Microscopic properties and physical behavior of materials and media.

题目：材料和介质的微观性质和物理行为摘要研究者和他的同事将考虑两个不同的研究领域：1)包含微观失配的非均匀材料的整体（均质）行为的估计，以及2)非均匀介质的直接和逆散射。众所周知，细尺度的材料相互作用（包括微观应变、应力、热场和塑性场等）在决定材料的整体行为方面起着至关重要的作用。虽然这种微观相互作用在许多情况下被很好地理解，但从理论和实验的角度来看，这一背景下的某些基本问题仍然难以捉摸，并且仍然没有完全解决。在这些方面提出的努力将寻求产生一种改进的理论理解，以及比以前可用于涉及各向异性和非线性材料的三维失配场的问题的更一般严格的求解方法。这样的求解器可以显著提高我们研究多晶塑性和弹性体磁流变等问题的能力。电磁波和声波在非均匀介质中的正散射和逆散射问题也将作为本研究工作的一部分进行研究。这里提出的工作将建立在研究者和他的同事最近对穿透和表面散射问题的快速高阶积分方程求解器的贡献之上。这些求解器能够以前所未有的精度和效率处理复杂的散射问题；建议的工作将特别强调最近开发的O(1)算法-它可以解决任意高频率的问题（随着频率的增加，通常需要越来越大的离散密度），具有规定的精度和固定的计算时间，与频率无关。提议的工作对社会感兴趣的各个领域产生影响，包括医学领域（光学相干断层扫描）、汽车（磁流变弹性体）、材料设计（致动器、材料强度）、军事（雷达）；通信（安装在飞机和航天器上的天线、无线通信）；关键尺寸计量（硅片光学监测）；下一代处理器中的电磁感应建模等。另一方面，就像这个PI以前的工作一样，目前的努力将导致培养研究生和本科层次的学生，以及博士后助理。特别是，在过去的五年里，这个PI指导了15个不同的研究项目，每个项目为期10个夏季周，每个项目都以一名本科生为中心。这些本科生包括7名美国公民、3名女性、2名黑人男性、2名西班牙裔和2名来自当地社区大学的学生。这些活动对所有参与者都有很高的回报，包括与本科生和该PI合作的研究生和博士后，以及一些提出具体工程应用的工业和实验室研究人员。由于这项工作，一些参与的研究生和博士后发现了在数学教学和指导方面的才能。大多数参与研究的本科生都进入了研究生院或四年制大学（视情况而定），并寻求从事数学相关领域的工作。将本科生纳入研究项目的努力将继续作为拟议工作的一部分。该奖项由数学科学部的应用数学和计算数学项目共同资助。

Proposal: DMS-0408040PI: Oscar P. BrunoInstitution: CaltechTitle: Microscopic properties and physical behavior of materials and mediaABSTRACTThe 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.