Modeling in Engineering—The Challenge of Multiple Scalesby Rob Ph i l l i p s

hether we consider thedesign of a new genera-tion of airliners such asthe Boeing 777 or the

development of the latest micro-p rocessors, engineering re l i e si n c reasingly on the use of mathe-matical models to characterizethese technologies. In the case ofthe 777, sophisticated models ofthe fluid mechanics of air flow overthe wings were an integral part ofthe design process, just as stru c-tural mechanics models ensure dthat flight in turbulence leads tonothing more grave than passengerd i s c o m f o rt .

Models of complex materialsthat make up our modern technolo-gies also pose a wide range of sci-entific challenges. Indeed, many ofthe most important recent advan-ces in the study of materials re s u l t-ing in entirely new classes of mate-rials such as the famed oxide high-t e m p e r a t u re superconductors or

f u l l e renes, and their structural part-ners known as carbon nanotubes,have engendered a flurry of model-ing eff o rts.

I m p o rtant problems that suchmodeling must confront are thoseof an intrinsically multiscale nature .What this means is that analysis ofa given problem re q u i res simulta-neous consideration of several spa-tial or temporal scales. This idea iswell re p resented in drawings madem o re than 500 years ago byL e o n a rdo da Vinci, in which theturbulent flow of a fluid is seen toinvolve vortices within vort i c e sover a range of scales. This sketch(see Fig. 1) serves as the icon forthe new Caltech center known asthe Center for Integrative Multi-scale Modeling and Simulation(CIMMS) [see article on page 10].CIMMS brings together facultymembers from several diff e re n tOptions and Divisions includingP rofessors K. Bhattachary a

(Mechanical Engineering), E. Candes (Applied & Compu-tational Mathematics), J. Doyle( C o n t rol & Dynamical Systems,Electrical Engineering, andBioengineering), M. Gharib( A e ronautics and Bioengineering),T. Hou (Applied & ComputationalMathematics), H. Mabuchi (Physicsand Control & Dynamical Systems),J. Marsden (Control & DynamicalSystems), R. Murray (Control &Dynamical Systems and Mechan-ical Engineering), M. Ort i z( A e ronautics and MechanicalEngineering), N. Pierce (Applied & Computational Mathematics), R. Phillips (Mechanical Engineer-ing and Applied Physics) and P. Schröder (Computer Science and Applied & ComputationalMathematics). The aim of multi-scale modeling is to construct mod-els of relevance to macro s c o p i cscales usually observed in experi-ment and tailored in the engineer-

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ing process without losing sight ofthe microscopic processes whichmay dictate processes at them a c roscale. Although the re l a t i o nbetween force and extension canbe observed macro s c o p i c a l l y, it isoften complex m i c ro s c o p i c p ro c e s s e sthat give rise to the macro s c o p i cf o rce-extension curves. Examplesinclude the breaking of hydro g e nbonds during protein deform a t i o n ,and the motion of defects in thed e f o rmation of crystalline solids.

key outcome of the use ofcomputers in science andengineering has been theability to solve pro b l e m sof ever- i n c reasing com-

p l e x i t y. Whereas the tools of nine-t e e n t h - c e n t u ry mathematicalphysics emphasized geometries ofhigh symmetry (such as sphere sand cylinders, each of which isaligned with a set of special func-tions such as the Legendre polyno-mials or Bessel functions), curre n tmodeling is aimed at consideringp roblems in their full thre e - d i m e n-

sional complexity. The key advanceenabling such calculations is high-speed computation. As a re p re s e n-tative case study of the high levelto which such models have beentaken, Fig. 2 shows the computa-tional grid (finite-element mesh)used to model a human kidneywhen subjected to ultrasonic shockwaves. The aim is to degrade kid-ney stones (shock-wave lithotripsy).As noted above, no assumptionsa re re q u i red concerning the sym-m e t ry of the body. The level of spa-tial resolution needed to constru c tmodels of systems of interest mayv a ry from one position in the sys-tem to another. Indeed, the finite-element method serves as a power-ful tool in the multiscale modelingarsenal. Eff o rts in the Phillips gro u pand that of Michael Ortiz are aimedat bringing these methods to bearon problems ranging from thed e f o rmation of dense metals suchas tungsten to the fragmentation ofhuman bone to the deformation ofindividual pro t e i n s .

