but are evaluated in a consistent mannerdue to an original solver located at the nodes. This nodal can be viewed as a a two-dimensional extension of the Godunov acoustic solver. The second order extension isderived using a MUSCL type approach.

• Rezoning step

The rezoning step is combined into a 3-step procedure. The first step of the procedureperforms the minimization of a quadratic objective function in order to smooth the grid.We have developped specific objective functions in order to adapt the grid motion tothe fluid flow. We improve the quality of the interface smoothing by repositioning itsnodes such that they are constrained to remain on a Bezier curve. Moreover, there areno numerical fluxes through the interfaces. This treatment preserve a quasi Lagrangianinterface tracking. The second step of the procedure is a local control of the admissiblesmoothing displacement of the nodes. This procedure allows the repositioning of thenodes such that the velocity displacement of the smoothed node is lower than the virtualvelocity displacement of the Lagrangian node. The third step of the procedure performsa global control and an improvement of the geometric quality of the grid, when previousprocedures cause the grid to become tangled or non-convex. The need of such a procedurealso exits when the Lagrangian step creates non valid elements in a grid.

• Remapping step

The remapping step is an interpolation procedure of mass, momentum and total energy,from the Lagrangian grid, to the rezoned one. The method we use is an unstructured anda cell-centered extension of the swept displacement face flux computation. This approachdoes not need the computation of the intersections of the old grid and the correspondingrezoned one, which makes this approach much more efficient. The fluxes are reconstructedusing a second order linear reconstruction.

Many numerical tests are presented. They are representative test cases for ALE sim-ulations and demonstrate the robustness and the accuracy of this method.

1

A Cell-Centered Anisotropic Diffusion Scheme onTwo-Dimensional Unstructured Meshes

Pierre-Henri Maire1, Jerome Breil11 UMR CELIA CEA–CNRS–Universite Bordeaux I, 33405 Talence, France

• Introduction

The goal of this presentation is the description and investigation of a new finite volumescheme for solving anisotropic diffusion equations on two-dimensional unstructured grids.Our scheme is primarily intended for use in applications where occur a strong couplingwith a cell-centered hydrodynamic scheme. Therefore, we have developed a robust, cell-centered diffusion scheme, which provides accurate results even on highly distorted grids.

• Isotropic scheme

The main feature of this scheme lies in the introduction of two normal fluxes and twotemperatures on each edge. A local variational formulation written for each corner cellprovides the discretization of the normal fluxes. This discretization yields a linear relationbetween the normal fluxes and the temperatures defined on the two edges impinging on anode. The continuity of the normal fluxes written for each edge around a node leads to alinear system. Its resolution allows to eliminate locally the edge temperatures as functionof the mean temperature in each cell. In this way, we obtain a small symmetric positivedefinite matrix located at each node. Finally, by summing all the nodal contributions oneobtains a linear system satisfied by the cell-centered unknowns. This system is character-ized by a symmetric positive definite matrix. We show numerical results for various testcases which exhibit the good behavior of this new scheme. It preserves the linear solutionson a triangular mesh. It reduces to a classical five-point scheme on rectangular grids. Fornon orthogonal quadrangular grids we obtain an accuracy which is almost second orderon smooth meshes.

• Anisotropic scheme and applications

The anisotropic extension is straightforward since the discretization is based on a localvariational formulation. The scheme is derived in the same way as in the isotropic case.We show on several numerical test cases the good behavior of the scheme. We also showthat our scheme can deal with the anisotropic Braginskii conductivity, which is used tomodelize electronic heat conduction in a magnetized plasma.

1

Collisions and Breakup of Droplets in a Thick spray

Julien Mathiaud(1)

(1) CEA, DIF, Bruyeres-le-Chatel, Francejulien.mathiaud@cea.fr

keywords: Sprays; DSMC; inelastic collisions; breakup; TAB model

Sprays are complex flows where dispersed particles (droplets) coexist witha fluid phase. We use an Eulerian-Lagrangian description, in which thedroplets are described by a particle distribution function, solution of a kineticequation (of Vlasov-Boltzmann type), while the surrounding gas is describedthanks to macroscopic quantities and standard equations of fluid mechanics(Euler or Navier-Stokes). In so-called thick sprays, the coupling between thephases is made through the volume fraction of droplets, and through a dragforce.

The kinetic equation for the droplets writes

∂tf(t, x, v, r, e) + v · ∇xf(t, x, v, r, e) +∇v(F (t, x, v, r) f(t, x, v, r, e))

+∇e(q(t, x, r, e) f(t, x, v, r, e)) = Q(f)(t, x, v, r, e),

where f(t, x, v, r, e) is the density of droplets which at time t and point xmove with velocity v, have radius r and internal energy e, where F and q arethe drag and energy transfer from the gas to the droplets, and Q is a kernelfor all the complex phenomena.

