Ebook ASCI alliance center for simulation of dynamic response in materials FY 2000 annual report Part 1

(BQ) Part 1 book ASCI alliance center for simulation of dynamic response in materials FY 2000 annual report has contents: Introduction and overview, integrated simulation capability, high explosives.

CALTECH ASCI TECHNICAL REPORT 076 caltechASCI/2000.076 ASCI Alliance Center for Simulation of Dynamic Response in Materials California Institute of Technology FY 2000 Annual Report Michael Aivazis, Bill Goddard, Dan Meiron, Michael Ortiz, James C.T Pool, Joe Shepherd, Principal Investigators ASCI Alliance Center for Simulation of Dynamic Response in Materials California Insitute of Technology FY 2000 Annual Report Michael Aivazis, Bill Goddard, Dan Meiron, Michael Ortiz, James C T Pool, Joe Shepherd Principal Investigators Contents Introduction and Overview 1.1 Introduction 1.2 Administration of the Center 1.3 Overview of the integrated simulation capability 1.4 Highlights of Research Accomplishments 1 Integrated Simulation Capability 2.1 Introduction 2.2 Algorithmic integration 2.3 Fluid dynamics algorithms 2.4 Solid Mechanics algorithms 2.5 Software integration 8 14 20 22 High Explosives 3.1 Overview of FY00 Accomplishments 3.2 Personnel 3.3 Material Properties and Chemical Reactions 3.4 Engineering Models of Explosives 3.5 Eulerian-Lagrangian Coupling Algorithms 3.6 Reduced Reaction Modeling 28 28 29 29 30 31 33 Solid Dynamics 4.1 Overview of FY 00 Accomplishments 4.2 Personnel 4.3 Nanomechanics 4.4 Mesomechanics 4.5 Macromechanics 4.6 Polymorphic Phase Transitions 4.7 Eulerian Elastic-Plastic Solver 4.8 FY 01 objectives 48 48 48 49 52 54 59 60 64 Materials Properties 5.1 Overview of FY 00 Accomplishments 5.2 Personnel 71 71 72 i CONTENTS 5.3 5.4 5.5 ii Materials properties for high explosives Materials Properties for Solid Dynamics Materials properties methodology development 73 76 80 Compressible Turbulence 6.1 Introduction 6.2 Overview of FY 00 Accomplishments 6.3 Personnel 6.4 Pseudo-DNS of Richtmyer-Meshkov instability 6.5 LES of Richtmyer-Meshkov instability 6.6 Implementation of CFD Euler solvers within GrACE 6.7 DNS of Rayleigh-Taylor instabilities 6.8 FY 01 objectives 82 82 83 84 85 87 90 94 99 Computational Science 7.1 Overview of FY 2000 Accomplishments 7.2 Personnel 7.3 Scalability 7.4 Visualization 7.5 Scalable I/O 7.6 Algorithms 101 101 101 102 104 106 108 Chapter Introduction and Overview 1.1 Introduction This annual report describes research accomplishments for FY 00 of the Center for Simulation of Dynamic Response of Materials The Center is constructing a virtual shock physics facility in which the full three dimensional response of a variety of target materials can be computed for a wide range of compressive, tensional, and shear loadings, including those produced by detonation of energetic materials The goals are to facilitate computation of a variety of experiments in which strong shock and detonation waves are made to impinge on targets consisting of various combinations of materials, compute the subsequent dynamic response of the target materials, and validate these computations against experimental data An illustration of the simulations that are to be facilitated by the Center’s Virtual Test Facility (VTF) are shown in Figure 1.1 The research is centered on the three primary stages required to conduct a virtual experiment in this facility: detonation of high explosives, interaction of shock waves with materials, and shock-induced compressible turbulence and mixing The modeling requirements are addressed through £ve integrated research initiatives which form the basis of the simulation development road map to guide the key disciplinary activities: Modeling and simulation of fundamental processes in detonation, Modeling dynamic response of solids, First principles computation of materials properties, Compressible turbulence and mixing, and Computational and computer science infrastructure CHAPTER INTRODUCTION AND OVERVIEW Figure 1.1: Illustrations of three key simulations performed using the Virtual Test Facility Top left: High velocity impact generated by the interaction of a detonation wave with a set of solid test materials Top right: High velocity impact generated by a ¤yer plate driven by a high explosive plane wave lens Bottom: con£guration used to examine compressible turbulent mixing 1.2 Administration of the Center 1.2.1 Personnel Overview The center activities are guided by £ve principal investigators: J Shepherd M Ortiz W A Goddard D Meiron J C T Pool M Aivazis High Explosives Solid Dynamics Materials Properties Compressible Turbulence Computational Science Computational Science and Software Integration In FY 00 the center personnel numbered as follows: • 16 Caltech faculty (including the center steering committee) • 12 external faculty af£liated with the center via sub-contracts • 18 Caltech graduate students • 24 research staff and postdoctoral scholars • 10 administrative staff (primarily part-time support from the Caltech Center for Advanced Computing Research (CACR)) CHAPTER INTRODUCTION AND OVERVIEW Figure 1.2: Diagrammatic representation of the VTF software architecture Detailed personnel listings are provided in the beginning of each chapter detailing the activities of each disciplinary effort within the center 1.2.2 Sub-contracts In addition to participants based at Caltech the Center has associated with it several sub-contractors who are providing additional support in a few key areas In the table below we list the contractors, their institutional af£liation and their area of research: R Phillips R Cohen