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Computational Mechanics:Where Is It Going?Iva BABUSKA . J. TINSLEY ODEN

1. Introduction

Mechanics, the discipline of applied science concerned with the study of mo-tion, defonnation, and flow of materials, solids and fluids, under the actionof forces, has been essential in the development of technological advances inthe industrialized world for over two centuries. During the last four decades,a dramatic, qualitative change in mechanics has occurred owing to the adventof electronic computation. Computers have made possible the transformationof purely qualitative theoretical mechanics into an indispensable tool used incountless engineering and scientific applications.

The rapid development of computers has originated the new discipline ofcomputer science, the discipline concerned specifically with the computer it-self as a tool. The term, computational science, on the other hand, refers tothe disciplines concerned with the development of techniques of mathematicalmodeling, mathematical analysis, numerical algorithms, software, and applica-tions to specific problems in science and engineering. Computational mechanicsresides in the computational sciences. Its borders are rapidly widening, as muchof the domain of classical and modern physics, biology, and chemistry are nowfurnishing new ground for work in computational mechanics.

To predict trends in this discipline, we must recognize the rapid and remark-able advances in computer hardware. Using the VAX/I 1/780 as unit measuringcapabilities only around a decade ago (I Megabyte of memory, 0.5 Gigabytedisk, I cpu with a speed of 0.1 megaflops), in 1992 the IBM RS580 had 64 timesthe memory of the VAX, a storage that was an order of magnitude greater, andit's CPU speed was 1000 times greater, this representing an increase in perfor-mance only over a three year period. In 1997, the SGI Power Challenge had1000 times the memory of the VAX, 150 times the disk storage, 6 cpu's givinga theoretical speed of 18,000 times the VAX; IBM's recently announced "BlueGene" will deliver a petaflop: 1015 operations/second. The great challenge incomputational mechanics will be to develop the underlying mathematics, algo-rithms, and data structures to take advantage of these dramatic developmentsin hardware. Even so, the intelligent and efficient use of these new capabilitieswill represent a challenge of equal weight to the development of the tools them-selves. As the old adage observes: "It is not enough to have a big hammer; onemust also hit the nail on the head."

In broad disciplines of mechanics, advances in both computer and compu-tational science have led to increased confidence in the use of computationalmodels to predict natural events and to design engineering systems. Today it

24 I.BABUSKA . J. TINSLEY ODEN

is possible to avoid many costly experiments using computer simulations, todevelop new materials, analyze multiscale phenomena, and make importantpredictive calculations in engineering design, in biology and medical applica-tions, environmental systems, the flows of rivers, oceans and estuaries and theirinteraction with the earth and the atmosphere, and many other application ar-eas. Still, computational mechanics is in a relatively early stage of development.Dramatic new advances can be expected. Here, we ask, what can we expect inthis field in the future?

2. Mathematical Modeling

It is first important to appreciate that computational mechanics is intrinsicallyconnected to mathematics and mathematical models. Theoretical mechanics in-volves theories which are communicated in the language of mathematics: par-tial differential equations, ordinary differential equations, integral equations;and their implementation draws from fundamental results in algebra, combi-natorial and discrete mathematics, geometry and topology. To bring moderncomputational methods to bear on this rich mathematical structure, a naturaldiscretization is used to transform a mathematically formulated problem intoone that can be digested and processed by modern digital devices. The mathe-matical model itself involves abstractions of physical laws together with vari-ous simplifications and various information that is used to depict the system athand. The mathematical problem may also be called a mathematical model, forit transforms a physical concept into a collection of mathematical expressionsand data.

A typical example is phenomena characterized by linear partial differen-tial equation, defined on a domain D, with specific boundary conditions andsource terms on the "right hand side." To render this mathematical model intoa computational setting, the input data must involve some description of thedomain D, the coefficients in the equation, the source data, boundary condi-tions or initial conditions. In specific applications, the domain D may be takenfrom CAD data or imaging data and the coefficients are related to materialproperties which could be determined experimentally or from other types ofmodels. Boundary conditions represent ~bstractions of the interaction of theboundary of the domain under considenltibn with the surrounding universe, allreduced to mathematical abstractions by the analyst. In applications, the goalof the computation is to obtain data of interest which are directly related to themathematical solution of the problem, typically, in the form of linear and non-linear functionals. These values are then used to make engineering decisions orscientific predictions.