One of the precepts which pre-sides over the field of computation-

al science and engineering isM o o re ’s law, which calls for a dou-bling in the number of transistorsper integrated circuit every 18months. For those of us who exploitcomputers to solve complex pro b-lems, this enables ever- i n c re a s i n gcomputational re s o u rces. Fro mmany perspectives, Moore ’s lawshould be seen as an expression ofunbridled optimism which has setthe agenda respected in the semi-conductor technology ro a d m a p(h t t p : / / p u bl i c . i tr s . n e t). It serves as aguide to understanding the way inwhich the re s o u rces of computa-tional scientists have incre a s e dsince the first models were solvedon primitive vacuum-tube computers.

On the other hand, for thosei n t e rested in bru t e - f o rce atomic-level calculation of the pro p e rties ofmaterials (or any of a wide range ofother problems occurring in fluidmechanics, meteoro l o g y, computa-tional biology, etc.), Moore ’s lawpaints an altogether more gloomyp i c t u re. To see this, we need onlyremark that the number of atoms ina cubic micron of material is ro u g h-ly 101 0 since about 3,000 atoms willfit onto each edge of such a cube.Calculations of this size are at leastt h ree orders of magnitude larg e rthan the 10 million atoms re a c h e don today’s best supercomputers inthe case of the simplest materials.Worse yet, this is but one facet ofthe problem. Just as the maximumsize accessible by direct numerical

Fi g u re 1. Ske tch by Le o n a rdo da Vinci illus-

t rates the sense in which turbulent flow of a

fluid is a multiscale phenomenon. Pa rcels of

fluid in a turbulent flow with a net ro t at i o n ,

vo rt i ce s, a re org a n i zed hiera rc h i cally in such a

way that there are vo rt i ces within vo rt i ce s.

A

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calculation is too small, so too arethe intervals of time being simulat-ed, with the current standard beingthat a nanosecond worth of simula-tion time (10- 9 seconds) re p re s e n t sl o n g simulation time. To drive homethis point, we note that if our inter-est is in the simulation of semicon-ductor processing, we will need tosimulate micron size regions fortimes much in excess of the

nanosecond simulation timesdescribed above. Similarly, shouldour interest be in simulating thep ro p e rties of the basic buildingblocks of life, what Francis Crickre f e rred to as the “two great poly-mer languages,” nucleic acids andp roteins, there too we are facedwith the simulation of scales inboth space and time that will con-tinue to defy our current bru t e -f o rce computational schemes.

As an antidote to this scourg eon the face of computational sci-ence, workers from a host of diff e r-ent fields ranging from appliedmathematics to meteorology tocomputational biology are engagedin work that has been dubbed“multiscale modeling.” From a com-putational perspective, the pre m i s eof multiscale modeling is that newmethods must be developed inwhich alternatives to the full bru t e -f o rce ideas described above areexamined. Though this vibrant fieldhas been hyped by giving it a spe-

cial name, I suggest that multiscalemodeling is really as old as scienceitself and was being practiced byNewton when he treated the Eart has a point mass, by Hooke when het reated a spring as an elastic con-tinuum, by Bernoulli in the devel-opment of the kinetic theory ofgases, by Lorentz in his earlyand primitive models of theabsorption of light in cry s-talline solids, and by Einsteinin his treatment of bothB rownian motion in liquidsand specific heats of cry s t a l l i n esolids. What all of these modelinge ff o rts have in common is the ideaof starting with a picture of thematerial of interest which iso p p ressively complex and finding away to replace that complexity

with a “coarse grained” model. Saidd i ff e re n t l y, such models can bethought of as viewing the pro b l e mof interest with lower re s o l u t i o n .An example from everyday experi-ence is gained by looking out thewindow of an airplane when flyingat 30,000 feet. At this re s o l u t i o n ,f o rests are smeared out and thevarious topographical features witha scale less than several meters areno longer observable. Nevert h e l e s s ,

f rom the perspective of understand-ing the overall forestation andtopography of a given re g i o n ,understanding at this level of re s o-lution is likely m o re useful than am o re accurate rendering with re s o-lution at the meter scale.

