computational heat transfer in complex systems: a review

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Computational Heat Transfer in Complex Systems:A Review of Needs and OpportunitiesJayathi Y. MurthyBirck Nanotechnology Center, Purdue University, [email protected]

Sanjay R. MathurPurdue University, [email protected]

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Murthy, Jayathi Y. and Mathur, Sanjay R., "Computational Heat Transfer in Complex Systems: A Review of Needs and Opportunities"(2012). Birck and NCN Publications. Paper 1239.http://dx.doi.org/10.1115/1.4005153

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Jayathi Y. Murthye-mail: [email protected]

Sanjay R. Mathure-mail: [email protected]

School of Mechanical Engineering and

Birck Nanotechnology Center,

Purdue University,

West Lafayette, IN 47907-1288

Computational Heat Transfer inComplex Systems: A Review ofNeeds and OpportunitiesDuring the few decades, computational techniques for simulating heat transfer in com-plex industrial systems have reached maturity. Combined with increasingly sophisticatedmodeling of turbulence, chemistry, radiation, phase change, and other physics, powerfulcomputational fluid dynamics (CFD) and computational heat transfer (CHT) solvershave been developed which are beginning to enter the industrial design cycle. In this pa-per, an overview of emerging simulation needs is first given, and currently-availableCFD techniques are evaluated in light of these needs. Emerging computational methodswhich address some of the failings of current techniques are then reviewed. New researchopportunities for computational heat transfer, such as in submicron and multiscale heattransport, are reviewed. As computational techniques and physical models becomemature, there is increasing demand for predictive simulation, that is, simulation which isnot only verified and validated, but whose uncertainty is also quantified. Current work inthe area of sensitivity computation and uncertainty propagation is described.[DOI: 10.1115/1.4005153]

Keywords: finite volume, multiscale, sensitivity analysis, immersed boundary,uncertainty quantification, phonons, Boltzmann transport equation

1 Introduction

Over the last few decades, computational fluid dynamics (CFD)and computational heat transfer (CHT) have increasingly becomean integral part of the industrial design and analysis cycle. Today,nearly every major company in every major industry routinelyuses CFD software. Comprehensive simulation systems encapsu-lating computer-aided design (CAD), mesh generation, CFD, andstructural analysis form the analysis backbone in industries asdiverse as automotive, aerospace, chemical processing, electronicscooling, energy, food processing, materials processing and manyothers. Increasingly, simulations are large-scale and sophisticated,routinely involving millions of degrees of freedom. Coupled flow,heat transfer, turbulence, chemistry, and radiation computationsare common. Massively-parallel computing on commodity clus-ters is increasingly the norm, and sophisticated graphics and post-processing support the engineer in making technical assessmentsof his=her designs. To the unpracticed eye, it would seem that in ascant two or three decades, the dream of having computer experi-ments replace difficult, expensive, and time-consuming laboratorytesting has finally come true.

Yet, despite all its success, it would be difficult today to find anindustrial practitioner or a university researcher who would bewilling to use CFD exclusively without supporting experimentalconfirmation. In many industries, CFD is used only broadly, as agoad to intuition, but not for quantitative decision-making. Fur-thermore, though CFD is used in analysis, its inclusion in optimi-zation and design is at a nascent stage, hindered by still-dauntingcomputational cost. In this paper, we review current and emergingneeds in complex system simulation, and critically assess popularcomputational methodologies in meeting these needs. We reviewemerging computational methodologies which seek to address theshortcomings of existing methods. Opportunities for research inemerging areas are identified. Increasingly, the industrial user isinterested not only in single-point simulations, but in understanding

the sensitivity of his=her system outputs to inputs, and how errorbands and lack of knowledge in system inputs and models translateto uncertainties in his=her predictions. Since most CHT problemsinvariably involve CFD and use similar numerical techniques, wespeak of the two interchangeably in the rest of the paper.

2 Emerging Simulation Needs and Opportunities

As CFD and CHT simulations become more ambitious, existingCFD approaches must be improved or augmented. Some criticalissues are identified below.

2.1 Mesh Generation for Complex Geometries. With cur-rent automatic mesh generation methods, it is not uncommon forthe industrial user to spend as much as 80% of engineer time onmesh generation, and as little as 20% on actually obtaining the nu-merical solution. Good quality mesh generators are critical. Alter-natively, the onus of addressing complex geometries must shiftfrom the user to the algorithm, through the use of meshless meth-ods [1] or immersed boundary methods [2], or other similar tech-niques. If Moore’s Law remains valid, computers will double inspeed every 2 years whereas humans will not. Thus, the user maybe willing to incur an increase in computational time if theapproach reduces human time.

2.2 Integrated Solvers. Knowledge domains remain all tooseparate, even today. Though CFD and CHT are well-integrated,future complex system simulations will require integration withother knowledge domains. For CHT, integration with structuralanalysis solvers for thermal stress analysis, or solvers for chargetransport and electrical potential to compute Joule heating, are al-ready necessary in many applications. In many instances, looseintegration will suffice, but in others, such as those involvingfluid-structure interaction [3] or fluid-structure-electrostatics inter-actions in micro-electro-mechanical systems (MEMS) [4], forexample, tight coupling between domain-based solvers is essen-tial. The implications for conservation of heat, mass, momentum,and charge must be carefully considered when the various

Contributed by the Heat Transfer Division of ASME for publication in the JOURNALOF HEAT TRANSFER. Manuscript received September 1, 2010; final manuscript receivedJuly 27, 2011; published online January 18, 2012. Editor: Yogesh Jaluria.

Journal of Heat Transfer MARCH 2012, Vol. 134 / 031016-1CopyrightVC 2012 by ASME

Downloaded From: http://heattransfer.asmedigitalcollection.asme.org/ on 09/11/2013 Terms of Use: http://asme.org/terms

domain-based solvers are based on different numerical approaches(finite volume versus finite element, for example). Alternatively,unified multiphysics techniques addressing different knowledgedomains may be developed. Finite element solutions of fluids andstructures are now available [3], and finite volume solvers forstructural analysis have also been published [5]. Node and cell-based finite volume solvers for semiconductor device simulationhave long been a staple of the micro-electronics domain [6].

2.3 Sensitivities, Adjoints, and Uncertainty Quanti-fication. In nearly all problems of industrial interest, the user isinterested in the variation of his=her outputs with changes in inputparameters. These sensitivities allow better design of experimentsand computations. The availability of accurate derivative informa-tion also allows more efficient optimization. Furthermore, innearly every practical application, inputs are never preciselyknown or measurable. The user seeks to know which inputs arethe most important, and how the uncertainty in inputs affects out-put predictions. At present, most commercial CFD solvers provideonly single-point simulation capabilities, i.e., the ability to per-form a CFD simulation for a single combination of inputs.Recently, a variety of advances in automatic code differentiation[7] and uncertainty propagation [8,9] have been published whichpromise enormous dividends for the CHT community. The exten-sion and application of these techniques to thermal transport incomplex systems is a fruitful area for new research.

2.4 Large-Scale Parallel Algorithms. At the U.S. weaponslaboratories, petascale platforms are now available [10] and dis-cussions of future exascale platform specifications are alreadyunderway. Current capacity in many industry and university set-tings is typically in the multiteraflop range, and petascale capacityis expected over the next decade. Multiprocess multicore environ-ments are the norm for commodity clusters.

During the last few years, graphics processing units (GPUs),which were originally designed for rendering and visualization,have emerged as massively-parallel coprocessors to the centralprocessing unit (CPU) [11]. Indeed, GPUs today outperformCPUs, both in floating point performance and memory bandwidthby a factor of ten [12] and the gap is rapidly growing, as shown inFig. 1. Today it is possible to configure desktop supercomputerswith hundreds of stream processors using GPUs to deliver teraflopperformance, all for a few thousand dollars. Recent CFD compu-tations exploiting GPUs have reported acceleration factors of50–100 for large-enough problems (of the order of tens of millionsof grid points) on commodity architectures [13]. Furthermore,many of the bottlenecks restricting the use of GPUs for CFD haverecently been removed. Unlike old fixed-function pipelines whichmade general purpose programming difficult, GPUs now allowlow-level access to hardware, combined with a high-level pro-gramming environment through programming models such as

Stanford University’s BrookGPU [14] and NVIDIA’s ComputeUnified Device Architecture (CUDA) [15].

