INDEX

AAE, see Auxiliary algebraic evaluations for manufacturing processes modeling, 739 Absorbing boundaries, 260, 267 overrelaxation with, 68 Absorbing media (Monte Carlo methods), 281–283 for structured meshes, 33 Accuracy: for two and three dimensions, 82–83 of discretization, 926–929 AMBER force field, 666 of numerical solution procedures, 17 AMG method, see Algebraic Acoustic mismatch model (AMM), 646, 649 AMM, see Acoustic mismatch model Adaptive grids, 483–489 Amorphous silicon, crystallization of, 680 coarsening, 486 AMR, see Adaptive mesh refinement data compression and mining, 487 AMRC, see Adaptive mesh-refinement and hybrid, 487–488 mesh-coarsening moving boundary problems, 486–487 Anderson method, 670 parallel extensions, 488–489 Anisotropic grids, 480–481 Adaptive mesh refinement (AMR), 484–485, Anisotropic scattering, Monte Carlo 488–490 application to, 285 Adaptive mesh-refinement and mesh-coarsening ANN, see Artificial neural networks (AMRC), 486, 488 Annealing, thermal process for, 748 Added-dissipation schemes, 41, 42 Adiabatic (zero normal derivative) boundary Anomalous diffusion, 631–632, 639–641 conditions, 78 A posteriori series convergence acceleration ADI technique, see Alternating direction implicit schemes, 500 technique Apparent rate of convergence, 423, 424 Adjoint forms of equations, 932 Approximate factoring, 83 Adjoint method: Area coordinates, 103 sensitivity analysis, 458–460 Artificial compressibility methods: stochastic inverse problems, 534–535 for continuity and momentum equations, 43 Algebraic multigrid (AMG) method, 36–37 for incompressible flow, 85 All-speed pressure-based algorithms, 383–341 Artificial neural networks (ANN), 802–804 Alternating direction explicit schemes, 76 Artificial supertriangle approach, 477–478 Alternating direction implicit (ADI) technique: Artificial viscosity, 28

946 INDEX 947

Astronomy, high-performance computing Block iterative methods, 66–67 for, 900 Block-structured meshes (grids), 8, 9, 476 Autohesion (thermoplastics), 792 Blood and blood vessels, see Bioheat transfer Auxiliary algebraic evaluations (AAE), 433–435 Blood perfusion rate, 855 Axisymmetric contraction, 704 Body-fitted meshes, 8 Axisymmetric expansion, 704 for irregular geometry, 314 velocity storage for, 41 Backdiffusion, 610–613 Boltzmann transport equation (BTE), 672, 936 Backward-facing step flow, DNS of, 713–716 Boundary conditions: Backward-implicit-, 114 absorbing, 260 Bayesian approach (inverse heat conduction), adiabatic (zero normal derivative), 78 542–553 appropriate, 56 with Gibbs sampler, 546–547 convective, 79 inverse problem transformation in, 544–546 for direct problems, 525–526 numerical results with, 548–553 finite-difference methods, 55–56 and stochastic inverse heat-conduction problem, finite-element methods, 96–97 542–544 of the first kind, 55 work using, 553 floating random walk, 267–268 Beam-warming scheme, 30 flux, 78–79 BEM, see Boundary-element method high-performance computing, 912–913 Beowulf computer, 898 manufacturing processes modeling, 738, Bernoulli distribution, 294 740–741 Bernoulli trials, 294 molecular dynamics method, 669–670 Biased Monte Carlo methods, 276, 278–281, 294. in Monte Carlo method, 260 See also Importance sampling moving, see Moving-boundary problems BI-CGSTAB, 38 with multigrid method, 71–72 Biconjugate gradients, 38 partially absorbing, 260 BIE (boundary integral equation), 128 pressure-based algorithms, 343–348 BIEM (boundary-integral equation method), 128 radiative heat transfer, 299–300 Bilinear element, 97–98 and random walk, 258 Binary eutectic alloy: reflecting, 260 benchmark problem, 613–617 of the second kind, 56 diagram, 609 in sensitivity analysis, 452–457 Bioheat transfer, 851–887 of the third kind, 56 background for analysis of, 852–854 unsteady conduction, 75–79 blood vessel tissue-interactive numerical thermal Boundary-element method (BEM), 125–159, models, 864–866 924 burn injury modeling, 868–870 discretization of, 10 Chen and Holmes bioheat equation, 857–860 error estimation with, 158 continuum models of heat transfer, 861 for large-scale problems, 156–157 effective thermal conductivity, 855–856 major advantage of, 225 freezing of biological materials, 872–884 nomenclature related to, 158–159 bioheat equation for, 877–882 radiative heat transfer, 155–156 imaging-assisted numerical analysis, for steady heat conduction, 126–134 882–884 for steady heat conduction with generation, heating of living tissue, 868–872 134–139 hyperthermic treatment of cancer modeling, for transient conduction, 139–155 870–872 Boundary integral equation (BIE), 128 imaging-assisted numerical analysis, 882–884 Boundary-integral equation method (BIEM), 128 medical imaging interactive numerical analysis, Boundary problems (IHCP), 526 866–868 Boundary treatments: nomenclature related to, 885–887 control-volume-based finite-difference methods, Pennes equation, 856–857 212–215 finite-difference solution of, 863–864 control-volume-based finite-element methods, finite-element solution of, 861–863 215–217 Weinbaum-Jiji bioheat equation, 860–861 Boussinesq approximations, 735 948 INDEX

Brick element, 110–112 adaptive mesh-refinement and mesh-coarsening, BTE, see Boltzmann transport equation 486, 488 Buoyancy effects: phase-change problems, 611–613 in continuous processing, 771 Coating process, optical fiber, 765, 767–768 data center airflow modeling, 826 Codes for numerical heat transfer, 936–941 in manufacturing processes modeling, commercial, 940–941 734–735 data exchange standards for, 938 under microgravity conditions, 775 data structures for, 936–937 Burn injury modeling, 868–870 programming languages and paradigms for, 937 public domain software resources for, 939 scripting language interface for, 938 C (programming language), 903 Verification of, 419–422, 438, 939–940 C++ (programming language), 903 Collaborative Adaptive Sensing of the Atmosphere CAD software, 474, 487 (CASA), 906–907 Calculations, Verification of, 419–422 Collocated equal-order CVFDMs/DVFEMs, 193 Cancer, hyperthermic treatment of, 870–872 Collocated interpolation schemes, 41–42 Capacitance matrix, 113 Collocated SIMPLE algorithm, 334–336 Carbon nanotubes, 681–689 Columnar solidification in static casting, 600–602, with fluids, 683–684 606 formation of, 681–682 Combustion systems, high-performance computing heat conduction of, 684–687 for, 900 thermal boundary resistance between surrounding Commercial codes, 940–941 materials and, 687–689 Complex-step method, 450–451 Carman-Kozeny relationship, 600, 603 Complex systems, Monte Carlo methods for, 274 Carravetta-Clementi (CC) potentials, 664–665 Composite materials: Carrier phase (two-phase flows), 699–700 polymer-matrix, 786–801 CASA, see Collaborative Adaptive Sensing of the thermoplastic-matrix, 792–801 Atmosphere consolidation stage, 792 Casting process, numerical modeling of, 767–770 fabrication of, 792–793 Cattaneo-Vernotte thermal wave model, 625–626, heat transfer, 793–797 634 prepregging stage, 792 CC potentials, see Carravetta-Clementi potentials spherulitic crystal microstructure evolution, Cell(mesh),7,9 797–801 Cell-based schemes, 9. See also Finite-volume thermosetting-matrix, 786–792 methods liquid composite molding, 767–789 Cell-centered FVMs, 192–193 pultrusion, 789–792 Cell centroid, 7 Compressible flow: Central limit theorem, 294 finite-difference methods, 85–88 Central moment, 294 multidomain , 709–713 Certainty, level of, 422 pressure-based algorithms, 337–342 CFD, see Computational fluid dynamics at subsonic inlet, 345–347 CGM, see Conjugate gradient method at subsonic outlet, 347–348 CGSTAB, 38 Computational fluid dynamics (CFD), 4 Checkerboarding, 39, 41, 47 false diffusion in, 318 Chemically reactive flows, numerical modeling of, gradient-search techniques in, 37–38 736–738 iterative solution methods in, 31 Chemical vapor deposition (CVD), 732, 771–774 Computational grids, 907 Chen and Holmes (CH) bioheat equation, 857–860 Computational heat transfer (CHT), 4 Choice of solvers, 927 Computer room air conditioning (CRAC), 823, 825, CHT (computational heat transfer), 4 831 CIM, see Continuous interface method Computing: Circumcircle test, 476–477 grid, 899, 906–911 Clausius-Clapeyron relation, 702 improvement in, 896–897. See also Coarse-grid correction schemes, 69 High-performance computing Coarsening, 473 parallel, 896 adaptive grids, 486 Condensation coefficient, 674–675 INDEX 949