i s t o ry is replete withbeautiful examples inwhich multiscale model-ing ideas have been usedto characterize a range of

p roblems. One such example isrelated to the following question:given that a gas is a collection ofatoms, is it possible to re p l a c emodels of the gas which acknowl-edge the underlying graininess ofmatter by those in which the atom-ic degrees of freedom are smeare dout into continuous fields such asd e n s i t y, temperature, and pre s s u re ?Of course, it is well known that theanswer to this query can be positedin the aff i rmative. Furt h e r, it ist h rough the multiscale vehicle ofthe kinetic theory of gases that thist r a n s f o rmation in perspective ismade.

As illustrated in Fig. 3, a gasmay be thought of as a collection ofmolecules, each engaged in its ownjiggling dance until, by chance, onemolecule collides either with anoth-er molecule or the surro u n d i n gwalls. The realization of the earlyt h e rmodynamicists was that theaccumulation of all such collisionsper unit time corresponds to ourm a c roscopic impression of thep re s s u re exerted on the walls by allof the gas molecules. Through awell-defined statistical form a l i s m ,statistical mechanics and thekinetic theory of gases instruct us

Fi g u re 3. Il l u s t ration of the re l ation be twe e n

molecular and co ntinuum descriptions of the

i nte rnal state of a gas. This figure comes fro m

the original paper of Daniel Be rn o u l l i , one of

the deve l o pers of the multiscale mod e l i n g

p a radigm kn own as the kinetic theory of

g a s e s.

Fi g u re 2. Co m p u t ational mesh used to eva l u-

ate the mechanical re s ponse of a ki d n ey to

u l t rasonic shock waves (co u rtesy of Kerstin

We i n be rg and Michael Ort i z ) .

H

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how to compute the macro s c o p i caverage quantities measured in thelab as a function of the underlyingmolecular coordinates. For thep resent argument, the key point isthat by evaluating the molecularmechanics of the various collisionsbetween molecules, it is possible tocompute parameters such as vis-c o s i t y, which show up in higherlevel continuum descriptions of thefluid. The existence of simpleparameters (such as viscosity) cap-t u re the details of the underlyingm i c roscopic collisions and allow usto replace these micro s c o p i cdetails with continuum notions, anexample of multiscale modeling atits best.

Work in the same vein as thekinetic theory of gases has contin-ued unabated and now re p resents ac o rnerstone of the modern ap-p roach to understanding materialsranging from steel to proteins. Inthe remainder of this article, weexamine one corner of this vastfield which has understanding asits first objective and, later, design-ing and controlling the response ofmaterials when they are subjectedto an applied force.

ne of the key ways tounderstand diff e rent mate-rials is to subject them tod i ff e rent external stimuli

and watch their attendant re s p o n s-es. One classic example of thisstrategy is embodied in the form u-lation of the laws of elasticity.Using experimental apparatus likethat shown in Fig. 4, Robert Hookem e a s u red the extension of materialbodies as a function of the imposedload and thereby formulated hisjustly famous law which hee x p ressed as an anagram C E I I-I NO S S ITTU V, which when

unscrambled reads Ut te n s i o , s i cv i s—“As the extension, so is thef o rce.” In modern parlance, this iswritten σ = Eε : s t ress is pro p o rt i o n-al to strain with the constant ofp ro p o rtionality given by theYo u n g ’s modulus, E. This basicidea jibes with our intuition: theh a rder you pull on something, them o re it stretches. Similar pro p o r-tionalities have been formulated formaterial response in other settingssuch as the relation between cur-rent and voltage (Ohm’s law) andthat between diffusion and thechemical gradient (Fick’s law). In

each of these cases, the basic ideacan be couched in the followingt e rm s :