G Miller R Ward C Kesselman D Reed M Parashar 1.3 1.3.1 Brown University Carnegie Institute of Washington U C Davis Univ Tennessee Univ So California, ISI Univ Illinois Rutgers University Quasi-continuum methods for plasticity High pressure equation of state of metals Multi-phase Riemann solvers Large scale eigenvalue algorithms Metacomputing, Globus Scalable I/O Parallel AMR Overview of the integrated simulation capability VTF software architecture The VTF software architecture is illustrated in Figure 1.2 The top layer is a scripting interface written in the Python scripting language which sets up all aspects of the simulation and coordinates the interaction of the simulation with the operating system and platform Also associated with the scripting environment is a materials properties database The database provides information to the solvers regarding equation of state, CHAPTER INTRODUCTION AND OVERVIEW reaction rates, etc The next layer consists of the VTF computational engines These engines are packaged as shared objects for which Python bindings are then generated At present the VTF architecture supports two such engines, a 3-D parallel Eulerian CFD solver which is used for simulations of high explosive detonation and simulations of compressible turbulent mixing, and a 3-D Lagrangian solid dynamics solver The solid dynamics solver is now fully parallel as of this writing At the next layer we have designated some of the lower level functionality of the engines For example, the CFD solver ultimately will have the ability to perform 3-D simulations using patch based parallel AMR Similarly the solid dynamics solver will ultimately also possess a capability to perform parallel adaptive meshing Finally at the lowest level are services used to facilitate various low level aspects of the simulations such as the ability to access distributed resources via meta-computing infrastructures such as Globus, and facilities for parallel communication and scalable disposition of the large date sets generated during the computation The philosophy of this software architecture is to enable a multi-pronged approach to the simulation of high velocity impact problems and the associated ¤uid-solid interaction For example it is well known that such simulations can be performed using a purely Lagrangian approach, a purely Eulerian approach, or some mixture of the two (such as ALE) The objective of the VTF architecture is to provide a ¤exible environment in which such simulations can be performed and the results of the differing approaches can be assessed As of the end of FY00 we have completed a fully three dimensional coupled simulation of a detonation interacting with a Tantalum target The simulation was run on all three ASCI platforms The present simulation utilizes a parallel ¤uid mechanics solver and a fully parallel solid mechanics solver In addition a full implementation of the Python based problem solving environment has been completed Details of the implementation can be found in Chapter 1.4 1.4.1 Highlights of Research Accomplishments High Explosives Material properties and chemical reactions The detailed reaction mechanism and rate constants were completed for HMX (C4 H8 N8 O8 , cyclotetramethylene-tetranitramine) and RDX (C3 H6 N6 O6 , cyclotrimethylenetrinitramine) gas phase decomposition This was a continuation of work begun in previous years Quantum mechanical computations were used to compute potential energy surfaces and thermal rate constants Molecular dynamics was used to examine the transfer of mechanical to thermal energy immediately behind a shock wave A new method of implementing reactive force £elds was developed and applied to RDX reactions initiated by a shock front in a crystal lattice at £nite temperature The Intrinsic Low Dimensional Manifold (ILDM) method was used to compute a reduced reaction mechanism for hydrogen-oxygen-nitrogen combustion The ILDM was implemented in a two-dimensional, Adaptive Mesh Re£nement (AMR) solution for a propagating detonation CHAPTER INTRODUCTION AND OVERVIEW Engineering models of explosives Improved models of the equation of state for a Plastic-Bonded Explosive (PBX) were developed and implemented in the Virtual Test Facility (VTF) The structure of the Zel’dovich-von Neumann Do¨ring (ZND) solution for the Johnson-Tang-Forest (JTF) model of PBX detonation was computed Eulerian-Lagrangian Coupling Algorithms The ghost ¤uid Eulerian-Lagrangian (GEL) coupling algorithm was implemented using the Grid Adaptive Computational Engine (GrACE) library and used for parallel AMR simulations of shock and detonation wave propagation in yielding con£nement simulated with a linear elastic £nite-element model A two-dimensional test problem with an exact numerical solution was developed and used to evaluate GEL schemes 1.4.2 Solid Dynamics Solid mechanics engine Accomplishments during FY 00 include the serial implementation of adaptive mesh re£nement (subdivision) and coarsening (edge collapse); and the fully parallel implementation of the solid dynamics engine within the VTF3D (without mesh adaption) Nanoscale At the nanoscale we have continued to carry out quasi-continuum simulations of nanoindentation in gold; and mixed continuum/atomistic studies of anisotropic dislocation line energies and vacancy diffusivities in stressed lattices Microscale At the microscale we have developed a phase £eld model of crystallographic slip and the forest hardening mechanism; we have re£ned our mesoscopic model of Ta by investigating the strengths of jogs resulting from dislocation intersections, the dynamics of dislocation-pair annihilation, and