Ebook ASCI alliance center for simulation of dynamic response in materials FY 2000 annual report Part 2

(BQ) Part 2 book ASCI alliance center for simulation of dynamic response in materials FY 2000 annual report has contents: Solid dynamics, materials properties, compressible turbulence, computational science.

Solid Dynamics

4.1 Overview of FY 00 Accomplishments

Accomplishments during FY 00 include the serial implementation of adaptive meshre£nement (subdivision) and coarsening (edge collapse); and the fully parallel imple-mentation of the solid dynamics engine within the VTF3D (without mesh adaption) As

in previous years, we have structured our material modeling efforts in terms of scale At the nanoscale we have continued to carry out quasicontinuum simulations

length-of nanoindentation in gold; and mixed continuum/atomistic studies length-of anisotropic location line energies and vacancy diffusivities in stressed lattices At the microscale

dis-we have developed a phase £eld model of crystallographic slip and the forest ening mechanism; we have re£ned our mesoscopic model of Ta by investigating thestrengths of jogs resulting from dislocation intersections, the dynamics of dislocation-pair annihilation, and by importing a variety of fundamental constants computed the

hard-MP group At the macroscale we have focused on various enhancements of our neering material models including the implementation and veri£cation of a Lagrangianarti£cial viscosity scheme for shock capturing in the presence of £nite deformationsand strength; the implementation of an equation of state and elastic moduli for Ta com-puted from £rst principles by Ron Cohen (MP group); and the implementation of theSteinberg-Guinan model for the pressure dependence of strength

eluci-of analyses the indentor sizes which may be considered are eluci-often considerably smallerthan experimentally employed values, which may in turn cause premature dislocationnucleation relative to observation Likewise, the size of the computational domain isnecessarily limited and the dislocations soon run up against arti£cial boundaries Inaddition, within a strict atomistic simulation it is dif£cult to account for the effect oflong-range elastic stresses such as might be present, e g., in a thin £lm/substrate sys-tem.

These limitations of straight atomistic simulation may be overcome by recourse tothe theory of the quasicontinuum [110,111,102,109,60] A full three dimensionalquasicontinuum analysis of the early stages of nanoindentation in gold thin £lms hasbeen carried out by Knap and Ortiz The surface of the £lm is a(001)-plane and thematerial obeys Johnson’s EAM potential [56,57] The calculations are based on a

model of a spherical indentor proposed by Kelchner et al [58] In this model, the dentor is regarded as an additional external potential interacting with atoms in the £lm.The computational domain is 2×2×1 µm in size and encompasses the full thickness

in-of the £lm The number in-of representative atoms in the initial mesh is1, 853, or an

X Y Z

Figure 4.1: Quasicontinuum calculation of a 2×2×1 µm (001)-gold thin £lm under

a 70 nm spherical indenter The total number of atoms in the sample is2.4 × 1011

(Knap and Ortiz, 2000) (a) Detail of computational mesh at 5.0 ªA containing90, 272representative atoms (initial mesh contains only1, 853) (b) View of the dislocationpattern, the color coding shown in the £gure identi£es: partial-dislocation core atoms(red); stacking fault atoms (yellow), surface atoms (blue)

eight-order of magnitude reduction from the total number of atoms (2.4 × 1011) in thesample We may note in passing that hundred-billion atom samples are well outsidethe scope of straight atomistic methods at present

The computational mesh for an indentor radius of70 nm at an indentation depth

of 5.0 ªA is shown in Fig 4.1 As may be seen from the £gure, the variation adaption criterion causes the mesh to be re£ned under the indentor, with theresult that the zone of full atomistic resolution grows steadily, as required The meshcontains90, 272 representative atoms This problem size is still modest compared tothat which is demanded by straight atomistics However, the adaptive character of themethod ensures that a suf£ciently large fully-resolved atomistic region lies beneaththe indentor at all times for dislocations to nucleate and grow into The dislocationpatterns predicted by the analysis for an indentor radius of70 nm at 5.0 ªA indentationare shown in Fig.4.1 The pattern is initially symmetric and involves slip on four{111}planes The symmetry of this pattern is eventually broken, and elongated dislocationloops propagate on selected{111} planes Away from the indentor the behavior ofthe crystal is ostensibly linear elastic, and captures the long-range elastic £eld of theindentor It should be carefully noted that, even in this region, all material behavior,

displacement-e g., the effective anisotropic elasticities of the crystal, emanates directly from the

behavior is entirely seamless.

Another mechanism which can be elucidated by direct atomistic modeling concernsvacancy migration through a stressed lattice in the presence of dislocations Olmstedand Phillips (2000) have calculated vacancy migration enthalpies in Al using EAM po-tentials The core structures obtained by Olmsted and Phillips are shown in Fig.4.2a.The cores are dissociated and vary in structure and energy according to the direction

of the dislocation line The dislocations split approximately into Shockley partials, though no extended region of stacking fault appears between the partials The partialsthemselves are suf£ciently spread out as to overlap The basic Shockley partial struc-ture is clear in the slip distributions, where the different partials have different widths.The screw partial in the30◦dislocation has the smallest width, whereas the edge partial

al-in the60◦dislocation is the widest Olmsted and Phillips have found that both the coreand the line-energy anisotropy are accurately predicted by linear elasticity Fig.4.2bcollects activation enthalpies for two vacancy migration directions as a function of anapplied uniaxial stress The objective of these calculations is to ascertain the effect ofstress on vacancy diffusivities The results of the calculations suggest that the variation

in enthalpy due to applied stress is linear, albeit anisotropic, to a £rst approximation.4.4 Mesomechanics

In the forest-dislocation theory of hardening, the motion of dislocations, which are theagents of plastic deformation in crystals, is impeded by secondary –or ‘forest’– dislo-cations crossing the slip plane As the moving and forest dislocations intersect, theymay result in a variety of reaction products, including jogs and junctions [83,94,4,

112,92,62,51,101,126] Cuiti˜no et al [26] have noted that the complex dislocationpatterns which develop during this process, the intricate interactions between disloca-tions and obstacles, and the resulting kinetics, are amenable to an ef£cient phase-£eldrepresentation In essence, the value of the phase £eld at a point on a slip plane countsthe number of dislocations which have passed over the point In this representation,the individual dislocation lines are recovered as the level contours of the phase £eld atintegral values

