Macroscopic Properties of Heterogeneous Media and Development of the Applied Mathematics Curriculum

数学科学：职业：异质介质的宏观性质与应用数学课程的发展[摘要]grabovsky本项目研究的统一思想是由弥尔顿引起的均匀化的一般希尔伯特空间方法。本文将研究基于此方法的三个主要问题。一个是探索g闭包的凸性性质，目的是为难以捉摸的集合开发新的数值逼近。这种方法可能会在一些尚不知道g闭包的基本情况下得到新的解析结果。第二个问题是利用弥尔顿w变换的有效张量的新公式来研究随机介质的有效行为。该公式为随机复合的场和有效张量提供了一个新的级数展开式。新的扩展显示，即使是相对高对比度的介质也能迅速收敛。最后，我们将阐述Lurie最近提出的一个令人兴奋的时空复合概念。时空复合材料是指电磁性能随时间快速变化的复合材料。这些时间振荡可能是复合试样高频振动的结果，也可能是材料本身活性的结果。本项目的研究重点是复杂智能材料的性能研究。这些材料的例子包括复合材料（用于航天器、飞机、汽车、天空、高尔夫球杆等）、随机介质（如岩石、粘土或骨头）、用于传感器和执行器的新型活性材料。这项拨款为各种研究和教育活动提供资金，其中包括一个小型弹性演示实验室。实验室的主要目的是通过实验和演示来提高学生对数学的兴趣。该实验室还将促进天普大学科学与技术学院的本科生参与数学研究和跨学科互动。这笔拨款还将支持一个多管齐下的应用研究项目，该项目由本研究中使用的通用数学工具统一。该项目的目标之一是预测复合材料的性能。在这个方向上的成功可以通过提供良好的理论预测来减少或消除昂贵和耗时的复合材料弹性性能测量。另一个应用是研究高对比无序介质，在水文学中对基于局部测量的大体积沉积岩的渗透率进行预测。渗透性是土壤的一个关键特性，它决定了地下水、石油或污染物如何在地下传播。另一个应用是时空复合材料的新领域：复合材料的成分处于快速的相对运动中，或者其成分能够在没有机械运动的情况下迅速改变其性质。这部分项目的成功对于设计和理解能够快速响应不断变化的环境的新一代智能材料至关重要。

NSF Award Abstract - DMS-0094089Mathematical Sciences: CAREER: Macroscopic properties of heterogeneous media and development of the applied mathematics curriculumAbstract0094089 GrabovskyThe unifying idea of this project's research is the general Hilbert space approach to homogenization due to Milton. There are three major issues based on this approach that will be investigated. One concerns the exploration of the convexity properties of G-closures with the goal of developing new numerical approximations to the elusive set. This approach may lead to the new analytic results in several basic cases, where the G-closure is still not known. The second issue is to make use of the new formula for effective tensors in terms of the W-transformation of Milton for studying effective behavior of the random media. That formula provides a new series expansion for the fields and effective tensors of the random composites. The new expansion is shown to converge rapidly even for a relatively high contrast media. Finally, we will develop a new and exciting idea of space-time composites proposed recently by Lurie. The space time composites refer to composites whose electromagnetic properties change rapidly in time. These temporal oscillations may be a result of high frequency vibrations of the composite specimen or of an active nature of the material itself. This project's research is centered on the investigation of properties of complex and smart materials. Examples of such materials include composites (used in spacecraft, airplanes, cars, skies, golf clubs, etc.), random media like rock, clay or bone, novel active materials that are used in sensors and actuators. This grant provides funding for a variety of research and educational activities that includes a small elasticity demonstration lab. The main purpose of the lab will be to heighten students' interest in mathematics through the inclusion of experiments and demonstrations. The lab will also facilitate involvement of undergraduates in mathematical research and interdisciplinary interaction throughout the College of Science and Technology at Temple University. The grant will also support a multi-prong applied research program unified by the common mathematical tools used in this research. One of the goals of this project is the prediction of properties of composite materials. Success in this direction may make it possible to reduce or eliminate expensive and time-consuming measurements of elastic properties of composite materials by providing good theoretical predictions. Another application is to the study of high-contrast disordered media, of interest in hydrology for prediction of the permeability of large volumes of sedimentary rock based on local measurements. The permeability is a crucial property of soil that determines how ground water, oil or pollutants propagate underground. Yet another application is to the novel field of space-time composites: composite materials whose components are in rapid relative motion or whose components have the ability to change their properties rapidly in time without mechanical motion. The success of this part of the project will be essential for designing and understanding the new generation of smart materials that are able to respond quickly to a changing environment.