Conditional filtered diffusion, 173 Control-volume-based finite-difference methods Conduction: (CVFDMs): carbon nanotubes, 684–687 boundary treatments, 212–215 inverse problem for, 529–542 as cell-centered FVMs, 192 meshless methods for, 235–238 discretized equations: molecular dynamics method, 671–672 momentum, 208 Monte Carlo methods, 254 for pressure, 210 in solids, 672 for θ, 205–206 steady, 61–73 domain discretization, 194–196 boundary-element method, 126–139 features of, 192 discrete equations for, 458–460 formulations of, 192–193 finite-difference method, 61–73 governing differential equations, 193–194 finite-element method, 92–102 integral conservation equations, 197 Monte Carlo method, 263–265 interpolation functions, 198–201 unsteady, 73–79 mass-conserving velocity field equation, adiabatic boundary conditions, 78 209–210 convective boundary condition, 79 nomenclature related to, 220–222 Dirichlet boundary conditions, 77–78 sequential iterative variable adjustment implicit schemes, 77–79 procedure, 217–219 simple explicit scheme, 74–76 steady-state conduction, 62–64 specified boundary flux condition, 78–79 unsteady conduction, 73–74 in two and three dimensions, 81–84 Control-volume-based finite-element methods unsteady conduction equation, (CVFEMs): 73–74 boundary treatments, 215–217 Confidence, level of, 422 discretized equations: Conformal meshes, 9 momentum, 208–209 Conjugate gradient method (CGM), 38 for pressure, 211–212 in high-performance computing, 914 for θ, 206–207 for stochastic inverse heat-conduction problem, domain discretization, 196 536–538 as element-based vertex-centered FVMs, 193 Conservation principles, 54 features of, 192, 193 Consistency of numerical solution procedures, 17 formulations of, 192 Consistent mass matrix, 113 governing differential equations, 193–194 Consolidation (thermoplastics), 792 integral conservation equations, 197 Constant, 294 interpolation functions, 201–205 Constitutional supercooling, 874 mass-conserving velocity field equation, 211 Constitutive relations models, 490, 624 nomenclature related to, 220–222 Continuity equation, 5 sequential iterative variable adjustment discrete, 42 procedure, 217–219 pressure-based methods for solving, 43–45 unequal-order interpolation of pressure-velocity, solution methods, 43–45 41–42 Continuous distribution function, 294 as vertex-centered FVMs, 193 Continuous interface method (CIM), 569–572, Control-volume equations (phase-change problems): 581–586 general equations, 602–604 assignment of material properties in phases, 570 using finite-difference approximations, 605, 606 immersed-boundary treatment, 570–572 Control volume methods, 59. See also Finite-volume for impact of liquid drop on polycarbonate methods surface, 585–586 Convection-diffusion equation: for interfacial characteristics of static drops, discretization of, 18–22 581–585 finite-element method, 115–117 Continuous processing, numerical modeling of, for natural convection in porous media, 513–518 769–772 transient radiative transfer equation vs., 300–301 Continuum stochastic sensitivity method (CSSM), Convection operators: 533–534 control of spatial oscillations, 28–31 Control-angle overlap (radiative heat transfer), higher-order schemes for, 25–28 316–317 upwind-weighted higher-order schemes, 26–28 950 INDEX

Convective boundary condition: Data centers, 821–849 one-dimensional steady conduction, 96–97 cooling strategy for, 823, 824 two-dimensional steady conduction, 99–102 CRAC modeling, 831 Convective heat transfer: mesh generation, 832 finite-difference method, 84–88 model data center, 832–839 finite-element method, 115–119 global results, 834–836 meshless methods, 238–244 local results, 836–839 Monte Carlo methods, 274–275 parametric results, 839 Convergence: modeling framework for, 825–827 acceleration of, 929–931 modeling issues for, 823, 825 analytical filtering solutions for, 499 nomenclature related to, 847–849 apparent rate of, 423, 424 objectives of modeling, 825 degraded convergence rate, 426 performance metrics for, 840–842 and diagonal dominance, 64 plenum modeling, 830, 831 with Jacobi and Gauss-Seidel techniques, rack-level modeling, 820 34–36 racks in, 823 with multigrid method, 72 servers, 822–823 noisy convergence rate, 425–426, 429–430 system-level thermal modeling, 827–829 of numerical solution procedures, 18 thermal design methodology for, 839–847 observed rate of, 423–425 cooling configurations, 843–847 in original SIMPLE algorithm, 333 cooling scheme map for, 847 oscillatory, 425 performance metrics, 840–842 Correction storage (CS) scheme, 69 Data compression and mining (adaptive grids), 487 Coupling, 17, 45 Data exchange standards, 938 for compressible flow, 87–88 Data grids, 907 for convective flows, 84–85 Data structures (numerical heat transfer), 936–937 pressure-velocity, 931 Decimation of grids, see Coarsening thermal-solutal, 607–609 Definite integrals (Monte Carlo), 253–254 two-way, 702 Degraded convergence rate, 426 CRAC, see Computer room air conditioning Crank-Nicolson-Galerkin method, 114 Degree, vertex, 476, 482 Crank-Nicolson scheme, 74, 77, 82–83 Delaunay triangulation, 476–480, 483, 489 Cray computers, 897–898 Delta form of equation, 69 Crystal growing, 775 Dense plant canopies, turbulent flow through, Crystallization: 394–395 of amorphous silicon, 680 Density-based methods, 45–46 in thermoplastic-matrix composites, 797–801 for continuity and momentum equations, 43 CS (correction storage) scheme, 69 and incompressible flows, 47 CSSM, see Continuum stochastic sensitivity Density functional theory (DFT), 668 method Design optimization, recommendations for, 931–936 Cubic nonlinear eddy viscosity model, 382–395 DFT (density functional theory), 668 Cumulative distribution function, 250, 295 Diagonal dominance, convergence and, 64 Cure (thermosetting polymers), 786 Diagonally dominant matrices, 16 Cut-cell technique, see Sharp interface method Diagonally dominant systems, 65 CVD, see Chemical vapor deposition Differentiation of analytical solutions, 444–448 CVFDMs, see Control-volume-based Diffusion: finite-difference methods anomalous, 631–632, 639–641 CVFEMs, see Control-volume-based finite-element in irregular geometries, 510–513 methods moving-boundary problems, 566–569 Czochralski process, 732, 775 time-dependent, 112–115 Diffusion equation, 55. See also Unsteady Damped Newton descent scheme, 482 conduction equation Darcy flow, 390, 514 Diffusive mismatch model (DMM), 646, 649 Darcy-Forchheimer flow, 390 Diffusivity, temperature-dependent, 152–153 Darcy’s law, 390, 391 Dimensionless sensitivity coefficients, 445–447 Darcy source treatment, 600 Direct microstructure (DMS) simulations, 595–596 INDEX 951

Direct numerical simulation (DNS), 698, 702–703, control-volume-based finite-element methods, 927 206–207 backward-facing step flow, 713–716 Dispersed phase (two-phase flows), 700–702 defined, 167 Distributed heat source, temperature distribution homogeneous shear flow, 721–724 with, 259 homogeneous turbulence, 703–709 Divergence, oscillatory, 425 inhomogeneous turbulence, 709–716 DMM, see Diffusive mismatch model in turbulent HMT computational analysis, 168 DMS simulations, see Direct microstructure vortical structures preservation, 929 simulations Direct problems, 525–526 DNS, see Direct numerical simulation Direct simulation. See also Direct numerical Domain discretization, 7–9 simulation and boundary-element method, 135 defined, 276, 295 control-volume-based finite-difference methods, DSMC (using Monte Carlo technique), 661 194–196 Monte Carlo methods, 276–278 control-volume-based finite-element methods, Direct solution methods: 196 for continuity and momentum equations, 43 moving boundary problems, 562–563 linear equations, 16 radiative heat transfer, 306 Dirichlet boundary condition, see Fixed-value two-dimensional, 97 boundary conditions DO method, see Discrete-ordinates method Dirichlet tessellation, 478 Donor-cell scheme, 204 Discrete distribution function, 295 Double-decomposition methodology models, Discrete equations (steady conduction), 458–460 393–394 Discrete level assumption, 608 DRM, see Dual-reciprocity methods Discrete numerical methods, 493–494 Droplets: Discrete-ordinates (DO) method, 298–299, 303, 305 liquid, on polycarbonate surface, 585–586 Discrete random variable, 295 on platinum solid surface, 676–677 Discretization, 923–924. See also Domain radius of, 673–674 discretization; Meshes static, interfacial characteristics of, 581–585 accuracy of, 926–929 Drug design, high-performance computing of convection-diffusion equation, 18–22 for, 900 finite-difference methods, 10–11 DSMC (direct simulation using Monte Carlo finite-element methods, 11–14 technique), 661 finite-volume methods, 14–15, 59–61 Dual-phase-lag model, 633 of governing equation, 10–15 anomalous diffusion, 639–641 isoparametric, 130 thermal lagging, 636–639 pressure-based algorithms: Dual-reciprocity methods (DRM): for multifluid flows, 332 steady heat conduction with generation, for single-fluid flow, 329–332 135–136 radiative heat transfer, 306–313 for transient heat conduction, 154–155 subparametric, 130 transient heat conduction in solids, 142–143 superparametric, 130 DuFort-Frankel scheme, 76 Discretized equations: momentum: EAM, see Embedded-atom method control-volume-based finite-difference Earthquake simulation, high-performance computing methods, 208 for, 900 control-volume-based finite-element methods, Earth Simulator, 901 208–209 Eckert number, 744 for pressure: Eddy viscosity models (EVMs): control-volume-based finite-difference cubic nonlinear, 382–395 methods, 210 zero-equation, 392 control-volume-based finite-element methods, Edges (mesh), 7 211–212 Effective pair potential for water, 663–665 for θ: Effective thermal conductivity (bioheat transfer), control-volume-based finite-difference 855–856 methods, 205–206 Electron heating, 628 952 INDEX