CFD algorithms must be tailored to take advantage of theseemerging platforms. Sequential implicit pressure-based algo-rithms [16] have long been the staple of the CHT community, butthis class of algorithm was tailored for low-memory serial plat-forms. Because the governing equations are solved sequentially,the computational and memory load on a single core is low, andintercore communication dominant, leading to poor scaling. Thepresence of large numbers of cores increases the surface-to-vol-ume ratio of partitions, and renders implicit algorithms effectivelyexplicit across partition boundaries, with serious consequences forthe convergence of incompressible flow solvers. Point-coupledalgorithms offer the possibility of better scalability [17,18] byincreasing the work per core. All CFD solvers must contend withthe low-memory bandwidth of multicore designs. The scalabilityof linear solvers on multicore architectures is critical, especiallyfor pressure-based methods and low-Mach number flows. Here,the convergence of the pressure field is frequently a bottleneck tooverall convergence, and is the most taxing for linear solvers aswell. For GPUs, most published CFD algorithms are at presentbased on explicit time-stepping techniques [13] because of the suit-ability of this type of algorithm to streaming, and the paucity of ef-ficient sparse linear solvers on these platforms. This presents asignificant departure from the algorithms widely used in the CHTcommunity, where implicit schemes have been the norm. Develop-ing robust algorithms which scale over tens of thousands of coresand on new GPU architectures will become critical for simulatinglarge complex multiphysics systems over the next decade.

2.5 Improvements to Physical Models. Ultimately, the suc-cess of CFD and CHT in complex system simulation will dependon the fidelity of the predicted results. Though modeling chal-lenges exist in nearly every flow problem of industrial interest, wefocus here on three areas with widespread impact.

Turbulence modeling remains an enduring bottleneck in com-plex system simulation, even in single phase nonreacting flows.Direct numerical simulations remain out of reach for most practi-cal industrial Reynolds numbers, though large-eddy simulationis now increasingly used. Nevertheless, Reynolds-averagedNavier—Stokes modeling remains the mainstream approach inindustry, and will continue to limit solution fidelity across a wideswath of industrial applications for the foreseeable future [19].

Recent interest in energy-related CFD and CHT spurredrenewed interest multiphase flow simulation [20,21]. Liquid–vapor multiphase flows in the presence of phase changeform the backbone of flow computations in many industries.Eulerian–Eulerian multiphase flow approaches are widely used,but at present, regime transition is done empirically for the mostpart. Large density differences between the vapor and liquidphases and the presence of evolving free surfaces make for verychallenging CFD problems. Thus, numerous opportunities existfor the development of improved physical models and computa-tional techniques, particularly in the direct simulation of complexevolving interfaces. Furthermore, integrated solvers computingsingle and multiphase flow, heat transfer, turbulence, stress analy-sis, and materials modeling in very complex domains are neces-sary to support the enormously diverse computational needs inindustry [22].

Significant new opportunities for modeling and simulation existin emerging technological areas, for example in the energy sectorand in microtechnology and nanotechnology. Thermal transport atsubmicron scales has recently become the focus of much researchin the heat transfer community [23–26]. Here too, opportunitiesfor the development novel computational approaches abound, par-ticularly in the integration of atomistic descriptions with meso-scale and macroscale approaches [27,28].

In this paper, we first review the state of the art in existing andwidely used CFD techniques to provide a context for later

Fig. 1 Evolution of peak floating point performance gapbetween GPUs and CPUs [12]

031016-2 / Vol. 134, MARCH 2012 Transactions of the ASME


sections. We then present results from recent work in three areasof relevance to the CHT community: (i) the development ofimmersed boundary techniques which alleviate the burden ofmesh generation, and facilitate the simulation of complex geomet-rical domains and moving=deforming bodies, (ii) computationaltechniques for thermal transport in emerging microelectronics,and (iii) the development of CFD and CHT solvers capable ofcomputing sensitivities and propagating uncertainty.

3 Unstructured Finite Volume Methods (FVMs)

Over the last two decades, cell [29,30] and node-based [31,32]unstructured finite volume methods have become the mainstay ofindustrial CFD and CHT computations. A overview of this classof techniques may be found in Ref. [33]. In this section, we brieflyreview a basic cell-based unstructured FVM to set the stage forthe sections that follow.

In a typical cell-based FVM, the computational domain is di-vided into arbitrary unstructured convex polyhedral called cells orcontrol volumes on which the conservation principle is enforced.A typical cell is shown in Fig. 2. General nonconformal interfaces,such as the face abc, are admitted automatically by consideringthe cell C1 to be four-sided; such flexibility makes it easy toperform hanging-node mesh adaptation. Also, hybrid meshescomposed of cells with different shapes are admitted.

The governing equation for the transport of a general scalarvariable / is given by

@

@tðq/Þ þ r � ðqV/Þ ¼ r � C/r/þ S/: (1)

Integrating Eq. (1) over the control volume C0 in Fig. 2, yieldsthe cell-balance equation

@

@tðq/Þ

� �0

rV0 þXf

Ff/f ¼Xf

Df þ S/DV0: (2)

Here, Ff is the mass flow rate out of the cell C0 across face f, DV0

is the volume of the cell C0, and Df is the diffusive transportthrough the face f into the cell C0.

Details of the discretization procedures may be found in Ref.[29]. Briefly, second-order accurate spatial operators are used todiscretize the convection, diffusion, unsteady and source terms inEq. (2). Cell gradients, necessary to determine secondary gradientterms and higher-order convection schemes, are computed using acell-based linear least-squares procedure. The resulting discreteequations are solved using unstructured iterative linear solverssuitable for sparse matrices, such as algebraic multigrid schemes[34] or Krylov subspace solvers [35]. For fluid flow, colocatedschemes are now the norm, with added- dissipation schemes beingemployed to mitigate checker-boarding [36].

For low-speed incompressible flows, sequential pressure-basedmethods such as SIMPLE and its variants are the most commonly

used [16]. Increasingly, though as memory becomes cheaper,coupled schemes, particularly those based on the multigrid idea,are being explored [17,37,38]. Over the next decade or two, wemay see a shift away from sequential solvers to these techniques.Coupled schemes promise better performance (5–10 times se-quential) over serial computations, albeit with a significantincrease in storage. For parallel platforms, performance gains mayresult from an increase in per-processor work compared to se-quential implementations. However, coupled algorithms based onthe multigrid idea must contend with the relatively poor perform-ance of multigrid methods on distributed memory parallel plat-forms. Coupled solvers result in block-sparse matrices which maybe stiff, and depending on the nature of the underlying coupling,may not be diagonally dominant. The search continues for robustscalable linear solvers for these types of coupled systems.

For complex multiphysics systems, unstructured finite volumemethods form a general basis to address a wide swath of physicsbased on conservation principles. Finite volume methods for radi-ative transport, stress analysis, charge transport, submicron heattransfer, and many others have already appeared in the literature[5,39–42].

4 Immersed Boundary Method (IBM) for Flow and

Heat Transfer

As mentioned in a previous section, mesh generation remainsone of the most time-consuming activities in the overall simula-tion cycle. There is great interest in CFD techniques which eithereliminate the mesh generation step altogether, as with meshlessmethods, or render it extremely easy, as with IBM [2] employingCartesian background meshes. IBM also provides a viable solutionto fluid-structure problems in which extreme mesh deformationmakes conventional arbitrary Lagrangian–Eulerian (ALE) techniques[3] difficult to use. These types of problems occur in in-cylindercombustion, rotor-body interaction, parachute and airbag deploy-ment, in mixing tanks, and in MEMS-based contact switches, amongothers. In many of these applications, interactions between the fluidand structure may cause extreme deformation and displacement,which may, in turn, significantly change the fluid field.

Another application of IBM is in the development of higher-order computational schemes for direct numerical simulations(DNS) and large-eddy simulations (LES) of turbulence [49,51].Higher-order methods for unstructured finite volume schemes arestill in development. One quick pathway to obtaining higher-orderschemes on arbitrary geometries is through IBM.