response = material parameter x stimulus

H o w e v e r, as one might guess,once the driving force (i.e., thestimulus) becomes too large, thesimple linear relation between forc eand response breaks down andcalls for more sophisticated analy-sis. A particularly compellingexample of these ideas is pre s e n t e din the emerging field of single-mol-ecule biomechanics in which thef o rce-extension curves for individ-ual molecules such as the pro t e i ntitin found in muscle are measure dusing the atomic-force micro s c o p e .An example of such a curve isshown in Fig. 5. The vertical axis inthis curve shows the applied forc e( m e a s u red in piconewtons) whilethe horizontal axis shows theextension of the molecule (meas-u red in nanometers). What is

remarkable is that the moleculegoes through a series of pro c e s s e sin which the load increases (corre-sponding to the elastic stre t c h i n gof the various domains) followed bya precipitous drop in the load (cor-responding to the breaking of col-lections of hydrogen bonds in oneof the globular domains of the pro-t e i n ) .

A second example of this sametype of massively nonlinear defor-mation is revealed by the pro c e s sused to create the tungsten fila-ments that light our homes everyevening. In this case, a cylindricalspecimen of tungsten, roughly a

p r o g r e s s r e p o r t

Fi g u re 4. E x pe ri m e ntal apparatus used by

Ro be rt Hoo ke in his elucidation of the laws of

e l a s t i c i ty.

o

meter long and several centimetersin diameter, is put through a seriesof deformation steps in which thetungsten is pro g ressively elongat-ed. By the end of this process, thetungsten rod of original length onthe order of a meter has now beens t retched to a length of h u n d re d s o fkilometers. This process takesplace without changing the overallvolume of the rod. We leave it tothe reader to work out what thisimplies about the final diameter ofthe tungsten filament.

The nonlinear deformation ofeither proteins or tungsten (andmost everything in between) is anintrinsically multiscale pro b l e mbecause in each case the macro-scopic force response is engen-d e red by microscopic processes. In the case of the deform a t i o nof a protein like that shown inFig. 5, it is the breaking ofp a rticular sets of hydro g e nbonds that give rise to steepd rops in the forc e - e x t e n s i o nc u rve, bonds which are character-ized by a length scale of 10-10 mand not the 10-8 m typical of them e a s u red force-extension curv e s .S i m i l a r l y, in the deformation oftungsten, it is the motion of atom-ic-scale defects known as disloca-tions that give rise to the overallplastic deformation. As a result, inboth of these cases a bridge isre q u i red which allows for a model-ing connection to be madebetween the “micro s c o p i c ”p rocesses such as bond bre a k i n gand the macroscopic observ a b l e ssuch as the force-extension curv e .

E ff o rts in the Phillips group andthat of Michael Ortiz have beenaimed at constructing multiscalemodels which are sufficiently gen-eral to be able to treat the forc e -extension curves in materials rang-ing from proteins to tungsten.

n intriguing alternative tothe atom-by-atom simula-tion of forc e - e x t e n s i o nc u rves like those dis-

cussed above has been the devel-opment of new techniques inwhich high resolution is kept onlyin those parts of the material whereit is really needed. We close thisessay with a brief exposition of theuse of these methods to examinethe way in which defects give riseto plastic deformation in strainedmaterials, and how by virtue ofentanglements of these defects,such materials are hard e n e d .Without entering into a detailedexposition of the character ofdefects that populate materials, wenote again that the plastic defor-

mation of materials is often mediat-ed by defects known as disloca-tions. Roughly speaking, disloca-tions are the crystal analog of thetrick one might use to slide ane n o rmous carpet. If we imaginesuch a carpet and we wish to slideit a foot in some direction, one wayto do so is by injecting a bulgef rom one side as shown schemati-cally in Fig. 6. Hence, rather thanhaving to slide the whole carpeth o m o g e n e o u s l y, we are facedinstead with only having to slide alittle piece with a width equal tothat of the bulge. Nevertheless, thenet result of this action is overalltranslation of the carpet. This samebasic idea is invoked in the settingof stressed crystals where the slid-ing of one crystal plane withrespect to another is mediated by aline object (like the bulge describedabove) on which atomic bonds arebeing re a rranged.