Electronics cooling, 827–829. See also Data centers immersed-boundary treatment, 570–572 Element (mesh), 7 for impact of liquid drop on polycarbonate Element-based vertex-centered FVMs, 193 surface, 585–586 Element equations, 96 for interfacial characteristics of static drops, Element libraries, 487 581–585 Elliptic partial differential equations, 54 diffusion, 566–569 Embedded-atom method (EAM), 667–669 domain discretization, 563 Emitting media, Monte Carlo methods for, 283–285 interface in, 562 Energy conservation equation, 5–6 moving-boundary problems, 561, 569–586 ENIAC, 897 continuous interface method, 569–572, EnSight, 917 581–586 Entropy generation, 828 sharp interface method, 572–586 Equal-order interpolation schemes, 41 sharp interface method, 572–586 Equilibrium problems, 54 computational procedure, 580–581 Errors, 418–419 finite-volume formulation, 575–580 defined, 295 for impact of liquid drop on polycarbonate incomplete iteration, 424 surface, 585–586 modeling, 17, 489–490 inclusion of immersed boundaries, 573–575 pollution, 435 interface as discontinuity in, 561 standard, 295 for interfacial characteristics of static drops, Error bounds, 422 581–585 Error estimation: topological changes in, 563 auxiliary algebraic evaluations, 433–435 Euler-Maruyamma approximation, 175 boundary-element method, 158 Events, 295 error bar in, 422 EVMs, see Eddy viscosity models error evaluation vs., 418 Ewald sum method, 669 F :U empirical correlation, 428, 431, s 95 Exodus method, 273 437–438 Expectation: Grid-Convergence Index vs., 430–431 defined, 295 grid-convergence studies for, 423 in Monte Carlo simulations, 286 Monte Carlo methods, 286–288 Explicit schemes, 21 single-grid, 431–435 for solution of continuity and momentum and superconvergent results, 932 equations, 46–47 surrogate estimators, 433 unsteady conduction, 74–76 Zhu-Zienkiewicz estimators, 433–437 Error transport equation, 432 Extended simple point charge (SPC/E), 663–665 Ethernet, 897 Euler equations, 46 Euler-Galerkin method, 114 Faces (mesh), 7 Eulerian-Eularian methods, 177, 178 False diffusion, 318 Eulerian-Lagrangian methods, 697–726 False scattering, 313, 318–319 direct numerical simulation (DNS) studies, FAS (full approximation storage), 36 702–703 FDF, see Filtered density function for homogeneous turbulence, 703–709 FDMs, see Finite-difference methods for inhomogeneous turbulence, 709–716 Feature-based adaptations, 436 governing equations, 699–702 Federal High-Performance Computing Program interface representation in, 560 (HPCP), 899 large-eddy simulation, 177, 178 FEMs, see Finite-element methods moving-boundary problems, 562 FIELDVIEW, 917 nomenclature related to, 724–726 Filtered density function (FDF), 171–177 stochastic modeling, 716–724 determining, 171–172 Eulerian methods: numerical solution of, 175–177 boundary condition in, 562 specific capability of, 172 continuous interface method, 572, 581–586 Filtering solutions, 498–500 assignment of material properties Fine-grained density, 171 in phases, 570 Fine-grid solution, 69 INDEX 953

Finite-difference methods (FDMs), 53–90, 923. meshless methods vs., 225–228 See also Control-volume-based finite-difference node-based, 9 methods radiative heat transfer, 304–305 conservation principles, 54 sharp interface method, 575–580 convective flows, 84–88 steady conduction, 62–64 discretization, 10–11 Fixed-grid methods, see Eulerian methods domain discretization, 225 Fixed random walk, 268–269. See also Random finite-volume discretization, 59–61 walk forming finite differences, 56–59 Fixed-value (Dirichlet) boundary condition, 55, 56 high-performance computing, 912 and optimum overrelaxation factor, 71 microscale heat transfer, 641–645 overrelaxation with, 67–68 nomenclature related to, 88–90 unsteady conduction, 75, 77–78 partial differential equations, 54–56 FLO, see Flow-oriented Pennes equation, 863–864 Floating-point operations (FLOPS), 156, 896 stability analysis, 79–81 Floating random walk, 263–268, 295 steady conduction, 61–73 FLOPS, see Floating-point operations unsteady conduction, 73–79 Flow-oriented upwind scheme (FLO), 202–204 unsteady conduction in two and three Fluid dynamics, high-performance computing for, dimensions, 81–84 900 Finite-difference methods (parameter space), Flux boundary conditions (unsteady conduction), 448–450 78–79 Finite-element methods (FEMs), 91–120, 923, 924. Flux limiters, 29–30 See also Control-volume-based finite-element Flux method (radiative heat transfer), methods 302–303 application example, 119–120 Flux-splitting methods, 46 cell and element shapes, 9 Fokker-Planck equation, 173 convection-diffusion equation, 115–117 Forchheimer flow, 390, 391 convective heat transfer, 115–119 Forchheimer-modified Darcy law, 406–407 discretization, 10–14 FORTRAN, 903 domain discretization, 225 Forward problems, 526 high-performance computing, 912 Fourier diffusion, 625, 626 isoparametric transformations, 106–109 Fourier , 703 meshless methods vs., 225–228 Fractional-step finite-element method, 119 Navier-Stokes equations, 117–119 Free energy, functional of, 630–631 node-based, 9 Free-space Green’s function, 127, 144 numerical integration, 109–110 Freezing of biological materials, 872–884 one-dimensional steady-state conduction, 92–97 bioheat equation for, 877–882 Pennes equation, 861–863 biophysics of, 873–877 rectangular elements, 104–105 imaging-assisted numerical analysis, 882–884 θ method, 113–115 Fs :U95 empirical correlation, 428, 431, three-dimensional elements, 110–112 437–438 time-dependent heat diffusion, 112–115 Full approximation storage (FAS), 36 triangular elements, 102–104 Full-width-at-half-time (FWHM), 647 two-dimensional steady-state conduction, 97–102 Fully turbulent flow, 390 Finite integral methods, 229 Functional of the free energy, 630–631 Finite series representations, 229 Fundamental solution, 127 Finite-volume (FV) methods, 298, 923–924. See also Future time method, 528 Control-volume-based finite-difference methods; FV methods, see Finite-volume methods Control-volume-based finite-element methods FWHM (full-width-at-half-time), 647 cell and element shapes, 9 cell-based, 9 discretization, 10, 14–15 Galerkin approximation: domain discretization, 225 obtaining, 95 and finite-difference methods, 53. See also semidiscrete, 113 Finite-difference methods Galerkin finite-element method, 12–14 high-performance computing, 912 Gauss grid, 711, 712 954 INDEX

Gauss-Seidel methods, 16–17 generation of, 472–473 convergence behavior of, 34–36 and geometric design, 473 coupled, 45 hybrid, 487–488 line, 32–33 isotropic, 480 with multigrid method, 68–72 modification and smoothing of, 481–483 red-black, 913 Grid computing, 899, 906–911 steady conduction, 65–66 benefits of, 907 steady conduction, with overrelaxation, 66–68, definition of, 907 71–72 enabling applications for, 908–910 GCBA (geometric conservation-based algorithms), limitations of, 910 353 TeraGrid, 910–911 GCI, see Grid-Convergence Index Grid-Convergence Index (GCI), 423–431 Generalized integral transform technique (GITT), confirmations of, 426–428 495–498, 502 error estimation vs., 430–431 computational procedure and convergence noisy and degraded convergence, acceleration, 498–500 425–426 Mathematica results vs., 506–510 observed rate of convergence, 424–425 General kernel reproduction methods (GKR), 229 oscillatory convergence, 425 General Purpose Solver (GPS), 905, 915 and Richardson Extrapolation, 431 General scalar transport equation: single-grid error estimators vs., 434–435 conservative form, 6–7 within solution adaptation, 435–437 discretization of, 10–15 statistical significance, 429 domain discretization, 7–9 theoretical expectations for, 431 nonconservative form, 7 U95 correlation for error estimators, 437–438 for pressure-based algorithms, 329 U95 empirical correlation, 428 solution of, 15–17 for unstructured grids and noisy convergence Geometric conservation-based algorithms (GCBA), rate, 429–430 353 Grid generation, 472–473 Geometric conservation law, 564–568 with CAD software, 474 Geometric design, 473 and geometric design, 473 Geometric multigrid approach, 36 radiative heat transfer, 306 Gibbs sampler, 546–547 solution-adaptive, 435–437 GITT, see Generalized integral transform technique structured, 475–476 GKR (general kernel reproduction methods), 229 unstructured, 476–480 Glass, 746, 747. See also Optical fiber drawing Grid independence, 419 Globus, 908–909 Group iterative methods, 66–67 Globus Toolkit, 908, 909 Gurtin-Pipkins internal energy relaxation model, GMRES, 38, 156 634 “Golden rule” of computational physics, 68 GPS, see General Purpose Solver HAL (computer), 897 Gradient calculation: Healing (thermoplastics), 792 spectral methods for stochastic inverse problems, Heat flux profile: 535 impulse, 540–541 unstructured mesh technique, 22–25 triangular, 538–540, 548–551 Gradient-search techniques, 37–38 Heating of living tissue, 868–872 Gradient theorem, 22–23 Hermite polynomials, 540 Grand challenge problems (HPC), 899–901 Heterogeneous nucleation, 678 Grashof number, 744 Hexahedral elements, 487–488 Green-Kubo formula, 671 HIFU (high-intensity focused ultrasound), 871 Green’s-function method, 128 High-energy physics, high-performance computing Grids, 471–472 for, 900 adaptive, 483–489 Higher-order schemes, 924–925 adaptive meshing and modeling, 489–490 boundary-element method, 133 anisotropic, 480–481 convection operators, 20, 25–28 computational, 906–911 CVFDM/CVFEM interpolation functions, defects in, 474 200–201 INDEX 955