IBM is a fixed-grid method wherein the complex-shapedboundary between the solid and fluid is immersed in a simplebackground mesh, as shown in Fig. 3. The body-shape does notconform to the background mesh, but intersects it arbitrarily. Arecent review of the field has been given in Ref. 2 and its applica-tion to turbulent flows has been reviewed in Ref. [43]. IBM wasoriginally developed by Peskin [44] to simulate flow through heartvalves. In the original formulation, the deformable valve wall wasrepresented as a set of nodal forces which were incorporated inthe fluid momentum equations as line forces. A number of var-iants of the method have since been developed [2]. In recentwork, Mohd-Yusof [45] and Fadlun et al. [46] dispensed with theidea of incorporating equivalent forces in the fluid momentumequation to represent the action of the interface. Instead, fluidnodes closest to the interface are identified, and a velocityinterpolated from the interface and an appropriate interior fluidneighbor is imposed on these near-interface nodes. More recently,Gilmanov and Sotiropolous [47] represented the interface usingtriangular unstructured meshes to facilitate the identification of asharp interface. This idea was combined with the material pointmethod to solve for the solid stress and deformation field in Ref.[48], coupled to an IBM treatment of fluid-structure interaction.A recent application of IBM to turbulent flow simulation andconjugate heat transfer using local hanging-node mesh adaptationmay be found in Ref. [49].

Fig. 2 Typical cell (C0) and neighbors for cell-based finite vol-ume scheme [29]

Journal of Heat Transfer MARCH 2012, Vol. 134 / 031016-3


We have developed general-purpose IBM methods suitable forunstructured meshes (and therefore by default, for structuredmeshes also) [50]. Our specific interest is in the computation oflow-Re viscous flows in MEMS and microfluidics applications, inwhich the accurate representation of near-wall viscous stressesand heat diffusion is critical. We employ the colocated cell-centered pressure-based formulation described in Ref. [29]. In ourmethod, the solid is superposed on a background unstructuredmesh, as shown in Fig. 3(a). Cells are marked as “fluid,” “solid,”or “immersed boundary (IB) cell;” IB cells contain the interface.Cell faces separating IB cells and fluid cells are marked as IBfaces. In the colocated formulation used in our work, a neighbor-hood of fluid centroids, and points on the solid surface are identi-fied, and the face velocity vector Vb (Fig. 3(b)) is found eitherthrough linear least-squares or quadratic least-squares interpola-tion. Scalars such as temperature are treated in similar fashion.These quantities define the flux of mass, momentum and scalars tothe cell C0. A solver based on the SIMPLE algorithm is developedto solve for the flow field.

To establish the accuracy of the method, we consider the prob-lem of thermal transport in a square domain in the presence of adecaying vortical flow [51]. The flow field is assumed to be

ux ¼ � cosðpxÞ sinðpyÞe�2p2t=Re

ux ¼ sinðpxÞ cosðpyÞe�2p2t=Re

The temperature satisfies

@T

@tþ @uxT

@xþ @uyT

@y¼ a

@2T

@x2þ @2T

@y2

� �þ Sðx; y; tÞ

where

Sðx; y; tÞ ¼ 0:5p ux sinð2pxÞ þ uy sinð2pyÞ� �

e�4p2at

The corresponding exact solution for temperature is

Tðx; y; tÞ ¼ 0:25 cosð2pxÞ þ cosð2pyÞ½ �e�4p2at

where a is the diffusivity of the scalar. Figure 4 shows the geo-metric arrangement of fluid cells and the immersed solid. Theexact solution is imposed on the solid, and on the external do-main boundaries. The objective is to compute the temperaturefield in the fluid and to establish the order of accuracy of ourcomputational scheme. Figures 5(a) and 5(b) show the order ofconvergence obtained for two different Peclet numbers, respec-tively. Two different numerical schemes (upwind and central)are used in the fluid for either linear least-squares or quadraticleast-squares interpolation for the IBM face values. The figuresshow that consistent second-order accuracy may be obtainedusing IBM.

Figure 6 shows the computation of a steady flow past a station-ary cylinder immersed in an bounded uniform flow at Re¼ 1.0.The problem is first solved using FLUENT with a body-fittedmesh of 4356 triangular cells. It is then solved by the immersedboundary method using a Cartesian mesh of 5000 cells, as shownin Fig. 6(a). Figure 6(b) shows a comparison of the velocity mag-nitude between FLUENT [52] and our IBM technique on differ-ent vertical lines in the domain. The velocity is scaled by inletvelocity U and the length is scale by cylinder diameter D. TheRMS difference between the FLUENT and IBM results is lessthan 1.4%.

Our computations indicate that the immersed boundary methodcan yield reasonable-quality results, and can be easily integratedinto colocated pressure-based solvers that form the mainstay ofmany commercial CFD codes. Thus, the user may generate onlyeasy-to-generate Cartesian meshes if desired, or more complexbody-conforming meshes, depending on the competing demandsof set-up speed and accuracy; the solver would handle either orboth situations in a unified way. Work is underway to generalizethe unstructured IBM formulation to more complex boundary con-ditions involving velocity and temperature slip in microfluidicsand submicron thermal transport, and to integrate the techniquewith other knowledge domains, including structural mechanicsand charge transport.

Fig. 3 Immersed boundary method. (a) An arbitrarily-shapedsolid is immersed in a background fluid mesh. The markings ofcells as “fluid,” “solid,” and “IB cell” are shown. (b) Cell no-menclature near IB face.

Fig. 4 Computational domain for decaying vortex problem



5 Modeling and Simulation of Submicron Thermal

Transport

Over the last decade, significant attention has been focused onsubmicron thermal transport phenomena in applications such asmicroelectronics, nanocomposites, development of thermoelectricmaterials, thermal interface materials, and many others. In nearlyall these applications, atomistic phenomena on the scale of 1–100nm must be integrated with mesoscale descriptions (tens to hun-dreds of nanometers) and eventually, with macroscale descriptions(tens of microns and greater). Similarly ranges in timescales(picoseconds to seconds) are also encountered. Physical modelsand computational techniques which can address this spread inlength and time-scales are a rich area for research. Moreover, dif-ferent carriers (electrons, phonons, gas molecules) may be primar-ily responsible for transport; interactions between carriers, forexample, between electrons and phonons in Joule heating, or atmetal-dielectric interfaces, is also common.

Transport modeling based on the Boltzmann transport equation(BTE) provides a unifying framework within which to model car-rier transport in the mesoscale regime, when wave effects may beignored; furthermore, the BTE yields the continuum equations—drift-diffusion equation for electrons, Fourier conduction for pho-nons and the Navier–Stokes equations for gas molecules—in thecontinuum limit. If atomistic effects are incorporated as appropri-ate properties and interfacial conditions in the BTE, a single uni-fied framework from the mesoscale to the macroscale may beformulated. Finite volume methods, such as that described previ-ously, are well-suited to address this entire range of physics. Wedescribe below recent work in the development of thermal trans-

port models based on the BTE, and solution acceleration schemesto facilitate convergence across a large range of Knudsennumbers.

5.1 Phonon BTE. The steady-state, nongray BTE for a pho-non band of frequency x and polarization p under the relaxationtime approximation is given in energy form by

r � vx;pe00x;p

� � ¼ e0x;p � e00x;psx;p

: (3)

where e00x;p is the volumetric energy density per unit solid angle ata given frequency x and polarization p, and e0x;p is the corre-sponding equilibrium energy density given by the Bose-Einsteindistribution. The relaxation time corresponding to (x,p) is sx;pand the corresponding group velocity is given by vx;p.

The phonon spectrum is divided into frequency bands, as shownin Fig. 7, and Eq. (3) is written for each band for each polarizationunder the assumption of an isotropic Brillouin zone; a fully-anisotropic treatment has been considered in Ref. [62]. Dispersioncurves are computed using the procedures described in Ref. [53]using the environment-dependent interaction potential (EDIP)[54] and the group velocity is computed from these dispersioncurves. The most significant unknown in Eq. (3) is the relaxationtime sx;p. This may be computed by fitting experimental data,based on simplified models of Umklapp, impurity and boundaryscattering, as in Refs. [55] and [59]. More accurate representationsemploying detailed atomic structure and Fermi’s Golden Rulewith strict enforcement of energy and momentum conservationrules have also been published by us and other researchers

Fig. 6 Low Re flow over a stationary cylinder in a channel.Comparisons of the velocity magnitude are made with FLUENTalong three vertical lines in the domain

Fig. 5 Order of convergence of IBM for decaying vortex prob-lem for (a) Pe5 1, and (b) Pe520



[56,57]. Recently, Henry and Chen published mode-wise single-mode relaxation times using molecular dynamics simulations[58]. Figure 8 shows a comparison of the single-mode relaxationtimes for Si longitudinal acoustic (LA) phonons using Fermi’sGolden Rule, and those from Ref. [58] using molecular dynamics;the comparison is reasonable.