An example of the dislocation pattern evolution predicted by the theory under cyclicsingle slip, and the resulting stress-strain and dislocation density curves are shown inFigs.4.3 The phase-£eld representation enables the tracking of complex geometri-cal and topological transitions in the dislocation ensemble, including dislocation loopnucleation, bow-out, pinching, and the formation of Orowan loops The theory alsopredicts a range of behaviors which are in qualitative agreement with observation, in-cluding: hardening and dislocation multiplication in single slip under monotonic load-ing; Taylor scaling, both under monotonic loading and, in an appropriate rate form,under cyclic loading; the Bauschinger effect under reverse loading; the fading memoryeffect, whereby reverse yielding gradually eliminates the in¤uence of previous loading;the evolution of the dislocation density under cycling loading, leading to characteristic

‘butter¤y’ curves; and others

By way of speci£c example, Fig.4.3e shows the effective cyclic response predicted

by the theory in single slip The overall trends are in good agreement with the

ex-considerable interest, Fig.4.3f Upon unloading the dislocation density decreases as aresult of the elastic relaxation of the dislocation lines The dislocation density bottomsout — but does not vanish entirely — upon the removal of the applied stress, point

’b’, as some dislocations remained locked in within the system in the residual state.The dislocation density increases again during reverse loading, segment b–c, and the

cycle is repeated during reloading, segment c–a, giving rise to a dislocation density vs.

slip strain curve in the form of a ‘butter¤y’ This type of behavior which is indeed served experimentally (Morrow, unpublished tests results), it also arises in models ofthe stored energy of cold work [10], and is in analogy to the hysteretic loops exhibited

ob-by magnetic systems [99,28]

Key inputs into this and similar theories which may be gleaned from atomisticsare: dislocation energies as a function of segment orientation; Peierls stresses; andthe strength of dislocation-dislocation reaction products The core structure and ener-getics of screw dislocation segments in bcc crystals has been extensively investigated[71,119,125,52] bcc edges have been investigated by Wang et al [119,118] Forinstance, for Ta they have calculated a ratio of edge to screw energies of1.77 Olmstedand Phillips [75] have used the embedded-atom method potential, as £tted by Erco-lessi and Adams [35] to the results of their £rst-principles calculations, to map out theentire range of energies of dissociated dislocation cores in aluminum Their resultsdemonstrate that the energies computed from atomistics can be reproduced almost ex-actly using linear elasticity theory provided that dissociation into partials is accountedfor and an appropriate stacking-fault energy is used, which again attests to the predic-tive ability of informed continuum models Duesbery and Xu [32] have calculated thePeierls stress for a rigid screw dislocation in Mo to be 0.022µ, where µ is the h111ishear modulus, whereas the corresponding Peierls stress for a rigid edge dislocation is0.006µ, or about one fourth of the screw value Wang et al [119,118] have calculated

a value of 0.03µ for the Peierls stress of screws in Ta, which is in the expected ballpark.The strength of dislocation jogs and junctions has recently been computed usingatomistic and continuum models [83,94,119,118,4,101] Thus, for instance, Rod-ney and Phillips [94] used the quasicontinuum method to simulate three-dimensionalLomer-Cottrell junctions, and determined that this type of junction may be unzipped

under stress Interestingly, Shenoy et al [101] subsequently showed that essentiallyidentical results may be obtained with a anisotropic elastic model provided that dislo-cation dissociation into partials is accounted for, which attests to the predictive power

of informed continuum models Shenoy et al [101] went on to map out the completestress-strength diagram for junctions, i e., the locus of points in stress space corre-

sponding to the dissolution of the junction Likewise, Wang et al [119,118] haveexhaustively cataloged the jogs and kinks of bcc crystals and computed their structuresand energies

We (Cuiti˜no, Ortiz and Stainier, 2000) have also developed a mesoscopic model

of the hardening, rate-sensitivity and thermal softening of bcc crystals The model ispredicated upon the consideration of an ‘irreducible’ set of unit processes, consistingof: double-kink formation and thermally activated motion of kinks; the close-range

the percolation motion of dislocations through a random array of forest dislocationsintroducing short-range obstacles of different strengths; dislocation multiplication due

to breeding by double cross-slip; and dislocation pair-annihilation Each of these cesses accounts for–and is needed for matching–salient and clearly recognizable fea-tures of the experimental record In particular, on the basis of detailed comparisonswith the experimental data of Mitchell and Spitzig [70], the model is found to capture:the dependence of the initial yield point on temperature and strain rate; the presence

pro-of a marked stage I pro-of easy glide, specially at low temperature and high strain rates;the sharp onset of stage II hardening and its tendency to shift towards lower strains asthe temperature increases or the strain rate decreases; the initial parabolic hardeningfollowed by saturation within the stage II of hardening; the temperature and strain-ratedependence of the saturation stress; and the orientation dependence of the hardeningrates

The choice of analysis tools which we have brought to bear on the unit processes

of interest, e g., transition-state theory, stochastic modeling, and simple linear-elasticmodels of defects and their interactions, is to a large extent conditioned by our desire

to derive closed-form analytical expressions for all constitutive relations As notedthroughout the paper, many of the mechanisms under consideration are amenable to

a more complete analysis by recourse to atomistic or continuum methods However,

at this stage of development, direct simulation methods, be it atomistic or continuumbased, tend to produce unmanageable quantities of numerical data and rarely result inanalytical descriptions of effective behavior The daunting task of post-processing thesedata sets and uncovering patterns and laws within them which can be given analyticalexpression is as yet a largely unful£lled goal of multiscale modeling

This larger picture notwithstanding, one concrete and workable link between cromechanical models and £rst-principles calculations concerns the calculation of ma-terial constants A partial list relevant to the present model includes: energy barriersand attempt frequencies for double-kink formation, kink migration, dislocation unpin-ning, cross-slip, and pair annihilation; dislocation-line and jog energies; and junctionstrengths Other properties which have yielded to direct calculation include the volu-metric equation of state (EoS), the pressure dependence of yield, and the pressure andtemperature dependence of elastic moduli As noted earlier, these results provide a suit-able basis for future extensions of the present model to higher temperatures, pressuresand strain-rates

mi-4.5 Macromechanics

As part of the operation of the VTF, sharp shocks develop in the solid components,including the canister and payload materials, which need to be handled appropriately.Under the conditions envisioned here, this invariably requires the introduction of arti-

£cial viscosity into the formulation When the solids are modeled within a Lagrangianframework, the problem arises of devising effective arti£cial viscosities which performwell for arbitrary unstructured tetrahedral meshes, and in the presence of large plasticdeformations and rigid-body rotations

We have developed and veri£ed an arti£cial viscosity method which is formulated

of the method has been veri£ed by comparing the numerical and exact solution for theRiemann problem in perfect gases in two dimensions and by verifying the jump con-ditions and shock propagation velocity in Ta For purposes of this study, we modelthe behavior of Ta up to extreme loads of the order of200 GPa by recourse to the

equation of state of Cohen et al., derived from ab initio quantum mechanical

calcula-tions [90] and by a Steinberg-GuinanJ2-isotropic large-deformation plasticity modelthat accounts for rate dependency and thermal softening effects The model is used tosimulate a plate impact experiment with an impact speedV = 2000 m/s