spatial oscillation control, 28 integral transform with Mathematica, unstructured meshes, 28 502–510 upwind-weighted, 26–28 Mathematica system, 501 High-intensity focused ultrasound (HIFU), 871 Monte Carlo, 273 High-performance computing (HPC), 896–919 nomenclature related to, 518–520 achieving petaflop performance, 905–906 numerical-analytical, 494–500 architecture for, 901–906 computational procedure and convergence for fluid flow, 916 acceleration, 498–500 future of, 917–918 integral transform method, 495–498 governing equations in, 911–915 Hyglac (computer), 898 grand challenge problems in, 899–901 Hyperbolic two-step model, 629–630 grid computing, 899, 906–911 Hyperbolid partial differential equations, 54 benefits of, 907 Hypersonic flows, 341, 342 definition of, 907 Hyperthermic treatment of cancer modeling, enabling applications for, 908–910 870–872 limitations of, 910 TeraGrid, 910–911 Identification problems (IHCP), 526 history of, 897–899 IHCPs, see Inverse problems of heat meshing methods in, 915 conduction MIMD vs. SIMD in, 902–904 ILLIAC IV, 897 nomenclature related to, 918–919 Ill-posed problems, 526, 527 parallel computers in, 901–902 ILU (incomplete lower-upper) decomposition, 33 programming languages for, 903–905 Imaging: systems for, 896–897 high-performance computing for, 900 visualization in, 916–917 medical, see Medical imaging HiPco, 683 Immersed-boundary treatment: Homeland security, high-performance continuous interface method, 570–572 computing for, 900 sharp interface method, 573–575 Homogeneous flows: Immersed objects, fixed-grid methods for, 569 assessment of shear flow, 721–724 Implicit schemes, 21 classification of, 704–706 alternating direction implicit technique: Homogeneous medium, Monte Carlo for manufacturing processes modeling, 739 methods for, 273 overrelaxation with, 68 Homogeneous nucleation, 677–678 for structured meshes, 33 Homogeneous turbulence, direct numerical for two and three dimensions, 82–83 simulation of, 703–709 solution of continuity and momentum HPC, see High-performance computing equations, 47 HPCP (Federal High-Performance Computing unsteady conduction, 77–79 Program), 899 Importance factors, 461, 464 Hsu method, 335 Importance sampling, 271–273, 275–276, 278–281 HVAC, 827–829. See also Data centers Impulse heat flux profile, 540–541 Hybrid adaptive grids, 487–488 Incomplete iteration error, 424 Hybrid grids, 487–488 Incomplete lower-upper (ILU) decomposition, 33 Hybrid methods, 192, 493–520. See also Incompressible flow: Eulerian-Lagrangian methods finite-difference methods, 85 control-volume-based FDMs, see high-performance computing, 916 Control-volume-based finite-difference SIMPLE algorithm for, 85, 336–337 methods at subsonic inlet, 344–347 control-volume-based FEMs, see at subsonic outlet, 347–348 Control-volume-based finite-element turbulent upward bubbly flow in a pipe, 356–358 methods Independent sample, 295 convection-diffusion for natural convection in Independent variable, 295 porous media, 513–518 Information technology management, see Data diffusion in irregular geometries, 510–513 centers mixed symbolic-numerical algorithms, 494, Inhomogeneous turbulence, direct numerical 500–510 simulation of, 709–716 956 INDEX

Initial conditions: triangular, 102–104 manufacturing processes modeling, 738 Isotropic grids, 480 molecular dynamics method, 670 Isotropic scattering: sensitivity analysis, 453, 457 probability functions for, 285 transient heat conduction in solids, 150–151 radiative heat transfer, 285 Integral conservation equations, 197 Iterative methods: Integral transform method (ITM), 495–498 Gauss-Seidel procedure, 65–66 algorithm optimization schemes, 517–518 linear equations, 16–17 with Mathematica, 501–510 multigrid, 68–73 with WorldData routine, 512–513 steady conduction, 64 Interactive effects of uncertainties, 808–809 ITM, see Integral transform method Interface definition (moving-boundary problems), 560, 562 Interfacial characteristics of static drop, 581–585 Jacobi technique, 34–36 Interfacial resistance (microscale heat transfer), Jayatilleke pee-function, 373 645–651 Internal heat generation in solids, 152 Karhunen-Loeve` (KL) expansion, 530 Interpolation functions: Kirchoff transform, 133 control-volume-based finite-difference methods, KL (Karhunen-Loeve)` expansion, 530 198–201 control-volume-based finite-element methods, 201–205 Lagrangian elements, 104 Intimate contact process (thermoplastics), 792 Lagrangian-Lagrangian methods, 177 Inverse problems, 525–526 Lagrangian methods: Mach number for particular exit velocity profile, boundary condition in, 562 933–936 combined with Eulerian methods, see recommendations for future research, 931–936 Eulerian-Lagrangian methods Inverse problems of heat conduction (IHCPs), diffusion, 566–569 526–555 domain discretization, 562–563 Bayesian approach, 542–553 geometric conservation law, 564–568 with Gibbs sampler, 546–547 interface in, 562 inverse problem transformation in, 544–546 interface movement in, 560 numerical results with, 548–553 moving-boundary problems, 561, 563–569 and stochastic inverse heat-conduction time-stepping scheme, 566 problem, 542–544 topological changes in, 563 work using, 553 transformed governing equations, categories of, 526 563–564 methodologies for, 526–529 Lagrangian Monte Carlo schemes, 175–177 nomenclature related to, 554–555 Laplace equation, 54, 55, 69 spectral stochastic approach, 529–542 Laplace transforms, 140–142 adjoint problem, 534–535 Large eddy simulation (LES), 167–180, developments in use of, 541–542 927 gradient calculations, 535 filtered density function (FDF), 171–177 numerical implementation, 536–538 determining, 171–172 numerical results with, 538–541 numerical solution of, 175–177 representation of random variables, 530–531 specific capability of, 172 stochastic inverse heat-conduction problem, governing equations, 169–170 531–534 high-performance computing, 916 IPSA, 353 nomenclature related to, 178–180 Irregular geometries: numerical solution of FDF, 175–177 diffusion in, 510–513 subgrid scale modeling, 170–171 radiative heat transfer, 313–316 in turbulent HMT computational analysis, Isoparametric discretizations, 130 168 Isoparametric elements: two-phase flows, 177–178, 698, 699 rectangular, 104–105 vortical structures preservation, 929 transformations of, 106–109 Large numbers, law of, 295 INDEX 957

Large-scale problems, boundary-element method for, Macroscopic pressure gradient, 406–408 156–157 Macroscopic turbulence models, 393 LASL algorithms, 353 Magnetic resonance angiography, 867–868 Latin hypercube sampling (LHS), 419, 462–466, Manufacturing processes, 729–778. See also 809 Materials processing Lattice-Boltzman model, 177 boundary conditions, 738, 740–741 Law of large numbers, 295 buoyancy effects, 734–735 κ − ε model, 392–393, 395 casting, 767–770 Least-squares approach: chemically reactive flows, 736–738 gradient in unstructured meshes, 23–25 chemical vapor deposition, 771–774 grid convergence, 427–428 continuous processing, 769–772 Lennard-Jones model, 675–676 general equations, 733–734 Lennard-Jones potential, 662–664 governing equations, 733–738 LEPP (longest-edge propagating path), 483 governing parameters, 744 LES, see Large eddy simulation idealizations and simplifications, 739–744 Level-set approaches, 487 initial conditions, 738 LGS, see Line Gauss-Seidel technique link between transport processes and material LHS, see Latin hypercube sampling characteristics, 747–750 LIF, see Local-instantaneous filtering with moving material or source, 741–743 Limiter functions, 29–30 and new materials, 730 Linear equations, large systems of, 64 nomenclature related to, 777–778 Linear solvers, 15–17, 31, 924 optical fiber drawing, 732, 759, 761–768 direct, 16 coating, 765, 767–768 gradient-search techniques, 37–38 feasible domain, 765, 766 iterative, 16–17 molecular-level changes in, 748–749 line Gauss-Seidel technique, 32–33 neck-down region, 761–764 multigrid methods, 33–37 numerical modeling of, 759, 761–768 Line-by-line TDMA, 32–33 preform diameter, 759 Line Gauss-Seidel (LGS) technique, 32–33 viscous dissipation in, 759 LINUX, 898 with phase change, 735–737 Liquid composite molding, 787–780 polymer extrusion, 732, 750–760 Liquid drop on polycarbonate surface, impact of, chemical conversion in, 756–758 585–586 feasibility of, 759–760 Liquid-vapor interface, 673–675 residence time distribution in, 750–753 Living tissue, heat transfer in, see Bioheat transfer twin-screw, 753–756 Lobatto grid, 711, 712 solution of governing equations, 738–739 Local-instantaneous filtering (LIF), 499–500 system simulation, design, and optimization for, Local radial point interpolation methods (LRPIM), 775–776 230 thermal materials processing, 730–733 Loki (computer), 898 and variable properties of materials, 744–747 Longest-edge propagating path (LEPP), 483 with very viscous flow, 743–744 Low-Reynolds-number models, 370 viscous dissipation in, 735 Low-temperature hyperthermia (LTH), Many-body potential for carbon and silicon, 870–871 666–668 Low-temperature scanning electron microscopy MAP, see Maximum a posteriori estimator (LTSEM), 877 Marching problems, 55–56 LRN κ-ε model, 382 compressible flow, 86–87 LRN model, 382–385 time-marching, 86 LRPIM (local radial point interpolation methods), high-performance computing, 912 230 RBFs in, 233–234 LTH, see Low-temperature hyperthermia transient flow in solids, 146, 149–150 LTSEM (low-temperature scanning electron Markov chain, 254 microscopy), 877 Markov Chain Monte Carlo (MCMC), 542, 546–547 Markov random field (MRF), 545 Macro-micro approach, 595–596. See also Mass conservation-based algorithms (MCBA), 353, Phase-change problems 354 958 INDEX

