Structures, Metamaterials, Scattering, and Inverse Problems

这个项目的意义是多方面的。它研究张力下的索网可以承受的力，这对土木工程，特别是桥梁或建筑设计应该是重要的。它研究了三维打印超材料可能的弹性响应，这可能有助于设计引导声波和弹性波的结构，控制振动，并可能用于隐身对抗声纳。它开辟了边界场等式的领域，推广了守恒定律的概念，并可用于计算非均质物体响应的基准数值算法。它开发了一种新的方法来散射包裹体上的电磁波和弹性波，这可以帮助人们了解任意形状的包裹体可以散射波的程度。它发展了新的界限，即使人们不知道两相复合材料的详细微观结构，它也可以用于预测两相复合材料的电磁响应。这应该有助于识别最能吸收能量的复合材料和纳米粒子。它探索了通过电流流动、与磁场的相互作用和振动的耦合效应，在超材料中可以实现哪些新的响应。这可能导致新类型的磁场传感器和耦合变形和磁效应的新型器件。对于生物医学、工程和反恐应用，从非侵入性检测中了解人体内部的东西至关重要，如果一个人能几乎肯定地说出事情就更好了。该项目将提供获得体内包裹体所占体积的精确下限和上限的新方法。这可能应用于确定某些肿瘤的大小，或身体中的空洞，或骨质疏松骨骼中的孔隙率。它研究了一类新的理论上的非均质体，对场有精确的解，这可能有助于对数值算法进行基准测试，并深入了解如何在非均质体中操纵场。最后，对两名博士后进行跨学科研究。该项目提供了来自结构、超材料、散射和反问题四个领域的思想的交叉融合。其中一些在复合材料理论中发展起来的想法将首次应用于逆问题和散射问题，在这些问题中，人们试图通过边界测量来确定物体内部的东西，而在散射问题中，人们试图了解随着散射体形状的变化而可能的散射响应的范围。关于边界场方程的工作源于复合材料中的精确关系理论，它试图探索经典守恒定律的推广，即物体内部无散度的场通过表面的净通量为零。对于包含某些大类非均匀介质的物体，这些等式提供了由Dirichlet到Neumann映射所满足的精确恒等式，当人们试图从边界测量中提取关于物体内部的信息时，Dirichlet到Neumann映射起着核心作用。关于3-d打印材料弹性张量的工作试图接近一个具有挑战性的问题，即在一种给定材料和空隙的混合物中，什么弹性张量是可能的。该提案解决了通过在时间(而不是频率)域中的边界测量来估计一个机构中包含内容的大小的相反问题。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The significance of this project is multifold. It studies the forces that cable networks under tension can support, and this should be important to civil engineering, in particular to bridge or building design. It studies the possible elastic responses of 3-d printed metamaterials, and this could be helpful in designing structure that guide acoustic and elastic waves, for controlling vibrations and potentially for cloaking against sonar. It opens the field of boundary field equalities, that generalizes the notion of conservation laws, and which can be used for benchmarking numerical algorithms for calculating the response of inhomogeneous bodies. It develops a new approach to scattering of electromagnetic and elastic waves off an inclusion, that can help one understand the extent to which an arbitrarily shaped inclusion can scatter the waves. It develops new bounds that can be useful for predicting the electromagnetic response of two-phase composites even if one does not know the detailed microstructure. This should help in the identification of the most energy-absorbing composites and nano-particles. It explores what novel responses can be achieved in metamaterials, through coupled effects of electrical current flow, interaction with magnetic fields, and vibrations. This may lead to new types of magnetic field sensors and to novel devices coupling deformation and magnetic effects. For biomedical, engineering and counterterrorism applications it is vitally important to know what is inside a body from non-invasive testing, and it is better if one can say things with near certainty. The project will provide new methods of obtaining precise lower and upper limits on the volume occupied by an inclusion in a body. This may have applications to determining the size of certain tumors, or voids in a body, or the porosity in an osteoporotic bone. It studies new classes of theoretical inhomogeneous bodies for which there is an exact solution for the field and this may be useful for benchmarking numerical algorithms, and for gaining insight into how fields can be manipulated in inhomogeneous bodies. Finally, it trains two postdocs in interdisciplinary research.The project provides a cross-fertilization of ideas from the four areas of Structures, Metamaterials, Scattering, and Inverse Problems. Some of these ideas, developed in the theory of composites, will be applied for the first time to inverse problems where one seeks to determine what is inside a body from boundary measurements, and to scattering problems where one seeks to understand the range of possible scattering responses as the shape of the scatterer is varied. The work on boundary field equalities, that stems from the theory of exact relations in composites, seeks to explore generalizations of the classic conservation law that a field which is divergence-free inside a body has zero net flux through the surface. For bodies containing certain wide classes of inhomogeneous media these equalities provide exact identities that are satisfied by the Dirichlet to Neumann map, which plays a central role when one seeks to extract information about what is inside a body from boundary measurements. The work on elastic tensors of 3-d printed materials seeks to bring a close to the challenging question of what elastic tensors are possible in mixtures of one given material plus void. The proposal addresses the inverse problem of estimating the size of an inclusion in a body, through boundary measurements in the time (rather than frequency) domain.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.