We have developed finite volume schemes for the solution ofEq. (3) [42,59,60] based on the discretization proceduresdescribed in the sections above. In this approach, the frequencyspectrum is discretized into bands, the angular space into controlangles, the spatial domain into control volumes and time into dis-crete time steps. Eq. (3) is integrated over space, time, angle, andfrequency to yield discrete balances of phonon energy over eachcontrol volume for the phonon group under consideration. Thediscrete equation set is solved sequentially and iteratively as adefault, using an standard algebraic multigrid scheme. Advances

from the CFD literature, such as unstructured meshes, higher-order discretization schemes, solution adaptivity, parallel process-ing, and the like, are thus automatically inherited by thisapproach. These features vastly amplify our ability to addresscomplex geometries, such as those occurring in nanocomposites,nanoporous materials or nanoparticle beds.

5.2 Electron-Phonon Coupling in MOSFETs. We now con-sider the problem of self-heating and submicron thermal transport inmodern metal oxide semiconductor field effect transistors (MOS-FETs). A typical bulk Si NPN field effect transistor (FET) is shownin Fig. 9. Joule heating due to electron–phonon coupling causes theformation of hot spots in the channel region of the transistor; thesehot spots are of the order of about 10–20 nm. Subthreshold leakage,increasingly a contributor to self-heating in microelectronics, scalesexponentially with channel temperature [61]. This requires the care-ful resolution of thermal transport at this scale.

Electron–phonon coupling in silicon FETs transfers energy toselect phonon groups, typically optical and acoustic phonons atthe Brillouin zone edge [62]. These phonon groups have lowgroup velocities, and short scattering time scales, on the order ofpicoseconds. The intensity and size of the hotspot depends onwhich phonon groups receive energy from electron–phonon scat-tering, how fast they move, and how quickly they scatter energyto other faster-moving phonon groups. Thus, careful resolution ofnongray effects is important.

Coupled electro-thermal simulations in MOSFETs haverecently been published [63,64]. Here, the self-heating term iscomputed from an electron Monte Carlo simulation and includedin a split-flux model for phonon transport. One-way couplingbetween a nongray description of phonon transport and an elec-tron Monte Carlo simulation has been considered in [59] and [62].

Figure 9 shows the computational domain employed in Ref.[59]. A two-dimensional rectangular domain encompassing thesource, drain, and channel regions is considered, with the dimen-sions shown in Fig. 9. The device is an NPN FET with source=drain doping of 1� 1020 cm�3. The channel region of the sub-strate is doped to 1� 1018 cm�3 while the rest of the substrate isdoped to 1� 1016 cm�3. The source=drain regions are 40 nm� 30nm in the top right and left corners, respectively, and the gate ox-ide is 2 nm thick. Source=drain and gate voltages are set to be 1V. All boundaries are held at 300 K, except the top boundary,which is assumed diffusely reflecting. The heat source due toelectron–phonon coupling is shown in Fig. 9 and is computedusing an electron Monte Carlo (e-MC) simulation developed byAksamija and co-workers, and described in Ref. [62]. The distri-bution of the heat generation from the e-MC simulation is shownin Table 1, and is seen to be primarily in the longitudinal-optical(LO), transverse-optical (TO), and LA modes. Spatially most of

Fig. 7 Discretization of phonon frequency spectrum for sili-con. Dispersion curves in the [100] direction are shown [59].

Fig. 8 Comparison of single-mode relaxation times in Si withMD simulations. The solid lines are computed using Fermi’sGolden Rule [56] and the dashed lines are taken from the MDsimulations of Ref. [58].

Fig. 9 Schematic of bulk NPN FET showing source, drain, andchannel regions, and phonon generation rates taken from Aksa-mija and co-workers [62]



the heat generation is in the drain region, though the source alsosees some Joule heating.

The lattice temperature computed using Eq. (3) is shown inFig. 10. As expected, an intense hot spot occurs in the drainregion, where the greatest heat generation occurs. However, thetemperature rise of 60 K over the ambient is far smaller than pre-vious unrealistically high predictions using gray models. Thus, thecareful resolution of dispersion and polarization effects both inthe electron–phonon heat generation rate, as well as in phonontransport properties is critical. Furthermore, the temperature riseis higher than the temperature rise of 6.5 K predicted by a Fouriermodel [62] which is unable to capture the bottlenecks to transportbetween optical and acoustic modes. We emphasize, however,that “temperature” in this nanoscale context cannot be interpretedas the thermodynamic temperature; it is merely a measure of thephonon energy.

5.3 Solution Acceleration Schemes for the BTE. A criticalproblem in devising computational methods for the BTE whichspan mesoscale to macroscale is the issue of Knudsen number, Knvx;psx;p=L. In a typical nongray phonon transport simulation,band-wise Knudsen numbers may range over 3–4 orders of mag-nitude, as is evident from the relaxation times shown in Fig. 8. Asa result, sequential solution procedures for solving Eq. (3) fail toconverge, primarily because of interband and interdirectional cou-pling. In a typical sequential solution scheme, each direction ineach band is visited sequentially, and the lattice temperature isthen updated. The process is repeated until convergence. WhenKn is small, the scattering term in Eq. (3) dominates, and thisloose coupling with other bands and directions through a sequen-tial loop is inefficient. Similar difficulties arise in electron trans-port and rarefied gas dynamics solutions of the BTE. Similarproblems have also been encountered in the solution of the radia-tive transfer equation (RTE) for participating radiation [42,65,66].If we are to use Eq. (3) to span all scales, our numerical methodmust work efficiently across the entire range of Knudsen numbers.

A number of publications in the thermal radiation literaturehave addressed solution acceleration schemes; many of these alsohave relevance to the BTE. One popular strategy is to advance theangular-average of the radiative intensity as a way to improveinterdirectional coupling. Chui and Raithby [65] proposed a multi-plicative correction of the average intensity in the context of thefinite volume scheme. However, the scheme was not uniformlyconvergent. Fiveland and Jessee [66] proposed and evaluated anumber of acceleration strategies for the discrete ordinatesmethod. These included the successive over-relaxation method,the mesh rebalance method, and the synthetic accelerationmethod. The mesh rebalance method, which is similar to Ref.[65], was found to perform the best, but its performance deterio-rated as the mesh-based optical thickness decreased. The methodhad to be modified to perform rebalance on a coarser mesh thanthat for the actual solution, so as to keep the rebalance mesh opti-cal thickness greater than unity. Mathur and Murthy [18] proposeda point-coupled multigrid technique to significantly accelerate so-lution convergence. The method works well for isotropic scatter-ing, and is therefore directly applicable to the BTE in the

relaxation time approximation. More recently, Raithby and Has-sanzadeh [67] developed the QL algorithm, in which an equationfor the average radiative intensity was used to better couple direc-tional intensities. Significant solution acceleration was reportedfor radiative equilibrium problems. Mathur and Murthy [68] pro-posed modifications to the scheme based on a two-level angularmultigrid idea which alleviated the loss of overall energy balancein the QL algorithm. Though these new methods hold much prom-ise for the BTE well, we are not aware of any use of these acceler-ation schemes in the BTE literature.

Recently, we proposed a hybrid Fourier-BTE model whichshows great promise for problems with a large spread in Knudsennumber [60]. The central idea is to identify a cut-off Knudsennumber, Kncutoff. For phonon bands with Kn<Kncutoff, a modifiedFourier equation (MFE) is solved, while the nongray BTE (Eq. 3)is solved for phonon bands with Kn � Kncutoff. The modifiedFourier equation is given by

r: kx;prTx;p� �þ Cx;p

sx;pTL � Tx;p� � ¼ 0

Here, kx,p is the band thermal conductivity, Cx,p the band specificheat, TL the lattice temperature, and Tx,p the “temperature” associ-ated with the band (x,p). (We emphasize that these“temperatures” are not to be interpreted as thermodynamic tem-peratures, and are merely proportional to the corresponding pho-non energies.) Furthermore, the MFE is subject to slip boundaryconditions at thermalizing boundaries; details may be found inRef. [59].

This hybrid approach improves computational efficiency bytwo main mechanisms. First, for low Kn bands, only a singleMFE is solved in each band; there is no angular dependence, andtherefore the computational and storage load is low. Second, sincethe MFEs are relatively few in number, they may be solved in afully-coupled fashion with the lattice temperature, significantlyalleviating the coupling issues that plague sequential solvers. Atypical solution loop for the hybrid model is shown in Fig. 11.