In the present formulation the effective numerical viscosity is assumed to be of theform:

whereη is the physical viscosity and 4ηhis the added arti£cial viscosity This latterterm, is computed at each £nite-element Gauss point in accordance with the followingexpression

whereh is a measure of the element size, ρ0is the unshocked density,∆u is a measure

of the velocity variation across the element,a is the sound speed, and c1andcL arecoef£cients In a multidimensional simulation a precise meaning needs to be given

to the variables∆u and h in a way that renders the arti£cial viscosity formulation

material-frame indifferent A For each elementΩe:

∆un+1= hJn+1− Jn

It should be noted that the arti£cial viscous stress is computed simultaneously withthe constitutive relations, which facilitates matters of implementation, and that (4.4) isstrictly material-frame indifferent

In order to identify the coef£cientsc1 andcL, we follow [27] and note that theRankine-Hugoniot jump conditions de£ne an implicit relationD(∆u) between theshock speed and the jump in velocity which is often well-approximated by a linear

£t of the form

linear momentum jump condition for a steady shock

a quadratic equation for∆u is obtained By considering then the arti£cial viscosity as

an approximate Riemann solver, the following relations forc1andcLare obtained

c1 ≈ s1

Our experience indicates that these constants tend to perform well for strong shocksbut their performance deteriorates for weak shocks For the latter the shock tends to besmeared over more elements than necessary and overheating effects are exacerbated

By way of validation, we have applied the method to a Riemann problem, alsoknown as the ‘shock-tube problem’, for ideal gas equation of state on both sides of theinitial contact surface Adiabatic heating was assumed in this test Fig.4.4shows thepressure and density ratiosp/p1andρ/ρ1along the centerline of the two dimensionalcon£guration together with the analytical solution every∆t = 1.69 × 10−5s Thegood agreement between the analytical and numerical solutions is evident from the

£gure Both the Rankine-Hugoniot jump conditions across the shock and the values inthe expanded gas are well predicted The speed of the contact surface and the shock arealso accurately computed It is interesting to note that, due to the Lagrangian character

of the formulation, the contact surface is extremely sharp Some spurious overheating

is evident in the slight curvature of the density and pressure pro£les after the shockand the contact surface This is a pathology of arti£cial viscosity methods, and can beimproved by the introduction of arti£cial heat ¤ux

We have also considered a piston problem consisting of the sudden operation of apiston as a means of introducing a shock or a compression wave in the material ahead

of it In this veri£cation test we have speci£cally aimed to test the performance ofthe arti£cial viscosity method in conjunction with realistic equations of state for Ta

In particular, we adopt a Vinet equation of state for Ta £tted by Cohen [90] to principles calculations The internal energy per unit massU (T, v) for this material isgiven by the expression:

£rst-U (T, v) =

−2.218 × 1011+ 1.813 × 109exp(−119.816 v1/3) + 137.849 T −

−5.903 × 1010exp(−119.816 v1/3) v1/3+ (4.9)+T2¡−0.01243 + 653.806 v − 7.0925 × 106v2+ 4.363 × 109v3¢ ++T3¡1.301 × 10−6− 0.0505 v + 408.911 v2+ 2.193 × 106v3¢

where T is the absolute temperature and v the speci£c volume per unit mass The

−783.37 + 3.368 × 10 v − 2.151 × 10 v + 1.689 × 10 v +

+T ¡−0.0249 + 1307.61 v − 1.419 × 107v2+ 8.725 × 109v3¢ + (4.10)+T2 ¡1.952 × 10−6− 0.0757 v + 613.367 v2+ 3.289 × 106v3¢ ++137.849 log(T )

All constants correspond to SI units Equations (4.9) and (4.10) completely de£ne thethermodynamic behavior of the material

to 8 elements The absence of oscillations behind the shock is noteworthy

Table 4.2: Plasticity model material parameters (SI units)

The hydrodynamic approximation for the propagation of shock waves in solids byde£nition neglects the deviatoric response of the material However, there are situations

in which the strength and viscosity of certain materials strongly in¤uence the shock

propagation of shocks in materials exhibiting a deviatoric–possibly plastic–response,

i e., in solids with strength The suitability for three-dimensional applications is also

evaluated The test concerns a plate-impact con£guration on Ta in three dimensions.For purposes of reducing the problem size, the geometry is idealized as a cylinder coredfrom the plate in its thickness direction, Fig.4.6 The cylinder has a radiusR = 0.02 mand a length equal to the thickness of the platet = 0.1 m The cylindrical surface isconstrained to move in the cylinder’s axial direction

The volumetric response of the material is described by the same equation of state

as in the piston test described above In addition, the material is assumed to exhibit a viatoric elastic-plastic response We adopt a standard formulation of £nite-deformationplasticity based on a multiplicative decomposition of the deformation gradient intoelastic and plastic components The values for the cubic crystal elastic constants of thematerial as a function of volume and temperature were obtained from Cohen’s quan-tum mechanical calculations [89] and are shown in Fig 4.7 It can be observed inthese £gures that the temperature dependence of the cubic constants at a £xed volume

de-is small, which indicates that the temperature effects manifest mostly through thermalexpansion Therefore and as a £rst approximation, we neglect the explicit dependence

of the elastic constants on temperature at a £xed volume We also idealize the elasticresponse as isotropic with moduli:

Figs 4.8and4.9show the evolution of the pressure and effective plastic strainsfor an impact velocityV0 = 500 m/s Upon impact a sharp shock appears insidethe materials which propagates along the length of the cylinder and soon acquires asteady pro£le When the shock reaches the free end a shock release wave is pro-duced In addition, a dispersive plastic wave is generated inside the material whichpropagates along the length of the cylinder behind the elastic precursor wave Plasticdeformations of the order of6% are attained during the calculation Neither the three-dimensional discretization nor the arti£cial viscosity method perturb the predominantlyone-dimensional character of the problem The ability of the method to resolve theshock across a small number of elements and without oscillations is noteworthy.4.6 Polymorphic Phase Transitions

In our work over the last two years [12,115,13,116] we developed a theory for phase

transitions in solids under shock-loading We showed that a single, well-substantiated

critical condition captures the essence of the phenomena under consideration Ourmodel results in predictions in close quantitative agreement with a wide variety ofexperimental results and, unlike other models considered previously, it does not requireuse of £tting parameters