In [2] and [3] are described models in which those phenomena (like colli-sion, coalescence or breakup), are taken into account.

In particular, rigorously defined kernels are given, corresponding to theT.A.B. model (see [1]), and corresponding to inelastic collisions in whichinternal energy as well as kinetic energy are exchanged between the droplets.

As far as numerical simulation is concerned, a particle (Monte-Carlo)method is used for the droplets.

The distribution function of the droplets is approximated at each time bya discrete measure (“the numerical particles”)

f 'N∑

i=1

ωi(tn)δxi(tn),vi(tn),ri(tn),ei(tn),

1

where N is the total number of numerical particles, and ωi (the numericalweight) is the number of real particles represented by the numerical particle i.

The kinetic equation is solved thanks to an operator splitting betweenthe transport (Vlasov) term, the collision/coalescence term and the breakupterm. We consider that in each cell, the distribution function does not dependon the spatial variable x. So we solve the collision and breakup steps in eachcell independently.

At this point, several methods can be used for the collision step. Bird’smethod consists in sampling couples of particles. Its advantages are thatmass and momentum and energy are exactly conserved. It has howeverin many situations the drawback of being tractable only with constant nu-merical weights. The alternative Nanbu’s scheme consists in sampling onlyone particle, so that mass, etc. will be conserved only when averaging overmany realisations, but it is better suited when one wants to use non constantweights.

Those two methods are by all means combined with the so-called ”spuri-ous collisions” trick, that enables to decrease significantly the computationalcost.

We wish to present in detail the modeling and the simulation methoddescribed above, together with some results in a somewhat realistic context.

References

[1] A.A. Amsden, P.J. O’Rourke, The T.A.B. method for numerical calcu-lation of spray droplet breakup Los Alamos National Laboratory, LosAlamos, New Mexico 87545.

[2] C. Baranger, Modelling of oscillations, breakup and collisions fordroplets: the establishment of kernels for the T.A.B model, Math. Mod.and Meth. in Appl. Sci. Vol. 14, No. 5 (2004) 775-794.

[3] J. Mathiaud, These, ENS de Cachan, 2006

2

Automatic Mesh Relaxation Control Using Mesh Quality Measures 

Ian MacDonaldAWE Aldermaston, Reading RG7 4PR, UK

Email: ian.macdonald@awe.co.uk

AbstractThe multi­material  ALE method provides  the means  to  continue calculations past where high material deformations would cause a purely Lagrangian formulation to fail. The simplest strategy, when the Lagrangian mesh motion becomes too distorted, is to globally relax the mesh within the region of interest. This forces a multi­material treatment for all the material interfaces within the region, regardless of whether they individually merit  it.  The ideal situation would be for interfaces to become multi­material only where the mesh is sufficiently distorted to warrant it, thus maintaining the accuracy benefits of the Lagrangian interface tracking wherever possible. This work aims to develop an intelligent algorithm that will automatically restrict the mesh relaxation to when and where it is really necessary for preventing mesh tangling and maintaining solution integrity. The main step is to introduce a measure of element quality which, in conjunction with user specified quality thresholds, selects where to allow  the  mesh  to   relax.  A number  of  quality  metrics  have  been  considered,   the simplest being the reciprocal of the element shear.In practice a two­threshold strategy is used. If the element quality drops below the first higher threshold, then only the non­interface nodes are allowed to relax. Only if the mesh quality continues to drop, falling below the second lower threshold, are the interface nodes  allowed  to  relax.  This  approach attempts  to  repair  potential  mesh tangling by relaxing  the  mesh  internal   to  materials,  before  ultimately resorting  to relaxing interfaces.The   above   scheme   has   been   implemented   in   the   3D   ALE   code   PEGASUS.   Its performance   will   be   illustrated   for   a   series   of   projectile   penetration   simulations, where the aim is to automatically restrict the relaxation of the material interfaces to the high deformation region immediately surrounding the penetration. 

Time Evolving Volume Fractions in Mixed Zones inDuring a Lagrange Step

Douglas S. Miller1, George B. Zimmerman1

1 Lawrence Livermore National Laboratory, P.O. Box 808 L-38, Livermore, CA 94550

Many hydrodynamics codes use a “Lagrange plus remap” approach, in which first apure Lagrangian step is taken then a mesh relaxation step is performed. This is one way toobtain an “ALE” code (Arbitrary Lagrange-Eulerian), in which the mesh motion can bepurely Lagrangian, purely Eulerian, or anywhere in between. The quality of the Lagrangestep is crucial to getting a good solution. However, a good solution can be ruined if mixedzones (zones which contain two or more materials) are not treated carefully. During theLagrange step, a zone which has materials with different bulk moduli (air and solid metal,for example) can develop unphysical densities and pressures in one or both materials if thevolume fractions remain constant. This problem can be avoided by evolving the volumefractions of the materials during the Lagrange step in a physically reasonable way thattakes the differences in bulk modulus into account. We discuss four different methods andshow test calculations that demonstrate their virtues and weaknesses.