We have demonstrated that the hybrid model engenders no lossof accuracy with respect to an all-BTE solution. Figure 12 showsa comparison of the band-wise heat transfer rates for the hybridand all-BTE models for the problem of nongray submicron heatconduction in a two-dimensional silicon domain. Errors wellunder 1% are found in all bands. Significant solution acceleration,from 2 to 100 times, has been documented in Ref. [59], as shownin Fig. 13. Moreover, the hybrid model is able to converge wellfor large-scale domains in which the all-BTE solver cannot.

6 Sensitivity Analysis

An emerging area of research during the last decade has beensensitivity analysis [7,69]. The objective here is to determine theJacobian of the outputs of a CFD=CHT solver with respect to theinputs, i.e., to determine

@yj@xi, where yj is the jth output and xi is the

ith input. By knowing these gradients or sensitivities, the usermay better design experiments and computations, and performoptimization. The components of the Jacobian matrix may befound either by finite differences, complex variable approaches[70], by developing equations for the appropriate gradients fromthe governing equations themselves (i.e., a continuous approach)[71,72] or through automatic code differentiation (i.e., the discreteapproach) [73,74]. The finite difference approach essentiallydetermines the gradient using

@yj@xk

¼ y xk þ Dxkð Þ � y xkð ÞDxk

þ O Dxk

ð Þ

The advantage of this technique is that it is conceptually simpleand requires no modification of the underlying solver, enablinglegacy codes to be used. To compute derivatives of outputs with

Table 1 Percentage of heat generation in each phononpolarization

Phonon polarization % of heat generation

TA1 2.0TA2 2.4LA 16.0LO 59.0TO1 10.5TO2 10.1Total 100



respect to N inputs, we would perform Nþ 1 runs. However, wemust choose Dxk carefully—too small a value leads to round-off,and too large a value to large truncation errors. When used in con-junction with iterative schemes, the accuracy of such a techniquewould depend on the finite termination criteria that are typicallyused.

More recently, continuous approaches have been developed[71,72] wherein sensitivity versions of the partial differentialequation (pde) or its adjoint are developed, discretized and solvedto obtain sensitivities. Though accurate gradients may be com-puted in this way, there are significant disadvantages. The sensi-tivity version of the pde is not itself a conservation equation, andmay involve complicated source terms and boundary conditions.The method is intrusive, with each new physical model beinghandled anew, and is not suitable for use with legacy codes.

Discrete approaches employing automatic code differentiationpromise some of the most efficient ways to obtain discrete tan-

gents [73]. The idea is illustrated in Fig. 14. We are given two out-puts p and q, which are functions of two inputs x and y. Theobjective is to compute the output derivatives p0 and q0 withrespect to the input variables x and y. The algebraic expressionsfor p and q are first decomposed into elemental operationsinvolving the input variables x and y. Then, their derivatives arewritten in terms of the elemental derivatives, i.e., the derivativesof inputs x and y. By setting only one input derivative to unity andall other input derivatives to zero, the procedure computes theexact derivatives of all outputs with respect to the input corre-sponding to the nonzero derivative. Since a CFD code is nothingbut a concatenation of such operations, it is possible to performthis operation on the whole code if every variable in the code iscarried along with its derivative.

Researchers have exploited the features of the Cþþ language,using templating and operator overloading, to perform these oper-ations automatically [75]. By defining a Tangent class which car-ries the variable and its derivative, and by overloading operatorssuch as addition, subtraction etc. to operate on both the variableand its derivative, the CFD solver may automatically be differenti-ated. The idea works through loops, conditionals and iterations.Thus, a conventional single-point CFD code is easily convertedinto one yielding exact derivatives without a rewrite of the wholecode, and may be compiled in either single-point or tangent modeas desired.

Figure 15 shows the application of sensitivity analysis to the prob-lem of discrete hole cooling of turbine blades [76]. The domain ofinterest is shown, and represents the baseline case considered in[77]. Here hot gas enters the computational domain, and is cooled bya coolant stream inclined at a 30-deg angle as shown. The bottomwall of the domain is adiabatic, and the lateral walls are symmetryboundaries. The ratio of the cold and hot mass flow rates is 1.0 andthe density ratio is 1.2. The perfect gas law is assumed.

Figure 15(a) shows temperature contours on the boundaries ofthe domain for the case when the inlet hot gas temperature is setat 409.5 K, and the coolant is at 341.5 K. We see that most of thedomain is therefore at 409.5 K. The footprint of the coolant injec-tion on the adiabatic wall is clearly visible. Figure 15(b) showscontours of the derivative @T=@Ti, where Ti is the inlet tempera-ture of the hot gas. We see that increasing the hot gas temperatureincreases the temperature everywhere in the domain except in asmall region near the cold jet entrance, where derivatives arenearly zero. The region of the cold jet “protected” from the hotgas is quite small, however. Beyond about four hole diametersdownstream of the jet, sensitivity to inlet temperature becomesgreater than 0.75, indicating that the adiabatic wall temperature isstrongly influenced by the hot gas beyond this point.

Fig. 11 Solution loop for hybrid Fourier-BTE model

Fig. 10 Contours of lattice temperature (K) in NPN FET

Fig. 12 Fractional band-wise heat transfer rates for heat con-duction in a silicon domain of size L5100 nm. Comparison ismade between the hybrid model and an all-BTE solution.



In addition to spatially dependent sensitivities, we can alsodetermine the sensitivity of global or average quantities. For thecase in Fig. 15, the average adiabatic wall temperature Taw isdetermined to be 393.995 K, so that the average effectiveness is0.227. This value is in keeping with values measured by Dhungelet al. in Ref. [78]. The gradient @Taw=@Ti is computed to be0.7929. As expected, it is positive; it is also relatively high, almostcomparable to the value of unity seen in most of the domain. This

indicates that for this case, the coolant flow is not very effective incontrolling the temperature of the adiabatic wall.

The idea of operator overloading and templating to performautomatic code differentiation is very general, and applies to allmodels in the CFD solver automatically. Similar ideas have beenimplemented in F90=95 codes successfully as well.

7 Uncertainty Quantification

A critical component of emerging simulations is the ability toquantify the uncertainty in simulation outputs in a systematic andrigorous manner, in much the same way as has been done with

Fig. 13 Acceleration factors in CPU time for hybrid solver versus an all-BTE solver for thecase of a two-band problem, one solved with the BTE, and one with the MFE. The first row ofnumbers shows the Knudsen number of the MFE versus the BTE band. The second row ofnumbers shows the lattice ratio (Cx,p=sx,p) in the MFE versus the BTE band.

Fig. 14 Discrete tangent computation procedure. (a) The func-tions to be differentiated are decomposed into elemental opera-tions, and elemental derivatives computed. (b) Inputs x and yare prescribed, and also the input derivatives x0 and y0. If one ofthe input derivatives is set to unity and the other to zero (x0 51,y0 5 0, for example), the corresponding output derivatives arecomputed.

Fig. 15 Sensitivity to hot gas inlet temperature Ti. (a) Contoursof temperature on the domain boundaries and (b) contours of›T=›Ti .



experimental data. Verification, validation and uncertaintyquantification must thus become intrinsic components of any cred-ible CFD or CHT simulation [79]. Aleatoric uncertainty, associ-ated with randomness in simulation inputs, and epistemicuncertainty, associated with a lack of knowledge of the underlyingphysics, must be considered carefully in the propagation ofuncertainty from inputs to outputs.

Our recent work has focused on quantifying uncertainty in theprediction of microsystem performance. In these systems, a vari-ety of aleatoric and epistemic uncertainties come into play. Thesemay include aleatoric uncertainties in geometric parameters andmaterial properties, which are strong functions of the fabricationtechnique. Epistemic uncertainties may be associated with size-dependence of fluid and thermal transport equations, the failure ofcontinuum constitutive models at the mesoscale, as well as uncer-tainties in model forms and model constants in molecular dynam-ics simulations, among numerous others.

We consider, here, the problem of cantilever damping at themicroscale, as shown in Fig. 16. A fixed-free cantilever with anominal length of 300 lm, width of 20 lm and thickness of 2.5 lmvibrates above a substrate at a nominal distance of 2.0 lm. Vibra-tional structures like these are used in a variety of MEMS for sens-ing, switching, metrology and other applications. The vibration ofthe cantilever is damped by squeeze film damping by the surroundinggas. The mean free path of gas molecules in air at STP is approxi-mately on the order of 100 nm, so that for gaps in the micron range,slip and rarefaction effects would come into play [80]. The objectiveis to determine the damping factor of the cantilever [81] as a functionof the system pressure, or the system Knudsen number.

