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 action of forces, has been essential in the development of technological advances in the industrialized world for over two centuries. During the last four decades, a dramatic, qualitative change in mechanics has occurred owing to the advent of electronic computation. Computers have made possible the transformation of purely qualitative theoretical mechanics into an indispensable tool used in countless engineering and scientific applications. The rapid development of computers has originated the new discipline of computer science, the discipline concerned specifically with the computer itself as a tool. The term, computational science, on the other hand, refers to the disciplines concerned with the development of techniques of mathematical modeling, mathematical analysis, numerical algorithms, software, and applications to specific problems in science and engineering. Computational mechanics resides in the computational sciences. Its borders are rapidly widening, as much of the domain of classical and modern physics, biology, and chemistry are now furnishing new ground for work in computational mechanics. To predict trends in this discipline, we must recognize the rapid and remarkable advances in computer hardware. Using the VAX/I 1/780 as unit measuring capabilities only around a decade ago (I Megabyte of memory, 0.5 Gigabyte disk, I cpu with a speed of 0.1 megaflops), in 1992 the IBM RS580 had 64 times the memory of the VAX, a storage that was an order of magnitude greater, and it'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 had 1000 times the memory of the VAX, 150 times the disk storage, 6 cpu's giving \ a theoretical speed of 18,000 times the VAX; IBM's recently announced "Blue Gene" will deliver a petaflop: 1015 operations/second. The great challenge in computational mechanics will be to develop the underlying mathematics, algorithms, and data structures to take advantage of these dramatic developments in hardware. Even so, the intelligent and efficient use of these new capabilities will 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; one must also hit the nail on the head." In broad disciplines of mechanics, advances in both computer and computational science have led to increased confidence in the use of computational models 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, to mec develop new materials, analyze multiscale phenomena, and make important engi predictive calculations in engineering design, in biology and medical applica- of tJ tions, environmental systems, the flows of rivers, oceans and estuaries and their moe interaction with the earth and the atmosphere, and many other application ar- cert. eas. Still, computational mechanics is in a relatively early stage of development. corr Dramatic new advances can be expected. Here, we ask, what can we expect in this field in the future? Imp com icall 2. Mathematical Modeling effie sem It is first important to appreciate that computational mechanics is intrinsically glOr connected to mathematics and mathematical models. Theoretical mechanics in- noti volves theories which are communicated in the language of mathematics: par- mati tial differential equations, ordinary differential equations, integral equations; mea and their implementation draws from fundamental results in algebra, combi- natorial and discrete mathematics, geometry and topology. To bring modern moe computational methods to bear on this rich mathematical structure, a natural inpt discretization is used to transform a mathematically formulated problem into that one that can be digested and processed by modern digital devices. The mathe- of a matical model itself involves abstractions of physical laws together with vari- by c ous simplifications and various information that is used to depict the system at circl hand. The mathematical problem may also be called a mathematical model, for of t~ it transforms a physical concept into a collection of mathematical expressions solu and data. COIll A typical example is phenomena characterized by linear partial differen- pixe tial equation, defined on a domain D, with specific boundary conditions and som source terms on the "right hand side." To render this mathematical model into inde a computational setting, the input data must involve some description of the anal domain D, the coefficients in the equation, the source data, boundary condi- matI tions or initial conditions. In specific applications, the domain D may be taken of t~ from CAD data or imaging data and the coefficients are related to material the I properties which could be determined experimentally or from other types of I models. Boundary conditions represent ~bstractions of the interaction of the nev{ boundary of the domain under considenltibn with the surrounding universe, all and reduced to mathematical abstractions by the analyst. In applications, the goal mod of the computation is to obtain data of interest which are directly related to the mod mathematical solution of the problem, typically, in the form of linear and non- leve linear functionals. These values are then used to make engineering decisions or mod scientific predictions. This Traditionally it has been assumed that all the data of the problem are known alar perfectly, and of course this is never the case. Intrinsic in the definition and utilization of any model, mathematical and computational, is a level of uncer- ngo: tainty. Physical details are always lost in the mathematical abstraction itself, com uncertainties exist in the input and output, the sources, boundary conditions, stoo the material characterizations, and even in interpreting the qualitative features used of the output solutions. If the role of computer simulation in ,computational Computational Mechanics: Where Is It Going? 25 ations, to mechanics is to continue to grow and have significant impact on science and important engineering, it will be absolutely necessary to determine quantitative estimates J applica- of the influence of uncertainties in modeling. Moreover, the robustness of the ; and their model itself in a crucial consideration: its sensitivity or insensitivity to the un- cation ar- certainty in the input data. Otherwise, no practical meaning can be assigned to elopment. computed predictions. expect in One important area in which computer simulations are now having some impact is the study of heterogeneous materials, that is materials, fluids or solids, composed of a heterogeneous conglomerate of constituents. In these cases, typically a mathematical model involves differential equations with "rough" coefficients. Then gradients of the solution, fluxes and/or stresses, are extremely sensitive to perturbations in these coefficients, while some averages over a re- trin\;ically gion could be more or less insensitive in relation to changes in the data. The :hanics in- notion of reproducibility of the response of a system to stimuli must be captured atics: par- mathematically in the characterization of the model and its inherent stability to equations; meaningful changes in data. 'a, comb i- Another area of increasing interest in the development of mathematical Ig modern models in computational mechanics is the definition of the domain D itself as , a natural input data. Let us assume that the domain is obtained by an imaging process so lblem into that scanning a certain specimen presents us with only a "pixel" characterization 'he mathe- of a domain. If the physical problems involve a microstructure characterized with vari- by circular inclusions, the pixel representation will, of course, not be a perfect ,system at circle. Suppose that we were interested in the value of the solution at the center model, for of the circle. If Dp denotes the pixel domain and u is the solution to the actual xpressions solution to the problem defined on D, then it is possible for some boundary conditions to show that lu(O) -up Iflu I may be of order 20% for arbitrarily small I differen- pixels. Hence the problem is how to smooth the pixel domain or whether or not litions and some reformulation of the boundary conditions removes this paradox. This can model into I indeed be done. Here again we see the essential importance of the mathematical " tion of the analysis. The mathematical analysis not only becomes the basis for defining the ary condi- mathematical model, but it is also essential for determining geometric properties LY be taken 'i of the model, their sensitivity to input data, and to understand the properties of .0 material :JI the model itself. ~r types of One must also come to grips with the fact that the mathematical model can tion of the never be identified completely with reality because of the various simplifications niverse, all and uncertainties inherent in the modeling itself. We may, however, endow the lS, the goal model with various types of complexities and thereby create a hierarchy of lated to the models. Then we can make the purely mathematical assumption that the highest lr and non- level of sophistication in this hierarchy is identified with reality and any other ecisions or model in this hierarchy is an approximation for which en'or can be estimated. This lays the mathematical groundwork for selecting appropriate models from are known I a large class of possible abstractions of physical events. I inition and :j It is also commonly accepted in computational mechanics that the fully :1of uncer- rigorous mathematical analysis of very complex problems can be extremely ;tion itself, complicated, if not impossible. Thus, in computational mechanics, it is under- conditions, stood that numerical experiments have an important role and may have to be ve features used for understanding the problem itself. This fact puts a tremendous burden nputational I

