Structures, Metamaterials, Scattering, and Inverse Problems

这个项目的意义是多方面的。它研究了索网在张力作用下所能承受的力，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3d打印的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，用于控制振动，以及潜在的针对声纳的隐形。它打开了边界场等式的领域，推广了守恒定律的概念，并可用于计算非均匀体响应的基准数值算法。它开发了一种新的方法来散射电磁波和弹性波，这可以帮助人们了解任意形状的包涵可以散射波的程度。在不知道两相复合材料详细微观结构的情况下，提出了预测两相复合材料电磁响应的新边界。这将有助于识别最能吸收能量的复合材料和纳米颗粒。它探索了通过电流的耦合效应、与磁场的相互作用和振动，在超材料中可以实现什么新的响应。这可能会导致新型的磁场传感器和新的器件耦合变形和磁效应。对于生物医学、工程和反恐应用来说，通过非侵入性测试了解人体内部的物质是至关重要的，如果能够近乎确定地说出一些事情就更好了。该项目将提供新的方法来获得物体中包含物所占体积的精确下限和上限。这可能应用于确定某些肿瘤的大小，或体内的空洞，或骨质疏松的骨质疏松。它研究了理论非均匀体的新类别，其中有场的精确解，这可能对基准数值算法有用，并有助于深入了解如何在非均匀体中操纵场。最后，培养两名跨学科研究的博士后。该项目提供了来自结构、超材料、散射和逆问题四个领域的思想的交叉施肥。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，即人们试图通过边界测量来确定物体内部的东西，以及散射问题，即人们试图理解散射体形状变化时可能的散射响应范围。边界场等式的研究源于复合材料中的精确关系理论，旨在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。对于含有广泛非齐次介质的物体，这些等式提供了狄利克雷到诺伊曼映射所满足的精确恒等式，当人们试图从边界测量中提取物体内部的信息时，狄利克雷到诺伊曼映射起着核心作用。3d打印材料弹性张量的研究试图解决一个具有挑战性的问题，即在一种给定材料加上空隙的混合物中可能出现什么样的弹性张量。该建议解决了通过在时间（而不是频率）域的边界测量来估计物体中包含物大小的逆问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.