CFD predictions of damping in this system are inherentlyuncertain because significant aleatoric uncertainty exists in thecantilever dimensions, particularly for the smallest dimensions inthe geometry, the thickness and the gap height. We haveemployed the sparse grid collocation-based generalized polyno-mial chaos (gPC) approach [82,83] to propagate aleatoric uncer-tainty in a finite volume simulation of cantilever damping in aKnudsen number range of 0.03–0.3 based on gap height. In thisapproach, the output, in this case the damping factor, is expandedin a polynomial basis in the independent random variable x

f xð Þ ¼Xni¼0

ai/i xð Þ

where /i (x) are polynomial basis functions in the random inputvariable x, in this case Legendre polynomials, ai are unknowncoefficients, and n is the order of the expansion. The coefficientsai are determined by exploiting the orthogonality of /i (x) so that

ai ¼ÐfðxÞ/iðxÞdxÐ/2

iðxÞðxÞdx ¼ 1

h

ðfðxÞ/iðxÞdx

However, the damping factor fðxÞ is not known continuously overthe range of x. Sparse collocation essentially evaluates

ai ¼ 1hi

PNj¼1

f xj� �

wj/i xj� �

where xj are collocation points, N in number, and wj the corre-sponding weights. In our computations, Smolyak sparse grids [83]are used in conjunction with a second-order gPC expansion; theuse of sparse grids significantly reduces the number of expensiveCFD runs necessary to find f xj

� �.

The response surface represented by the polynomial expansionis shown for the case of cantilever damping in Fig. 16. The damp-ing ratio shows a strong dependence on the gap height h, and amilder dependence on cantilever thickness t. By assuming normaldistributions of gap height and thickness, the response surface issampled, and the output probability density function (pdf), meanand standard deviation of damping coefficient may be found.

Figure 17 shows the predicted mean value and standard devia-tion of damping factor for mode 1 vibration of the cantilever as afunction of pressure [84]. The unstructured finite volume schemeoutlined above is used for the computation of damping coefficient,in conjunction with a Navier–Stokes slip jump model for this in-termediate Knudsen number range [85]. The cantilever thicknesshas a mean value of 2.5 lm and a standard deviation of 15%,while the mean value of the gap is 2.0 lm and its standard devia-tion is 15%. We see from Fig. 17 that the predictions fall close toexperiments from [81]. The predicted uncertainty in damping fac-tor is as high as 48%, resulting primarily from the steep depend-ence of the damping factor on gap height for small values of thegap height, as seen in Fig. 16. Furthermore, computations at themean cantilever dimensions differ significantly from the mean ofthe computed damping factor distribution. Since CFD simulationsfrequently compute the former while experiments measure thelatter, it is no surprise that difficulties are encountered in CFDvalidation against experiments in these types of systems.

The collocation-based gPC approach has the virtue of beingnonintrusive, and is usable with legacy codes. Adaptive colloca-tion techniques have begun to appear in the literature which placecollocation points in random space based on error measures [86],to resolve steep gradients or discontinuities in random space. For

Fig. 16 Gas damping of micro-cantilever. The surface corre-sponding to a second-order generalized polynomial chaosexpansion of damping ratio versus cantilever thickness t andgap height h is shown.

Fig. 17 Damping ratio versus pressure for gas damping ofmicrocantilever. Comparison with the experimental results ofRef. [81] (squares) is shown. Triangles indicate computations atnominal dimensions and circles indicate the mean of the com-puted damping factor distribution. The error bars indicate one(computed) standard deviation on either side of the computedmean.



large numbers of random independent variables, however, collo-cation gPC may become uncompetitive with intrusive GalerkingPC techniques [9], and with Monte Carlo methods for yet largernumbers of independent variables. We are developing operatoroverloading and templating-based approaches to intrusive Galer-kin gPC which promise to convert single-point CFD and CHTsolvers relatively straightforwardly into those capable of uncer-tainty propagation [87], With this approach, CFD and CHT practi-tioners across a wide swath of industry, as well as those inemerging CHT areas, would inherit the ability to perform sensitiv-ity, optimization, and uncertainty quantification automatically.

8 Conclusions

In this paper, we have considered the current status of CFD andCHT and identified important bottlenecks and opportunities forfuture work. CFD and CHT have made impressive advances in thelast three decades, and have become indispensible tools in indus-try. This success has whetted the appetite for ever-larger and morecomplex simulations. The key driver for the next two decades isintegration: integration of multiple subsystems, multiple knowl-edge domains, and multiple scales. The size and complexity ofthese simulations will require algorithms scalable to petascale andexascale parallel platforms and which are capable of exploitingnew architectures. Furthermore, emerging areas such as micro-technology and nanotechnology, biotechnology, and the diverseenergy sector, will require consideration of new physics and thedevelopment of new algorithms, particularly those which can spanmultiple knowledge domains and scales seamlessly. In all theseareas, sensitivity and adjoint analyses will offer far more informa-tion than conventional single-point simulations and will facilitatebetter design of physical and computational experiments as wellas efficient design optimization. Methodologies for uncertaintyquantification will allow the user to systematically identify theprimary sources of uncertainty in his=her predictions, and to as-certain the effect of specific simulation inputs on final predictionuncertainty. They will also allow the user to quantify margins anduncertainties, and will provide actionable information suitable fordecision-making. These advances not only promise to expand anddeepen the role of CFD and CHT simulation in traditional analy-sis, design and optimization, but also to allow simulation to enterbroader decision-making processes.

Acknowledgment

Support of the author and Purdue’s PRISM Center from theDepartment of Energy (National Nuclear Security Administration)under Award Number DE-FC52-08NA28617 is acknowledged.Results reported in this paper are the product of collaborations witha number of researchers at PRISM. The contributions of Dr. LinSun, Dr. Jia Li, Dr. Dongbin Xiu, Dr. Chunjian Ni, and Dr. Jose‘Pascual-Gutierrez, and graduate students, James Loy, Dhruv Singh,Ben Pax, and Aarti Chigullapalli are gratefully acknowledged.

Nomenclatureai ¼ coefficients in polynomial expansion

Cx,p ¼ band volumetric specific heatD ¼ diffusion term

e00x,p ¼ phonon energy densitye0x,p ¼ equilibrium phonon energy density

F ¼ convection termh ¼ gap height

Kn ¼ Knudsen numberkx,p ¼ band thermal conductivityL ¼ lengthp ¼ phonon polarization

S/ ¼ source termS ¼ source termT ¼ temperature

Tx,p ¼ band temperatureTL ¼ lattice temperaturet ¼ time, cantilever thickness

ux,uy ¼ velocity componentsV ¼ velocity vector

vx,p ¼ phonon group velocitywj ¼ weight functionx,y ¼ Cartesian coordinates; input and output

Greek Symbolsa ¼ thermal diffusivity

DV ¼ cell volumeq ¼ density/ ¼ transported scalar; polynomial basis function

C/ ¼ diffusion coefficientf ¼ damping factorx ¼ phonon frequency

Subscriptsaw ¼ adiabatic wallf ¼ facei ¼ inlet

References[1] Li, S., and Liu, W. K., 2002, “Meshfree and Particle Methods and Their

Applications,” Appl. Mech. Rev., 55(1), pp. 1–33.[2] Mittal, R., and Iaccarino, G., 2005, “Immersed Boundary Methods,” Annu.

Rev. Fluid Mech., 37, pp. 239–261.[3] Tezduyar, T. E., Sathe, S., and Stein, K.., 2006, “Solution Techniques for the

Fully Discretized Equations in Computation of Fluid-Structure InteractionsWith Space-Time Formulations,” Comput. Methods Appl. Mech. Eng., 195, pp.5743–5732.

[4] De, S. K., and Aluru, N. R., 2004, “Full-Lagrangian Schemes for DynamicAnalysis of Electrostatic MEMS,” J. Microelectromech. Syst., 13(5),p. 737–758.

[5] Demirdzic, I., and Muzaferija, S., 1995, “Numerical Method for Coupled FluidFlow, Heat Transfer and Stress Analysis Using Unstructured Moving MeshesWith Cells of Arbitrary Topology,” Comput. Meth. Appl. Mech. Eng., 125, pp.235–255.

