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

NSF奖摘要- DMS-0094089数学科学：职业生涯：非均匀介质的宏观特性和应用数学的发展Grabovsky该项目研究的统一思想是由于米尔顿的均匀化的一般希尔伯特空间方法。将对基于这种方法的三个主要问题进行调查。 一个是关于G-闭包的凸性性质的探索，目的是开发新的数值逼近的难以捉摸的集合。这种方法可能会导致新的分析结果，在几个基本情况下，G-闭包仍然是未知的。 第二个问题是利用米尔顿W变换的有效张量公式研究随机介质的有效行为。该公式为随机复合体的场和有效张量提供了新的级数展开式。新的扩展显示收敛迅速，即使是一个相对较高的造影剂。最后，我们将发展一个新的和令人兴奋的想法的时空复合材料最近提出的Lurie。 时空复合材料是指电磁特性随时间快速变化的复合材料。 这些时间振荡可能是复合材料试样的高频振动或材料本身的活性性质的结果。本项目的研究重点是复杂智能材料的性能研究。 这种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，传感器和执行器中使用的新型活性材料。 该补助金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。 该实验室还将促进本科生参与坦普尔大学科学技术学院的数学研究和跨学科互动。该补助金还将支持一个多方面的应用研究计划，该计划由本研究中使用的常用数学工具统一起来。 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能会减少或消除昂贵和耗时的测量复合材料的弹性性能提供良好的理论预测。 另一个应用是高对比度无序介质的研究，在水文学中，根据当地测量结果预测大量沉积岩的渗透率。 渗透性是土壤的一个重要特性，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：复合材料的组成部分处于快速的相对运动中，或者其组成部分能够在没有机械运动的情况下迅速改变其性质。 该项目这一部分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.