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Discrete scattering by a pair of parallel defects.

ABSTRACT: Scattering of a time-harmonic anti-plane shear wave due to either a pair of crack tips or a pair of rigid constraint tips on square lattice is considered. The two problems correspond to the so-called zero-offset case of scattering due to a pair of identical Sommerfeld screens. The peculiar structural symmetry allows the reduction of coupled equations to two scalar Wiener-Hopf equations and a total of four geometrically reduced problems on lattice half-plane. Exact solution of each problem for incidence from the bulk lattice, as well as from an associated lattice waveguide, is constructed. A suitable superposition of the four expressions is used to construct the solution of the main problem. The discrete paradigm involving the wave mode incident from the waveguide is relevant for modern applications where an investigation of mechanisms of electronic and thermal transport at nanoscale remains an interesting problem. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 2)'.

SUBMITTER: Sharma BL

PROVIDER: S-EPMC6894522 | biostudies-literature | 2020 Jan

REPOSITORIES: biostudies-literature

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Discrete scattering by a pair of parallel defects.

Sharma Basant Lal BL Maurya Gaurav G

Philosophical transactions. Series A, Mathematical, physical, and engineering sciences 20191125 2162

Scattering of a time-harmonic anti-plane shear wave due to either a pair of crack tips or a pair of rigid constraint tips on square lattice is considered. The two problems correspond to the so-called zero-offset case of scattering due to a pair of identical Sommerfeld screens. The peculiar structural symmetry allows the reduction of coupled equations to two scalar Wiener-Hopf equations and a total of four geometrically reduced problems on lattice half-plane. Exact solution of each problem for in ...[more]

PMID: 31760901

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Project description:Microstructure-based design of materials requires an atomic level understanding of the mechanisms underlying structure-dependent properties. Methods for analyzing either the traditional diffraction profile or the pair distribution function (PDF) differ in how the information is accessed and in the approximations usually applied. Any variation of structural and microstructural features over the whole sample affects the Bragg peaks as well as any diffuse scattering. Accuracy of characterization relies, therefore, on the reliability of the analysis methods. Methods based on Bragg's law investigate the diffraction peaks in the intensity plot as distinct pieces of information. This approach reaches a limitation when dealing with disorder scenarios that do not conform to such a peak-by-peak basis. Methods based on the Debye scattering equation (DSE) are, otherwise, well suited to evaluate the scattering from a disordered phase but the structure information is averaged over short-range distances usually accessed by experiments. Moreover, statistical reliability is usually sacrificed to recover some of the computing-efficiency loss compared with traditional line-profile-analysis methods. Here, models based on Bragg's law are used to facilitate the computation of a whole PDF and then model powder-scattering data via the DSE. Models based on Bragg's law allow the efficient solution of the dispersion of a crystal's properties in a powder sample with statistical reliability, and the PDF provides the flexibility of the DSE. The whole PDF is decomposed into the independent directional components, and the number of atom pairs separated by a given distance is statistically estimated using the common-volume functions. This approach overcomes the need for an atomistic model of the material sample and the computation of billions of pair distances. The results of this combined method are in agreement with the explicit solution of the DSE although the computing efficiency is comparable with that of methods based on Bragg's law. Most importantly, the method exploits the strengths and different sensitivities of the Bragg and Debye theories.

Project description:PurposeTo establish a cardiac signal from scattering matrix or scattering coefficient measurements made on a 7T 8-channel parallel transmit (pTx) system, and to evaluate its use for cardiac gating.MethodsMeasurements of the scattering matrix and scattering coefficients were acquired using a monitoring pulse sequence and during a standard cine acquisition, respectively. Postprocessing used an independent component analysis and gating feature identification. The effect of the phase of the excitation radiofrequency (RF) field ( B1+ shim) on the cardiac signal was simulated for multiple B1+ shim configurations, and cine images were reconstructed from both the scattering coefficients and electrocardiogram (ECG).ResultsThe cardiac motion signal was successfully identified in all subjects with a mean signal-to-noise ratio of 33.1 and 5.7 using the scattering matrix and scattering coefficient measurements, respectively. The dominant gating feature in the cardiac signal was a peak aligned with end-systole that occurred on average at 311 and 391 ms after the ECG trigger, with a mean standard deviation of 13.4 and 18.1 ms relative to ECG when using the scattering matrix and scattering coefficients measurements, respectively. The scattering coefficients showed a dependence on B1+ shim with some shim configurations not showing any cardiac signal. Cine images were successfully reconstructed using the scattering coefficients with minimal differences compared to those using ECG.ConclusionWe have shown that the scattering of a pTx RF coil can be used to estimate a cardiac signal, and that scattering matrix and coefficients can be used to cardiac gate MRI acquisitions with the scattering matrix providing a superior cardiac signal. Magn Reson Med 80:633-640, 2018. © 2017 The Authors Magnetic Resonance in Medicine published by Wiley Periodicals, Inc. on behalf of International Society for Magnetic Resonance in Medicine. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Discrete scattering by a pair of parallel defects.

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Discrete scattering by a pair of parallel defects.

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