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on the proper use of numerical analysis. Misleading conclusions can easily should r

arise if numerical experiments are not well-designed and are not fully analyzed. lated to I Computational science, therefore, is a complement to mathematical analysis numeric and vice versa. It will not replace mathematics, but it will definitely change interpret the philosophy of mathematics and complement it in important applications. B.C In addition, computers will be used for creation of mathematically formulated a posteri conjectures, testing conjectures into computer-based proofs, a somewhat con- respect 1 troversial application but one which we predict will be more and more accepted The errc by the mathematical community. interval Thus, we predict a broader and deeper acceptance in pure mathematics of the C.A computational sciences and this will bring today's pure math closer to applied and redl mathematics and computation. problem We believe that one thing is clear: much of the intuition and heuristics of numc common to applied and computational mechanics in its early days will grad- model. ually give way to more powerful ideas, methods, and approaches. With fur- ther advances in computational mechanics, computer simulations will be used as a guide in many cases where intuition fails or cannot be used as a guide. 4.Whl The cost of computation has been declining for decades, while the human cost of the understanding and formulation of the mathematical problem and If the pr the computational model, of developing software and implementing it to solve bodies u practical problems, is continuing to rise. For this reason, we expect to see sig- science; nificant progress in the incorporation of automated methods, adaptivity, and dislocat! automatic decision making within mathematical algorithms. These technolo- systems gies will be applied to a broader class of phenomena, including, in particular of the a highly nonlinear phenomena that are viewed in both deterministic and stochastic millenni settings. Much more sophisticated models will be used in the future, including disciplir those which incorporate multiscale features, micromechanical aspects of mate- for appl rials in fluids, and which ultimately involve optimization in design and manu- geared t facturing. mechan The a rigoro ful mod 3. Numerical Mathematics in Computational Science heuristic reliabili To many users of the methods and technologies of computational mechanics, formati( the ultimate proof comes with the comparison of computed simulations with Hollywi behaviors a~tually observed in the laboratory or in nature. These comparisons Sec( are always accompanied by errors, error in the choice of the mathematical many ir model itself, error in the data furnished to characterize the model and error devices. in the numerical approximation. To distinguish between these errors, we first events a must develop confidence that the numerical errors are measurable and distinct made. F when compared to the error of the mathematical model. In the future, it will precise be necessary to control these eITors, to estimate them with some precision and directio to adaptively make changes in the model and the numerical approximation so of syste as to enhance the validity and reliability of the computed simulation. To obtain The this goal, one must deal with the following issues. materia A. With an increasing computational effort we haye to be able, at least the- icals, ar oretically, to solve the problem accurately, i.e. the computed data of interest subject Computational Mechanics: Where Is It Going') 27

