(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.
Chapter Solid Dynamics 4.1 Overview of FY 00 Accomplishments 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) As in previous years, we have structured our material modeling efforts in terms of lengthscale At the nanoscale we have continued to carry out quasicontinuum simulations of nanoindentation in gold; and mixed continuum/atomistic studies of anisotropic dislocation line energies and vacancy diffusivities in stressed lattices 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 dislocationpair annihilation, and by importing a variety of fundamental constants computed the MP group 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 4.2 Personnel • Faculty: – Thomas J Ahrens – Alberto Cuiti˜no (Rutgers University) – Michael Ortiz – Robert Phillips 48 CHAPTER SOLID DYNAMICS 49 – Deborah Sulsky (University of New Mexico) – Pierre Suquet (University of Marseille) • Research fellows: – Jarek Knap – Raul Radovitzky • Post-doctoral fellows: – Sylvie Aubry – David Olmsted (Brown University) • Graduate students: – Matt Fago – Marisol Koslowski – Adrian Lew – Jean-Francois Molinari 4.3 Nanomechanics Nanoindentation provides an effective experimental and conceptual device for elucidating the plastic behavior of materials at the nanoscale [128, 39, 73, 44] One of the critical questions that arise in this setting concerns the conditions attendant to dislocation nucleation Upon indentation, and after a preliminary elastic stage, the onset of permanent deformation is mediated by the nucleation and propagation of dislocations The dislocation nucleation event and the early stages of growth of the nascent dislocation loops are amenable to effective atomistic simulation [5, 58] However, in this type of analyses the indentor sizes which may be considered are often considerably smaller than experimentally employed values, which may in turn cause premature dislocation nucleation relative to observation Likewise, the size of the computational domain is necessarily limited and the dislocations soon run up against arti£cial boundaries In addition, within a strict atomistic simulation it is dif£cult to account for the effect of long-range elastic stresses such as might be present, e g., in a thin £lm/substrate system These limitations of straight atomistic simulation may be overcome by recourse to the theory of the quasicontinuum [110, 111, 102, 109, 60] A full three dimensional quasicontinuum analysis of the early stages of nanoindentation in gold thin £lms has been carried out by Knap and Ortiz The surface of the £lm is a (001)-plane and the material 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 indentor 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 of the £lm The number of representative atoms in the initial mesh is 1, 853, or an CHAPTER SOLID DYNAMICS 50 Z X (a) Y (b) 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 is 2.4 × 10 11 ª containing 90, 272 (Knap and Ortiz, 2000) (a) Detail of computational mesh at 5.0 A representative atoms (initial mesh contains only 1, 853) (b) View of the dislocation pattern, 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 × 10 11 ) in the sample We may note in passing that hundred-billion atom samples are well outside the scope of straight atomistic methods at present The computational mesh for an indentor radius of 70 nm at an indentation depth ª is shown in Fig 4.1 As may be seen from the £gure, the displacementof 5.0 A variation adaption criterion causes the mesh to be re£ned under the indentor, with the result that the zone of full atomistic resolution grows steadily, as required The mesh contains 90, 272 representative atoms This problem size is still modest compared to that which is demanded by straight atomistics However, the adaptive character of the method ensures that a suf£ciently large fully-resolved atomistic region lies beneath the indentor at all times for dislocations to nucleate and grow into The dislocation ª indentation patterns predicted by the analysis for an indentor radius of 70 nm at 5.0 A are 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 dislocation loops propagate on selected {111} planes Away from the indentor the behavior of the crystal is ostensibly linear elastic, and captures the long-range elastic £eld of the indentor It should be carefully noted that, even in this region, all material behavior, e g., the effective anisotropic elasticities of the crystal, emanates directly from the CHAPTER SOLID DYNAMICS 51 (a) (b) Figure 4.2: Quasicontinuum calculations of vacancy migration in Al (Olmsted and Phillips, 2000) Top: Dislocation core structure for four mobile (111) dislocations in Al using EAM potential Bottom: slip distribution across cores (red); £tted partial dislocations (solid); sum of £tted partials (dashed) Bottom: Activation enthalpies for vacancy motion in two directions as a function of uniaxial stress in 001 direction CHAPTER SOLID DYNAMICS 52 interatomic potential, and the transition from fully-resolved atomistics to continuum behavior is entirely seamless Another mechanism which can be elucidated by direct atomistic modeling concerns vacancy migration through a stressed lattice in the presence of dislocations Olmsted and Phillips (2000) have calculated vacancy migration enthalpies in Al using EAM potentials 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, although no extended region of stacking fault appears between the partials The partials themselves are suf£ciently spread out as to overlap The basic Shockley partial structure is clear in the slip distributions, where the different partials have different widths The screw partial in the 30◦ dislocation has the smallest width, whereas the edge partial in the 60◦ dislocation is the widest Olmsted and Phillips have found that both the core and the line-energy anisotropy are accurately predicted by linear elasticity Fig 4.2b collects activation enthalpies for two vacancy migration directions as a function of an applied uniaxial stress The objective of these calculations is to ascertain the effect of stress 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 the agents of plastic deformation in crystals, is impeded by secondary –or ‘forest’– dislocations crossing the slip plane As the moving and forest dislocations intersect, they may 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 dislocation patterns which develop during this process, the intricate interactions between dislocations and obstacles, and the resulting kinetics, are amenable to an ef£cient phase-£eld representation In essence, the value of the phase £eld at a point on a slip plane counts the 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 at integral values An example of the dislocation pattern evolution predicted by the theory under cyclic single slip, and the resulting stress-strain and dislocation density curves are shown in Figs 4.3 The phase-£eld representation enables the tracking of complex geometrical and topological transitions in the dislocation ensemble, including dislocation loop nucleation, bow-out, pinching, and the formation of Orowan loops The theory also predicts a range of behaviors which are in qualitative agreement with observation, including: hardening and dislocation multiplication in single slip under monotonic loading; Taylor scaling, both under monotonic loading and, in an appropriate rate form, under cyclic loading; the Bauschinger effect under reverse loading; the fading memory effect, 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- CHAPTER SOLID DYNAMICS 53 perimental cyclic stress-strain data for structural steels reported by [77], which was obtained from tests specially designed to exhibit the fading memory effect caused by reversed loading The evolution of the dislocation density during a loading cycle is of considerable interest, Fig 4.3f Upon unloading the dislocation density decreases as a result of the elastic relaxation of the dislocation lines The dislocation density bottoms out — 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 observed experimentally (Morrow, unpublished tests results), it also arises in models of the stored energy of cold work [10], and is in analogy to the hysteretic loops exhibited by magnetic systems [99, 28] Key inputs into this and similar theories which may be gleaned from atomistics are: dislocation energies as a function of segment orientation; Peierls stresses; and the strength of dislocation-dislocation reaction products The core structure and energetics 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] For instance, for Ta they have calculated a ratio of edge to screw energies of 1.77 Olmsted and Phillips [75] have used the embedded-atom method potential, as £tted by Ercolessi and Adams [35] to the results of their £rst-principles calculations, to map out the entire range of energies of dissociated dislocation cores in aluminum Their results demonstrate that the energies computed from atomistics can be reproduced almost exactly using linear elasticity theory provided that dissociation into partials is accounted for and an appropriate stacking-fault energy is used, which again attests