One of the key features ofd e f o rmed crystals is the fact thatthe defects described above canencounter other such defectswhich exist on diff e rent cry s t a lplanes. The net result is the form a-tion of a local entanglement knownas a dislocation junction. The for-mation of such entanglements hasthe observable consequence thatthe crystal is harder—the criticals t ress needed to perm a n e n t l y

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Fi g u re 5. S c h e m atic of the fo rce - exte n s i o n

c u rve measure m e nt proce d u re and the fo rce -

extension curve for the muscle pro tein titin.

As shown in (A), the molecule is stre tc h e d

using the ato m i c - fo rce micro s co pe and leads

to (B), a fo rce - extension spe ct rum which is a

m e c h a n i cal fingerp ri nt for the molecule of

i nte rest (co u rtesy of Julio Fe rn a n d ez) .

Fi g u re 6. The sliding of a ca rpet by inject i n g

a bulge is analogous to defo rm ation of crys-

tals by injecting dislocat i o n s.

a

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d e f o rm the material (i.e., the plastict h reshold) is raised by the pre s e n c eof junctions. Although this entan-glement is ultimately and intrinsi-cally a particular configuration ofthe various atoms that make up amaterial, by exploiting ideas fro melasticity theory it is possible tore p resent all of this atomic-levelcomplexity in terms of two inter-acting lines. For present purposes,

the replacement of the all-atomperspective by an elastic theore t i-cal surrogate is exactly the type ofmultiscale analysis argued for earli-er in this essay.

F i g u re 7 shows the stru c t u re ofsuch a dislocation junction as com-puted not by considering the atomsthat make up the material, butrather as a collection of interactinglines. Just as the various moleculesthat make up a gas can be elimi-nated from consideration by invok-ing an equation of state andexploiting hydrodynamics, so too inthe context of modeling the defor-mation of materials may we re p l a c e

defects that are intrinsicallyatomistic by elastic surro g a t e swhich allow us to answer themultiscale challenge of materialresponse. As a result of exploitingthe correspondence between theatomic-level and elastic descriptionof junctions, we have been able toevaluate the critical stress neededto disentangle the two dislocationsthat make up a given dislocation

junction. One example pre s e n t e dh e re (that of interactions betweendislocations), ferrets out the natureof the conspiracy between the vari-ous defects such as dislocations,grain boundaries, and cracks thatmake up materials and that areresponsible for observed macro-scopic material response. Some ofthe other problems we have exam-ined using multiscale models arethe nucleation of dislocations atcrack tips, the interactions of dislo-cations with grain boundaries, andthe response of proteins to extern a lf o rcing (Fig. 5).

his essay has attempted toconvey some of the excite-ment that has arisenbecause of the advent of

the ability to build models of sys-tems of interest to scientists andengineers that intrinsically involvemultiple scales in either space ortime or both. Though we havea rgued that multiscale modelinghas always been a part of the theo-retical arsenal used to investigatep roblems ranging from turbulentflow to the magnetic pro p e rties ofmaterials, high-speed computationhas led to a re s u rgence of intere s tin the construction of coarse-grained models. This re p resents anamusing twist of fate since naivelyone might have expected that suchcomputational re s o u rces wouldallow for the “first principles” simu-lation of processes without theneed for theoretical surrogates. Onthe other hand, I have argued thatas it has always been, the develop-ment of compelling models of theworld around us must be basedupon the realization of a tastefuldistinction between those feature sof a system which are really neces-s a ry and those that are not. Thisidea served as a cornerstone inmany of the great historical exam-ples of multiscale modeling ands e rves as an embodiment ofE i n s t e i n ’s dictum that “Thingsshould be made as simple as possi-ble—but not simpler. ”

Rob Phillips is Professor of M e ch a n i ca lE n g i n e ering and Applied Physics.

Visit Professor Rob Phillips’s web pages viaht t p : / / w w w. m e. ca l te c h . e d u / f a c u l ty

p r o g r e s s r e p o r t

Fi g u re 7. A junction be tween two disloca-

tions as modeled using the same theory of

e l a s t i c i ty first deve l o ped by Ro be rt Hoo ke

and deri ved using the ex pe ri m e ntal appara-

tus of Fi g. 4.

t