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Inverse Problems in Science and Engineering Volume 21, 2013 - Issue 2

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Original Articles

Application of the linear sampling method to identify cracks with impedance boundary conditions

Fahmi Ben HassenENIT-LAMSIN, Université de Tunis el-Manar, BP 37, 1002 Tunis, Tunisia

Yosra BoukariENIT-LAMSIN, Université de Tunis el-Manar, BP 37, 1002 Tunis, Tunisia;INRIA Saclay Ile de France/CMAP Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France

Houssem HaddarINRIA Saclay Ile de France/CMAP Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France

Pages 210-234 | Received 15 Jun 2011, Accepted 17 Apr 2012, Published online: 19 Jun 2012

Cite this article
https://doi.org/10.1080/17415977.2012.686997

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Figure 1. Exact geometry of the cracks.

Figure 2. Reconstruction of an L-shaped crack (, middle) with vertices (0.75, 0), (0, 0), (0, 0.75) for λ^± = 0.01(1 + i) (top), λ^± = 5(1 + i) (middle), λ^± = 1000(1 + i) (bottom).

Figure 2. Reconstruction of an L-shaped crack (Figure 1, middle) with vertices (0.75, 0), (0, 0), (0, 0.75) for λ± = 0.01(1 + i) (top), λ± = 5(1 + i) (middle), λ± = 1000(1 + i) (bottom).

Figure 3. Reconstruction of an arc-shaped crack (, top) with centre (−0.5, −0.5), radius 0.8 and angle varying from 0 to π/2 for λ^± = 0.01(1 + i) (top), λ^± = 5(1 + i) (middle), λ^± = 1000(1 + i) (bottom).

Figure 3. Reconstruction of an arc-shaped crack (Figure 1, top) with centre (−0.5, −0.5), radius 0.8 and angle varying from 0 to π/2 for λ± = 0.01(1 + i) (top), λ± = 5(1 + i) (middle), λ± = 1000(1 + i) (bottom).

Figure 4. Reconstruction of two segments with endpoints (0, 0.8), (0, 0) and (0, 0), (0.4, −0.8) (, bottom) for λ^± = 0.01(1 + i) (top), λ^± = 5(1 + i) (middle), λ^± = 1000(1 + i) (bottom).

Figure 4. Reconstruction of two segments with endpoints (0, 0.8), (0, 0) and (0, 0), (0.4, −0.8) (Figure 1, bottom) for λ± = 0.01(1 + i) (top), λ± = 5(1 + i) (middle), λ± = 1000(1 + i) (bottom).

Figure 5. Reconstruction of the normals in the case of L-shape (, middle) for λ^± = 0.01(1 + i) (top), λ^± = 5(1 + i) (middle), λ^± = 1000(1 + i) (bottom).

Figure 5. Reconstruction of the normals in the case of L-shape (Figure 1, middle) for λ± = 0.01(1 + i) (top), λ± = 5(1 + i) (middle), λ± = 1000(1 + i) (bottom).

Figure 6. The exact crack σ is represented in blue and L ⊂ σ in red.

Table 1. Reconstruction of the average of λ^± given by (43) on σ = [−0.5, 0.5] × {0} for n equidistant directions of plane waves starting from θ = 0 with the wavelength λ = 1.

Display Table

Table 2. Reconstruction of the average of λ^± given by (43) on the arc σ centred at (0, 0) with aperture for n equidistant directions of plane waves starting from θ = 0 till θ = 2π with the wavelength λ=1.

Display Table

Table 3. Reconstruction of the average of λ^± given by (43) on a perturbed geometry where the exact crack is σ=[−0.5, 0.5]×{0} for 100 equidistant directions of plane waves starting from θ=0 till θ=2π and a wavelength λ=1.

Display Table

Table 4. Reconstruction of λ^± given by (43) on a perturbed geometry where the exact crack is σ=[−0.5, 0.5] × {0} for 100 equidistant directions of plane waves starting from θ = 0 till θ = 2π and a wavelength λ = 4.

Display Table

Table 5. Reconstruction of the average of λ^± using formula (45) on L = σ = [−0.5, 0.5]×{0}.

Display Table

Table 6. Reconstruction of λ^± using formula (45) on the perturbed geometry where the exact crack is σ = [−0.5, 0.5]×{0} for the wavelength λ = 1 and β = 6.

Display Table

Table 7. Reconstruction of λ^± using formula (45) on the perturbed geometry where the exact crack is σ = [−0.5, 0.5]×{0}, β = 6 and the wavelength λ = 4.

Display Table

Table 8. Reconstruction of the average of λ^± using formula (45) when L = [−0.5, 0.5] × {0} is a part of the exact crack σ = [−2, 0.75] × {0} (see Figure 6, left).

Display Table

Table 9. Reconstruction of the average of λ^± using (45) on L = {0}×[−0.25, 0.25] which is a part of the exact L-shaped crack σ in Figure 6 for the wavelength λ = 1.

Display Table

Table 10. Reconstruction of the average of λ^± on L using formula (45) where L is the arc centred at (0, 0) with aperture which is a part of the exact crack σ (see Figure 6, right) for the wavelength λ=1.

Display Table

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Application of the linear sampling method to identify cracks with impedance boundary conditions

Table 1. Reconstruction of the average of λ^± given by (43) on σ = [−0.5, 0.5] × {0} for n equidistant directions of plane waves starting from θ = 0 with the wavelength λ = 1.

Table 2. Reconstruction of the average of λ^± given by (43) on the arc σ centred at (0, 0) with aperture for n equidistant directions of plane waves starting from θ = 0 till θ = 2π with the wavelength λ=1.