by importing a variety of fundamental constants computed the MP group Macroscale At the macroscale we have focused on various enhancements of our engineering material models including the implementation and veri£cation of a Lagrangian arti£cial viscosity scheme for shock capturing in the presence of £nite deformations and strength; the implementation of an equation of state and elastic moduli for Ta computed from £rst principles by Ron Cohen (MP group); and the implementation of the Steinberg-Guinan model for the pressure dependence of strength 1.4.3 Materials Properties High Explosives In simulations supporting high explosives, the MP team has completed the decomposition mechanism of RDX and HMX molecules using density functional theory, obtained a uni£ed decomposition scheme for key energetic materials, obtained a detailed reaction network of 450 reactions describing nitramines, developed ReaxFF, a £rst-principles based bond-order dependent reactive force£eld for nitramines, and pursued MD simulations of nitramines under shock loading conditions CHAPTER INTRODUCTION AND OVERVIEW Solid dynamics In simulations supporting solid dynamics, the MP team has developed a £rst-principles qEAM 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 simulated vacancy formation and migration energies, related to vacancy aggregation and spall failure We have run high-velocity impact MD simulations to investigate spall failure in materials We have simulated a thermal equation of state for Ta from density functional theory calculations, and have simulated the elasticity of Ta versus P to 400 GPa, and T to 10000 K Finally, we have begun work on Fe by examining the hcp phases of Fe Methodology 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 mechanical eigenproblem that uses a block-tridiagonal representation of a matrix to yield more ef£cient scaling of the eigensolver We have developed a variational quantum Monte Carlo program to yield more accurate simulations of metals at high temperature and pressure Materials Properties Database Finally, we have begun work on the materials properties database, to allow archival of QM and MD simulations, and automatic generation of the derived properties required by the HE and SD efforts 1.4.4 Compressible Turbulence Pseudo DNS Simulations of 3-D R-M instability This work is ongoing We have to date developed a simulation capability using the WENO scheme and have performed a simulation of R-M instability with reshock LES modeling was also included A key issue is the overall dissipative nature of the advection scheme which can contaminate the small scale behavior seen by the LES model Sub-grid modeling for LES of compressible turbulence We have to date implemented the LES model of Pullin along with the use of high order advection schemes such as WENO At present no further development of the sub-grid model has been contemplated since the main issues that need to be overcome is the proper interplay of high order advection schemes with turbulence modeling Development of 3-D AMR solver This work is ongoing We have successfully developed a 3-D solver for compressible ¤ow utilizing adaptive mesh re£nement under the GrACE computational framework We have begun the investigation of Richtmyer-Meshkov instability with reshock using the AMR capability High resolution 3-D DNS of R-M and R-T flows We have developed and examined two parallel codes, one a fully compressible multi-species DNS code with full physical viscosity utilizing Pad´e-base methods and the other a high order CHAPTER HIGH EXPLOSIVES 33 Figure 3.2: Detail of a failing detonation in a very deformable casing, showing isocontours of density for the solid explosive HMX (bottom part) and isocontours of the xcomponent of the stress tensor for the casing (upper part) The line across the reaction zone indicates the one-half mass fraction value in the explosive boundary from these simulations converge to the values predicted by the theory as the Eulerian and Lagrangian computational meshes are re£ned An example of these results is shown in Fig 3.4, where the loading shock is moving from left to right The dilatational, or p- wave, and the distortional, or s- wave, are easily identi£able in the Lagrangian domain Isocontours on the left hand side of the plot are an indication of a previous transient, or start-up solution, and of the £nite thickness of the solid 3.6 Reduced Reaction Modeling Reduced reaction modeling was used to simulate a propagating detonation in a hydrogenoxygen-argon mixture This work is an extension of studies carried out in previous years The prior work includes the simulation of one-dimensional unsteady detonations with detailed chemistry in the hydrogen-oxygen-argon system and with reduced chemistry computed using the Quasi-Steady State Analysis (QSSA) method The work carried out in FY00 was based on an alternative reduction method, Intrinsic Low Dimensional Manifold or simply ILDM, that was originally developed for use in lowspeed ¤ame simulations The ILDM method was used to compute a four-dimensional manifold for a hydrogenoxygen-argon mixture and the approximate solution of species evolution on the mani- CHAPTER HIGH EXPLOSIVES 34 Figure 3.3: Mesh evolution for the failing deformation (frames #5, 10, 20, 29): the green pro£le indicates the Lagrangian boundary 0.2 s-front p-front interface 0.1 y -0.1 shock -0.2 x 0.2 0.4 Figure 3.4: Detail of the superseismic shock at the ¤uid-solid interface CHAPTER HIGH EXPLOSIVES 35 fold was incorporated as the chemical reaction portion of a two-dimensional adaptive mesh re£nement solution of the Euler equations The