This work was performed under the auspices of the U.S. Department of Energy byUniversity of California, Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.

1

The Comoving-frame and Laboratory-frameNonequilibrium Grey Radiation Diffusion

Approximations in the Nonrelativistic Limit

Jarrod D. Edwards and Jim E. MorelTexas A&M University

College Station, Texas, USA

We contrast the comoving-frame and laboratory-frame non-equilibrium grey radia-tion diffusion approximations in the nonrelativistic limit. This limit corresponds to non-relativistic material motion, which we define as v ≤ 0.01c, where v is the material speedand c is the speed of light. All of the non-relativistic equations we consider are correct toO(v/c) unless otherwise stated. Our main results are as follows.

One need only neglect the time derivative of the flux in the laboratory-frame grey P1

equations to obtain the laboratory-frame diffusion approximation, but one must neglectseveral additional terms in the comoving-frame grey P1 equations to obtain the comoving-frame diffusion approximation.

The comoving-frame grey diffusion equation does not rigorously conserve laboratory-frame radiation energy. Conservation is only meaningful with respect to laboratory-frame quantities because the comoving frame is an accelerated reference frame. Thusthe comoving-frame grey diffusion approximation is not conservative. However, the erroris small. Furthermore, if one neglects the difference between the comoving-frame andlaboratory-frame radiation energy densities (a reasonable nonrelativistic approximation),the equation becomes conservative.

The comoving-frame P1 equations conserve the laboratory-frame radiation energy.Thus the lack of conservation in the diffusion approximation arises from the terms thatare dropped from the P1 equations to obtain the diffusion approximation.

In static media the equilibrium diffusion approximation is known to be asymptoticallycorrect through O(ǫ). We show that both the laboratory-frame and comoving-frame greydiffusion approximations preserve the asymptotic equilibrium diffusion limit through O(ǫ).This means that both approximations are fully valid in this limit.

The comoving-frame grey diffusion equation is considerably simpler than the laboratory-frame diffusion equation. A simplification to the laboratory-frame radiation energy andmomentum source terms results in an laboratory-frame grey diffusion equation thathas exactly the same form as the comoving-frame equation. The simplified equationis not correct to O(v/c), but it nonetheless preserves equlibrium solutions, preserves theequilibrium-diffusion limit, and is always conservative. Thus we believe that this equationis a viable alternative to the comoving-frame equation.

A Cell By Cell Anisotropic Adaptive Mesh ALEMethod

J. M. Morrell1, P. K. Sweby2, A. Barlow1

1 AWE, Aldermaston2 The University of Reading

In this work a cell by cell anisotropic adaptive mesh technique is combined with astaggered mesh Lagrangian plus remap Arbitrary Lagrangian Eulerian method.

Many features of interest, such as shocks, involve large variations in one dominantdirection. Anisotropic refinement of elements can increase the resolution in the directionof interest without wasting refinement in the other directions. The method developedhere combines the advantages of ALE with increasing the number of elements throughcell by cell anisotropic refinement. The use of local refinement avoids the prohibitivelylarge number of elements and nodes that would be required if the resolution was increasedthroughout the entire domain.

The quadrilateral elements may be subdivided anisotropically in only one direction, orisotropically in both directions. The elements are subdivided in their local directions, therefinement is aligned with the ALE mesh, which is often aligned with the flow directionsor features of interest. Anisotropic refinement on the ALE mesh is therefore particularlyefficient and beneficial.

Cell by cell refinement is used rather than selecting a group or cluster of elementsto refine as in structured Adaptive Mesh Refinement. An efficient solution procedure isdeveloped that solves only on the finest resolution existing on each part of the mesh,rather than solving on every refinement level. The solution is obtained on the DynamicMesh, which contains coarse, isotropic and anisotropically refined elements; this can beviewed as solving on an unstructured mesh where disjoint or hanging nodes are used atresolution transitions.

Results are presented for a range of test problems. The adaptive method achieves thesame accuracy as a uniformly fine calculation in a fraction of the time.

1

Title:Smoothed Particle Hydrodynamics as a tool for modeling material strengthand failure.

Author:J. Michael Owen ^1

^1 Lawrence Livermore National Laboratory P.O. Box 808 M/S L-38 Livermore, CA 94550 USA mikeowen@llnl.gov

Abstract:Meshless hydrodynamics methods such as Smoothed Particle Hydrodynamics(SPH) offer interesting advantages and challenges for studying problemsinvolving material strength, fragmentation, and failure. The primaryadvantages of SPH for studying the breakup of solids are that it is arobust Lagrangian technique and it does not assume a fixed topology.The robust Lagrangian nature allows SPH to evolve history variables tiedto the mass (such as the deviatoric stress, strain, and damage) withoutintroducing artificial diffusion of these properties due to advection orremapping, avoiding the complexities of numerically mixed materialproperties. The lack of a fixed topology also naturally allows for gapsto open in materials, proceeding to the formation of distinct fragmentswhich detach and evolve independently. I will discuss current work weare pursuing modeling the fragmentation and breakup of solids using SPH,including comparison of our results with some recent experiments.