Table 3. Reconstruction of the average of λ^± given by (43) on a perturbed geometry where the exact crack is σ=[−0.5, 0.5]×{0} for 100 equidistant directions of plane waves starting from θ=0 till θ=2π and a wavelength λ=1.

Table 4. Reconstruction of λ^± given by (43) on a perturbed geometry where the exact crack is σ=[−0.5, 0.5] × {0} for 100 equidistant directions of plane waves starting from θ = 0 till θ = 2π and a wavelength λ = 4.

Table 5. Reconstruction of the average of λ^± using formula (45) on L = σ = [−0.5, 0.5]×{0}.

Table 6. Reconstruction of λ^± using formula (45) on the perturbed geometry where the exact crack is σ = [−0.5, 0.5]×{0} for the wavelength λ = 1 and β = 6.

Table 7. Reconstruction of λ^± using formula (45) on the perturbed geometry where the exact crack is σ = [−0.5, 0.5]×{0}, β = 6 and the wavelength λ = 4.

Table 8. Reconstruction of the average of λ^± using formula (45) when L = [−0.5, 0.5] × {0} is a part of the exact crack σ = [−2, 0.75] × {0} (see Figure 6, left).

Table 9. Reconstruction of the average of λ^± using (45) on L = {0}×[−0.25, 0.25] which is a part of the exact L-shaped crack σ in Figure 6 for the wavelength λ = 1.

Table 10. Reconstruction of the average of λ^± on L using formula (45) where L is the arc centred at (0, 0) with aperture which is a part of the exact crack σ (see Figure 6, right) for the wavelength λ=1.

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Application of the linear sampling method to identify cracks with impedance boundary conditions

Figures & data

Table 1. Reconstruction of the average of λ± given by (43) on σ = [−0.5, 0.5] × {0} for n equidistant directions of plane waves starting from θ = 0 with the wavelength λ = 1.

Table 2. Reconstruction of the average of λ± given by (43) on the arc σ centred at (0, 0) with aperture for n equidistant directions of plane waves starting from θ = 0 till θ = 2π with the wavelength λ=1.

Table 3. Reconstruction of the average of λ± given by (43) on a perturbed geometry where the exact crack is σ=[−0.5, 0.5]×{0} for 100 equidistant directions of plane waves starting from θ=0 till θ=2π and a wavelength λ=1.

Table 4. Reconstruction of λ± given by (43) on a perturbed geometry where the exact crack is σ=[−0.5, 0.5] × {0} for 100 equidistant directions of plane waves starting from θ = 0 till θ = 2π and a wavelength λ = 4.

Table 5. Reconstruction of the average of λ± using formula (45) on L = σ = [−0.5, 0.5]×{0}.

Table 6. Reconstruction of λ± using formula (45) on the perturbed geometry where the exact crack is σ = [−0.5, 0.5]×{0} for the wavelength λ = 1 and β = 6.

Table 7. Reconstruction of λ± using formula (45) on the perturbed geometry where the exact crack is σ = [−0.5, 0.5]×{0}, β = 6 and the wavelength λ = 4.

Table 8. Reconstruction of the average of λ± using formula (45) when L = [−0.5, 0.5] × {0} is a part of the exact crack σ = [−2, 0.75] × {0} (see Figure 6, left).

Table 9. Reconstruction of the average of λ± using (45) on L = {0}×[−0.25, 0.25] which is a part of the exact L-shaped crack σ in Figure 6 for the wavelength λ = 1.

Table 10. Reconstruction of the average of λ± on L using formula (45) where L is the arc centred at (0, 0) with aperture which is a part of the exact crack σ (see Figure 6, right) for the wavelength λ=1.

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Table 1. Reconstruction of the average of λ^± given by (43) on σ = [−0.5, 0.5] × {0} for n equidistant directions of plane waves starting from θ = 0 with the wavelength λ = 1.

Table 2. Reconstruction of the average of λ^± given by (43) on the arc σ centred at (0, 0) with aperture for n equidistant directions of plane waves starting from θ = 0 till θ = 2π with the wavelength λ=1.

Table 3. Reconstruction of the average of λ^± given by (43) on a perturbed geometry where the exact crack is σ=[−0.5, 0.5]×{0} for 100 equidistant directions of plane waves starting from θ=0 till θ=2π and a wavelength λ=1.

Table 4. Reconstruction of λ^± given by (43) on a perturbed geometry where the exact crack is σ=[−0.5, 0.5] × {0} for 100 equidistant directions of plane waves starting from θ = 0 till θ = 2π and a wavelength λ = 4.

Table 5. Reconstruction of the average of λ^± using formula (45) on L = σ = [−0.5, 0.5]×{0}.

Table 6. Reconstruction of λ^± using formula (45) on the perturbed geometry where the exact crack is σ = [−0.5, 0.5]×{0} for the wavelength λ = 1 and β = 6.

Table 7. Reconstruction of λ^± using formula (45) on the perturbed geometry where the exact crack is σ = [−0.5, 0.5]×{0}, β = 6 and the wavelength λ = 4.

Table 8. Reconstruction of the average of λ^± using formula (45) when L = [−0.5, 0.5] × {0} is a part of the exact crack σ = [−2, 0.75] × {0} (see Figure 6, left).

Table 9. Reconstruction of the average of λ^± using (45) on L = {0}×[−0.25, 0.25] which is a part of the exact L-shaped crack σ in Figure 6 for the wavelength λ = 1.

Table 10. Reconstruction of the average of λ^± on L using formula (45) where L is the arc centred at (0, 0) with aperture which is a part of the exact crack σ (see Figure 6, right) for the wavelength λ=1.