results were compared to a solution available in the literature which used exactly the same conditions but a uniform grid and detailed chemistry The mathematics of the ILDM method were explored using exactly solvable systems and methods from dynamical systems theory 3.6.1 Intrinsic Low-Dimensional Manifolds A complete computational code had to be developed and implemented to compute the ILDM since there are no existing publically available codes The basic methodology is essentially the same as that of [67] The details are described in [34] and only a brief overview is given here The ILDM method is based on dynamical systems theory From numerical simulations of chemically reacting systems with detailed chemistry, the empirical observation can be made that these systems are rapidly attracted to low-dimensional manifolds in the chemical state space Fast chemical processes relax towards the manifold and slow processes represent movements tangential to the manifold If the equilibration of the fast processes and subsequent collapse onto the low-dimensional manifold occurs faster than the shortest timescale of interest in the ¤ow, then the chemical system can be approximated as lying only on the manifold This greatly reduces the number of degrees of freedom of the reactive system The location of the low-dimensional manifold is computed as follows The low-dimensional manifold is de£ned by the points in the state space for which the rate vector is perpendicular to the eigenvectors associated with the fastest relaxing timescales (most negative eigenvalues) The number of chemical degrees of freedom in the reduced mechanism is the user-prescribed dimension of the manifold, that is, the number of degrees of freedom desired in the reduced system This is the one parameter that must be provided by the user An appropriate choice is not always simple, but can be made by comparing the eigenvalues of the Jacobian at various states in a model problem with the timescales of the ¤ow in the £nal application In practice, the eigenvectors are often ill-conditioned, with several being almost degenerate To avoid the numerical dif£culties associated with near degenerate eigenvectors, we use an alternative basis given by the real Schur vectors of the Jacobian, as suggested by [67] The manifold equations were solved using one of two methods The £rst method utilized a one-dimensional arc-length continuation code Given a solution point on a one-dimensional manifold, the arc-length continuation package predicts the location of a neighboring solution point by approximating the local slope of the manifold, then uses Newton’s method to correct the point When a multi-dimensional ILDM was required, repeated calls were made with different parameterization equations successively omitted so that the continuation process advanced in different axis directions of the phase space The second method was used on the rare occasions when the approximate arc-length continuation process failed It is based on a pseudo time-stepping method proposed by [66] In this way, a Cartesian grid of solution points in the space was gradually mapped out The entire solution vector and required derivatives were stored in an indexed data CHAPTER HIGH EXPLOSIVES 36 structure For applications to computing detonations, the manifold was parameterized by selected compostion variables, energy, and density The solution of low-dimensional manifolds is also complicated by the fact that there are regions where (a) the solution is unphysical (such as negative quantities of some species), or (b) no manifold solution exists In general, neither of these regions is known a priori So the solution technique must identify the boundaries of these regions as it proceeds In addition, there will be user-prescribed domain boundaries, given some knowledge of what parts of the state space are likely to be visited in a practical application The £nal boundary of the computed manifold will be a combination of these physical, intrinsic, and user-prescribed boundaries 3.6.2 ILDM for Hydrogen-Oxygen-Diluent Mixtures The ILDM code was validated by computing one and two-dimensional manifolds of a CO–H2 –air system studied by [67] in their original work Following that, the constantvolume reaction of mixtures of hydrogen and air were examined The example selected for the constant-volume validation study was stoichiometric H2 –air at a density ρ = 4.58 kg/m3 and an internal energy e = 1.28 MJ/kg These thermodynamic conditions approximately correspond to the von Neumann state of a CJ detonation in the mixture, so the induction region will be similar to that in the £rst ZND validation study of the earlier QSSA model As for the QSSA model, the starting detailed reaction mechanism was from [65] A representative portion of the constant-volume ILDM for this mixture is shown in Fig 3.5 One of the dependent species, H, is shown as a function of the two parameterizing species, H2 O and H2 Similar plots could be made to visualize how each of the species depends on the parameterizing species Also plotted is the one-dimensional manifold (curve in three space) and six sample reaction trajectories computed from solving the full system of equations The one-dimensional manifold was computed separately by assuming that there was a one-species parameter representation of the ILDM at this particular mass and energy density Note that the one-species ILDM curve lies on the two-species ILDM surface In other words, the lower-dimensional ILDMs for a system are subspaces of the higher-dimensional ILDMs None of the trajectories collapse onto the one-dimensional ILDM until very late