Mass-conserving velocity field equation: MATLAB PDE toolkit, 479–480 control-volume-based finite-difference methods, Matrices, diagonally dominant, 16 209–210 MAW, see Mass-weighted upwind scheme control-volume-based finite-element methods, Maximum a posteriori (MAP) estimator, 545–547 211 Maximum likelihood estimator (MLE) framework, Mass lumping, 114 529 Mass-weighted upwind scheme (MAW), 204–205 Max-in angle (circumcircle) test, 476–477 Materials, manufacturing: Maxwell-Boltzmann velocity distribution, 670 link between transport processes and MCBA, see Mass conservation-based algorithms characteristics of, 747–750 MCBA-SIMPLE, 355 under microgravity conditions, 775 MCMC, see Markov Chain Monte Carlo new, 730 MC methods, see Monte Carlo methods thermal processing of, 730–733 MEAM (modified embedded-atom method), 668 variable properties of, 744–747 Mean, in Monte Carlo simulations, 286 Materials processing, 785–815 Mean value method, 460 model-based process control, 801–808 Median, 295 artificial neural networks, 802–804 Medical imaging: complex mold geometrics, 807–808 imaging-assisted numerical analysis, 882–884 controller architecture, 803–804 interactive numerical analysis, 866–868 example of, 805–806 Meshes. See also Grids; Unstructured meshes online flow-control schemes, 801–802 block-structured, 8, 9 optimization problem, 804 body-fitted, 8 racetracking in, 806 conformal, 9 nomenclature related to, 814–815 for domain discretization, 7–9 polymer-matrix composites, 786–801 finite-difference, 56–58 thermoplastic, 792–801 high-performance computing methods, 915 thermosetting, 786–792 nonconformal, 9 thermal, 730–733 regular, 8 thermoplastic-matrix composites, 792–801 stair-stepped, 8 consolidation stage, 792 structured, 9 fabrication of, 792–793 terminology used with, 7 heat transfer, 793–797 unstructured, 9 prepregging stage, 792 discretization of convection-diffusion spherulitic crystal microstructure evolution, equation, 18–22 797–801 generation of, 476–480 thermosetting-matrix composites, 786–792 gradient calculation, 22–25 liquid composite molding, 767–789 Grid-Convergence Index for, 429–430 pultrusion, 789–792 pressure-velocity staggering for, 41 under uncertainty, 808–813 Mesh agglomeration strategies, 37 Materials sciences, high-performance computing for, Mesh generation: 899 data centers, 832 MATHCAD, Monte Carlo evaluation with, 281, 282 recommendations for future research, 928 Mathematica system, 494, 501 resources for, 939 codes for fully transient two-dimensional case, use of, 926 516 Meshless local Petrov-Galerkin (MLPG) method, difference-differential equations, 653 226 DIVPAG subroutine, 516 Meshless methods, 225–246, 924 DSolve function, 503 FEM and FDM/FVM vs., 225–228 integral transform, 502–510 local radial point interpolation methods, 230 Integrate function, 505 meshless Petrov-Galerkin methods, 229–230 NDSolve function, 502, 505–510, 516, 645 in natural convection, 238–244 random number generation, 252 nomenclature related to, 245–246 Reimann sum approximation, 652 radial basis functions, 230–235 WorldData routine, 512–513 in simple heat-conduction problem, 235–238 Mathematica Technical Center, 501 smoothed particle hydrodynamics techniques, MathReader, 502 229 INDEX 959

Meshless Petrov-Galerkin (MLPG) methods, Modified embedded-atom method (MEAM), 668 229–230 Molecular dynamics method, 659–689 Mesh parameter, 95 boundary conditions, 669–670 Message Passing Interface (MPI), 904–905 control of temperature/pressure, 670 Metal alloy solidification models, theoretical basis crystallization of amorphous silicon, 680 for, 594–596 effective pair potential for water, 663–665 Method of Manufactured Solutions (MMS), 420–422 equations of motion and potential functions, Method of weighted residuals (MWR), 192 661–662 Mfree2D, 226 formation of clusters/fullerene/carbon nanotubes, Microchannels, turbulence modeling with, 393 681–682 Microgravity conditions, materials processing under, heat conduction and heat transfer, 671–672 775 heat transfer of carbon nanotubes, 682–689 Microscale heat transfer, 623–655 initial conditions, 670 anomalous diffusion, 631–632 integration of Newtonian equation, 669 difference-differential equations by Mathematica, Lennard-Jones potential, 662–664 653 liquid-vapor interface, 673–675 finite differencing, 641–645 many-body potential for carbon and silicon, functional of the free energy, 630–631 666–668 heat equations in, 624–632 nucleation: interfacial resistance, 645–651 and phase change, 677–678 molecular dynamics method in, 660. See also and stability of nanobubbles in confined Molecular dynamics method system, 678–680 nomenclature related to, 653–655 pair potential and EAM for solid metal, 667–669 phonon scattering, 626–628 for phase interface and phase change, 673–682 Reimann sum approximation by Mathematica, potential for larger molecules in liquid phase, 652 665–666 semianalytical computational methods, 636–641 solid-liquid-vapor interactions, 675–676 thermal lagging, 633–636 thermophysical and dynamic properties, 671 thermal waves, 625–626 Momentum equations, 5 two-step heating: discretized: hyperbolic, 629–630 CVFDM, 208 parabolic, 628–629 CVFEM, 208–209 MIMD (multiple instruction, multiple data), pressure-based methods for solving, 43–45 901–904 solution methods, 43–45 Minimod limiter, 29, 30 staggered control volumes for, 40 Mixed condition, 56 Momentum interpolation, 42 Mixed symbolic-numerical algorithms, 494, 500–510 Monotonicity-preserving schemes, 925 integral transform with Mathematica, 502–510 Monte Carlo (MC) methods, 249–291 Mathematica system, 501 for absorbing and emitting media, 281–285 MLE (maximum likelihood estimator) framework, biased, 278–281 529 convection problems, 274–275 MLPG (meshless local Petrov-Galerkin) method, 226 defined, 295 MLPG (meshless Petrov-Galerkin) methods, definite integrals, 253–254 229–230 direct simulation, 276–278 MLSQ (moving least-squares) methods, 924 estimation of error, 286–288 MMS, see Method of Manufactured Solutions estimation of thermophysical properties of Model-based process control, 801–808 complex systems, 274 artificial neural networks, 802–804 and exodus method, 273 complex mold geometrics, 807–808 historical background of, 250 controller architecture, 803–804 for homogeneous medium, 273 example of, 805–806 hybrid, 273 online flow-control schemes, 801–802 importance sampling, 271–273, 278–281 optimization problem, 804 nomenclature related to, 289–291 racetracking in, 806 numerical solution of FDF, 175–177 Modeling errors, 17, 489–490 for optically thick media, 286 Modification of grids, 481–483 radiation problems, 275–276 960 INDEX