In our FY 2000 work [12,115,13,116] we (1) Provided the general solution forRiemann problems which arise in our theory, we (2) Obtained closed-form approximatesolutions for a general piece-wise constant initial value problem, and we (3) Developed

a variety of numerical schemes, including Godunov-type schemes and £nite-differenceschemes which fully take into account non-elastic effects As the result of these efforts

we obtained close quantitative agreement with all experimental data available in thesystems studied, with no recourse to parameter £tting

4.6.1 Riemann Solver

The Riemann problem — which consists of two distinct spatially constant states on thesides of a jump discontinuity — has played a central role in our study of the dynam-ics of shock-induced phase transitions in solids As in the case of gas dynamics, thesolution of these Riemann problems consists of four constant states separated by threeself-similar waves The central wave is always a contact discontinuity The left- andright-facing waves, in turn, may be single rarefaction fans or single shock fronts, butthey may also be self-similar waves of more general types — including the two-wavestructures mentioned above The occurrence of such generalized self-similar wavesdistinguishes the Riemann problem arising in our theory from the Riemann problem ingas dynamics — where the left- and right-facing waves can only be rarefaction fans orshock fronts

The Riemann solver developed in [12] was used as a building block for solutions ofthe general initial and boundary value problems arising in experiment In [115], weutilized a certain “rarefaction discontinuity approximation”, in which rarefaction fansare substituted by rarefaction discontinuities This approximation allowed us to re-duce the entire solution of the impact problem to an approximate closed-form solution,which was constructed by piecing together the solutions of the sequence of Riemannproblems.

As an illustration, in Figure 4.10 we show the sequence of Riemann problems,

RP1, RP2, arising from the experimental setup corresponding to experiment 1 of [3]HereRP1denotes the initial Riemann problem which arises from the piece-wise con-stant data on the ¤yer-target system at the impact timet = t1, with left and right statesequal to the initial states of the ¤yer and the target, respectively The rest of the Rie-mann problems,RP2, RP3, , arise later as the waves generated in previous Riemannproblems reach material surfaces or collide with each other

We applied our theory to two widely-studied polymorphic phase changes: thegraphite–diamond transition [36] and theα–² transition in iron [3] In [115,116] wecompared our predictions with the corresponding results of experiment (a detailed dis-cussion of the material parameters of the associated equations of state and transforma-tion boundaries is also included in [115])

The sequence of Riemann problems arising from the experimental setup of [3] givesrise to many rarefaction waves which in¤uence the monitored interface (MI) within thetime-ranges of the experimental pro£les reported in [3] Consequently, the “rarefactiondiscontinuity approximation” approach used in [115] and the direct numerical simu-lation approach used in [116] produce the MI velocity pro£les which differ in somedetails We found (see Figure4.11) (center) that the closed-form solutions of [115]capture accurately the main features of the experimental pro£les

4.6.3 Full Numerical Solvers

In [116], on the other hand, no simplifying approximations were made In this proach, appropriate shock-capturing numerical schemes based on the Riemann solverwere used to obtain the solution of the impact problem In particular we produced afull phase-transition-capable Godunov solver for the purely non-linear elastic case, aswell as a phase-transition £nite difference scheme for systems containing dissipativepure phase EOSs A preliminary set of results provided by the full solver [116] areshown in Figure4.11(right), which shows the full scheme reproduces experiment to a

ap-£ne degree of detail

4.7 Eulerian Elastic-Plastic Solver

In this section we describe work accomplished this year on contract PC291132 for theCaltech Center for Simulation of Dynamic Response of Materials Originally, one ofthe Center’s main challenges was the simulation of problems involving the strong cou-pling of solid mechanics and gas dynamics Solid mechanics is traditionally solved in

a Lagrangian computational framework, whereas the best available gas dynamics

ap-pling was accomplished using an implementation of the so-called ghost ¤uid method.

An alternative strategy would be to attempt a fully Eulerian, or a fully Lagrangian lution This section describes progress on the development of one such alternative:the coupling of solid mechanics and gas dynamics within a single uniform Eulerianframework

so-Toward this end, We have worked on the development of a new Eulerian solid chanics method This work is described in the paper “A High-Order Eulerian GodunovMethod for Elastic/Plastic Flow in Solids”, coauthored by P Colella, which has beenaccepted for publication by the Journal of Computational Physics (preprints available

me-as LBNL report LBNL-45647 and Caltech report cit-me-asci-tr055)

More recently, we have worked on the development of a fully-conservative of-¤uid based interface method for the coupling of Eulerian solid mechanics and gasdynamics solvers Preliminary versions of this work were presented at the October 10

volume-2000 site visit Since then, the implementation has been improved and it now includes3D block-structured adaptive mesh re£nement (AMR)

A numerical method has been developed that provides high-order accurate tions to multi-material (solid-¤uid) shock-capturing problems in 3D, in parallel, andwith adaptivity The method employs a single uniform Eulerian grid: data manage-ment, domain partitioning, and visualization problems are thereby greatly simpli£ed.4.7.1 Eulerian Solid Mechanics

solu-A central variable in solid mechanics is the deformation tensorFij = ∂xi/∂aj thatdescribes the motion of a laboratory (Eulerian) reference frame {x} relative to thematerial (Lagrangian) reference frame{a} In some formulations the inverse of F,

gij = ∂ai/∂xj, is more convenient As gradients, these tensors are subject to theequality of mixed partial derivatives For example,

These constraints are the central issue in the development of an Eulerian mentation of solid mechanics When these constraints are not obeyed, the conservationform equations of solid mechanics may suffer eigenvector de£ciencies: the numericalsolutions to the PDEs of Eulerian solid mechanics may give inaccurate wave speeds andmay be computationally unstable Approaches may be constructed to numerically en-force (2), as traditionally done in the £eld of magnetohydrodynamics where the gaugecondition∇ · B = 0 holds on the magnetic £eld B However, associated with any nu-merical enforcement of (2) is a truncation errorτ > 0: (2) cannot be enforced exactly

imple-in any numerical scheme

straints They modi£ed the underlying PDEs in such a way that if the constraint isobeyed, the solution is the solution to the original system of equations However, if theconstrain is not obeyed, the new system of equation remains numerically well-behaved.With reference to solid mechanics, eigenvector de£ciencies and their accompanyingproblems would no longer occur Enforcement of (2) would then become a question ofaccuracy, not one of stability.