times, near equilibrium So clearly, a one-dimensional ILDM is insuf£cient to describe the chemistry in this system Three of the trajectories also take considerable time to collapse onto the two-dimensional ILDM and would probably require a higher number of dimensions for the trajectories to be well represented by projections onto an ILDM However, these three are all unusual contrived examples that start with a large amount of hydrogen radicals Practical initial compositions are more likely to contain only major species The diagonal ILDM boundary to the lower left of the £gure corresponds to the physical boundary of all hydrogen atoms being in the major species H and H2 O, so practical initial compositions will all lie on this line The three realistic trajectories that start from this line lie almost exactly on the manifold for all times, even at the start, so the two-species dimension ILDM appears to be suf£cient to describe the entire reaction for these trajectories The example selected for the constant-volume validation study at the start of this section is one of these trajectories, the one starting from the CHAPTER HIGH EXPLOSIVES 37 φH2O (mol/kg ) φH (mol/kg) 15 kg φ H 10 (m ol/ ) 15 Figure 3.5: Adiabatic constant volume reaction in mixtures with the same elemental composition as stoichiometric H2 –air, and with ρ = 4.58 kg/m3 , e = 1.28 MJ/kg Surface: two-dimensional slice of ILDM; —— one-dimensional ILDM; – – – sample reaction trajectories; · · · · · · vertical projection of reaction trajectories onto twodimensional ILDM; • equilibrium point lower right of the £gure Thus, we decided an ILDM that was two-dimensional in the number of species would be suf£cient in this validation study For a full detonation simulation, the total dimension of the manifold will be four since energy and mass density have to always be considered as independent parameters in the general case To show the domain of grid points found by the ILDM code, Fig 3.5 is repeated in Fig 3.6, this time viewed directly down on the two-dimensional plane formed by the parameterizing species The grid spacing chosen in this case was 0.2 mol/kg in each parameter direction Some of the intrinsic manifold boundaries are evident As in the previous £gure, the one-dimensional ILDM is also shown for comparison Only one of the sample reaction trajectories is shown, the one used in the constant-volume validation study Two signi£cant issues were found in the course of this study First, the reaction rates need to be interpolated carefully – particularly near the boundaries of the ILDM This was solved by extrapolating the ILDM table outside the boundaries and performing the interpolation using the logarithms of the reaction rates rather than the rates CHAPTER HIGH EXPLOSIVES 38 φ H2 (mol/kg) 15 10 0 φ H2O (mol/kg) 10 15 Figure 3.6: Adiabatic constant-volume reaction in mixtures with the same elemental composition as stoichiometric H2 –air, and with ρ = 4.58 kg/m3 , e = 1.28 MJ/kg Points: two-dimensional ILDM grid; ——one-dimensional ILDM; – – –sample reaction trajectory; • equilibrium point thermselves Second, the ILDM has to be supplemented at early times with some representation of the rapid process (chemical induction) that occurs just behind the shock Once the induction process has progressed far enough, the state can be projected onto the ILDM and integration in the reduced system commenced This was solved by either integrating the full rate equation system (detailed kinetics) in the case of zero-dimensional problems or else tabulating a supplemental induction manifold in the case of one- or two-dimensional simulations The switch from induction to the ILDM is determined by a threshold level in the amount of water The computation of the induction manifold is simpli£ed by observing that the processes in the induction region are approximately thermally neutral and quasi-steady Examination of simulation results with detailed chemistry in the induction region shows a strong correlation between all of the chemical species, indicating that the chemical composition lies roughly on a three-dimensional manifold, one species variable, density, and internal energy Constant-volume reaction simulations were used to determine the induction manifold The results of the constant-volume validation study are shown in Fig 3.7, which compares the reaction zone pro£les computed with detailed chemistry, the two-dimensional ILDM reduced mechanism, and a three-step QSSA reduced mechanism 3.6.3 Simulation of Detonation Propagation Simulations of detonation propagation in hydrogen-oxygen mixtures have been carried out in one and two space dimensions Computations using the ILDM method were CHAPTER HIGH EXPLOSIVES 39 60 3500 (a) (b) 55 3000 45 T (K) P (atm) 50 40 35 2500 2000 30 25 0.0 0.5 t ( µ s) 1500 0.0 1.0 0.25 0.5 t ( µ s) 1.0 0.5 t ( µ s) 1.0 0.004 (c) (d) 0.20 0.003 yH yH2O 0.15 0.002 0.10 0.001 0.05 0.00 0.0 0.5 t ( µ s) 1.0 0.000 0.0 Figure 3.7: Adiabatic constant-volume reaction in stoichiometric H2 –air, with ρ = 4.58 kg/m3 and e = 1.28 MJ/kg Solid lines: detailed reaction mechanism; Dashed lines: two-dimensional ILDM reduced mechanism, with φcutoff = 0.05 mol/kg on H2 O; Dotted lines: three-step QSSA reduced mechanism (a) Pressure; (b) Temperature; (c) Mass fraction of H2 O; (d) Mass fraction of H compared with results obtained with both detailed and QSSA reduced chemistry All the ILDMs used in this work for hydrogen systems had a reduced chemical dimension of two and were paremeterized with θ1 = φH2 O and θ2 = φH2 , the speci£c mole fractions of water and hydrogen Adding density and energy as parameters, the