Microscopic properties and physical behavior of materials and media.

California Institute Of Technology, Pasadena CA

Investigators

Abstract

Proposal: DMS-0408040 PI: Oscar P. Bruno Institution: Caltech Title: Microscopic properties and physical behavior of materials and media ABSTRACT The investigator and his colleagues will consider two distinct research areas: 1) Estimation of the overall (homogenized) behavior of inhomogeneous materials containing microscopic misfits, and, 2) Direct and inverse scattering by inhomogeneous media. As is well known, fine scale material interactions (including microscopic strains, stresses, thermal and plastic fields, etc.) play crucial roles in determining the overall behavior of materials. While such microscopic interactions are well understood in many cases, certain fundamental issues in this context have remained elusive from both theoretical and experimental standpoints, and have still not been fully resolved. The proposed efforts in these regards will seek to produce an improved theoretical understanding as well significantly more general rigorous solvers than previously available for problems involving three-dimensional fields of misfits in anisotropic and nonlinear materials. Such solvers should allow significant improvements in our capability to investigate problems such as polycrystalline plasticity and magneto-rheology of elastomers. Problems of direct and inverse scattering of electromagnetic and acoustic waves by inhomogeneous media will also be investigated as part of this research effort. The work proposed here will build upon recent contributions by the investigator and his colleagues on fast high-order integral-equation solvers for penetrable and surface scattering problems. These solvers can deal with complex scattering problems with unprecedented accuracy and efficiency; the proposed work will deal with special emphasis on recently developed O(1) algorithms - which can resolve problems of arbitrarily high frequency (which would usually require increasingly large discretization densities as frequencies increase) with a prescribed accuracy and within a fixed computational time-independent of frequency. The proposed work impacts upon a variety of areas of societal interest, including the medical field (optical coherence tomography), automotive (magneto-rheological elastomers), materials design (actuators, strength of materials), military (radars); communications (aircraft- and spacecraft-mounted antennas, wireless communications); critical dimension metrology (optical monitoring of silicon wafers); modeling of electromagnetic induction in next-generation processors; etc. As previous work by this PI, on the other hand, the present effort will result in training of students at both, graduate and undergraduate levels, as well as postdoctoral associates. In particular, over the last five years, fifteen different research efforts have been directed by this PI, each one for a period of ten summer weeks and each one centered around one undergraduate student. The undergraduate students included seven American citizens, three women, two black males, two Hispanic, and two students from a local community college. These activities have been highly rewarding for all involved, including the graduate students and postdocs that collaborated with the undergraduate students and this PI, as well as some of the industrial and lab researchers who proposed specific engineering applications. As a result of this work, some of the participating graduate students and postdocs discovered a talent for teaching and advising in mathematics. Most of the participating undergraduate students went on to graduate schools or four-year colleges, as appropriate, and have sought to pursue work on mathematically related fields. The efforts to integrate undergraduate students in research projects will continue as part of the proposed work. The award is funded jointly by the Applied Mathematics and Computational Mathematics programs of the Division of Mathematical Sciences.