On selective filtering of hourglass instability modesin lagrangian hydrodynamics.

Bernard Rebourcet CEA/DIFBruyeres-le-Chatel, France

One usual problem occuring in numerical computations of inviscid flows in the conflictbetween artificial dissipation due to numerical algorithms and its opposite, the dispersionassociated, for instance, with 2nd order accuracy scheme.

This fact is particularly important in lagrangian simulations where the action of theseproperties is intrinsic with regards to the scheme, the grid structure, the boundary shapeand the kinematic of the flow.

Dealing with physical problems admitting thresholds and requiring a high level ofconvergence, it is of tremendous importance to be able to control numerical instabilities.

We propose a selective algorithm devoted to damp short wave lengths and to preserveresolved physical waves and their kinetic energy.

1

Kershaw’s schem e on unstructured grid

Bernard RebourcetCEA/DIFFrance

bernard.rebourcet@cea.fr

Legacy numerical technics dating from the 60’ s-80’ s are described by way of finite difference standard that does not follow modern numerical analysis notations and reasoning. For instance structured mesh framework enslaves the key features of these methods and their basic principles remain surprisingly ignored by many readers accustomed to finite elements.This is true for papers related to lagrangian hydro schemes but also for those concerning diffusion operator discretization on distorted mesh.A good paradigm is Kershaw’ s scheme (1981) - which is still quite popular among ICF codes users - that we try to excavate to be able to expose its ability to deal with unstructured mesh in any dimension.With regards to the low cost of this algorithm and the sufficiently good results it provides on realistic grids, we think it deserves at least new description.

Mixed Finite Element Methods for Lagrangian Hydrodynamics

Robert N. Rieben1, 1Scientific B­Division, Lawrence Livermore National Laboratory 

7000 East Avenue, Livermore, CA 94550rieben1@llnl.gov  

Algorithms for the numerical simulation of hydrodynamics sometimes give rise to spurious unphysical modes which can lead to artificial grid distortion and symmetry breaking. Such spurious behaviour can be attributed to the way in which the acceleration of grid nodes is computed. Typically a control volume is used to define a finite difference approximation for the gradient operator. This approach is combined with an “hourglass­filter” which is used to damp  or   project   out   the   so   called  hourglass  modes.  We  present   results   concerning   the development and use of  mixed finite  element  methods for   the numerical  solution of  the hydrodynamic   equations   in   a   Lagrangian   frame.   Mixed   finite   element   methods   are   a promising   alternative   to   traditional   finite   difference   methods   for   discretizing   spatial differential operators such as the gradient and divergence, and can subsequently be used to define an acceleration operator which is inherently free of spurious modes. We utilize the Brezzi­Douglas­Marini (BDM) elements on quadrilaterals for discretizing the pressure and velocity. In this approach, the pressure is piece­wise constant in a zone (as is typically the case)  while   the  velocity   is  discretized  on  mesh  faces   (edges   in  2D)  using  a  divergence conforming basis set where the degrees of freedom are the normal projections of the nodal velocity on mesh faces. The BDM basis functions maintain coordinate system invariance by transforming covariantly. We present early results that suggest such an approach is much better at controlling (or altogether eliminating) spurious grid distortion on a set of canonical test problems. We also point out that the main drawback to this approach is the need to assemble and solve a global sparse linear system at each Lagrange time step.

Abstract proposed for presentation at ``Numerical methods for multi­material fluid flows,’’ Conference/workshop to be held at Czech Technical University in Prague, Czech Republic, September 10­14, 2007.

Consequences for scalability arising from multi­material modeling  

 J. Hu, S. J. Mosso, A. C. Robinson, Sandia National Laboratories*

T.A. Gardiner, Cray Inc.

J. E. Crepeau, Applied Research Associates, Inc.

Key to success of large scale computing is efficient use of computational resources.  Multi­material modeling by very definition implies a potential load imbalance with respect to computing on a parallel architecture.  We will present results in two different areas where material discontinuities impact scalability and performance.  In the first case we describe the impact that significant material discontinuities have on an H(curl) algebraic multigrid method.  In the second case we discuss scalability tradeoffs between interface reconstruction techniques of varying quality ranging from SLIC to a new 2nd order reconstruction algorithm. We will discuss how these results vary for two important large scale architectures including ASC Purple and Red Storm.

*Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy's National Nuclear Security Administration under contract DE­AC04­94AL85000.