lookup tables were four-dimensional Comparisons with ZND solutions were used to validate the results for stable, overdriven detonations while instability thresholds and periods were used in the cases of unstable but still overdriven detonations Mesh convergence and timing studies were carried out 3.6.4 One-Dimensional Unsteady Simulations A series of simulations were carried out with detailed chemical kinetics, three-step QSSA, and the ILDM method on overdriven stoichiometric hydrogen-oxygen detonations Three overdrives were considered, f = 1.4 (stable), f = 1.3 (unstable, one mode), and f = 1.2 (unstable, two modes) The results for the latter case are shown in Fig 3.8 Comparison of the QSSA and ILDM computations with detailed chemistry indicates that the approximate methods introduce a slight shift in the neutral stability boundary location (the overdrive at which oscillations set in) In the unstable cases, CHAPTER HIGH EXPLOSIVES 40 Ps (bar) 60 50 40 30 t ( µ s) 10 (a) Ps (bar) 80 70 60 50 40 30 t ( µ s) 10 15 (b) Ps (bar) 60 50 40 30 t ( µ s) (c) Figure 3.8: Shock pressure versus time, for a one-dimensional detonation in stoichiometric H2 –O2 , initially at atm and 300 K, with f = 1.2 Finest grid level contains 200 cells per ZND induction length (a) Detailed chemical kinetics, (b) Reduced kinetics, 3-step QSSA (c) Reduced kinetics, two-species ILDM comparison of amplitudes and periods of the fully saturated instability cycle indicate reasonably good agreement between all three methods with slight quantitative differences These comparisons were carried out in non-dimensional coordinates in order to account for the shift in the ZND reaction length introduced by the approximation method The necessity of scaling can be observed in the substantially longer period of the QSSA solution in Fig 3.8 as compared to either detailed chemistry or the ILDM solution For a grid convergence study of the nonlinear instability, the f = 1.3 detonation was examined for the detailed and QSSA cases but it is not suitable for the ILDM case since it was stable Instead, we decreased the overdrive factor to f = 1.27 and the detonation developed one unstable mode similar to the f = 1.3 detailed chemistry detonation The plots of oscillation period and pressure turning points both suggest that the ILDM reduced mechanism simulation is grid converged for as few as 75 £ne mesh cells per induction length This represents a substantial improvement over the detailed CHAPTER HIGH EXPLOSIVES 41 mechanism where 150 cells were required Comparison of relative CPU times for the various methods clearly demonstrates the advantages of the ILDM method For the detailed mechanism, the great majority of CPU time is spent integrating the chemical reaction equations The ILDM method reduces the chemistry time by about a factor of 27 and the total time by about a factor of 15 As a result, the computation is only about one order of magnitude slower than the one-step Arrhenius model, and many detonation simulations previously run with only a one-step model or two-step induction parameter model will now be viable with ILDM reduced mechanisms 3.6.5 Two-dimensional Unsteady Simulations The ILDM method was applied to the case of two-dimensional, unsteady detonation propagation The problem chosen was that studied by Oran et al [76] Using detailed chemistry, they simulated an unsupported (not overdriven) cellular detonation in a mixture of stoichiometric H2 –O2 with 70% Ar dilution, initially at 6.67 kPa and 298 K Their computation equivalent to that presented here was performed on a £xed, uniform grid and required about one day of CPU time on a 256-node CM-5 parallel Connection Machine [121] The large amount of work was justi£ed in providing a benchmark for validation of detonation simulations with simpler thermochemical models By contrast, the computation presented here with an ILDM reduced mechanism ran in £ve days on a single processor 750MHz Linux workstation The decrease in computational effort was a result of using adaptive mesh re£nement and reduced chemistry A CFL number of 0.3 was used in both Oran et al’s [76] simulation and our work here The detailed reaction mechanism used to generate the ILDM reduced mechanism was the same 8– species, 24–reversible reaction mechanism for H2 –O2 combustion [14] that was used in the benchmark study [76] As in the original detailed mechanism study, our computational domain was a channel of width 6.016 cm The £nest grid size was 0.015 cm in the x-direction and 0.0235 cm in the y-direction, matching the resolution of the benchmark [76] calculation This grid size corresponded to 256 mesh cells in the transverse direction across the channel and, with a ZND induction length of about 0.15 cm, 10 mesh cells per ZND induction length in the streamwise direction By repeating the simulation at £ner resolution, Oran et al [76] showed the benchmark resolution to be suf£cient for a grid converged solution in this mixture Two levels of mesh re£nement were used in the present work, each with a re£nement ratio of four, so there were 16 mesh cells across the channel in the coarsest grid Re£nement was performed on the gradient of density to £nely capture the leading and transverse shock waves as well as the triple point slip lines In addition, re£nement was performed on the gradient of H O mole number to resolve the reaction zone The gradients were examined in both the x- and y-axis directions To avoid the large amount of work required to resolve the trailing vortices behind the detonation which not affect the detonation front, re£nement was not performed more than 10 cm behind the front