[6] Laux, S. E., and Grossman, B. M., 1984, “A Generalized Control-Volume For-mulation for Modeling Impact Ionization in Semi-Conductor Transport,” IEEETrans. Comput.-Aided Des., 4(4) pp. 520–526.

[7] Griewank, A., and Walther, A., 2008, Evaluating Derivatives: Principles andTechniques of Algorithmic Differentiation 2nd ed., SIAM, Philadelphia.

[8] Ghanem, R., and Spanos, P. D., 1991, and Karniadakis, G. E., 2002, “TheWiener-Askey Polynomial Chaos for Stochastic Differential Equations,” SIAMJ. Sci. Comput. (USA), 24(2), p. 619–644.

[9] Xiu, D., and Karniadakis, G. E., 2002, “The WieneroAskey Polynomial Chaosfor Stochastic Differential Equations,” SIAM J. Sci. Comput. (USA), 24 (2),p. 619–644.

[10] “Historical Computing Trends,” The New York Times, Nov. 17, 2008.[11] Owens, J. D., 2007, “A Survey of General-Purpose Computation on Graphics

Hardware,” Comput. Graph. Forum, 26, pp. 80–113.[12] “NVIDIA CUDA Complete Unified Device Architecture Programming Guide,”

Version 2.0.[13] Senocak, I., Thibault, J., and Caylor, M., 2009, “Rapid-Response Urban

CFD Simulations Using a GPU Computing Paradigm on Desktop Super-computers,” Eighth Symposium on the Urban Environment, Phoenix, AZ,Jan. 10-15.

[14] http:==graphics.stanford.edu=projects=brookgpu=[15] http:==www.nvidia.com=object=cuda_home.html[16] Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere,

Washington DC.[17] Vanka, S. P., 1986, “Block-Implicit Multigrid Solution of Navier-Stokes Equa-

tions in Primitive Variables,” J. Comput. Phys., 65, pp. 138–158.[18] Mathur, S. R., and Murthy, J. Y., 1999, “Coupled Ordinate Method for Multi-

grid Acceleration of Radiation Calculations,” J. Thermophys. Heat Transfer,13(4), pp. 467–473.

[19] Pope, S., 2000, Turbulent Flows, 1st ed., Cambridge University Press,Cambridge, UK.

[20] Prosperetti, A., and Tryggvason, G., 2009, Computational Methods for Multi-phase Flow, 1st ed., Cambridge University Press, Cambridge, UK.

[21] Gidaspow, D., 1994,Multiphase Flow and Fluidization: Continuum and KineticTheory Descriptions, 1st ed., Academic Press, San Diego, CA.

[22] http://www.scidacreview.org=1001=html=energy.html[23] Tien, C. L., Majumdar, A., and Gerner, F., 1998, Microscale Energy Transport,

Taylor and Francis, Washington, D.C.



[24] Cola, B. A., Xu, X.., and Fisher, T.S., 2007, “Increased Real Contact Resistance inThermal Interfaces: A CarbonNanotube FoilMaterial,” Appl. Phys. Lett. 90, p. 093513.

[25] Dresselhaus, M., Chen, G., Tang, M. Y., Yang, R., Lee, H., Wang, D., Ren, Z.,Fluerial, J.-P., and Gogna, P., 2007, “New Directions for Low-DimensionalThermoelectric Materials,” Adv. Mater., 19, pp. 1043–1053.

[26] Seol, J. H., Jo, I., Moore, A. L., Lindsay, L., Aitken, Z. H., Pettes, M. T., Li, X.,Yao, Z., Huang, R., Broido, D., Mingo, N., Ruoff, R. S., and Shi, L., 2010,“Two-dimensional Phonon Transport in Supported Graphene,” Science, 328(5975), pp. 213–216.

[27] Hadjiconstantinou, N. G., 1999, “Hybrid Atomistic-Continuum Formulationsand the Moving Contact-Line Problem,” J. Comput. Phys.,” J. Comput. Phys,154(2), pp. 245–265.

[28] Choi, J. H., Bansal, A., Meterelliyoz, M., Murthy, J. Y., and Roy, K., 2007,“Self-Consistent Approach for Leakage Power and Temperature Estimation toPredict Thermal Runaway in FinFET Circuits,” IEEE Trans. Comput.-AidedDes., 26(1), pp. 2059–2068.

[29] Mathur, S. R., and Murthy, J. Y., 1997, “A Pressure-Based Method for Unstruc-tured Meshes,” Numer. Heat Transfer, 31(2), pp. 195–216.

[30] Davidson, L., 1996, “A Pressure-Correction Method for Unstructured MeshesWith Arbitrary Control Volumes,” Int. J. Numer. Methods Fluids, 22, pp.265–281.

[31] Baliga, B. R., 1997, “Control-Volume Finite Element Methods for Fluid Flowand Heat Transfer,” Advances in Numerical Heat Transfer W. J. Minkowycz andE. M. Sparrow, ed., Taylor & Francis, New York, Vol. 1, Chap. III, pp. 97–135.

[32] Schneider, G. E., and Raw, M. J., 1986, “Control-Volume Finite ElementMethod for Heat Transfer and Fluid Flow Using Co-located Variables—I. Com-putational Procedures,” Numer. Heat Transfer, 9, pp. 253–276.

[33] Acharya, S., Baliga, B. R., Karki, K., Murthy, J. Y., Prakash, C., and Vanka, S.P., 2007, “Pressure-Based Finite-Volume Methods for Computational FluidDynamics,” ASME J. Heat Transfer, 129, pp. 407–424.

[34] Hutchinson, B. R., and Raithby, G. D., 1986, “A Multigrid Method Based onthe Additive Correction Strategy,” Numer. Heat Transfer, 9, pp. 511–537.

[35] Saad, Y., Iterative Methods for Sparse Linear Systems, SIAM, 2003, Philadel-phia, PA.

[36] Rhie, C. M., and Chow, W. L., 1983, “A Numerical Study of the TurbulentFlow Past an Isolated Airfoil With Trailing Edge Separation,” AIAA J., 21, pp.1525–1532.

[37] Webster, R., 1994, “An Algebraic Multigrid Solver for Navier-Stokes Prob-lems,” Int. J. Numer. Methods in Fluids, 18, pp. 761–780.

[38] Mavriplis, D. J., 2000, “Viscous Flow Analysis Using a Parallel UnstructuredMultigrid Solver,” AIAA J., 38(11), pp. 2067–2076.

[39] Raithby, G. D., and Chui, E. H., 1990, “A Finite-Volume Method for PredictingRadiant Heat Transfer in Enclosures With Participating Media,” J. Heat Trans-fer, 112, pp. 415–423.

[40] Murthy, J. Y., and Mathur, S. R., 1998, “Finite Volume Method for RadiativeHeat Transfer Using Unstructured Meshes,” J. Thermophys. Heat Transfer,12(3), pp. 313–321.

[41] Mathur, S. R., and Murthy, J. Y., 2009, “A Multigrid Method for the Poisson-Nernst-Planck Equations,” Int. J. Heat Mass Transfer, 52, pp. 4031–4039.

[42] Murthy, J. Y., and Mathur, S. R., 2002, “Computation of Sub-Micron ThermalTransport Using an Unstructured Finite Volume Method,” J. Heat Transfer,124(6), pp. 1176–1181.

[43] Iaccarino, G., and Verzicco, R., 2003, “Immersed Boundary Technique for Tur-bulent Flow Simulation,” Appl. Mech, Rev., 50(3), pp. 331–347.

[44] Peskin, C. S., 1981, “The Fluid Dynamics of Heart Valves: Experimental, Theo-retical and Computational Methods,” Annu. Rev. Fluid Mech., 14, pp. 235–259.

[45] Mohd-Yusof, J., 1997, “Combined Immersed Boundaries=B-Spline Methodsfor Simulations of Flows in Complex Geometries,” Annual Research Briefs,Center for Turbulence Research, Stanford, CA, pp. 317–328.

[46] Fadlun, E. A., Verzicco, R., Orlandi, P., and Mohd-Yusof, J., 2000, “CombinedImmersed Boundary Finite-Difference Methods for Three-Dimensional Com-plex Flow Situations,” J. Comput. Phys., 161, pp. 37–60.

[47] Gilmanov, A., and Sotiropoulos, F., 2005, “A Hybrid Cartesian=ImmersedBoundary Method for Simulating Flows With 3D Geometrically Complex Mov-ing Bodies,” J. Comput. Phys., 207, pp. 457–492.