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                Conservative Formulation and Numerical Methods for                                  Multiphase Compressible Media                                                 E.Romenski, D.Drikakis      Aerospace Science Department, Fluid Mechanics & Computational Science Group                                     Cranfield University, Cranfield, MK43 0AL, UK

Although a substantial  progress has been made in  last  few decades in the theoretical and numerical modelling of multiphase media, there is no widely accepted mathematical model even for   two­phase compressible   flows.  The main challenge  in  the development of high­accuracy   numerical   methods   for   multiphase   compressible   flows   is   associated   with   the formulation of a mathematical model that satisfies important properties such as hyperbolicity, symmetric hyperbolic system in particular, fully conservative form of the governing equations and compatibility and consistency of the mathematical model with the thermodynamic laws. These properties provide a solid mathematical framework for the theory of different initial­boundary value problems and allow developing highly accurate numerical methods.

According   to   classical   theories,  multiphase  mulifluid   flows   are   considered  as   interacting continua governed by mass, momentum and energy balance laws for each phase. We propose a   new   approach   for   the   modelling   of   multiphase   flows   based   on   the   theory   of thermodynamically compatible systems and irreversible phenomenological thermodynamics.This approach allows us to formulate classes of hyperbolic conservation­form equations using generalized   potentials   and   variables.   Using   phenomenological   thermodynamic   laws   a structure of governing balance laws is derived according to which the mixture is assumed as a continuum in which multiphase character of flow is taken into account by the appropriate choice  of   parameters   of   state.    Thus   the  multiphase   flow  is   governed  by   the   additional differential equations in addition to the mass, momentum and energy conservation laws for the mixture. The full set of conservation­form hyperbolic equations can be derived by using the formalism of thermodynamically compatible systems. The most general model governs a multiphase compressible flow with different phase pressures and temperatures. Constitutive relationships such as equation of state (EOS) for the mixture and source terms responsible for phase interaction are required to close the system. The EOS for the mixture can be derived using the equations of state for each phase. 

The conservation form of the governing equations provides a straightforward basis for the development of high­order accurate numerical method. A second­order finite volume method 

based on the solution of the Riemann problem as obtained by the GFORCE method has been developed. A number of numerical examples for one­ and two­dimensional problem for two­phase flow are presented.  

Advances in multi-scale methods for Lagrangianshock hydrodynamics using Q1/P0 finite element

discretizations

G. Scovazzi1, E. Love1, M.J. Shashkov2

1 1431 Computational Shock- and Multiphysics, Sandia National Laboratories,Albuquerque, New Mexico 87185-1913

2 Theretical Divivsion, Group T-7, Los Alamos National Laboratory, Los Alamos, NewMexico 87545

A new multi-scale, stabilized method for Q1/P0 finite element computations of La-grangian shock hydrodynamics is presented. Instabilities (of hourglass type) are controlledby a stabilizing operator derived using the variational multi-scale analysis paradigm. Theresulting stabilizing term takes the form of a pressure correction. With respect to cur-rently implemented hourglass control approaches, the novelty of the method resides in itsresidual-based character. The stabilizing residual has a definite physical meaning, sinceit embeds a discrete form of the Clausius-Duhem inequality. Effectively, the proposedstabilization samples and acts to counter the production of entropy due to numerical in-stabilities. The proposed technique is applicable to materials with no shear strength, forwhich there exists a caloric equation of state. The stabilization operator is incorporatedinto a mid-point, predictor/multi-corrector time integration algorithm, which conservesmass, momentum and total energy. Encouraging numerical results in the context of com-pressible gas dynamics confirm the potential of the method.

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Relaxation approximationfor hyperbolic fluid systems

Frederic Coquel1, Edwige Godlewski1, Nicolas Seguin1

1 Universite Pierre et Marie Curie - Paris 6, CNRS, UMR 7598,Laboratoire J.-L. Lions, BC 187, 75252 Paris, France

This presentation is devoted to the relaxation approximation of systems of conservationlaws which fall into in the canonical frame of fluid systems proposed by Despres. Moreprecisely, such systems, when written in Lagrangian coordinates, verify the followingrequirements: Galilean invariance, reversibility for smooth solutions and their entropyflux is zero. Euler system of compressible gas dynamics, multi-species multi-temperaturemodels, models of ideal magnetohydrodynamics... are some of the systems which fulfillthese hypotheses.

The aim of this work is to develop a relaxation approximation of such systems, fromboth a theoretical and a numerical point of view. Such an idea has already been proposedby Jin and Xin, using a global linearization of the system. Here, we take advantage ofthe structure of the system, which allows us to separate the linearly degenerate part andthe fully nonlinear part of the system. Then, we perform a relaxation approximationon the nonlinear part. This approximation leads to a linearly degenerate system with arelaxation source term (see the works of Suliciu, of Coquel et al. and the Bouchut’s bookfor similar works in the case of gas dynamics).