Monte Carlo (MC) methods, (continued) Natural convection: random and pseudo-random numbers, 251–253 meshless methods in, 238–244 random walk: in porous media, convection-diffusion for, in two-dimensional geometries, 255–262 513–518 with variable step size, 263–268 Natural coordinates, 103 and shrinking boundary method, 273 Natural-element method (NEM), 924 simplicity of, 273 Navier-Stokes equations: stratified, 419 finite-element method, 117–119 in thermal conduction problems, 254 finite-volume discretization of, 575 transient problems: high-performance computing, 912 with fixed step size, 268–269 for high-performance computing, 915 with variable step size, 269–271 one-dimensional, 84–85 unbiased simulation, 276–278 three-dimensional, 513–514 in uncertainty analysis, 462 Near-net shape structured materials, thermal sprays Morphology-oriented models, 394, 398 for, 775 Moving-boundary problems, 559–589 NEM (natural-element method), 924 adaptive grids, 486–487 NEMD (nonequilibrium molecular dynamics), 671 combined Eulerian-Lagrangian methods, 562 Neumann condition, 56 domain discretization, 562–563 Newmark algorithm, 115 fixed-grid (Eulerian) methods, 561, 569–586 Newtonian equation, integration of, 669 continuous interface method, 569–572, Newton-Raphson scheme, 154 581–586 Nip region (thermoplastics), 792 sharp interface method, 572–586 NLEVMs (nonlinear eddy-viscosity models), 370 interface definition, 562 Node-based schemes, 9, Seen also Finite–element interfacial boundary conditions, 562 methods domain discretization, 562–563 Nodes (mesh), 7 movement and deformation of interface, Noise, 114 563 Noisy convergence rate, 425–426, 429–430 moving-grid (Lagrangian) methods, 561, Nonconformal meshes, 9 563–569 Nonequilibrium molecular dynamics (NEMD), 671 nomenclature related to, 587–589 Nonisothermal flows, direct numerical simulation of, Moving-grid methods, see Lagrangian methods 708–709 Moving least-squares (MLSQ) methods, 924 Nonlinear eddy-viscosity models (NLEVMs), 370 Moving materials or sources: Nose-Hoover´ method, 670 and continuous processing, 769–772 Nucleation: numerical modeling of, 741–743 in freezing of biological materials, 874 MPI, see Message Passing Interface heterogeneous, 678 MRF (Markov random field), 545 homogeneous, 677–678 MRM (multiple-reciprocity method), 135 and phase change, 677–678 Multidomain spectral element method (compressible and stability of nanobubbles in confined system, flow), 709–713 678–680 Multigrid methods, 33–37, 68–73 Numerical-analytical methods, 494–500 Multiple-reciprocity method (MRM), 135 Numerical diffusion, 318. See also False scattering Multiplicative generator, 252, 253 Numerical error, 418–419 Multiscale modeling, high-performance computing Numerical integration, 109–110 for, 900 Numerically exact (as term), 419 Multiscale simulation of thermal phenomena, 936 Numerical methods: Multiwalled carbon nanotubes (MWNTs): hybrid, 494–500 geometry of, 682 computational procedure and convergence water in, 683–684 acceleration, 498–500 “Mushy region” (bioheat freezing of tissue), 881 integral transform method, 495–498 MWNTs, see Multiwalled carbon nanotubes recommendations for future research, 928–931 MWR (method of weighted residuals), 192 Numerical scattering, 319 Numerical simulation, 4 Nanoscale heat transfer, molecular dynamics method Numerical solution procedures, 7–19 in, 660. See also Molecular dynamics method accuracy of, 17 INDEX 961

consistency of, 17 in hierarchic model systems, 490 convergence in, 18 truncation error for, 59 coupling, 17 types and classification of, 54–56 discretization of governing equation, 10–15 Partially absorbing boundaries, 260 domain discretization, 7–9 Particle-tracking algorithm, 713 for linear equations, 15–17 Path to solution (linear equations), 15 nomenclature related to, 48–49 PC clusters, 897–899 nonlinearity, 17 PCE, see Polynomial chaos expansion stability of, 18 PDEs, see Partial differential equations Nusselt number, 378–381 PDF, see Probability density function for three-dimensional cavity center, 517, 518 Peclet` number, 115 for two-dimensional vertical cavity, 516–517 Penalty finite-element method, 118–119 Pennes equation: finite-difference solution of, 863–864 Observations, 295 finite-element solution of, 861–863 Observed rate of convergence, 423–425 Petrov-Galerkin formulation, 116–117, 226 Octree graphs, 485, 487 Phase change: ODE solvers, see Ordinary differential equations manufacturing processes with, 735–737 solvers molecular dynamics of, 673–682 One-dimensional diffusion equation, 11–12 Phase-change problems, 593–619 One-dimensional steady-state conduction, 92–97 backdiffusion, 610–613 One-level rule, 485–486 benchmark problem, 613–617 OPLS potential, 665–666 coarsening, 611–613 Optical fiber drawing, 732 columnar solidification in static casting, coating, 765, 767–768 600–602 feasible domain, 765, 766 control-volume equations using finite-difference molecular-level changes in, 748–749 approximations, 605, 606 neck-down region, 761–764 general control-volume equations, 602–604 numerical modeling of, 759, 761–768 governing macroscopic equations, 596–602 preform diameter, 759 nomenclature related to, 618–619 viscous dissipation in, 759 numerical solution of, 602–613 Optically thick media, Monte Carlo methods for, 286 phase diagram, 609–610 Optimal order of integration, 109 theoretical basis for metal alloy solidification Ordinary differential equations (ODE) solvers, 644, models, 594–596 645 thermal-solutal coupling, 607–609 Oscillatory convergence, 425 Phase diagram, 609–610 Oscillatory divergence, 425 Phase interface, molecular dynamics of, Overrelaxation: 673–682 with alternating direction implicit technique, 68 Phonon Boltzmann transport equation, 672, 936 with Dirichlet boundary condition, 67–68, 71 Phonon scattering, 626–628 with Gauss-Seidel method, 66–67, 71 Physical models: with pressure-based algorithms, 330 recommendations for future research, 927–928 successive, 66–67 use of, 925–926 successive line, 66–67 Piecewise-linear interpolant, 92–94 with tridiagonal matrix algorithm, 67 PISO algorithm, 349–350, 353, 356 Plane strain, 704 Parabolic partial differential equations, 54 Planetary sciences, high-performance computing for, Parabolic two-step model, 628–629 899 Parallel computers (HPC), 896, 901–904 Plenum modeling (data centers), 830, 831 Parallel extensions (adaptive grids), 488–489 Poisson equation, 54 Parallel Virtual Machine (PVM), 904–905 Pollution errors, 435 Parameter uncertainty, 419, 436, 444, 460–462 Polycarbonate surface, impact of liquid drop on, Partial differential equations (PDEs): 585–586 analytical approaches to solving, 493 Polymer extrusion, 732, 750–760 in finite-difference methods, 54–56 chemical conversion in, 756–758 in grid smoothing, 482 feasibility of, 759–760 962 INDEX

Polymer extrusion, (continued) SIMPLE, 333–337 residence time distribution in, 750–753 SIMPLEC, 349 twin-screw, 753–756 SIMPLE variants, 348–349 Polymer-matrix composites, 786–801 subsonic inlet, 344–347 thermoplastic, 792–801 subsonic outlet, 347–348 consolidation stage, 792 supersonic inlet, 347 fabrication of, 792–793 supersonic outlet, 348 heat transfer, 793–797 wall boundary condition, 344 prepregging stage, 792 Pressure-correction equation, 354–356 spherulitic crystal microstructure evolution, PRESTO (pressure staggering option) scheme, 797–801 827 thermosetting, 786–792 PRIME algorithm, 353, 356 liquid composite molding, 767–789 Primitive variables formulation, 193 pultrusion, 789–792 Prior probability density function, 545 Polynomial chaos expansion (PCE), 530–531 Probability, 295 Population, 295 Probability density function (PDF), 171 Porous media: posterior, 545 natural convection in, 513–518 prior, 545 turbulent transport through, see Turbulent Probability distribution, 295 transport through porous media Probability distribution function, 250 Posterior probability density function (PPDF), 545 Programming languages, 903–905, 937 Potential functions: Pseudo-random numbers, 251–253 effective pair potential for water, 663–665 Pseudo-random sequence, 295 and equations of motion, 661–662 Public domain software resources, 939 for large molecules in liquid phase, 665–666 Pultrusion, 789–792 Lennard-Jones potential, 662–664 Pure congruential generator, 252 many-body potential for carbon and silicon, PVM, see Parallel Virtual Machine 666–668 molecular dynamics method, 661–669 Quadrature formulas, 109 pair potential and EAM for solid metal, 667–669 Quadrilateral elements, 487, 488 PPDF (posterior probability density function), 545 Quadtree graphs, 485 Prandtl number, 373, 744 QUICK scheme, 27, 201 Prepregging (thermoplastics), 792 Pressure, 210 discretized equations for: Racetracking (RTM), 806 CVFDM, 210 Racks (data centers), 823, 826 CVFEM, 211–212 Rack-level modeling (data centers), 820 storage of, 38–42, 931 Radial basis functions (RBFs), 136–137, Pressure-based algorithms, 325–364 230–235 all-speed single-fluid flow algorithm, 338–341 Radiative heat transfer, 297–320 boundary conditions, 343–348 boundary conditions, 299–300 continuity and momentum equations, 43–45 boundary-element method, 155–156 discretization: control-angle overlap, 316–317 of kth scalar equation for multifluid flow, 332 convection-diffusion equation, 300–301 for single-fluid flow, 329–332 discrete-ordinates method, 303 extension to compressible flow, 337–342 discretization equation, 306–313 general scalar equation, 329 domain discretization, 306 governing equations, 327–329 false scattering, 318–319 improving Rhie-Chow interpolation, 359–362 finite-volume method, 304–305 for multifluid flow, 353–354 flux method, 302–303 nomenclature related to, 362–364 governing equations, 299–300 performance assessment: importance sampling, 275–276 of multifluid flow algorithms, 356–359 irregular geometries, 313–316 of single-fluid flow algorithms, 351–353 Monte Carlo methods for, 275–276 PISO, 349–350 nomenclature related to, 319–320 pressure-correction equation, 354–356 ray effect, 317–318 INDEX 963