to the predictive ability of informed continuum models Duesbery and Xu [32] have calculated the Peierls stress for a rigid screw dislocation in Mo to be 0.022µ, where µ is the 111 shear modulus, whereas the corresponding Peierls stress for a rigid edge dislocation is 0.006µ, 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 using atomistic and continuum models [83, 94, 119, 118, 4, 101] Thus, for instance, Rodney and Phillips [94] used the quasicontinuum method to simulate three-dimensional Lomer-Cottrell junctions, and determined that this type of junction may be unzipped under stress Interestingly, Shenoy et al [101] subsequently showed that essentially identical results may be obtained with a anisotropic elastic model provided that dislocation 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 complete stress-strength diagram for junctions, i e., the locus of points in stress space corresponding to the dissolution of the junction Likewise, Wang et al [119, 118] have exhaustively cataloged the jogs and kinks of bcc crystals and computed their structures and 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 is predicated upon the consideration of an ‘irreducible’ set of unit processes, consisting of: double-kink formation and thermally activated motion of kinks; the close-range CHAPTER SOLID DYNAMICS 54 interactions between primary and forest dislocation, leading to the formation of jogs; the percolation motion of dislocations through a random array of forest dislocations introducing short-range obstacles of different strengths; dislocation multiplication due to breeding by double cross-slip; and dislocation pair-annihilation Each of these processes accounts for–and is needed for matching–salient and clearly recognizable features of the experimental record In particular, on the basis of detailed comparisons with 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 of a marked stage I 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 as the temperature increases or the strain rate decreases; the initial parabolic hardening followed by saturation within the stage II of hardening; the temperature and strain-rate dependence of the saturation stress; and the orientation dependence of the hardening rates 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-elastic models 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 noted throughout 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 continuum based, tend to produce unmanageable quantities of numerical data and rarely result in analytical descriptions of effective behavior The daunting task of post-processing these data sets and uncovering patterns and laws within them which can be given analytical expression is as yet a largely unful£lled goal of multiscale modeling This larger picture notwithstanding, one concrete and workable link between micromechanical models and £rst-principles calculations concerns the calculation of material constants A partial list relevant to the present model includes: energy barriers and attempt frequencies for double-kink formation, kink migration, dislocation unpinning, cross-slip, and pair annihilation; dislocation-line and jog energies; and junction strengths Other properties which have yielded to direct calculation include the volumetric equation of state (EoS), the pressure dependence of yield, and the pressure and temperature dependence of elastic moduli As noted earlier, these results provide a suitable basis for future extensions of the present model to higher temperatures, pressures and strain-rates 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 Lagrangian framework, the problem arises of devising effective arti£cial viscosities which perform well for arbitrary unstructured tetrahedral meshes, and in the presence of large plastic deformations and rigid-body rotations We have developed and veri£ed an arti£cial viscosity method which is formulated CHAPTER SOLID DYNAMICS 55 directly at the constitutive level and, therefore, independent of the dimension of space and of the particular choice of £nite element interpolation The analysis of the method has been restricted to two and three-dimensional quadratic elements The correctness of the method has been veri£ed by comparing the numerical and exact solution for the Riemann problem in perfect gases in two dimensions and by verifying the jump conditions and shock propagation velocity in Ta For purposes of this study, we model the behavior of Ta up to extreme loads of the order of 200 GPa by recourse to the equation of state of Cohen et al., derived from ab initio