1I1 easily should, not essentially change when the effort is sufficiently large. This is re- nalyzed. lated to convergence. Although rigorous mathematical proof is desired, careful analysis numerical tests often are the basic tools. Once more it is necessary to make and ( change interpret the' numerical results carefully. lications. B. Computation has to be reliable and the range of the accuracy obtained by rmulated a posteriori error estimation should be computed. The estimate has to be with ,hat con- respect to 'the data of interest or a norm which is relevant for the application. accepted The error· estimate should be guaranteed and a reliability interval which is the interval between upper and lower bounds should also be available. ics of the C. An adaptive procedure which leads to the solution in a prescribed range :> applied and reduces drastically the human effort and computational etfort for large problelT¥.'is essential. Here adaptive procedures are related not only to the elTor leuristics of nun,rical approximation but also to the selection of the mathematical viII grad- model .•. With fur- I be used , a guide. 4. Where Are We Going? Ie human blem and If the province of mechanics is truly the study of the behavior of systems and .t to solve bodies under the action of forces, then all types of such systems encountered in osee sig- science are fair game: subatomic systems, dynamics of systems of molecules and ivity, and dislocations, biological systems, quantum effects, chemical kinetics, geological technolo- systems, the systems of interest in astronomy and astrophysics, the behavior particular of the atmosphere and oceans, and many more. At this beginning of the new stochastic millennium, we observe that the scientific foundations of may of these diverse including disciplines has evolved to a remarkable level of maturity; the subjects are ripe s of mate- 1 for applications, mathematical analysis, computer modeling, and simulation I geared toward predictive capability. This is the new ground for computational .nd manu- t iI mechanics, I The challenges are formidable. Most important will be the development of i a rigorous and meaningful mathematical structure for characterizing the use- :i ful models of such phenomena. Today, there remains much empiricism and i heuristics in these fields. This must give way to precise definitions of the scope, j' reliability, and structure of a large body of mathematical models; else, the trans- lechanics, ~ formation of these to useful computational tools will be no more reliable than tions with Hollywood animations. mparisons Secondly, the enormous range of temporal and spatial scales over which thematical many important phenomena occur must be conquered by new methods and and error 'j devices. A clear delineation of which scales are important in models of specific 's, we first "i I events and of how phenomena determined by events at multiple scales must be nd distinct made. Progress can be measured when concrete approaches emerge that make Jre, it will precise the limitations of anyone class of models and which give concrete cision and 11 direction as to how these classes can be enlarged or reduced to capture features imation so J l of system response of interest. . To obtain • ; The payoff of the new computational mechanics could be enormous: new i materials, optimized to serve a variety of desirable functions, new drugs, chem- Itleast the- icals, and chemical processes, predictive surgical procedures reviewed for each of interest ~ subject on the basis of detailed personalized computational models, reliable

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predictions of weather, environmental flows, galactic phenomena, new sub- From atomic devices for thousands of applications, all done with a level of reliability unimaginable only a decade ago. JOHN C. BJ Finally, it is emphasized that there is much new mathematical work to do; these advancements will not be achieved without significant advances in mathematical modeling, numerical analysis, and computers and computer science, all components of the new computational mechanics.

1. Introd

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