We [68] developed this approach to write a form of the PDEs of solid mechanics

in which density, momenta, and energy, and internal state variables (those describingplasticity, for example) are in conservation form The equations of change for theinverse deformation gradientg are not in conservation form, but the nonconservativeterm is zero when (2) is obeyed

To enforce (2), we recognize that the PDEs may be arranged to give a conservationequation forG = ∇ × gT:

∂G

Numerically, then, from an initial condition in whichG = 0, we expect departures from

G = 0 to be a dipolar £eld that could be eliminated by diffusion of G

2D and 3D implementations of this approach were developed that use a split advection scheme [21,95] Resulting 2D and 3D simulation results show highorder of numerical accuracy (second-order), and excellent preservation of symmetry.4.7.2 Eulerian Solid-Fluid Coupling

non-operator-In solid-¤uid problems, the solid and ¤uid domains will not in general be aligned with

a regular rectangular Eulerian grid: some computational cellsijk will contain nonzerosolid volume fractionsΛS

ijk and nonzero ¤uid volume fractionsΛF

ijk The usual cretization of Eulerian conservation form dynamics is then subject to the time step limit

dis-∆t ∝ Λ1/3h where h is the width of a cell When ΛS 6= 0 and ΛF 6= 0, but ΛS → 0

orΛF → 0, the stable time step plummets and the calculation halts

An approach to this dilemma is the mass redistribution strategy of Chern andColella [19] They propose that one £rst perform a numerically stable but non-conservativeupdate The difference between the stable but non-conservative solution, and the un-stable but fully conservative solution, is a generalized mass differenceδMijk Theyreestablish global conservation by distributing this mass difference among neighboringfractionally- and fully-occupied cells This strategy has been used to model ¤ows withirregular interior boundaries [81], to track shock fronts [6], and to track reaction fronts[49,11,84]

We are adapting this approach to track solid-¤uid material interfaces The rithm may be decomposed into the four logical components:

algo-• The ¤uid subproblem The irregular domain occupied by the ¤uid is “extended”

by averaging or extrapolation to provide some sort of boundary condition data

in cells containing no ¤uid The extended ¤uid domain is then advanced in timeusing traditional Eulerian high-order Godunov methods In my implementation,

advance it in time using the high-order Eulerian Godunov approach we oped [68].

devel-• The interface subproblem A volume-of-¤uid [85] approach is used to construct,from volume fractions, a piece-wise-planar representation of the solid-¤uid in-terface Using the geometric description of the interface, an interface-normalsolid-¤uid Riemann problem may be set up and solved The solution to thisinterface Riemann problem contains the ¤ux of energy and momentum acrossthe material interface, and dictates the velocity of the interface itself An non-operator-split advection scheme after [7] is used to time step the material volumefractions

• Reestablishing conservation The solution to the multiphase problem is structed by combining the solutions to the extended single-phase problems, theupdated ¤uid fraction data, and the computed ¤uxes of momentum and energyacross the interface Global conservation is maintained via the approach of Chernand Colella

con-A 3D implementation of this approach has been written, and incorporated within

a framework for data management and MPI parallelism called Chombo [22] This plication implements block-structured adaptive mesh re£nement to solve the interfacesubproblem (and other tagged regions of the overall problem domain) on a £ne com-putational mesh, while solving the remainder of the problem on a coarser mesh.The method described above for solid-¤uid coupling using Eulerian Godunov meth-ods is a composite of a number of separable algorithms, each of which has an associ-ated stencil For example, the Godunov methods themselves have stencils of9 × 9 × 9(ghost cell width 4), and the volume-of-¤uid interface reconstruction method has astencil of5 × 5 × 5 (ghost cell width 2) In combination, the algorithms have a domain

ap-of dependence ap-of29 × 29 × 29 (14 ghost cells) A straightforward implementation ofthe above method, dealing with the overall domain of dependence as a single stencil,would be very expensive in terms of memory requirements and communication cost.One could implement each algorithm separately (requiring only small stencils like93

or53), but this approach is also expensive in that the number of variables stored pergrid point becomes quite large, and in parallel computations a large number of separatecommunications are required for each time step The current implementation strikes acompromise by using a stencil of133(ghost width 6) and one extra intra-step commu-nication

This program, together with the Eulerian solid mechanics solver, prove the ity to solve coupled ¤uid-solid problems conservatively, with high accuracy, and withadaptivity

abil-The main goal for integrated simulation capability in FY 01 is the completion of a3D couple Eulerian-Lagrangian simulation of HE detonation interacting with a Tanta-lum target on the ASCI terascale platforms on upwards of 1000 processors the solidmechanics engine will employ parallel meshing ad parallel mechanics but no adaptivere£nement A fully 3D ¤uid-solid interaction algorithm will be utilized We shall alsowork towards developing scalable adaptive remeshing algorithms, and the incorpora-tion into the VTF3D of fracture and fragmentation capability During FY 01 we willcontinue the development and implementation of improved models of single crystalplasticity and spall, including the development of ef£cient polycrystalline averagingschemes; and multiscale modeling of void nucleation, growth and coalescence leading

to spall

R lo a in g

L a

n lo a

in g

a c

b d

maximum load (c–d), and reloading (d–a) (e) Applied resolved shear stress vs average slip (f) Evolution of dislocation density vs average slip.

0 0.005 0.01 y[m]

Figure 4.7: Temperature and dependency of cubic elastic constants for Tantalum from[89].

1.51E+10 1.08E+10 6.46E+09 2.15E+09 Pressure [Pa]

(a)

1.51E+10 1.08E+10 6.46E+09 2.15E+09 Pressure [Pa]

(b)

1.51E+10 1.08E+10 6.46E+09 2.15E+09 Pressure [Pa]

(c)

1.51E+10 1.08E+10 6.46E+09 2.15E+09 Pressure [Pa]

(d)

Figure 4.8: Plate-impact test Contour plots of pressure at successive times

0.20 0.15 0.09 0.04

accumulated plastic strains

(a)

0.20 0.15 0.09 0.04

accumulated plastic strains

(b)

0.20 0.15 0.09 0.04

accumulated plastic strains

(c)

0.20 0.15 0.09 0.04

accumulated plastic strains

(d)

Figure 4.9: Plate-impact test Contour plots of accumulated plastic strain at successivetimes

−0.5 0.5 1.5

x/Specimen Thickness

0 0.2 0.4 0.6 0.8

Figure 4.11:α–² shock induced phase transition in iron Left: Experimental curves [3].Center: Approximate closed-form solutions of [115] Right: Full numerical solutions

of [116] Free-surface velocity pro£les are presented for experiments 14, 1, 17, 6,and 9 of [3] (The corresponding impactor velocities are: 0.6127 km/s, 0.9916 km/s,1.307 km/s, 1.567 km/s, and 1.900 km/s.) As in [3], the time axes are normalized bydividing the actual time from impact by the specimen thickness