The top and bottom boundary conditions were re¤ecting walls with inviscid slip conditions The upstream and downstream boundary conditions were linearly extrapolated in¤ow and out¤ow respectively By not overdriving the detonation with a rear CHAPTER HIGH EXPLOSIVES 42 piston condition, we simulate a self-propagating detonation that would theoretically travel at the CJ velocity if it were hydrodynamically stable The initial condition for the simulation was the detailed chemistry ZND pro£le for a CJ detonation interpolated onto the computational grid Although the transverse instability would eventually grow from the numerical roundoff error, this growth may require computation for a long propagation distance To accelerate the growth, the transverse instability was triggered by an applied perturbation behind the initial detonation front This perturbation consisted of a stationary pocket of ¤uid with the same composition as the unreacted freestream mixture, and with temperature and pressure equal to seven times that of the freestream The pocket had an axial dimension of 1.05 cm and a transverse dimension of 1.41 cm, starting 0.3 cm behind the initial shock location and centered in the channel The chosen perturbation was identical to that used by Oran et al [76] Figure 3.9 shows various ¤ow£eld properties at a late time, t = 422.3 µs The cellular structure is very regular, as expected for a heavily Ar diluted mixture [37] The measured cell size is 3.0 cm width and 5.5 cm length, with variations of only ±0.1 cm for different cells The computational results of the two-dimensional cellular detonation with an ILDM reduced mechanism can be compared directly with the previous results of [76] using detailed chemistry In their work, the £nal con£guration reached was also a mode four detonation, although it brie¤y transitioned to a mode six detonation before settling into the mode four con£guration about 280 µs into the computation The cell size was 3.0 cm in width and 5.5 cm in length which agrees perfectly with the present result One minor difference was that Oran et al [76] observed weak vertical striations in the ¤ow that they attributed to longitudinal instability However, our one-dimensional detonation simulations with this chemical system were hydrodynamically stable, even when computed with full detailed chemistry This suggests a slight difference in the respective ¤ow solvers and is not related to the use of reduced chemistry A more signi£cant difference was the angle each transverse wave makes with the channel centerline In Oran et al’s [76] calculation, the angle was considerably smaller The intersection point of two transverse waves at the instant that they extended from adjacent triple point collisions was 5.2 cm behind the leading shock in Fig 3.9(b), while it was 11.4 cm in the equivalent pressure contour plot of Oran et al [76] A quantitative comparison can be made between the centerline velocities of the leading shock The period and shape of the £nal periodic pro£les agree very well, but the range between the maximum and minimum velocities was slightly greater in Oran et al’s [76] detailed chemistry calculation Their velocity plot contained considerable noise, but the range in U/UCJ was about 0.88 to 1.36, compared with 0.93 to 1.26 in our calculation This difference is consistent with the observation made in the onedimensional validation studies that the ILDM reduced mechanism detonation is slightly more stable than a detonation computed with the parent detailed mechanism 3.6.6 Summary Reduced reaction mechanisms have been demonstrated as a viable option for gaseous detonation simulations when more accuracy is desired than an empirical one- or two- CHAPTER HIGH EXPLOSIVES 43 (a) Numerical schlieren–type image (b) Pressure (c) Streamwise velocity Figure 3.9: Flow£eld at t = 422.3 µs, for a two-dimensional CJ detonation in 2H + O2 + 7Ar, initially at 6.67 kPa and 298 K CHAPTER HIGH EXPLOSIVES 44 step reaction model The simple technique of QSSA was used to develop a three-step reduced mechanism for H2 –O2 –diluent systems suitable for detonation simulations across a wide range of conditions and mixtures The mechanism was found to predict ZND induction lengths to within a factor of two, and give reasonable agreement with detailed chemistry in one-dimensional unsteady detonation simulations However, due to the signi£cant quantitative errors as well as a number of implementation dif£culties when trying to improve the computational ef£ciency of the model or apply it to a large chemical system, we decided to pursue a more advanced reduction technique The ILDM method was discussed and a code for computing manifolds of arbitrary dimension was developed The code was veri£ed against published homogeneous combustion results and then used to compute four-dimensional manifolds for H –O2 – diluent detonation systems Implementation of the method into ignition-type applications was demonstrated to be feasible with the use of a separate induction manifold to represent the chemical reactions in the induction zone where the system had not yet collapsed onto the low-dimensional manifold This method permitted accurate reproduction of the induction time while still maintaining the excellent computational ef£ciency of the scheme Without the use of an induction manifold, a much higher dimension ILDM would have been necessary to capture the induction region in the H2 –O2 –diluent examples The ILDM reduced mechanism coupled with the induction manifold was found to reproduce detailed chemistry constant-volume combustion and steady ZND detonation almost exactly, a major