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Computational Mechanics: Where Is It Going?

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Computational Mechanics: Where Is It Going? r 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 it- self 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 applica- tions 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 remark- able 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, algo- rithms, 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 compu- tational 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, typ- ically a mathematical model involves differential equations with "rough" co- efficients. 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.
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