quantum mechanical calculations [90] and by a Steinberg-Guinan J2 -isotropic large-deformation plasticity model that accounts for rate dependency and thermal softening effects The model is used to simulate a plate impact experiment with an impact speed V = 2000 m/s In the present formulation the effective numerical viscosity is assumed to be of the form: ηh = η + ηh (4.1) where η is the physical viscosity and ηh is the added arti£cial viscosity This latter term, is computed at each £nite-element Gauss point in accordance with the following expression ηh = max 0, − 34 hρ0 (c1 ∆u − cL a) − η ∆u < ∆u ≥ (4.2) where h is a measure of the element size, ρ0 is the unshocked density, ∆u is a measure of the velocity variation across the element, a is the sound speed, and c1 and cL are coef£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 element Ωe : h≡ d!|Ω| (4.3) ∂J ∂t (4.4) ∆u ≡ h where d is the spatial dimension, |Ω| is the element volume and (4.4) is evaluated at each Gauss point in turn The right-hand side of (4.4) may further be discretized in time in the form: ∆un+1 = h Jn+1 − Jn ∆t (4.5) It should be noted that the arti£cial viscous stress is computed simultaneously with the constitutive relations, which facilitates matters of implementation, and that (4.4) is strictly material-frame indifferent In order to identify the coef£cients c and cL , we follow [27] and note that the Rankine-Hugoniot jump conditions de£ne an implicit relation D(∆u) between the shock speed and the jump in velocity which is often well-approximated by a linear £t of the form D = s0 + s1 ∆u (4.6) CHAPTER SOLID DYNAMICS 56 where s0 and s1 are constants By replacing this relation into the Rankine-Hugoniot linear momentum jump condition for a steady shock ∆p = ρ0 D∆u (4.7) a quadratic equation for ∆u is obtained By considering then the arti£cial viscosity as an approximate Riemann solver, the following relations for c1 and cL are obtained c1 ≈ s1 cL ≈ s0 /a (4.8) Our experience indicates that these constants tend to perform well for strong shocks but their performance deteriorates for weak shocks For the latter the shock tends to be smeared over more elements than necessary and overheating effects are exacerbated By way of validation, we have applied the method to a Riemann problem, also known as the ‘shock-tube problem’, for ideal gas equation of state on both sides of the initial contact surface Adiabatic heating was assumed in this test Fig 4.4 shows the pressure and density ratios p/p1 and ρ/ρ1 along the centerline of the two dimensional con£guration together with the analytical solution every ∆t = 1.69 × 10 −5 s The good agreement between the analytical and numerical solutions is evident from the £gure Both the Rankine-Hugoniot jump conditions across the shock and the values in the expanded gas are well predicted The speed of the contact surface and the shock are also 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 shock and the contact surface This is a pathology of arti£cial viscosity methods, and can be improved by the introduction of arti£cial heat ¤ux We have also considered a piston problem consisting of the sudden operation of a piston 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 of the 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 £rstprinciples calculations The internal energy per unit mass U (T, v) for this material is given by the expression: U (T, v) = −2.218 × 1011 + 1.813 × 109 exp(−119.816 v 1/3 ) + 137.849 T − −5.903 × 1010 exp(−119.816 v 1/3 ) v 1/3 + (4.9) +T −0.01243 + 653.806 v − 7.0925 × 10 v + 4.363 × 10 v + +T 1.301 × 10−6 − 0.0505 v + 408.911 v + 2.193 × 106 v where T is the absolute temperature and v the speci£c volume per unit mass The CHAPTER SOLID DYNAMICS 57 entropy S(T, v) follows as: S(T, v) = −783.37 + 3.368 × 106 v − 2.151 × 109 v + 1.689 × 1014 v + +T −0.0249 + 1307.61 v − 1.419 × 107 v + 8.725 × 109 v + (4.10) +T 1.952 × 10−6 − 0.0757 v + 613.367 v + 3.289 × 106 v + +137.849 log(T ) All constants correspond to SI units Equations (4.9) and (4.10) completely de£ne the thermodynamic behavior of the material p2 [GPa] v2 [10−5 m3 /Kg] T2 [K] D [m/s] Theoretical 194.86 3.953 5263 5852 Numerical 194.8 ± 0.1 3.953 ± 0.001 5260 ±30 5855 ±30 Table 4.1: Comparison between theoretical and numerically obtained values of the jump conditions The label indicates the state after the shock D is the shock velocity The reference con£guration is the rectangle [0, 10] cm × [0, 1] cm in R The material is initially at rest with homogeneous temperature 293 K The initial density is ρ0 = 16650 Kg/m3 At time t0 a velocity