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Research Papers

Reduced Boundary Sensitivity and Improved Contrast of the Regularized Inverse Problem Solution in Elasticity

[+] Author and Article Information
Yue Mei

Mechanical Engineering Department,
Texas A&M University,
College Station, TX 77843

Sergey Kuznetsov

National University of Science and Technology,
Materials Modeling
and Development Laboratory,
Moscow 119049, Russia

Sevan Goenezen

Mechanical Engineering Department,
Texas A&M University,
College Station, TX 77843
e-mail: sgoenezen@tamu.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 30, 2015; final manuscript received October 30, 2015; published online December 8, 2015. Assoc. Editor: Harold S. Park.

J. Appl. Mech 83(3), 031001 (Dec 08, 2015) (10 pages) Paper No: JAM-15-1456; doi: 10.1115/1.4031937 History: Received August 30, 2015; Revised October 30, 2015

We observe that posing the inverse problem as a constrained minimization problem under regularization leads to boundary dependent solutions. In this paper, we propose a modified objective function and show with 2D examples that our method works well to reduce boundary sensitive solutions. The examples consist of two stiff inclusions embedded in a softer unit square. These inclusions could be representative of tumors, which are in general stiffer than their background tissues, thus could potentially be detected based on their stiffness contrast. We modify the objective function for the displacement correlation term by weighting it with a function that depends on the strain field. In a simplified 1D coupled model, we derive an analytical expression and observe the same trends in the reconstructions as for the 2D model. The analysis in this paper is confined to inclusions of similar size and may not overlap when projected on the horizontal axis. They may, however, vary in position along the vertical axis. Furthermore, our analysis holds for an arbitrary number of inclusions having distinct stiffness values. Finally, to increase the overall contrast of the tumors and simultaneously improve the smoothness, we solve the regularized inverse problem in a posterior step, utilizing a spatially varying regularization factor.

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Figures

Grahic Jump Location
Fig. 1

Given are the following target shear modulus distributions: (a) two horizontally positioned inclusions with μ=5 in a homogeneous background of μ=1, (b) two horizontally positioned inclusions with μ=5 in the left inclusion, μ=10 in the right inclusion, and μ=1 in the background, (c) two inclusions positioned on the diagonal of the unit square, with μ=5 in both inclusions and μ=1 in the background

Grahic Jump Location
Fig. 2

(Top) Shear modulus reconstructions for the problem domain in Fig. 1(a) with uniform compression for two regularization factors. (Bottom) Plot of the shear modulus values along the horizontal centerline through both inclusions.

Grahic Jump Location
Fig. 3

(Top) Shear modulus reconstructions for the problem domain in Fig. 1(a) with linear compression for two regularization factors. (Bottom) Plot of the shear modulus values along the horizontal centerline through both inclusions.

Grahic Jump Location
Fig. 4

(Top) Shear modulus reconstructions for the problem domain given in Fig. 1(b) with uniform compression for two regularization factors. We note that the target shear modulus in the left inclusion is 5 and the right inclusion is 10. (Bottom) Plot of the shear modulus values along the horizontal centerline through both inclusions.

Grahic Jump Location
Fig. 5

(Top) Shear modulus reconstructions for the problem domain given in Fig. 1(c) with linear compression for two regularization factors. The inclusions are located along the diagonal. (Bottom) Plot of the shear modulus values along the diagonal line through the center of both inclusions.

Grahic Jump Location
Fig. 6

(Left) Shear modulus reconstruction for the target shear modulus distribution from Fig. 1(a) with varying compression boundary. (Right) Shear modulus plot versus the horizontal line through the center of both inclusions for the reconstructed and target shear modulus distribution.

Grahic Jump Location
Fig. 7

(Left) Shear modulus reconstruction for the target shear modulus distribution from Fig. 1(b) with uniform compression boundary. (Right) Shear modulus plot versus the horizontal line through the center of both inclusions for the reconstructed and target shear modulus distribution.

Grahic Jump Location
Fig. 8

(Left) Shear modulus reconstruction for the target shear modulus distribution from Fig. 1(c) with linear compression boundary. (Right) Shear modulus plot versus the diagonal line through the center of both inclusions for the reconstructed and target shear modulus distribution.

Grahic Jump Location
Fig. 9

(Left) Shear modulus reconstruction utilizing the methodology introduced in Sec. 2.3. The target shear modulus distribution is given in Fig. 1(a). (Right) Plot of the shear modulus along the diagonal centerline passing through the center of both inclusions for the target and reconstructed values.

Grahic Jump Location
Fig. 10

(Left) Shear modulus reconstruction utilizing the methodology introduced in Sec. 2.3. The target shear modulus distribution is given in Fig. 1(b). (Right) Plot of the shear modulus values along the horizontal centerline passing through the center of both inclusions for the target and reconstructed values.

Grahic Jump Location
Fig. 11

(Left) Shear modulus reconstruction utilizing the methodology introduced in Sec. 2.3. The target shear modulus distribution is given in Figure 1(c). (Right) Plot of the shear modulus along the diagonal centerline passing through the center of both inclusions for the target and reconstructed values.

Grahic Jump Location
Fig. 12

One-dimensional analogue of previous shear modulus inclusions, represented by two nonhomogeneous bars

Grahic Jump Location
Fig. 13

Error plot of the reconstructed shear modulus in the left inclusion over the exact shear modulus in the right inclusion μ¯in2 for different shear modulus values in the left inclusion μ¯in1 ( = 3, 4, 5, 6, 7, 8, 9, 10). The conventional method has been used.

Grahic Jump Location
Fig. 14

Error plot of the reconstructed shear modulus in the left inclusion over the exact shear modulus in the right inclusion μ¯in2 for different shear modulus values in the left inclusion μ¯in1 (= 3, 4, 5, 6, 7, 8, 9, 10). The new spatially weighted method has been used.

Grahic Jump Location
Fig. 15

Error plot of the reconstructed shear modulus in the left inclusion over the exact shear modulus in the right inclusion μ¯in2 for different shear modulus values in the left inclusion μ¯in1 (= 3, 4, 5, 6, 7, 8, 9, 10). The conventional method has been used for the case that the compression is applied uniformly.

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