improvement on the three-step QSSA mechanism It was also found to give reasonably good agreement with detailed chemistry in unsteady one-dimensional detonation simulations, although, as for the QSSA method, the neutral stability limit was slightly shifted Finally, the ILDM reduced mechanism was used to simulate a two dimensional cellular detonation in 70% Ar diluted stoichiometric H2 –O2 The agreement with previously published detailed chemistry results for this mixture was excellent, showing signi£cant improvements on earlier induction parameter models The predicted mode number of the detonation was the same as in the detailed chemistry simulation The detailed chemistry and ILDM reduced chemistry results were consistent with experimental data, verifying the ability to accurately simulate gaseous detonations with an inviscid Euler code and a suf£ciently advanced chemical model, at least in the case of regular detonations and the absence of signi£cant wall losses Numerical simulations of a one-dimensional unsteady H2 –O2 detonation on a £xed mesh took about four times less CPU time with ILDM reduced chemistry than with the QSSA reduced mechanism and 15 times less than with detailed chemistry The improvement over detailed chemistry would be even greater when noting that a coarser mesh could be used for a grid converged solution with the reduced mechanism The ILDM simulations were about an order of magnitude more expensive than a simple one-step Arrhenius model and presumably a few times slower than a two-step perfect gas induction parameter model, but the greatly improved accuracy warrants this extra expense in many situations The true value of ILDM reduced mechanisms will come in simulations of larger chemical systems such as hydrocarbons or nitramines For these systems, detailed chemistry simulations of multi-dimensional detonations are completely infeasible, but with only a few extra manifold dimensions, ILDM reduced chemistry could be applied to these systems at an expense not too much greater CHAPTER HIGH EXPLOSIVES 45 0.7 0.6 0.4 0.3 H O mole fractions 0.5 0.2 0.1 0 0.1 0.2 0.3 0.4 H mole fractions 0.5 0.6 0.7 Figure 3.10: Sample reaction trajectories of a chemical system projected onto the H2 − H2 O plane for a constant volume explosion Equilibrium point is shown by the asterisk than the hydrogen simulations presented here 3.6.7 ILDM method and Invariant Manifolds The original development of the ILDM method was motivated by the observation that reaction trajectories collapse into lower-dimensional subspace of the original phase spaces long before equilibrium is reached For example, several numerically integrated reaction trajectories of a Hydrogen-Air combustion system are shown in Fig 3.10 The method was £rst shown to be viable for detonation systems in [34] with the use of auxiliary ”Induction Manifolds” to capture the initial fast transients Theoretical justi£cation of the algorithm was put forth by [93] but in their work, they applied the ILDM algorithm to dynamical systems that are written in singular perturbation form The ILDM algorithm identi£es the low-dimensional manifold as the zero level-set of a function f (x) = 0, where f : Rm → Rn The inverse function theorem (see for example [46]) gives the condition on f that determines whether the level-set is a manifold It was proven in [93] that the set identi£ed by the ILDM algorithm is an invariant manifold (see, for example, [45]) but a counter example is easy to £nd Early results indicate that the invariant manifold identi£ed is ”close” to the invariant manifold and the distance is a function of the spectral gap of the system Consider the following system from [124] This system has an unstable nonhyperbolic £xed point at the origin and an asymptotically stable £xed point at (1, 1) Sample orbits (solid lines) and numerically computed ILDM (heavy dashed line) are shown in Fig 3.11 x˙ = x2 y − x5 y˙ = −y + x2 (3.1) CHAPTER HIGH EXPLOSIVES 46 1.5 0.5 −0.5 −1 −1.5 −2 −1.5 −1 −0.5 0.5 1.5 Figure 3.11: Sample orbits of dynamical system given by Eq (3.1) It appears in Fig 3.11 that the numerically computed ILDM (dashed line) identi£ed the correct lower-dimensional invariant subspace of the system shown in Eq (3.1) If the ILDM is truly an invariant subspace, then the ILDM (written as the graph (x, φ(x)) satis£es Eq (3.2) y(x, ˙ φ(x)) dφ − =0 x(x, ˙ φ(x)) dx (3.2) This inspires the error measure for two dimensional systems given in Eq 3.3 Since the invariant set of the system is not known a priori, the error measure should not depend on it In Eq (3.3), the error of the ILDM at a point x is zero when it satis£es the original dynamical system A plot of the error as a function of x is shown in Fig 3.12 It can be seen that the errors are at the two equilibrium points x = 0, and non-zero everywhere else It is observed that the spectral gap determines how fast these errors grow away from equilibrium points; this work is still in progress φ = y(x, ˙ φ(x)) dφ − x(x, ˙ φ(x)) dx (3.3) CHAPTER HIGH EXPLOSIVES 0.5 x 10 47 −3 −0.5 −1 Error −1.5 −2 −2.5 −3 −3.5 −4 −4.5 −0.2 0.2 0.4 x 0.6 0.8 Figure 3.12: Error from the ILDM identi£cation ... 10 1 10 1 10 1 10 2 10 4 10 6 10 8 Chapter Introduction and Overview 1. 1 Introduction This annual report describes research accomplishments for FY 00 of. . .ASCI Alliance Center for Simulation of Dynamic Response in Materials California Insitute of Technology FY 2000 Annual Report Michael Aivazis, Bill Goddard,... Principal Investigators Contents Introduction and Overview 1. 1 Introduction 1. 2 Administration of the Center 1. 3 Overview of the integrated simulation capability 1. 4