Electromagnetic Crack Detection Inverse Problems Using Terahertz Interrogating Signals [report]

H. T. Banks, Nathan L. Gibson, William P. Winfree
2003 unpublished
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 Maxwell's equations. Using simulations as forward solves, our Newton-based, iterative optimization scheme resolves the dimensions and
more » ... imensions and location of the defect. Numerical results are given in tables and plots, standard errors are calculated, and computational issues are addressed. Problem Description We interrogate an (infinitely long) slab of homogeneous nonmagnetic material by a polarized, windowed signal (see [2] for details) in the Thz frequency range (see Figure 1 ). We start with a wave normally incident Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT 18. NUMBER OF PAGES 59 19a. NAME OF RESPONSIBLE PERSON a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18 where D(t, z) is the electric flux density, µ 0 is the magnetic permeability of free space, σ is the conductivity, and J s is a source current density (determined by our interrogating signal). We take the partial derivative of Equation (1a) with respect to z, and the partial of Equation (1b) with respect to t. Equating the ∂ 2 H ∂z∂t terms in each, and thus eliminating the magnetic field H, we have: (where denotes z derivatives and˙denotes time derivatives). Note that we have neglected magnetic effects and we have let the total current density be J = J c + J s , where J c = σE is the conduction current density given by Ohm's law. For our source current, J s , we want to simulate a windowed pulse, i.e., a pulse that is allowed to oscillate for one full period and then is truncated. Further, we want the pulse to originate only at z = 0, simulating an infinite antenna at this location. Thus we define where ω is the frequency of the pulse, t f = 2π/ω is fixed, I [0,t f ] (t) represents an indicator function which is 1 when 0 ≤ t ≤ t f and zero otherwise, and δ(z) is the Dirac delta distribution. Remark 1 Computationally, having a truncated signal introduces discontinuities in the first derivatives which are not only problematic in the numerical simulations (producing spurious oscillations), but are also essentially non-physical. Therefore in our implementation we actually multiply the sine function by an exponential function (see Figure 3 ) rather than the traditional indicator function. The exponential is of the form
doi:10.21236/ada446719 fatcat:p6yplqt6vjcdlkolj7d5qnggky