Traditionally it has been assumed that all the data of the problem are knownperfectly, and of course this is never the case. Intrinsic in the definition andutilization of any model, mathematical and computational, is a level of uncer-tainty. Physical details are always lost in the mathematical abstraction itself,uncertainties exist in the input and output, the sources, boundary conditions,the material characterizations, and even in interpreting the qualitative featuresof the output solutions. If the role of computer simulation in ,computational

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Computational Mechanics: Where Is It Going?

mechanics is to continue to grow and have significant impact on science andengineering, it will be absolutely necessary to determine quantitative estimatesof the influence of uncertainties in modeling. Moreover, the robustness of themodel itself in a crucial consideration: its sensitivity or insensitivity to the un-certainty in the input data. Otherwise, no practical meaning can be assigned tocomputed predictions.

One important area in which computer simulations are now having someimpact is the study of heterogeneous materials, that is materials, fluids or solids,composed of a heterogeneous conglomerate of constituents. In these cases, typ-ically a mathematical model involves differential equations with "rough" co-efficients. Then gradients of the solution, fluxes and/or stresses, are extremelysensitive to perturbations in these coefficients, while some averages over a re-gion could be more or less insensitive in relation to changes in the data. Thenotion of reproducibility of the response of a system to stimuli must be capturedmathematically in the characterization of the model and its inherent stability tomeaningful changes in data.

Another area of increasing interest in the development of mathematicalmodels in computational mechanics is the definition of the domain D itself asinput data. Let us assume that the domain is obtained by an imaging process sothat scanning a certain specimen presents us with only a "pixel" characterizationof a domain. If the physical problems involve a microstructure characterizedby circular inclusions, the pixel representation will, of course, not be a perfectcircle. Suppose that we were interested in the value of the solution at the centerof the circle. If Dp denotes the pixel domain and u is the solution to the actualsolution to the problem defined on D, then it is possible for some boundaryconditions to show that lu(O) -up Iflu Imay be of order 20% for arbitrarily smallpixels. Hence the problem is how to smooth the pixel domain or whether or notsome reformulation of the boundary conditions removes this paradox. This canindeed be done. Here again we see the essential importance of the mathematicalanalysis. The mathematical analysis not only becomes the basis for defining themathematical model, but it is also essential for determining geometric propertiesof the model, their sensitivity to input data, and to understand the properties ofthe model itself.

One must also come to grips with the fact that the mathematical model cannever be identified completely with reality because of the various simplificationsand uncertainties inherent in the modeling itself. We may, however, endow themodel with various types of complexities and thereby create a hierarchy ofmodels. Then we can make the purely mathematical assumption that the highestlevel of sophistication in this hierarchy is identified with reality and any othermodel in this hierarchy is an approximation for which en'or can be estimated.This lays the mathematical groundwork for selecting appropriate models froma large class of possible abstractions of physical events.

It is also commonly accepted in computational mechanics that the fullyrigorous mathematical analysis of very complex problems can be extremelycomplicated, if not impossible. Thus, in computational mechanics, it is under-stood that numerical experiments have an important role and may have to beused for understanding the problem itself. This fact puts a tremendous burden

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26 I.BABUSKA . J. TINSLEY ODEN

on the proper use of numerical analysis. Misleading conclusions can easilyarise if numerical experiments are not well-designed and are not fully analyzed.Computational science, therefore, is a complement to mathematical analysisand vice versa. It will not replace mathematics, but it will definitely changethe philosophy of mathematics and complement it in important applications.In addition, computers will be used for creation of mathematically formulatedconjectures, testing conjectures into computer-based proofs, a somewhat con-troversial application but one which we predict will be more and more acceptedby the mathematical community.

Thus, we predict a broader and deeper acceptance in pure mathematics of thecomputational sciences and this will bring today's pure math closer to appliedmathematics and computation.

