Structures, Metamaterials, Scattering, and Inverse Problems

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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.