V0 pointing towards the positive x-direction (longitudinal axis) is imposed at x = The velocity V0 is chosen so as to generate a strong shock The arti£cial viscosity constants are c = 1.29 and cL = 0.99 theoretical and numerical values of the variables ahead of the shock are collected in Table 4.1 Fig 4.5 shows pressure and velocity values at t = 1.36 µs The shock is spread over to elements The absence of oscillations behind the shock is noteworthy p p ˙0 m n σy Tref Tmelt α β × 10−4 × 10−3 10 5 × 108 293 1343 1 Table 4.2: Plasticity model material parameters (SI units) The hydrodynamic approximation for the propagation of shock waves in solids by de£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 CHAPTER COMPUTATIONAL SCIENCE 105 10000 S ec / 25 tim e s teps 1000 3K x 2K Initia l G rid P erfe ct S ca ling 100 10 10 C P Us (log2) refinem ent level= refine fac tor = 2, refine ever y step s Figure 7.3: RM3D + Grace Scaling on ASCI Blue Mountain 24K x 16K Effective Grid Points Using Parallel, Distributed, Adaptive Mesh Re£nement VolPro 500) under a programmable (batch job capable) graphic visualization system Results include static images, animations, as well as interactive 3D visualizations under a semi-immersive system (i.e Responsive Workbench and Immersa Desk II) • Outlined module based graphic wrapper for VTF simulation to work as independent layer over PYRE (Python based) framework for simulation startup, monitoring, steering, halting and re-staring • Created a prototype of a parallel volume-rendering cluster capable of rendering a 5123 volume at interactive rates (5-7 frames per second) Further work of cluster will extend its capabilities to be able to volume render a 1024 volume at rates of 12-15 frames per second within two years Done in collaboration with Compaq/Tandem and RTViz • Created and incorporated a 3D OpenGL, Python, and GTK geometry viewer into VTF simulation for monitoring • Collaborated with Caltech’s Computer Graphics group as part of an NSF grant for Large Data Visualization, to which ASCI/VTF simulation results are relevant • Collaborated with researchers in the TeraVoxel NSF grant for Large Real Data Acquisition, Storage, and visualization Future generation of visualization cluster will be partly funded under this grant Results also very relevant to ASCI efforts in general • Continued the visualization group biweekly meetings covering topics relevant to scientists and collaborators’ needs, such as available volume rendering tech- CHAPTER COMPUTATIONAL SCIENCE 106 niques and software, data segmentation work at Caltech, and remote collaborative techniques for visualization, among others 7.5 Scalable I/O The scalable I/O development was focused on designing a simple, ¤exible and extendible checkpoint system for the VTF codes The working assumption was that the internal state representation of the executing applications should be stored in collections of self-describing £les which can be transferred to and interpreted on any ASCI machine For improved performance and scalability, parallel I/O mechanisms should be used whenever available The check-pointing library should also be easily portable to ASCI platforms and provide means for convenient management and access to the collected data The top-level interface of our library was based on two cooperating object classes: data objects and I/O handlers Data objects are collections of named data buffers with their attributes, such as pointers to memory locations, corresponding memory region sizes and identifying labels A data object itself must also be assigned a name (handle), which is later used to derive paths for the checkpoint £le hierarchy Member buffers contained in a data object can be attached and detached in any order and referred to by index or chosen label In order to perform data storage or retrieval, data object needs to be associated with an I/O handler, which maintains information about mapping between the data buffers and checkpoint £les (and data structures within them), performs actual I/O operations, veri£es data validity and handles additional £le attributes and user hints Each checkpoint is represented by a tree of £les, whose root is a master meta-data £le The master £le keeps track of relative locations of the data £les and stores global attributes (such as elapsed time, iteration number, physical constants, etc.) Typically, each data buffer is stored in a separate £le; when running in parallel, the corresponding data structures from different processing nodes can be combined into one physical £le, which results in much more manageable