From the theoretical point of view, this approximation verifies a Gibbs principle and wecan show that it falls into the general theory of relaxation developed in the last few years(see for instance the Yong’s works). From the numerical point of view, this approach en-ables us to construct conservative, entropy satisfying and positive Finite Volume schemes.Several examples of applications will be presented with explicit constructions of numericalschemes.

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On the Numerical Simulation of Plasma Flowswith Mixing

Remi Sentis 1, C. Baranger 1, G. Carre 1, D. Paillard 1.1 CEA/Bruyeres - BP 12 - 91680 Bruyeres - F

In the simulation of multicomponant plasma flows at very high temperature(for instance in the plasma produced by laser beams), mixing of two differentfluids can occur. A classical model for such phenomena consists in a system ofsix equations which corresponds to the conservation of mass, momentum andenergy for each fuid, besides an equation for the electronic energy. The twofluids are assumed to occupy the same volume and the global pressure is thesum of the pressure of the two fluids. If the friction coefficient σ0 betweenthe two fluids (which depends on the internal energy) is assumed to be largeenough, we can made a simplification of this model owing to a closure concerningthe enthalpies of the two fluids. The resulting model consists of the classicalconservation equations of mass, momentum and energy for the averaged fuidcoupled with an equation for the concentration c and an equation for the relativekinetic energy K. If one denotes ρ,u, ε the density, the averaged velocity andthe averaged internal energy, the concentration obey the non linear diffusionequation

ρDtc−∇.

(c(1− c)

D0

σ0ε∇c

)= 0. (1)

where Dt is the Lagrange derivative (with velocity u) and D0 is a boundedsmooth function of c. Moreover, the kinetic energy K obey an equation of thefollowing type

ρDtK + 2ρK∇.u + 2σ0ρ2K = source term. (2)

To solve this system, one uses a classical numerical Lagrange scheme ofWilkins type ; the new variables c,K are evalueted at the center of each cell ;for the momentum equation the mixing pressure 2ρK is added to the materialpressure. The diffusion equation (??) is solved by an iterative way. Notice thatif the initial value cin of c is a Heavyside function, then it remains an unstablesolution of (??).

In the framework of Inertial Confinement Fusion, we will present numericalresults for a case where two fluids with a large relative velocity collide. Aftersome time, a mixing between the two fluids occurs. We sees that when themixing model is taken into account, the value of the density is quite lower fromthe one obtained with the pure mono-fluid model.

References.A. Decoster, Modeling of collisions, P.A. Raviart ed. Masson, Paris (1997).

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Closure Models for Multimaterial Cells in ArbitraryLagrangian-Eulerian Hydrocodes

Mikhail Shashkov1

1 Los Alamos National Laboratory, T-7, MS B284, Los Alamos, NM 87545, U.S.A.

High-speed multimaterial flows with strong shear deformations occur in many prob-lems of interest. Due to the nature of shock wave propagation in complex materials, theArbitrary Lagrangian-Eulerian (ALE) Methods are currently the only proven technologyto solve such problems. In ALE methods, the mesh does not move with the fluid, and soit is unavoidable that mixed cells containing two or more materials will appear.

Multimaterial cells are introduced in ALE methods to represent material interfacesthat undergo high deformation. The main difficulties in this case are how to accuratelydetermine the thermodynamic states of the individual material components and the nodalforces that such a zone generates, despite the lack of information about the velocitydistribution within multimaterial cells.

A separate set of material properties is normally maintained for all the materials ineach multimaterial cell along with the volume fractions that define the fraction of the cell’svolume occupied by each material. The volume fractions also can be used to reconstructmaterial interfaces inside mixed cell.

A subcell model is then required to define how the volume fractions and states of theindividual materials evolve during the Lagrangian step. This subcell model is required toclose the governing equations, which otherwise are underdetermined.

In my presentation I will describe different closure models and present numerical com-parison of different models in Lagrangian calculations with mixed cells.

This work was carried out under the auspices of the National Nuclear Security Ad-ministration of the U.S. Department of Energy at Los Alamos National Laboratory underContract No. DE-AC52-06NA25396 and the DOE Office of Science Advanced ScientificComputing Research (ASCR) Program in Applied Mathematics Research.

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Review of Radiation­hydrodynamics Research at AWE

Richard P. Smedley­StevensonAWE, Aldermaston, Reading, RG7 4PR, UK

In this paper we present a review of the various computational capabilities of the multi­material radiation­hydrodynamics codes at AWE. The computational physics group at AWE has   adopted   a   dual   route   strategy   for   the   modelling   of   the   various   physical   processes involved   in   modelling   complex   plasma   physics   experiments.   Specifically,   we   have developed both Eulerian and ALE hydrodynamics algorithms in two and three­dimensions, the details of which are described elsewhere.

