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

NSF奖摘要-DMS-0094089数学科学：事业：非均匀介质的宏观特性和应用数学课程的发展摘要0094089 Grabovsky由于米尔顿的贡献，这个项目研究的统一思想是一般的希尔伯特空间方法。基于这一方法，将调查三个主要问题。一个是关于G-闭包的凸性性质的探索，目的是开发对难以捉摸的集合的新的数值逼近。在G-闭包未知的几种基本情况下，这种方法可能会得到新的解析结果。第二个问题是利用基于弥尔顿W变换的有效张量的新公式来研究随机介质的有效行为。该公式为随机复合材料的场和有效张量提供了新的级数展开式。结果显示，即使是对比度相对较高的介质，新的膨胀也会迅速收敛。最后，我们将发展Lurie最近提出的一个新的、令人兴奋的时空合成概念。时空复合材料是指电磁性能随时间变化很快的复合材料。这些时间振荡可能是复合材料试件的高频振动或材料本身的活跃性质的结果。本项目的研究主要集中在复杂智能材料的性能研究上。这类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，用于传感器和执行器的新型活性材料。这笔赠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。该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整个坦普尔大学科学与技术学院的数学研究和跨学科互动。这笔赠款还将支持一项多管齐下的应用研究计划，该计划统一了这项研究中使用的常见数学工具。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通过提供良好的理论预测来减少或消除昂贵且耗时的复合材料弹性性能测量。另一个应用是研究高对比度的无序介质，这在水文学中很有意义，可以根据当地的测量来预测大量沉积岩的渗透率。渗透性是土壤的一个重要属性，它决定了地下水、石油或污染物在地下的传播方式。还有另一个应用是时空复合材料的新领域：其成分处于快速相对运动或其成分能够在没有机械运动的情况下随时间快速改变其性质的复合材料。该项目这一部分的成功对于设计和理解能够对不断变化的环境做出快速反应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.