representative works, 319 data center airflow modeling, 825 scattering phase function, 300 high-performance computing, 916 spatial differencing schemes, 313 Reynolds averaged simulation (RAS), 167–168 transient radiative transfer equation, 299–301 defined, 167 Radiative transfer equation (RTE), 301, 303–305 in turbulent HMT computational analysis, 168 Random event, 295 Reynolds number, 391, 744 Random flight, 250 Reynolds transport theorem, 54 Random numbers, 251–253, 295 Rhie-Chow interpolation, 331, 333–336, Random variables: 359–362 discrete, 295 Richardson Extrapolation (RE), 423, 431, 436 in Monte Carlo methods, 250–251 RKPM, see Reproducing kernel particle methods in spectral methods for stochastic inverse Robins condition, 56 problems, 530 RTD, see Residence time distribution Random vector, 295 RTE, see Radiative transfer equation Random vortex model, 177 RTM process, see Resin transfer molding Random walk, 250 process defined, 295 Runge-Kutta method, 46–47 fixed, 268–269 floating, 263–268, 295 SAGE, 897 and importance sampling, 271–273 Sample, 295 for transient problems, 269–271 Sampling error (Monte Carlo method), 287–288 in two-dimensional geometries, 255–262 Sampling methods (sensitivity analysis), 462–466 with variable step size, 263–268 SAR, see Specific absorption rate RANS, see Reynolds-averaged Navier-Stokes Scalar filtered mass density function (SFMDF), RAS, see Reynolds averaged simulation 172–173 Ray effect, 317–318 Scalar transport equation: Rayleigh-Benard convention, 171 conservative form of, 14–15 RBFs, see Radial basis functions general, 6–7 RE, see Richardson Extrapolation conservative form, 6–7 Rectangular elements, 104–105 discretization of, 10–15 Red-black Gauss-Seidel method, 913 domain discretization, 7–9 Reflecting boundaries, 260, 268 nonconservative form, 7 Regression analysis, 463–464 for pressure-based algorithms, 329 Regularity conditions, 526 solution of, 15–17 Regular meshes, 8 Scaled sensitivity coefficient, 458–459 Reimann sum approximation by Mathematica, 652 Scattering phase function, 285, 300 Relaxation parameter, 113 Scripting language interface, 938 Reliability recommendations for future research, SEM, see Sensitivity equation method 931–936 Semianalytical computational methods, 636–641 Representative elementary volume (REV), 391, Semidiscrete Galerkin approximation, 113 595–596 Sensitivity, 419 Reproducing kernel particle methods (RKPM), 229, Sensitivity analysis, 443–460, 462, 466 924 adjoint method, 458–460 Residence time distribution (RTD), 750–753 complex-step method, 450–451 Resin transfer molding (RTM) process: continuum stochastic sensitivity method, and model-based process control, 801–807 533–534 thermosetting-matrix composites, 787–789 differentiation of analytical solutions, 444–448 Resolved-volume simulations, 178 finite-difference methods, 448–450 Resource management grids, 907 sampling methods for, 462–466 Retrospective problems (IHCP), 526 sensitivity coefficient computation, 444–460 REV, see Representative elementary volume complex-step method, 450–451 Reynolds-averaged governing equations: differentiation of analytical solutions, conservation, 6 444–448 gas-phase momentum, 5 finite-difference methods, 448–450 mixture continuity, 5 and importance of parameters, 460 Reynolds-averaged Navier-Stokes (RANS), 716 scaled, 458–459 964 INDEX

Sensitivity analysis, (continued) SIMPLEC algorithm, 349, 353, 356, 359, 834 sensitivity equation method, 452–457 Simple explicit scheme (unsteady conduction), sensitivity equation method for 74–76 multidimensional problems, 458 Simple implicit scheme (unsteady conduction), 74 software-differentiation method, 451 SIMPLEM algorithm, 356 sensitivity equation method, 452–457 Simple Monte Carlo simulation, 276–278 sensitivity equation method for multidimensional Simple point charge (SPC), 663–665 problems, 458 Simple random variable, 295 software-differentiation method, 451 SIMPLEST algorithm, 356, 359 Sensitivity equation method (SEM), 452–457 SIMPLEX algorithm, 353, 356, 359 adjoint methods vs., 458–460 Single-grid a posteriori error estimators, 431–435 for multidimensional problems, 458 Single-walled carbon nanotubes (SWNTs): Sequential iterative variable adjustment procedure applications for, 682 (SIVA), 217–219 formation of, 681–682 Sequential solution procedure, 17 geometry of, 682–683 Serendipity elements, 105 heat conduction along, 684–687 Servers (data centers), 822–823 hydrogen absorption with, 683 SFC, see Space-filling curves thermal boundary resistance of, 687–689 SFMDF, see Scalar filtered mass density function water in, 683–684 SGI-Cray computers, 898 SIP, see Strongly implicit procedure SGS closure problem, see Subgrid scales closure SIVA, see Sequential iterative variable adjustment problem procedure SHAKE algorithm, 669 Skew upstream differencing scheme (SUDS), Shape functions: 200–201 finite-element methods, 12, 97–98 SLOR, see Successive line overrelaxation isoparametric transformations of, 106–109 Smagorinsky based SGS viscosity closure, 170 one-dimensional steady conduction, 93–94 Smoothing of grids, 481–483 Sharp interface method (SIM), 572–586 Smooth particle hydrodynamics (SPH), 229, 924 computational procedure, 580–581 Software-differentiation method (sensitivity finite-volume formulation, 575–580 analysis), 451 for impact of liquid drop on polycarbonate Solidification: surface, 585–586 columnar, in static casting, 600–602, 606 inclusion of immersed boundaries, 573–575 metal alloy models of, 594–596 interface as discontinuity in, 561 microscopic phenomena associated with, 747 for interfacial characteristics of static drops, Solid-liquid-vapor interactions, 675–676 581–585 Solution-adaptive grid generation, 435–437 Shear flow, homogeneous, 704, 721–724 SOR, see Successive overrelaxation Shrinking boundary method, 273 Space-filling curves (SFC), 488–489 SIHCP, see Stochastic inverse heat-conduction Sparse Cholesky matrix solver, 905 problem Sparse coefficient matrix, 64–65 SIM, see Sharp interface method Spatial differencing schemes (radiative heat transfer), SIMD (single instruction, multiple data), 901–904 313 SIMPLE algorithm, 43–45, 333–337 Spatial-multiblock procedure (radiative heat collocated, 334–336 transfer), 314–315 for double-decomposition turbulence models, Spatial oscillations (steady problems), 28–31 393–394 SPC, see Simple point charge for incompressible flow, 85, 336–337, 356–358 SPC/E, see Extended simple point charge for inviscid transonic dusty flow, 359 Species transport, 6, 395–396 multifluid version of, 355 Specific absorption rate (SAR), 871–872 original, 333–334 Spectral methods: performance assessment of, 351–353, 356–359 multidomain spectral element method for and pressure-based algorithms, 326 compressible flow, 709–713 for pressure-velocity coupling, 392 for stochastic inverse problems, 529–542 for turbulent flow field within spatially periodic adjoint problem, 534–535 array, 392 developments in use of, 541–542 variants of, 348–349, 353 gradient calculations, 535 INDEX 965

numerical implementation, 536–538 Stochastic modeling: numerical results with, 538–541 Eulerian-Lagrangian methods, 716–724 representation of random variables, 530–531 inverse, 529 stochastic inverse heat-conduction problem, materials processing under uncertainty, 531–534 809–811 for turbulent convection in porous medium, 395 Stochastic process, 254 SPH, see Smooth particle hydrodynamics techniques Stokes hypothesis, 710 Spherulitic morphology evolution Storage of pressure and velocity, 38–42, 931 (thermoplastic-matrix composites), 797–801 Stratified Monte Carlo, 419 SQUIDs (superconducting quantum interference Strongly implicit procedure (SIP): devices), 906 in high-performance computing, 913 Stability analysis, 79–81, 641–644 for structured meshes, 33 Stability constraint, 75, 82 Strouhal number, 744 Stability of numerical solution procedures, 18 Structured grid generation, 475–476 Staggered storage of pressure and velocity, 39–41 Structured meshes, 9 Stair-stepped meshes, 8 Sturm-Liouville system, 510 Standard error, 295 Subgrid scales (SGS) closure problem, 168, “Standard Man” data, 861–862 170–171 Starches, chemical conversion of, 747–748 Submicron thermal transport, 936 Static casting, columnar solidification in, 600–602 Subparametric discretizations, 130 Static drop, interfacial characteristics of, 581–585 Subsonic inlet, pressure-based algorithms for, Statistical significance, 429 344–347 Steady conduction: Subsonic outlet, pressure-based algorithms for, boundary-element method, 126–139 347–348 discrete equations for, 458–460 Successive line overrelaxation (SLOR), 66–67 finite-difference method, 61–73 Successive overrelaxation (SOR), 66–67 direct methods, 64 SUDS, see Skew upstream differencing scheme Gauss-Seidel procedure, 65–66 Superbee limiter, 29–31 Gauss-Seidel procedure with overrelaxation, Supercomputers, 898–899. See also 66–68 High-performance computing iterative methods, 64 Superconducting quantum interference devices multigrid method, 68–73 (SQUIDs), 906 obtaining algebraic representations, 61–64 Superconductivity, high-performance computing for, solving system of algebraic equations, 64–73 899 finite-element method, 92–102 Superparametric discretizations, 130 one-dimensional conduction, 92–97 Supersonic flows, pressure-based algorithms for, two-dimensional conduction, 97–102 341, 342 Monte Carlo method, 263–265 inlet, 347 Steady-state transient radiative transfer equation outlet, 348 (RTE), 301, 303–305 Surrogate estimators, 433 Steepest descent, method of, 38 SWNTs, see Single-walled carbon nanotubes Stochastic inverse heat-conduction problem Symbolic-numerical algorithms, see Mixed (SIHCP), 529 symbolic-numerical algorithms Bayesian approach to, 542–544 Systems biology, high-performance computing for, continuum stochastic sensitivity method, 900 533–534 System-level thermal modeling (data centers), problem definition, 531–533 827–829 spectral approach to, 529–542 adjoint problem, 534–535 developments in use of, 541–542 Tangent forms of equations, 932 gradient calculations, 535 Taylor series expansions: numerical implementation, 536–538 finite-difference methods, 10–11, 57, 58 numerical results with, 538–541 upwind scheme, 26–27 representation of random variables, 530–531 TDMA, see Tridiagonal matrix algorithm stochastic inverse heat-conduction problem, TE, see Truncation error 531–534 Technology infrastructure facilities, see Data centers 966 INDEX

