Nathan L. Gibson

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We consider a simplified model of methane hydrates which we cast as a nonlinear evolution problem. For its well-posedness we extend the existing theory to cover the case in which the problem involves a measurable family of graphs. We represent the nonlinearity as a subgradient and prove a useful comparison principle, thus optimal regularity results follow.(More)
We apply an inverse problem formulation to determine characteristics of a defect from a perturbed electromagnetic interrogating signal. A defect (crack) inside of a dielectric material causes a disruption, from reflections and refractions off of the interfaces, of the windowed interrogating signal. We model the electromagnetic waves inside the material with(More)
We study the stability properties of, and the phase error present in, a finite element scheme for Maxwell’s equations coupled with a Debye or Lorentz polarization model. In one dimension we consider a second order formulation for the electric field with an ordinary differential equation for the electric polarization added as an auxiliary constraint. The(More)
In this paper we employ the periodic unfolding method for simulating the electromagnetic field in a composite material exhibiting heterogeneous microstructures which are described by spatially periodic parameters. We consider cell problems to calculate the effective parameters for a Debye dielectric medium in the cases of circular and square microstructures(More)
The stability properties, and the phase error present in higher order (in space) staggered finite difference schemes for Maxwell’s equations coupled with a Debye polarization model are analyzed. We present a novel expansion of the symbol of finite difference approximations, of arbitrary (even) order, of the first order spatial derivative operator. This(More)
We compare an inverse problem approach to parameter estimation with homogenization techniques for characterizing the electrical response of composite dielectric materials in the time domain. We first consider an homogenization method, based on the periodic unfolding method, to identify the dielectric response of a complex material with heterogeneous(More)
We present results from our computational efforts to determine defect characteristics via solution of inverse problems which involve the 2D wave equation in a domain which presents geometric complexities. We develop a technique for determining “optimal” placement of a receiver based on the geometry of the interrogation problem which involves a plane wave(More)
1 Project Description The past decade has seen a plethora of activity in the field of Terahertz imaging. The timing is mainly due to the recent development of highly efficient Terahertz generation and detection devices [17] which allow more researchers to have access to experimentation; the popularity is due to the wide range of applications the technology(More)