5.1 Overview of FY 00 Accomplishments

The Materials Properties team is tasked with providing parameters for a full physics,full chemistry, 3D description of materials in the detonation of high explosives, solidssubjected to severe dynamic loading, and compressive and turbulent mixing The teamalso must develop the technology required to predict these properties solely from £rst-principle, validate the accuracy of these properties by comparison to relevant exper-iments, implement this technology on massively parallel computers, and incorporatethe parameters into a materials properties database

In simulations supporting high explosives, the MP team has completed the position mechanism of RDX and HMX molecules using density functional theory, ob-tained a uni£ed decomposition scheme for key energetic materials, obtained a detailedreaction network of 450 reactions describing nitramines, developed ReaxFF, a £rst-principles based bond-order dependent reactive force£eld for nitramines, and pursued

decom-MD simulations of nitramines under shock loading conditions

In simulations supporting solid dynamics, the MP team has developed a £rst-principlesqEAM force-£eld for Ta We have used this force £eld to simulate the melting curve

of Ta in shock simulations up to 300 GPa We have also investigated properties related

to single-crystal plasticity, particularly core energies for screw and edge dislocations,Peierls energies for dislocation migration, and kink nucleation energies We have sim-ulated vacancy formation and migration energies, related to vacancy aggregation andspall failure We have run high-velocity impact MD simulations to investigate spallfailure in materials We have simulated a thermal equation of state for Ta from densityfunctional theory calculations, and have simulated the elasticity of Ta versus P to 400GPa, and T to 10000 K Finally, we have begun work on Fe by examining the hcpphases of Fe

In methodological developments and software integration, we have developed theMPI-MD program, which allows parallel computations of materials with millions of

71

mechanical eigenproblem that uses a block-tridiagonal representation of a matrix toyield more ef£cient scaling of the eigensolver We have developed a variational quan-tum Monte Carlo program to yield more accurate simulations of metals at high temper-ature and pressure Finally, we have begun work on the materials properties database,

to allow archival of QM and MD simulations, and automatic generation of the derivedproperties required by the HE and SD efforts

5.2 Personnel

The Materials Properties team consists of senior researchers:

• William A Goddard, III, Co-PI,

and long-term visitors

• Peter Schultz (Sandia National Laboratories),

• Darren Segall (MIT ASCI-2 Center)

Reaction mechanism of RDX and HMX thermal decomposition in gas phase Wehave recently completed a DFT investigation of the detailed chemistry of RDX [15]and HMX (manuscript submitted) and are now extending it to incorporate effects incondensed phases (solvation and cage effects) using a hybrid QM/MM method [82].Development of the reactive force £eld (ReaxFF) for initiation of reactions in RDXand HMX We have recently formulated bond order dependent approach for calculatingreaction products under shock conditions We have performed MD simulations of thetype indicated below where two ¤yer plates of RDX material were impacted againsteach other.

MD simulations of energy coupling between shock wave and molecular modesWhat are the precise circumstances required for an irreversible coupling to occur be-tween mechanical energy stored in a hot spot and the release of exothermic chemicalenergy? How does the shock induce melting in RDX and HMX? What is the mecha-nism and the effect of the solid-to-solid phase transition in HMX? When phase trans-formations take place, issues of de-wetting are important i.e., how coherent is a secondphase? MD is ideal for studying cohesion strength and friction properties of an inter-face between phases How these properties depend on temperature and strain rate is notwell understood either Once such properties are known, then continuum £nite elementanalysis can provide answers on possible void behavior in an energetic material.The following sections outline more detailed results of QM and MD simulations.5.3.1 Quantum Mechanical Calculations of Reaction Mechanisms in RDXand HMX

Although gas phase unimolecular decomposition experiments [127] suggest at leasttwo major pathways, there is no mechanistic understanding of the reactions involv-ing RDX In order to address this issue, we have completed the gas phase reactionmechanism of RDX using DFT at the B3LYP/6-31G* level [15] For the unimoleculardecomposition of RDX, we £nd two additional pathways: (i) concerted decomposition

of the ring to form three CH2NNO2 (M=74) molecules, (ii) homolytic cleavage of an

NN bond to form NO2 (M=46) plus RDR (M=176) which subsequently decomposes toform various products Experimental studies[127] suggest that the concerted pathway

is dominant while theoretical calculations have suggested that the homolytic pathwaymight require signi£cantly less energy, and (iii) successive HONO elimination to form

3 HONO (M=47) plus stable 1,3,5-triazine (TAZ) (M=81) with subsequent sition of HONO to HO (M=17) and NO (M=30) and at higher energies of TAZ intothree HCN (M=27) (iv) Formation of a 4-membered oxy-ring which then decomposes

decompo-to CH2O and N2O

We examined all pathways for barriers for all low-lying products and found:

• a threshold at ≈40 kcal/mol for which HONO elimination leads to TAZ plus 3HONO while NN homolytic cleavage leads to RDR plus NO2 and the concertedpathway is not allowed;

Figure 5.1: Uni£ed decomposition mechanism of RDX and HMX from ab initio culations.

cal-• above ≈52 kcal/mol the TAZ of the HONO elimination pathway can decomposeinto 3 HCN while the HONO can decompose into HO + NO;

• above ≈60 kcal/mol the concerted pathway opens to form CH2NNO2;

• at a threshold of ≈65 kcal/mol the RDR of the NN homolytic pathway can compose into other products

de-These predictions are roughly consistent with previous experimental results andshould be testable with new experiments on RDX

We have now extended these calculations to HMX where three distinct lar decomposition channels have been identi£ed through ab initio calculations:

unimolecu-1 Homolytic cleavage of N-N bond to form NO2 (M=46) and HMR (M=250)which subsequently decomposes to form various products;

2 Successive HONO eliminations to give four HONO (M=47) plus a stable mediate (M=108);

inter-3 O-migration from one of the NO2 groups of HMX to neighboring C-atom lowed by the decomposition of intermediate (M=296) to INT222 (a ring openedRDX structure) and MN-oring (M=74), which can undergo dissociation to smallermass fragments

fol-certed decomposition of HMX to four MN (M=74) molecules is not a favorable way, as found in RDX decomposition experimentally and theoretically The uni£edscheme is presented in Figure5.1.

path-5.3.2 Molecular Modeling of Shocked HE Using Reactive Force Fields

We have developed a radically new force £eld (ReaxFF) which allows bond breakingand formation and models transition states in non ionic reactions It was originallydeveloped for hydrocarbons and we have now extended the method to work on anyclass of materials for which quantum mechanical calculations can be done Using the

QM calculations we have parameterized ReaxFF to handle RDX and HMX It is the

£rst fully atomistic reactive force £eld for these important HE materials It is based oncalculating a bond-order vs distance correlation (equation 1) from QM calculations ofsingle, double and triple bonds (where relevant) and parameterizing the force £eld toreproduce these ab initio results

