Progress Report - Modeling in Engineering: The Challenge of Multiple Scales

hether we consider the design of a new generation of airliners such as the Boeing 777 or the development of the latest microp rocessors, engineering re l i e s i n c reasingly on the use of mathematical models to characterize these technologies. In the case of the 777, sophisticated models of the fluid mechanics of air flow over the wings were an integral part of the design process, just as stru ctural mechanics models ensure d that flight in turbulence leads to nothing more grave than passenger d i s c o m f o rt . Models of complex materials that make up our modern technologies also pose a wide range of scientific challenges. Indeed, many of the most important recent advances in the study of materials re s u l ting in entirely new classes of materials such as the famed oxide hight e m p e r a t u re superconductors or f u l l e renes, and their structural partners known as carbon nanotubes, have engendered a flurry of modeling eff o rts. I m p o rtant problems that such modeling must confront are those of an intrinsically multiscale nature . What this means is that analysis of a given problem re q u i res simultaneous consideration of several spatial or temporal scales. This idea is well re p resented in drawings made m o re than 500 years ago by L e o n a rdo da Vinci, in which the turbulent flow of a fluid is seen to involve vortices within vort i c e s over a range of scales. This sketch (see Fig. 1) serves as the icon for the new Caltech center known as the Center for Integrative Multiscale Modeling and Simulation (CIMMS) [see article on page 10]. CIMMS brings together faculty members from several diff e re n t Options and Divisions including P rofessors K. Bhattachary a (Mechanical Engineering), E. Candes (Applied & Computational Mathematics), J. Doyle ( C o n t rol & Dynamical Systems, Electrical Engineering, and Bioengineering), M. Gharib ( A e ronautics and Bioengineering), T. Hou (Applied & Computational Mathematics), H. Mabuchi (Physics and Control & Dynamical Systems), J. Marsden (Control & Dynamical Systems), R. Murray (Control & Dynamical Systems and Mechanical Engineering), M. Ort i z ( A e ronautics and Mechanical Engineering), N. Pierce (Applied & Computational Mathematics), R. Phillips (Mechanical Engineering and Applied Physics) and P. Schröder (Computer Science and Applied & Computational Mathematics). The aim of multiscale modeling is to construct models of relevance to macro s c o p i c scales usually observed in experiment and tailored in the engineer-