checkpoint hierarchies If access to different I/O environments (HDF5, DMF) is required during data object life span, several I/O handlers may be used interchangeably with the same data object To avoid the development cost of low-level £le system interface and yet another semi-portable type description mechanism, the underlying I/O operations are accomplished through the HDF5 library The additional bene£ts offered by the HDF5 are low-level data conversion routines, B-tree structured £les, transparent meta-data management, API simplifying speci£cation of various dataset layouts and several £le system drivers, including parallel I/O via MPI-IO The checkpoint library was written almost exclusively in C++, which makes the incorporation into existing Python framework quite simple and enables generic exceptionbased approach to error handling The users may also modify the default behavior of library objects by supplying hints, which, among other parameters, control automatic buffer storage and cache ¤ushes when data object is destroyed, memory allocation for data buffers, printout of debugging messages or usage of collective I/O for parallel £les The VTF-speci£c wrappers are derived from generic data object class modi£ed to support automatic buffer attachment, data-types and selections, labeling, customized CHAPTER COMPUTATIONAL SCIENCE 107 try { HDFhandler ioh(communicator); RM3dState s("Fluid_checkpoint", ioh, ×tep, &time, ¶ms, &data); s.store(); } catch (IOerror err) {io_error_handler(err);} Figure 7.4: Checkpoint invocation for ¤uid data; params and data are structures containing physical constants and description of ¤uid data arrays, respectively memory to £le mapping and global attribute processing An example illustrating the ease of use of our interface can be seen in £gure 7.4 The wrappers include support for both ¤uid and solid runtime structures The ¤uid data contribute the most of total checkpoint volume Their structure is highly regular (3- and 4-dimensional arrays) thus allowing recreation of global data-space in parallel mode when storing a checkpoint or simpli£ed partitioning of saved data across processing nodes during restart Additionally, disk space is conserved due to elimination of ghost cell boundary (allocated by the ¤uid solver) thanks to hyper-slab extraction support The solid data comprises several much less structured entities (solid meshes, mesh boundaries, parallel mechanics setup) and hence a meaningful combination of items imported from different computing nodes is much more dif£cult Currently, the buffers are stored verbatim with some clustering applied to reduce the number of resultant £les 7.5.1 Integration with Globus The main objective of this project is the development of set of interfaces and modules in Python, which would allow convenient access to Grid metacomputing functionality In particular, the following goals were delineated: • Interface to globusrun enabling a user to start process on any remote resource, • Remote I/O capability via GASS (Global Access to Secondary Storage), • Rehostable version of Python in which interpreter with core libraries can be loaded and executed on remote machine, • Extraction of remote host resource and availability information Currently, about half of these goals are met The remote execution module has been developed The interface allows the user to specify the most important parameters required to correctly synthesize an RSL request required by remote Globus server These parameters include: target machine, the number of nodes to run on, input, output and error stream redirections, target machine, job manager type, type and path to the executable and associated command line CHAPTER COMPUTATIONAL SCIENCE 108 arguments Execution of MPI jobs and Python scripts (assuming that the interpreter is properly installed on the remote host) is fully supported Another module provides remote I/O functionality It is based on wrappers for GASS open and close calls and modi£cation of internal £le representation, so that proper actions can be triggered if a remote £le is being accessed ¿From within a Python script, the only difference seen by the user is that in the remote case a URL replaces a local £le path The subsequent read or write access may be then performed transparently using standard read and write methods on the £le object returned by open Since the remote I/O requires a GASS server to be operational on remote machine, a preliminary version of module supporting setup of GASS was also made available In addition to the code development, a proper test environment was set