Additional terms in the force £eld address angle, torsion, charge and van der Waalsterms, and quantum mechanical effects such as radicals, under- and over-coordination,and resonance Initial parameterization was based on the 3 primary reactions of RDX,and now we have added some secondary reactions and energies of diatomic, triatomicand other smaller species in order to provide more constraints We have done pre-liminary MD simulations of ¤yer plate experiments using 2 impact speeds, below andabove the initiation threshold At impact speed of 2km/sec no reaction is initiated andthe ¤yer plates bounce back with some conformational changes in the RDX molecules

- some of the nitro groups ¤ip between the equatorial and axial positions However, atimpact speed of 3km/sec several primary reactions are initiated at molecules at the rar-efaction edge Preliminary results in RDX are extremely encouraging in that they showthat only above threshold impact there is initiation of reactions and a large fraction ofthe product molecules observed are in fact observed in experiments There are somefragments observed that are unlikely in such an early stage and more constraints, in theform of additional DFT calculations on secondary product channels, small molecules(O3, CH2N etc.), are being added to improve the £t between ReaxFF and ab initiopotential energy surface

5.3.3 Mechanical Coupling of the Shock Wave to the Molecular ModesThe £rst event that has to happen when the shock front passes is the transfer of energyfrom this mechanical wave to the molecular vibrational modes The mechanical wave

is long wavelength (low frequency;<200 cm-1) while the £rst bonds to break [15](N-N or even C-N) are at much higher frequency (>1000 cm-1) The exact mechanism

by which the energy in the shock wave is transferred to the molecular modes is clear Dlott has proposed a vibrational up-pumping mechanism where low frequencymodes like torsion and ring breathing might couple in to the lattice mode and combine

un-in various ways to concentrate the energy un-in the bond that breaks £rst [18,31,59,123].However, no direct experimental evidence exists for this mechanism In an effort to un-

4608 atoms) and analyzed the material in slices to see what are the characteristic lengthand time scales in the energy transfer process The system is sliced into 12 regions andthe total kinetic energy (KE) is partitioned into different components (KE of the slice,

KE of molecules within each slice, KE of atoms within each molecule) progressivelyand the velocity and temperature of the individual molecules as well as atom typesare monitored What we observe is that the inter-molecular KE (relative motion of 2molecules) has a rapid rise time (200 fs) while the intra-molecular temperature (vi-brational energy) rises much more slowly Additionally the motion of the O and Catoms are the most energetic There is an important difference between DMNA andRDX/HMX - the methyl rotors are unconstrained, whereas the methylene bridges inthe ring are constrained We expect this difference to cause a different rise time for theatomic motions in the cyclic nitramines

5.4 Materials Properties for Solid Dynamics

We developed a many-body Force Field (qEAM FF) based on ab-initio quantum chanical calculations [105] to study, using Molecular Dynamics (MD), a variety ofprocesses that govern the mechanical and thermal properties of Tantalum We studiedspall failure, dislocation properties (core energy and structure, Peierls stress and poten-tial, kink propagation and formation energies, etc.), melting as a function of pressure,vacancy formation and migration energies, and thermal expansion

me-5.4.1 Spall failure

Using MD with the qEAM FF we studied the rapid expansion of Ta metal followingthe high compression (50 to 100 GPa) induced by high velocity (2 to 4 km/s) impact[106] Figure5.2shows snapshots of the process at different times for Ta with impactvelocity of2 km/s The dots represent atomic positions projected on a [100] plane.Figure5.2(a) shows the initial state of the system Figure5.2(b) shows the system at

2 ps in compression Figure5.2(c) shows time5.25 ps when the system is expanding.Figures5.2(d)-(e) show timest = 6.5 ps and t = 7.25 ps, where we can see thespall plane and void growth The projectile containsNp = 8192 atoms per periodicsimulation cell [16 × 16 × 16 unit cells, ∼ (53.253 A)ª 3] and the targetNt = 16384(32 × 16 × 16 unit cells)

We found that catastrophic failure in this system coincides with a critical behaviorcharacterized by a void distribution of the formN (V ) ∝ V−τ, withτ ∼ 2.2 Thiscorresponds to a threshold in which percolation of the voids results in tensile failure

We de£ne an order parameter (φ, the ratio of the volume of the largest void to the totalvoid volume) which changes rapidly from∼0 to ∼1 when the metal fails and scales

as φ ∝ (ρ − ρc)β with exponent β ∼ 0.4, where ρ is the total void fraction Wefound similar behavior for FCC Ni suggesting that this critical behavior is a universalcharacteristic for failure of solids in rapid expansion

Figure 5.2: Snapshots of the spall process at different times Ta withvimp= 2.5.4.2 1/2a<111> screw dislocation in Ta

Using our qEAM FF for Ta we have carried out MD simulations to investigate the corestructure, core energy and Peierls energy barrier and stress for the 1/2a<111> screwdislocation We found that the equilibrium dislocation core has three-fold symmetryand spreads out in three<112> directions on {110} planes Core energy per Burgersvector b was determined to be 1.36 eV/b We studied dislocation motion and annihi-lation via Molecular Dynamics simulations of a periodic dislocation dipole cell, with

<112> and <110> dipole orientation In both cases the dislocations move in zigzag

on primary 110 planes Atoms forming the dislocation cores are distinguished based

on their atomic energy In this way we can accurately de£ne the core energy and itsposition not only for equilibrium con£gurations but also during dislocation motion.Peierls energy barrier was computed to be∼0.07 eV/b with a Peierls stress of ∼0.03 µ,whereµ is the bulk shear modulus of perfect crystal [120] The process of double kinkformation is very important in determining the dislocation mobility We calculated the

formation energy of six single kinks and obtained values in the range from 0.10 to

also studied the process of double kink formation and migration with low temperature

MD simulation of a dislocation dipole annihilation (70 b long dislocations) We helpthe nucleation of the double kink by introducing a vacancy in the path of one dislo-cation Thus by nucleating a double kink the activation energy for dislocation motiondecreases from∼0.07 eV/b (for a straight dislocation) to 0.016 eV We also studieddislocation dipole annihilation at £nite temperatures (from 20 K to 300 K) From our

MD simulations we obtain that the hopping rate for the dislocation follows the nius law; with an activation energy for dislocation motion of0.0053 eV, this leads to

Arrhe-an estimated Peierls stress of∼ 140 MPa in good agreement with experimental results.5.4.3 Melting curve