[48] Gilmanov, A., and Acharya, S., 2008, “A Hybrid Immersed Boundary and Ma-terial Point Method for Simulating 3D Fluid-Structure Interaction Problems,”Int. J. Numer. Methods Fluids, 56, pp. 2151–2177.

[49] Kang, S., Iaccarino, G., and Ham, F., 2009, “DNS of Buoyancy-DominatedTurbulent Flows on a Bluff Body Using the Immersed Boundary Method,” J.Comput. Phys., 228(9), pp. 3189–3208.

[50] Sun, L., Mathur, S. R., and Murthy, J. Y., 2009, “An Unstructured FiniteVolume Method for Incompressible Flows With Complex ImmersedBoundaries,” ASME Paper No. IMECE2009-12917.

[51] Kang, S., Iaccarino, G., Ham, F., and Moin, P., 2009, “Prediction of Wall-Pressure Fluctuation in Turbulent Flows With an Immersed Boundary Method,”J. Comput. Phys., 228, pp. 6753–6772.

[52] Ansys, 2009, “FLUENT 12 User Manual,” Ansys Inc., Canonsburg, PA.[53] Ashcroft, N.W., and Mermin, N. D., 1976, Solid State Physics, Harcourt, Inc.,

Orlando, FL.[54] Bazant, M. Z., Kaxiras, E., Justo, J. F., 1997, “Environment-Dependent Intera-

tomic Potential for Bulk Silicon,” Phys. Rev. B 56, pp. 8542–8552.[55] Mingo, N., and Yang, L., 2003, “Predicting the Thermal Conductivity of Si and

Ge Nanowires,” Nano Lett., 3(12), pp. 1713–1716.

[56] Pascual-Gutierrez, J. A., Murthy, J. Y., and Viskanta, R. V., 2009, “ThermalConductivity and Phonon Transport Properties of Silicon Using PerturbationTheory and the Environment-Dependent Interatomic Potential,” J. Appl. Phys.,106(6), p. 063532.

[57] Broido, D., Ward, A., and Mingo, N., 2005, “Lattice Thermal Conductivity ofSilicon From Empirical Interatomic Potentials, Phys. Rev. B, 72, p. 014308.

[58] Henry, A. S., and Chen, G., 2008, “Spectral Phonon Transport Properties of Sil-icon Based on Molecular Dynamics Simulations and Lattice Dynamics,”J. Comput. Theor. Nanosci., 5, pp. 1–12.

[59] Loy, J., 2010, “An Acceleration Technique for the Solution of the PhononBoltzmann Transport Equation,” M.S. thesis, Purdue University, West Lafay-ette, IN.

[60] Loy, J., Singh, D., and Murthy, J. Y., 2009, “Non-Gray Phonon Transport Usinga Hybrid BTE-Fourier Solver,” ASME Paper No. HT2009-88056.

[61] Choi, J. C., Bansal, A., Meterelliyoz, M., Murthy, J. Y., and Roy, K., 2006,“Leakage Power Dependent Temperature Estimation to Predict Thermal Run-away in FinFet Circuits,” ICCAD, San Jose, CA, Nov. 5–9.

[62] Ni, C., 2009, “Phonon Transport Models for Heat Conduction in Sub-MicronGeometries With Applications to Microelectronics,” Ph.D thesis, PurdueUniversity, West Lafayette.

[63] Rowlette, J., and Goodson, K. E., 2008, “Fully-Coupled Non-equlilbrium Elec-tron-Phonon Transport in Nanometer-Scale Silicon FETS,” IEEE Trans. Elec-tron Devices, 55, pp. 220–232.

[64] Sinha, S., Pop, E., and Goodson, K. E., 2006, “Non-equilibrium Phonon Distri-butions in Sub-100 nm Silicon Transistors,” ASME J. Heat Transfer, 128, pp.638–647.

[65] Chui, E. H., and Raithby, G. D., 1992, “Implicit Solution Scheme to ImproveConvergence Rate in Radiative Transfer Problems,” Numer. Heat Transfer, 22,pp. 251–272.

[66] Fiveland, W. A., and Jessee, J. P., 1996, “Acceleration Schemes for the DiscreteOrdinates Method,” J. Thermophys. Heat Transfer, 10(3) pp. 445–451.

[67] Hassanzadeh, P., Raithby, G. D., and Chui, E. H., 2008, “Efficient Calculationof Radiation Heat Transfer in Participating Media,” J. Thermophys. Heat Trans-fer, 22(2), pp. 129–139.

[68] Mathur, S. R., and Murthy, J. Y., 2009, “An Acceleration Technique for theComputation of Participating Radiative Heat Transfer,” ASME Paper No.IMECE2009-12923.

[69] Andrea Saltelli, Editor, 2009, “Special Issue on Sensitivity Analysis,” Reliab.Eng. Syst. Saf., 94(7), pp. 1133–1244.

[70] Martins, J., Kroo, I. M., and Alonso, J. J., 2000, “An Automated Method forSensitivity Analysis Using Complex Variables,” Paper No. AIAA-2000-0689.

[71] Jameson, A., 2003, “Aerodynamic Shape Optimization Using the Adjoint Meth-od,” Aerodynamic Drag Prediction and Reduction (VKI Lecture Series), vonKarman Institute of Fluid Dynamics, Rhode St Genese, pp. 3–7.

[72] Nadarajah, S. K., and Jameson, A., 2005, “A Comparison of the Continuousand Discrete Adjoint Approach to Automatic Aerodynamic Optimization,”AIAA Paper 2005–324.

[73] Jemcov, A., and Mathur, S. R., 2004, “Algorithmic Differentiation of GeneralPurpose CFD Code: Implementation and Verification,” ECCOMAS 2004,Jyvaskyla, Finland., 24–28 July.

[74] Jemcov, A., and Mathur, S., 2005, ” Sensitivity Analysis and Uncertainty Prop-agation in Compressible Inviscid Flows,” 6th World Congress of Structural andMultidisciplinary Optimization, Rio de Janeiro, Brazil, 30 May–03 June.

[75] Mathur, S., 2009, Personal Communication.[76] Mathur, S., Murthy, J. Y., and Chai, J. C., 2009, unpublished work.[77] Heidmann, J. D., 2008, “A Numerical Study of Anti-Vortex Film Cooling Designs

at High Blowing Ratio,” ASME Paper No. GT2008-50845.[78] Dhungel, S., Phillips, A., Ekkad, S., and Heidmann, J. D., 2007,

“Experimental Investigations of a Novel Anti-Vortex Design,” ASME PaperGT2007-27419.

[79] ASME, 2009, “Standard for Verification and Validation in Computational FluidDynamics and Heat Transfer,” No. ASME V&V 20, ASME, New York.

[80] Guo, X., and Alexeenko, A., 2009, “Compact Model for Squeeze Film Damp-ing Based on Rarefied Flow Simulations, J. of Micromech. Microeng. 19, p.045026.

[81] Ozdoganlar, O. B., Hansche, B. D., and Carne, T. G., 2005, “ExperimentalModal Analysis for Micromechanical Systems,” Experimental Mechanics,45(6), pp. 498–506.

[82] Xiu, D., and Hesthaven, J., 2005, “Higher-Order Collocation Methods for Dif-ferential Equations With Random Inputs,” SIAM J. Sci. Comput., 27(2), pp.1118–1139.

[83] Smolyak, S., 1963, “Quadrature and Interpolation Formulas for TensorProducts of Certain Classes of Functions,” Soviet Math. Dokl. v4., pp.240–243.

[84] Murthy, J., Mathur, S., Pax, B., Chigullapalli, A., Li, J., and Xiu, D., 2009,unpublished results.

[85] Karniadakis, G., Beskok, A., and Aluru, N. R., 2005, Microflows and Nano-flows, Springer, New York.

[86] Agarwal, N., and Aluru, N. R., 2009, “A Domain Adaptive Stochastic Colloca-tion Approach for Analysis of MEMS Under Uncertainties,” J. Comp. Phys.,228(20), pp. 7662–7668.

[87] Mathur, S., Chigullapalli, A., and Murthy, J.Y., “A Unified Unintrusive Dis-crete Approach to Sensitivity Analysis and Uncertainty Propagation in FluidFlow Simulations,” ASME Paper No. IMECE2010-37789.



computational heat transfer in complex systems: a review

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