up for local computing resources The Globus servers were upgraded to the most recent version (1.1.3) on the Origin 2000, HP V-class and Beowulf cluster A number of thin clients was con£gured on workstations to facilitate access to Grid resources without the need for remote logons 7.6 7.6.1 Algorithms The Closest Point and Distance Transform For an overview of the closest point transform, (CPT), see Chapter Here we discuss the implementation and performance of the Characteristics/Scan-Conversion, (CSC), algorithm The CSC algorithm was implemented for computing the closest point transform to a triangular mesh in 3D The CSC algorithm is embarrassingly concurrent The ¤uid grid on which we compute the CPT is distributed over a number of processors The CPT is computed up to a distance d away from the solid boundary At each time step each ¤uid processor receives the entire b-rep, (boundary representation) Each ¤uid processor then selects the portion of the b-rep that is relevant for its domain Then each ¤uid processor simply executes the sequential CSC algorithm with its portion of the b-rep and its portion of the ¤uid grid We examine the performance of the CSC algorithm as we vary the grid size To verify that the algorithm has linear computational complexity in the grid size, we examine execution time as we re£ne the grid We compute the closest point transform to a tessellation of the unit sphere with 2048 faces on the domain (−2, 2) × (−2, 2) × (−2, 2) to a distance of 0.05 for grid sizes from 103 to 2003 Figure 7.5 shows a log-log plot of execution time versus grid size along with the line of linear scalability This shows the linear computational complexity Initially there is super-linear scalability due to coarser inner loops in the algorithm As the grid is re£ned, the polyhedra contain more grid points Scan converting polyhedra containing many points is more ef£cient than scan converting polyhedra containing few points Next we examine the performance of the Characteristics/Scan Conversion algorithm as we vary the mesh size To verify that the algorithm has linear computational complexity in the mesh size, we examine execution time as we re£ne the mesh We compute the closest point transform on a 100 × 100 × 100 grid to tessellations of the unit sphere on the domain (−1.2, 1.2) × (−1.2, 1.2) × (−1.2, 1.2) to a distance of 0.1 execution time (sec) execution time (sec) CHAPTER COMPUTATIONAL SCIENCE 10 109 10 10 10 0 10 10 10 grid size (a) 10 10 number of faces 10 (b) Figure 7.5: (a) Log-log plot of execution time versus grid size The line of linear scalability.(b) Log-log plot of execution time versus the number of faces The line of linear speed-up for mesh sizes from 32 to 131072 faces Figure 7.5 shows a log-log plot of execution time versus mesh size along with the line of linear scalability This shows the linear computational complexity Initially there is super-linear scalability because the total volume of the polyhedra decreases as the mesh is re£ned 7.6.2 Efficient eigensolutions for computational chemistry We investigated the problem of computing eigenvalues and eigenvectors of an irreducible symmetric block tridiagonal matrix B E1 E1 B E2 ∈ Rn×n , E B (7.1) Mp := Ep−1 Ep−1 Bp where the blocks Bi ∈ Rki ×ki (i = 1, 2, , p) along the diagonal are symmetric, and the off-diagonal blocks Ei ∈ Rki+1 ×ki (i = 1, 2, , p − 1) have rank one: Ei = σ i u i v i with ui = vi = and σi > The block sizes ki have to satisfy ≤ ki < n p and i=1 ki = n The self-consistent-£eld procedure in Quantum Chemistry yields computational matrix eigenproblems with the property that the magnitudes of the matrix elements rapidly decrease as they move away from the diagonal; thus, they can be approximated by matrices of the above form Although the off-diagonal blocks Ei of the matrices arising in these problems are in general not rank-one matrices, it is possible to approximate them with rank-one matrices, and in many cases, the approximations may be suf£cient for the desired accuracy CHAPTER COMPUTATIONAL SCIENCE 110 We have developed an extension of the divide-and-conquer algorithm to compute the full eigensystem of block tridiagonal matrices of the above form The algorithm has produced very attractive results in terms of ef£ciency and accuracy The implications of unequally sized blocks have been studied, and a reliable method for determining a merging order whose associated arithmetic complexity is at least close to optimal for any given sequence of block sizes has been stated Details of the algorithm and 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