We calculated the melting temperature of Ta using the qEAM FF as a function of sure up to 300 GPa The zero pressure isTmelt= 3150 ± 50 K in very good agreementwith the experimental values in the range3213 − 3287 K High pressure values agreewell with shock melting experiments [105]

pres-5.4.4 Vacancy formation and migration energies

Vacancy formation, migration and coalescence play an important role in understandingspall failure in metals [74] We calculated the vacancy formation energy (2.95 eV) andmigration energy (1.09 eV) in perfect bcc Ta Vacancy mobility can be enhanced bythe presence of dislocations (pipe diffusion) We calculated the energy cost to make avacancy in the core of a 1/2a<111> screw dislocation; we obtained 2.45 eV, lower thanthe bulk value Furthermore, the presence of the dislocation also lowers the migrationbarrier to 0.8 eV

5.4.5 Tantalum Thermal Equation of State

The tantalum thermal equation of state has been computed up to 400 GPa and 10,000 Kusing £rst-principles methods Analysis of the results was completed and a paper sub-mitted to Phys Rev B [20] The Linearized Augmented Plane Wave (LAPW) methodand the GGA were used to compute the static and electronic thermal contributions tothe free energy, and a mixed basis pseudopotential method was used to compute phononfree energies with the particle in a cell model, which should be a good approximation

at high temperatures We found a simple functional form to £t the free energies as afunction of volume and temperature, and with this function all of the equation of stateparameters can be obtained The free energy function has been implemented and tested

in the VTF We £nd excellent agreement with the experimental Hugoniot Results areshown in Fig.5.3

5.4.6 Tantalum Thermoelasticity

Using the same approach as above for the elastic constants, we have computed theelastic constants of Ta as a function P and T Results are shown in Fig.5.4 Theseresults have also been included in the VTF The zero temperature results were alsoused to help develop an accurate potential model for Ta that can be used with molecular

100.0 150.0 200.0 250.0 300.0

Pressure (GPa) 0

2000 4000 6000 8000 10000

Volume (au) 0.0

5.4.8 Pressure induced electronic transition in zinc

We have studied the pressure induced electronic topological transition in Zn, and foundthat there is no lattice parameter anomaly as had previously been seen in experimentsand theory Our results are in agreement with hydrostatic high pressure experiments.Our paper on this is in press in Phys Rev B [103]

We have computed the Raman frequencies of iron as functions of pressure We obtaingood agreement with experiment Most interestingly, we £nd two Raman peaks inantiferromagnetic hcp iron These two peaks are seen in experiments, indication thathcp Fe is magnetic Previously the second Raman peak was not understood.

5.5 Materials properties methodology development

In methodological developments and software integration, we have developed the

MPI-MD program, which allows parallel computations of materials with millions of atoms

on hundreds of processors We have developed an algorithm for the quantum cal eigenproblem that uses a block-tridiagonal representation of a matrix to yield moreef£cient scaling of the eigensolver We have developed a variational quantum MonteCarlo program to yield more accurate simulations of metals at high temperature andpressure Finally, we have begun work on the materials properties database, to allowarchival of QM and MD simulations, and automatic generation of the derived proper-ties required by the HE and SD efforts

mechani-The MPI-MD program is a fully parallel version of the MP team’s MPSim program.This program enables ef£cient simulation of million-atom metal MD simulations onhundreds of calculations The program achieves near-perfect scaling up to 100 proces-sors, and then experiences some degradation between 100-1000 processors The causes

of the parallelization degradation are well-understood and will soon be overcome.During this year we have also developed an improved diagonalization scheme for

QM simulations The eigenproblem is the rate-determining step for large QM tions We have examined the structure of the relevant matrices, and have determinedthat a block-tridiagonal decomposition of this matrix is appropriate for molecular sys-tems We have used this structure in conjunction with the divide-and-conquer algorithm

simula-to enable faster scaling of the eigensolver

The materials properties team is also developing a materials properties database toarchive materials properties for the HE and SD simulations of the Virtual Test Facility.This database will allow these continuum calculations to be assured the most up-to-date MP numbers at any given time Moreover, this database will allow archival andrecomputation of any of the fundamental QM and MD calculations that go into theparameters used by the VTF; this capability will enable recalculation of each parameter

as soon as new simulations data becomes available

0.0 100.0 200.0 300.0 400.0 500.0

Pressure (GPa) 0.0

tem-Compressible Turbulence

6.1 Introduction

In this chapter we describe the accomplishments to date of our research program incompressible turbulence The origin of compressible turbulence in the virtual facilitylies in the fact that under the appropriate loading conditions strong shocks can prop-agate and interact with solid targets consisting of layers of distinct materials with theresult that material strength is no longer relevant Such interactions also arise in thisfacility if the target materials are already in the ¤uid or gaseous state and consist ofdensity strati£ed layers of material with superposed perturbations An example of such

a con£guration is shown in Figure 6.1 Upon interaction with the shock, the rial interfaces are impulsively accelerated and the resulting baroclinic generation ofvorticity due to the misalignment of pressure and density gradients gives rise to thewell-known Richtmyer-Meshkov instability and ultimately produces turbulent mixingthat can contaminate or dilute the materials bordering the interfaces The modelingand simulation of these Richtmyer-Meshkov instabilities and the resulting inhomoge-neous anisotropic turbulence is a major thrust of the center’s research program Theinstability process as well as the modeling of the resulting turbulence lies at the heart

mate-of many ASCI applications An understanding mate-of compressible turbulence and mixing

is essential, for example, in important ASCI applications in which shock-driven plosion is a key step In order to develop an understanding of the relevant phenomenaand a modeling capability the compressible turbulence effort has concentrated on thefollowing objectives:

im-• Direct numerical simulation (DNS) of strong shock Richtmyer-Meshkov bility The ultimate objective here is to develop an LES model which can coexistwith shock capturing schemes and thus provide reliable turbulence modeling inthe presence of shock waves

insta-• Development of a DNS database for decaying and driven compressible lence The objective is to have a basis of comparison to verify our turbulence

turbu-82

Detonator Membrane

Figure 6.1: A caricature of the proposed facility which gives rise to compressible lence In the con£guration shown above, the materials (in the gaseous or ¤uid state) areexposed directly to the loading arising from the point detonation of the high explosive

turbu-modeling efforts The speci£c ¤ows under consideration are

– Decaying compressible turbulence

– Richtmyer-Meshkov instability initiated by weak shocks

Our accomplishments to date are itemized brie¤y below:

Pseudo DNS Simulations of 3-D Richtmyer-Meshkov instability

This work is ongoing We have to date developed a simulation capability using theWENO scheme and have performed a simulation of R-M instability with reshock LESmodeling was also included A key issue is the overall dissipative nature of the advec-tion scheme which can contaminate the small scale behavior seen by the LES model