RUI: Time-dependent Composites and Inverse Problems

San Francisco State University, San Francisco CA

Investigators

Abstract

Composite materials find numerous applications in engineering and manufacturing for their unique properties that are superior to those of conventional materials. The search for composites that guarantee higher levels of performance requires the development of mathematical models able to accurately predict the behavior of such materials. Particularly interesting are time-dependent composites: passive materials, such as viscoelastic composites, exhibit a time-dependent response when subject to static loadings, due to the dissipation of energy carried out by the viscous component; on the other hand, active materials, such as space-time composites, in which energy accumulates or dissipates depending on the change of the applied field, provide entirely new ways to channel waves. The first goal of the project is to provide accurate models to predict the static response of passive time-dependent composites and to describe wave propagation in active composites. The second goal of the project concerns inverse problems: given the response of an inhomogeneous medium, not necessarily a composite, the aim is to develop reconstruction formulas to determine the internal geometry of the medium, a fundamental problem in geotechnical and medical applications for which one wants to recover the internal structure of a body. For instance, for a homogeneous elastic body with a viscoelastic inclusion, the objective is to determine the volume and the shape of the inclusion by applying suitable loadings and by measuring the corresponding response at a specific moment of time. San Francisco State University is a Research Undergraduate Institution; several undergraduate students will be involved in this research and will be trained through their participation. The nature of the project will bring forward mathematical advancements in several areas with focus on the following research avenues. (i) Wave equations with time-dependent coefficients: the project aims at exploring the behavior of solutions of such equations in the presence of dissipation, dispersion, nonlinearity, and more than one spatial dimension. The ultimate goal is to address the issue regarding the implementation of space-time composites to design one-way propagation devices and to engineer filtering devices and absorbers. (ii) The theory of Herglotz-Nevannlina-Pick functions: such theory plays a crucial role in the description of the behavior of passive and causal materials and one of the objective of the project is to broaden its applications to bound the response of two- and three-dimensional passive time-dependent composites in time, by using the analyticity of the effective properties tensor, and to solve inverse problems regarding lossy inhomogeneous media, by using the analyticity of the Dirichlet-to-Neumann map. (iii) Boundary integral formulations: one of the objective of the project is to determine the elastic fields in a two-dimensional unbounded isotropic medium with an elastic inclusion of arbitrary shape, by applying the single layer potential technique and by choosing a suitable set of functions for the series expansion of the elastic fields. The goal is to investigate how to derive reconstruction formulas for the shape of the inclusion. 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.