We believe that one thing is clear: much of the intuition and heuristicscommon to applied and computational mechanics in its early days will grad-ually give way to more powerful ideas, methods, and approaches. With fur-ther advances in computational mechanics, computer simulations will be usedas a guide in many cases where intuition fails or cannot be used as a guide.The cost of computation has been declining for decades, while the humancost of the understanding and formulation of the mathematical problem andthe computational model, of developing software and implementing it to solvepractical problems, is continuing to rise. For this reason, we expect to see sig-nificant progress in the incorporation of automated methods, adaptivity, andautomatic decision making within mathematical algorithms. These technolo-gies will be applied to a broader class of phenomena, including, in particularhighly nonlinear phenomena that are viewed in both deterministic and stochasticsettings. Much more sophisticated models will be used in the future, includingthose which incorporate multiscale features, micromechanical aspects of mate-rials in fluids, and which ultimately involve optimization in design and manu-facturing.

3. Numerical Mathematics in Computational Science

To many users of the methods and technologies of computational mechanics,the ultimate proof comes with the comparison of computed simulations withbehaviors a~tually observed in the laboratory or in nature. These comparisonsare always accompanied by errors, error in the choice of the mathematicalmodel itself, error in the data furnished to characterize the model and errorin the numerical approximation. To distinguish between these errors, we firstmust develop confidence that the numerical errors are measurable and distinctwhen compared to the error of the mathematical model. In the future, it willbe necessary to control these eITors, to estimate them with some precision andto adaptively make changes in the model and the numerical approximation soas to enhance the validity and reliability of the computed simulation. To obtainthis goal, one must deal with the following issues.

A. With an increasing computational effort we haye to be able, at least the-oretically, to solve the problem accurately, i.e. the computed data of interest

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Computational Mechanics: Where Is It Going')

should, not essentially change when the effort is sufficiently large. This is re-lated to convergence. Although rigorous mathematical proof is desired, carefulnumerical tests often are the basic tools. Once more it is necessary to make andinterpret the' numerical results carefully.

B. Computation has to be reliable and the range of the accuracy obtained bya posteriori error estimation should be computed. The estimate has to be withrespect to 'the data of interest or a norm which is relevant for the application.The error· estimate should be guaranteed and a reliability interval which is theinterval between upper and lower bounds should also be available.

C. An adaptive procedure which leads to the solution in a prescribed rangeand reduces drastically the human effort and computational etfort for largeproblelT¥.'is essential. Here adaptive procedures are related not only to the elTorof nun,rical approximation but also to the selection of the mathematicalmodel .•.

4. Where Are We Going?

If the province of mechanics is truly the study of the behavior of systems andbodies under the action of forces, then all types of such systems encountered inscience are fair game: subatomic systems, dynamics of systems of molecules anddislocations, biological systems, quantum effects, chemical kinetics, geologicalsystems, the systems of interest in astronomy and astrophysics, the behaviorof the atmosphere and oceans, and many more. At this beginning of the newmillennium, we observe that the scientific foundations of may of these diversedisciplines has evolved to a remarkable level of maturity; the subjects are ripefor applications, mathematical analysis, computer modeling, and simulationgeared toward predictive capability. This is the new ground for computationalmechanics,

The challenges are formidable. Most important will be the development ofa rigorous and meaningful mathematical structure for characterizing the use-ful models of such phenomena. Today, there remains much empiricism andheuristics in these fields. This must give way to precise definitions of the scope,reliability, and structure of a large body of mathematical models; else, the trans-formation of these to useful computational tools will be no more reliable thanHollywood animations.

Secondly, the enormous range of temporal and spatial scales over whichmany important phenomena occur must be conquered by new methods anddevices. A clear delineation of which scales are important in models of specificevents and of how phenomena determined by events at multiple scales must bemade. Progress can be measured when concrete approaches emerge that makeprecise the limitations of anyone class of models and which give concretedirection as to how these classes can be enlarged or reduced to capture featuresof system response of interest.

The payoff of the new computational mechanics could be enormous: newmaterials, optimized to serve a variety of desirable functions, new drugs, chem-icals, and chemical processes, predictive surgical procedures reviewed for eachsubject on the basis of detailed personalized computational models, reliable

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28 I.BABUSKA . J. TINSLEY ODEN

predictions of weather, environmental flows, galactic phenomena, new sub-atomic devices for thousands of applications, all done with a level of reliabilityunimaginable only a decade ago.

Finally, it is emphasized that there is much new mathematical work to do;these advancements will not be achieved without significant advances in math-ematical modeling, numerical analysis, and computers and computer science,all components of the new computational mechanics.

FromJOHN C. BJ

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