By   employing  operator   splitting   techniques,   these  hydrodynamics   algorithms  have  been coupled   to   accurate   deterministic   and   Monte   Carlo   solutions   of   the   thermal   radiation transport equations. The sophistication of the deterministic models ranges from the basic equilibrium grey diffusion approximation with a single temperature, to full multi­frequency discrete ordinate transport simulations with up to 10,000 unknowns per hydro cell.

The aim of this paper is to provide an overview of these transport algorithms, focussing on the   issues   associated  with  obtaining   accurate   solutions   for  multi­material   problems.  We include a comprehensive set of results for problems ranging from simplified test problems to high fidelity models of complex plasma physics experiments designed to provide a stringent test of the accuracy of the various radiation­hydrodynamics models.

Extensions of the Multi-material DEM ModelDavid E. Stevens^1^1 Lawrence Livermore National Laboratory 8000 East Avenue, L-98 Livermore, California, 94551

The proper representation of multiphase phenomena is important for the simulation of many non-ideal flows. The Discrete Equation Method (DEM) of Chinnayya et al builds up a complete multiphase solution by summing up a series of single-phase contact problems between phases. This allows the usage of extremely accurate single-phase Riemann solvers and the incorporation of additional effects such as granular stresses. Thus, a multiphase solution with the accuracy of the underlying single-phase solver can be utilized.

This presentation will discuss extensions of the DEM method beyond just simple particle-gas problems. The extensions of interest are flows with deviatoric stresses, such as granular effects, three-phase flows with solid obstacles in addition to particulate, and the incorporation of advanced interface reconstruction techniques. This last topic is of interest in that DEM has multi-material capabilities beyond just particle-laden flows. References:

Chinnayya, A., Daniel, E., and Saurel, R., Modeling detonation waves in heterogeneous energetic media, J. Comput. Phys, 196, 490-538, 2004.

Conservative Rezoning of Domain Boundaryin ALE Simulations

Pavel Vachal1, Richard Liska1

1 Czech Technical University in Prague, Brehova 7, 115 19 - Praha 1, Czech Republic

The Arbitrary Lagrangian-Eulerian (ALE) method is very popular for simulation ofphenomena with large-scale changes of volume and shape of the computational domain,such as the high-velocity impact problem. After the Lagrangian step, the mesh mustbe rezoned to preserve sufficient precision of further computation. Special care has tobe taken of the boundary nodes, where an inadequate movement may lead to unwantedchanges of domain shape and volume.

We suggest a method based on constrained numerical optimization, which adapts themesh boundary while a priori preserving the volume of the entire domain and if possiblealso of the particular cells. The method tends to reduce the amount of quantities to beremapped between the meshes as low as possible. Several criteria of local mesh qualityand their combinations are tested and studied.

The presented method can be also used as a boundary preprocessor for some of theexisting techniques, such as Winslow smoothing or Reference Jacobian method. Also,since volume is preserved (completely or to a large extent), the same approach can beapplied to treatment of the material interfaces.

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Testing Multimaterial Treatment of Mixed Elementsin ALEGRA

William J. Rider, V. Gregory Weirs, Heath Hanshaw, Ed Love, and Michael WongSandia National LaboratoriesP. O. Box 5800, MS 0378

Albuquerque, NM 87185-0378USA

We present a series of test problems for verifying the accuracy of the treatment ofmixed elements in Multimaterial Arbitrary Lagrangian Eulerian (ALE) codes under avariety of physical conditions. A mixed element is one in which more than one materialis present and the materials are assumed to occupy distinct subvolumes of the element;the location of the interfaces between materials may or may not be explicitly computed.Mixed elements arise during the remap step, or from the discretization of the initialconditions.

The deformation of an element changes the thermodynamic and stress states of thematerials in the element. In ALEGRA and many other hydrocodes, mixture relationsspecify the distribution of the element deformation, stress and temperature to each ma-terial. The simplest mixture assumption apportions based only on the volume fractionsof the materials. This can lead to aphysical material states when e.g. one material ismuch more compressible than another; the deformation of the stiffer material is greatlyoverestimated, often leading to extremely high stresses or temperatures which underminethe simulation. More complex mixture rules have been shown to produce more physicallyreasonable material states in some cases, but a generally satisfactory method for mixedelements remains an elusive objective.

We have developed test problems specifically to quantitatively assess different meth-ods for treating mixed elements. These test problems feature exact solutions, enablingquantitative error analysis and code verification, but are also motivated by real-world ap-plications. The test problems are applicable to methods which involve explicit interfacetracking, as well as the mixing rules implemented in ALEGRA.

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Mar-tin Company, for the United States Department of Energy’s National Nuclear SecurityAdministration under Contract DE-AC04-94AL85000.

