Multiscale Analysis as an Approximation Scheme

Abstract

The development of microstructure and the attendant effective or macroscopic behavior of materials may often be understood as the material’s ultimately futile attempt to minimize a certain functional which has no minimizer. In the pursuit of this unattainable objective, the material develops fine nested structures often spanning the atomistic and many continuum scales. A number of ‘multiscale’ analytical and computational tools have been devised in order to study these material systems. For the most part, such schemes are computational in nature and ad hoc, in the sense that they offer no clear notion and guarantee of convergence to an exact solution. For systems with well-separated lengthscales, a well-developed mathematical theory, the ‘modern calculus of variations’, offers scientists and engineers a radically different and powerful way of thinking about what it means to ‘solve’ multiscale problems and approximate their ‘solutions’. In particular, within this modern calculus of variations framework multiscale analysis is a rigorous approximation scheme with a well-defined notion of convergence. I plan to illustrate the challenges and opportunities that this analysis framework offers by way of selected examples of application, including: the atomistic-to-continuum transition in molecular statics; discrete-to-continuum transition in dislocation dynamics; subgrain dislocation structures in polycrystals; and the time-evolution of surface roughness in thin films.

Brief Bio

Professor Ortiz received a BS degree in Civil Engineering from the Polytechnic University of Madrid, Spain, and MS and Ph.D. degrees in Civil Engineering from the University of California at Berkeley. From 1984-1995 he held a faculty position in the Division of Engineering of Brown University, where he carried out research activities in the fields of mechanics of materials and computational solid mechanics. He is currently the Dotty and Dick Hayman Professor of Aeronautics and Mechanical Engineering at the California Institute of Technology, where he has been in the faculty since 1995 and currently serves as the director of Caltech’s DoE/PSAAP Center on High-Energy Density Dynamics of Materials. Professor Ortiz has been a Fulbright Scholar, a Sherman Fairchild Distinguished Scholar at Caltech, Midwest and Southwest Mechanics Seminar Series Distinguished Speaker, a Fellow and an elected member-at large of the US Association for Computational Mechanics, and an elected Fellow of the American Academy of Arts & Sciences. Professor Ortiz is the recipient of the Alexander von Humboldt Research Award for Senior U.S. Scientists, the IACM International Computational Mechanics Awards for Research, the USACM Computational Structural Mechanics Award, the ISI Highly Cited Researcher Award, the inaugural 2008 Rodney Hill Prize conferred every four years by the IUTAM and the Hans Fischer Senior Fellowship of the Institute of Advanced Studies of the Technical University of Munich. Professor Ortiz has served in the University of California Office of the President Science and Technology Panel and in the Los Alamos National Laboratory T-Division Review Committee. He currently serves in the Lawrence Livermore National Laboratory Predictive Science Panel, the Sandia National Laboratories Engineering Sciences External Review Panel, the Lawrence Livermore National Laboratory Chemistry, Materials, Earth and Life Sciences Directorate Review Committee, the National Research Council Panel for the Evaluation of QMU and he chairs the Lawrence Livermore National Laboratory Engineering Directorate Review Committee. He has been editor of the Journal of Engineering Mechanics of ASCE and of the Journal of Applied Mechanics of the ASME, associate editor of the journal Modeling and Simulation in Materials Science and Engineering and of the Journal for Computational Mechanics, and is presently associate editor of the Journal for the Mechanics and Physics of Solids and of the Archive for Rational Mechanics and Analysis.

Add bookmark at 

• Google
• delicious
• Webnews
• Linkarena
• Mr. Wong
• Stumbleupon
• Yigg