TECPLOT, 917 Monte Carlo methods for: TeraGrid, 910–911 with fixed step size, 268–269 Ternary eutectic alloy diagram, 610 with variable step size, 269–271 Tersoff potential, 666–667 in solids, 139–155 Tetrahedral elements, 110, 111, 487, 488 dual-reciprocity method, 142–143 Thermal boundary resistance: Laplace transforms, 140–142 between carbon nanotube and surrounding time-dependent fundamental solution, materials, 687–689 144–155 in small systems, 672 Transient radiative transfer equation (TRTE), Thermal cycle in materials, 749–750 299–301 Thermal lagging, 633–639 convection-diffusion equation vs., 300–301 dual-phase-lag model, 633–636 linearized, 308–309 general behavior of, 636–639 steady-state form of, 301 Thermal materials processing, 730–733 Transition flow, 390 Thermal-solutal coupling, 607–609 Transonic flows, 341, 342 Thermal sprays, for near-net shape structured Transport equations: materials, 775 development approaches for, 410–412 Thermal therapy (TT), 870–871 filtered density function, 172 Thermal waves, 625–626 time averaging, for porous medium model, Thermoplastic-matrix composites, 792–801 392–393 consolidation stage, 792 for truncation error, 432 fabrication of, 792–793 Triangular elements, 102–104, 487, 488 heat transfer, 793–797 Triangular heat flux profile, 538–540, 548–551 Triangulation, Delaunay, 476–480, 483, 489 prepregging stage, 792 Tridiagonal matrix algorithm (TDMA), 32 spherulitic crystal microstructure evolution, line, 32 797–801 overrelaxation with, 67 Thermoplastic polymers, 786 Trilinear hexahedron, 110–112 Thermosetting-matrix composites, 786–792 TRTE, see Transient radiative transfer equation liquid composite molding, 767–789 Truncation error (TE), 17 pultrusion, 789–792 finite-difference methods, 58–59 Thermosetting polymers, 786 transport equation for, 432 θ, discretized equations for: TT, see Thermal therapy CVFDM, 205–206 Turbulence modeling, 369–386. See also Turbulent CVFEM, 106–107 transport through porous media θ method, 113–115 and grid convergence, 427 3D: heat transfer through porous media, 407–409 discretization of surfaces in, 225 high-performance computing, 916 hexahedra and tetrahedra with, 488 low-Reynolds-number, 370 Three-dimensional elements, 110–112 nomenclature related to, 385–386 Tikhonov regularization, 527, 542 UMIST-A scheme, 375–381 Time-advancement strategy, 17 capabilities of, 378 Time-dependent problems: flows with intense wall heating, 376 AAE applied to, 434–435 kinetic energy dissipation rate next to wall, heat diffusion, 112–115 377 transient heat conduction in solids, 144–155 upflow in vertical pipe, 378–381 Time-marching problems, 86 UMIST-N scheme, 381–385 high-performance computing, 912 LRN and cubic nonlinear EVM vs., 382–395 RBFs in, 233–234 mildly heated disk spinning about its own transient flow in solids, 146, 149–150 axis, 382–385 Time-splitting method, 739 skewing of velocity profile across sublayer, Time-stepping scheme, 566 381–382 TIP4P, 664–665 viscosity-affected sublayer, 370–375 Tissue engineering, 873 wall-function approaches, 375 Transient conduction: Turbulent flows: boundary-element method for, 139–155 Eulerian-Lagrangian simulations for, 697–726 INDEX 967

large eddy simulation for, 167–180 GCI estimation of, 430–431 Reynolds-averaged governing equations for, 5 importance factors, 461, 464 Turbulent Prandtl number, 373 in inverse problems, 528 Turbulent transport through porous media, materials processing computation under, 389–414 808–813 macroscopic pressure gradient determination, optimization, 809–813 406–407 stochastic model for, 809–811 models for, 400–405 numerical vs. parameter, 419 based on double-decomposition, 399 parameter, 436, 444 based on morphology, 398 sampling methods for, 462–466 based on time averaging, 396 use of term, 418 based on volume averaging, 394, 397, 398 Unequal-order interpolation schemes, 41 comprehensive synthesis of, 396–406 Uniform distribution, 295 double-decomposition methodology, 393–394 UNIVAC, 897 heat transfer, 407–409 UNIX, 898 morphology-oriented, based on Unsteady conduction, 73–79 volume-averaged theory, 394, 398 adiabatic boundary conditions, 78 volume-averaging Reynolds-averaged convective boundary condition, 79 governing equations, 391–392 fixed-value boundary conditions, 77–78 nomenclature related to, 412–414 implicit schemes, 77–79 species transport, turbulent equation for, simple explicit scheme, 74–76 395–396 specified boundary flux condition, 78–79 through dense plant canopies, 394–395 in two and three dimensions, 81–84 time averaging the transport equations for porous unsteady conduction (diffusion) equation, 73–74 medium model, 392–393 Unsteady conduction (diffusion) equation, 73–74 transport equations development approaches, Unstructured meshes (grids), 9 410–412 discretization of convection-diffusion equation, Twin-screw extrusion, 753–756 18–22 Two-dimensional steady-state conduction, 97–102 generation of, 476–480 Two-level rule, 484 gradient calculation, 22–25 Two-phase (two-fluid) flows: Grid-Convergence Index for, 429–430 carrier phase, 699–700 pressure-velocity staggering for, 41 dispersed phase, 700–702 radiative heat transfer, 315–316 Eulerian-Lagrangian simulations for, 697–726 Upwind differencing, 116 large eddy simulation, 177–178 Upwind scheme, 20 two-way coupling, 702 flow-oriented, 202–204 Two-step heating: higher-order schemes, 26–28 hyperbolic, 629–630 mass-weighted, 204–205 parabolic, 628–629 in pressure-based algorithms, 340–341 second-order, 27 UMIST-A scheme, 375–381 third-order, 27 capabilities of, 378 flows with intense wall heating, 376 kinetic energy dissipation rate next to wall, 377 Valence, vertex, 476, 482 upflow in vertical pipe, 378–381 Validation, 419–420, 438. See also Verification and UMIST-N scheme, 381–385 Validation LRN and cubic nonlinear EVM vs., 382–395 Variance, in Monte Carlo simulations, 286 mildly heated disk spinning about its own axis, Variance of distribution, second central moment, 382–385 295 skewing of velocity profile across sublayer, VAT, see Volume-averaged theory 381–382 VAX computers, 897 UMIST scheme, 201 Velocity: Unbiased Monte Carlo simulation, 276–278, in original SIMPLE algorithm, 333 295 storage of, 38–42, 931 Uncertainty, 444, 460–466 Velocity-scalar filtered density function (VSFDF), conservatism of estimators, 429 173–175 968 INDEX

Velocity-scalar filtered mass density function in optical fiber drawing, 759 (VSFMDF), 171–175 in polymer extrusion, 750 Verification and Validation (V&V), 418–439 Viscous flow, manufacturing process modeling of, calculating observed rate of convergence, 743–744 424–425 Visualization, in high-performance computing, Calculation Verification, 422–423 916–917 Code Verification, 420–422 Volume-averaged theory (VAT), 304, 398 degraded convergence rate, 426 Von Karman constant, 372 Fs :U95 empirical correlation, 428, 431, Von Neumann necessary condition, 81 437–438 Von Neumann stability analysis, 79–81, Grid-Convergence Index, 423–424, 429–431, 641–644 434–435 Voronoi diagram, 478 confirmations of, 426–428 VSFDF, see Velocity-scalar filtered density function error estimation vs., 430–431 VSFMDF, see Velocity-scalar filtered mass density for unstructured grids and noisy convergence function rate, 429–430 VSL, see Viscosity-affected sublayer (VSL) grid-convergence studies, 423 V&V, see Verification and Validation Method of Manufactured Solutions, 420–422 noisy convergence rate, 425–426 Wall boundary condition, 344 oscillatory convergence, 425 Wall functions: resources, 939–940 turbulence modeling, 370–375 Richardson extrapolation, 431 UMIST-A scheme, 375–381 sequence of, 418–420 UMIST-N scheme, 381–385 single-grid error estimators, 431–435 Wave equation, 55 statistical significance, 429 Weight function, 12 terminology of, 418–419 Weinbaum-Jiji (WJ) bioheat equation, 860–861 Validation, 438 Well-posed problems, 526–527 Zhu-Zienkiewicz estimators, 433–437 WJ equation, see Weinbaum-Jiji bioheat equation Vertex-based schemes, 9 Work units, 71 Vertex-centered FVMs, 193 Vertex valence (degree), 476, 482 Young-Laplace equation, 673 Vertices (mesh), 7 Viscosity, artificial, 28 Viscosity-affected sublayer (VSL), 370–375 Zero-equation eddy viscosity model, 392 Viscous dissipation: Zero normal derivative boundary conditions, 78 in manufacturing processes, 735 Zhu-Zienkiewicz (ZZ) estimators, 433–437