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A Level-Set Particle Method with Remeshing forMultifluids Simulations

Lisl Weynans1, Georges-Henri Cottet2, Bernard Rebourcet3

1 Laboratoire Jacques-Louis Lions, UMPC, 175, rue du Chevaleret, 75013 Paris, France2 Laboratoire Jean Kuntzmann, Tour IRMA, BP 53, 38041 Grenoble Cedex 9

3 CEA/DAM, BP 12, 91680 Bruyeres-le-Chatel

We present a particle-grid method applied to the system of compressible Euler equa-tions. The fluid is divided into particles, that is, masses concentrated on points. Theseparticles carry the conservative quantities of the fluid: mass, momentum and total energy,and move in a lagrangian way, at the velocity of the fluid.

At the beginning of each time step, particles are located at the centers of the cells ofa underlying grid. Euler equations are solved on this grid, and particles locations evolvedaccording to the local velocity. In order to preserve their uniform distribution, particlesare then “remeshed” on the grid by a conservative interpolation process, using high-orderinterpolation kernels, which represents the key element of this method.

A level-set-like technique is then adapted to the particle method: the level-set func-tion φ is discretized on the particles, advected and remeshed in the same way as theother variables. We present numerical results obtained with this method for severalhydrodynamic instabilities: Kelvin-Helmholtz instability, shock-bubble interaction, andRichtmeyer-Meshkov instability. A multilevel technique applied to the variable φ allowus to improve the interface resolution and the conservation of partial masses for a smalladditional cost.

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Multiphase Realizations of Turbulence ModelsRobin Williams^1^1 AWE Aldermaston, Reading, RG7 4PR

Multiphase and turbulent flows are important in a range of engineeringapplications. In both cases, model equations can be determined fromaveraging the equations of hydrodynamics, with closure terms enteringto truncate the approximation heirarchy. Forms for these closureterms may be derived by a variety of means, such as experiment or highresolution numerical simulation. Certain general desiderata alsoexist, such as the stability and dissipativity of the model equations.

The present paper investigates the relationship between turbulent andmultiphase models. In particular, writing a simple k-epsilonturbulence model as a hyperbolic relaxation system naturally limitsthe strength of turbulent diffusion terms and demonstrates that thesystem is globally dissipative. The stability of this treatment isinvestigated using an extension of Whitham's method to the case ofmultiple finite damping constants, and extensions to an adaptivetreatment of multiphase turbulent flow discussed.

Methods for Computation of Thermodynamic States of Mixed Cells in Lagrangian Gas Dynamics

Goncharov E.A., Kolobyanin V.Yu., Sadchikov V.V.,Yanilkin Yu.V.Institute of Theoretical and Mathematical Physics,

Russian Federal Nuclear Center All­Russian Research Institute of Experimental Physics,  Sarov, Russia

The   problem   of   correct   computations   of   mixed   cells   containing   two   and   more materials arises during computations in Lagrangian­Eulerian and Eulerian variables. One of the important problems here concerns correct computations of the thermodynamic state of components in Lagrangian gas dynamics being an integral part of the Lagrangian­Eulerian technique. The paper considers several computational methods for thermodynamic states of mixed cells   in Lagrangian gas dynamics differing in  their closing relations.  The methods based on the following assumptions are studied:

1. one and the same compressibility of components;2. equal pressures of components;3. equal pressure increments of components;4. equal velocities of components;5. parameters of materials are determined as a result of solving Riemann 

problem.The paper presents computation results for several problems that allow comparison of 

the methods with regard to their efficiency and accuracy. It is shown that each of the methods of interest has its own applicability area and the choice of what method is preferable is made depending on the physical problems to be solved.

Geometrical Interface Reconstruction on Arbitrary Meshes

Jin Yao

Lawrence Livermore National Laboratory 7000 East Avenue, California 94550, U.S.A. 

A  purely  geometrical  method   is   developed   to   construct  material   interfaces   on   arbitrary meshes using volume fractions. The method is an extension of the Youngs method on regular meshes.   The   orientation   and   slope  of   the   interface   facets   contained   in  mixed   cells   are obtained by properly marking nodes of cells and matching volume fractions in neighbour cells. A simple, universal rule for defining the topology of intersections between arbitrary shapes is used to define the facets. Instead of the planar facets used in Youngs method, the new method derives   the shape  of  a   facet  based on volume  fractions   therefore   improves connectivity of the interface across cell walls. Curvature of the interface is then naturally obtained. Thus, the new method improves the accuracy of Youngs method by an order. In principle, corners can be detected by extrapolating facets around potion of the interface with large curvature, then adjusted according to the volume fractions. 

Under the assumption that an accurate initial interface geometry is available. One is able to track the interface geometry over each time step and use the new method to match volume fractions to fine tune the interface geometry. Change of interface topology is detected by looking for neighbouring facets with considerably different slopes.

With the help of this geometrical method, the difficulties with disjoint facets across cell faces and T/Y intersections in Youngs method may be effectively dealt with.