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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.
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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.
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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.
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