The factorization method for cracks in elastic scattering

Jun Guo; Qinghua Wu; Guozheng Yan

doi:10.3934/ipi.2018016

Article Contents

2018, Volume 12, Issue 2: 349-371. Doi: 10.3934/ipi.2018016

This issue Previous Article Mumford-Shah-TV functional with application in X-ray interior tomography Next Article Enhancing D-bar reconstructions for electrical impedance tomography with conformal maps

The factorization method for cracks in elastic scattering

1.
South-Central University For Nationalities, Wuhan 430074, China
2.
School of Mathematics and Statistics, Hubei Engineering University, Xiaogan 43200, China
3.
Central China Normal University, Wuhan 430079, China

* Corresponding author: Guozheng Yan
* Corresponding author: Guozheng Yan

Received: January 31, 2017

Revised: October 31, 2017

Published: March 2018

This research is supported by National Natural Science Foundation of People's Republic of China, No.11571132 and No.11601138

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Abstract

This paper is concerned with the scattering problems of a crack with Dirichlet or mixed impedance boundary conditions in two dimensional isotropic and linearized elasticity. The well posedness of the direct scattering problems for both situations are studied by the boundary integral equation method. The inverse scattering problems we are dealing with are the shape reconstruction of the crack from the knowledge of far field patterns due to the incident plane compressional and shear waves. We aim at extending the well known factorization method to crack determination in inverse elastic scattering, although it has been proved valid in acoustic and electromagnetic scattering, electrical impedance tomography and so on. The numerical examples are presented to illustrate the feasibility of this method.

Keywords:

Mathematics Subject Classification: Primary: 35R30; Secondary: 35Q60.

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Figure 2. Inversion example for DIP: reconstruction of the semi-circle (52) for $\mu = 2$, $\lambda = 1, \mathbf{p} = [1/2, \sqrt{3}/2]^\top$, noise level = $5\%$ with different $\omega$.

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Figure 4. Inversion example for DIP: reconstruction of the line (53) for $\omega = 4$, $\mu = 1$, $\lambda = 2$, $\mathbf{p} = [-1/2, \sqrt{3}/2]^\top$ with different noise levels.

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Figure 6. Inversion example for DIP: reconstruction of the curve (54) for $\omega = 4$, $\mu = 1$, $\lambda = 2$, noise level = $1\%$ with different polarization directions $\mathbf{p}$.

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Figure 3. Inversion example for MIP: reconstruction of the semi-circle (52) for $\mu=1$, $\lambda=2$, $\eta=2$, $\mathbf{p}=[0,1]^\top$, noise level=$1\%$ with different $\omega$.

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Figure 5. Inversion example for MIP: reconstruction of the line (53) for $\omega = 5$, $\mu = 1$, $\lambda = 3$, $\eta = 1$, $\mathbf{p} = [\sqrt{3}/2, 1/2]^\top$ with different noise levels.

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Figure 7. Inversion example for MIP: reconstruction of the curve (54) for $\omega = 5$, $\mu = 1$, $\lambda = 2$, $\eta = 1$, noise level = $1\%$ with different polarization directions $\mathbf{p}$.

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Figure 1. The exact objects: the shape of (52) (left), the shape of (53) (middle) and the shape of (54) (right)

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