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February  2016, 10(1): 165-193. doi: 10.3934/ipi.2016.10.165

A divide-alternate-and-conquer approach for localization and shape identification of multiple scatterers in heterogeneous media using dynamic XFEM

1. 

Mechanical & Aerospace Engineering Department, University of Miami, Coral, FL 33124, United States

2. 

Civil & Environmental Engineering Department, University of California, Los Angeles, CA 90095, United States

Received  November 2014 Revised  September 2015 Published  February 2016

A numerical method for localization and identification of multiple arbitrarily-shaped scatterers (cracks, voids, or inclusions) embedded within heterogeneous linear elastic media is described. An elastodynamic implementation of the extended finite element method (XFEM), which is endowed with a spline-based parameterization of the scatterer boundaries, is employed to solve the forward (wave propagation) problem. This particular combination enables direct, sensitivity-based, and computationally efficient manipulation of the scatterers' boundaries over a stationary background mesh during the inversion process. The inverse problem is cast as a formal optimization problem whereby the discrepancy between the measured wave responses and those from the estimated scatterers is minimized. The solution is achieved through a gradient-based procedure that is steered by a divide-alternate-and-conquer strategy. The divide-and-conquer segment of the search algorithm seeks the global minimizer among potentially multiple solutions, whereas the alternate-and-conquer segment adaptively refines the shapes of identified scatterers. The results of several synthetic experiments with various types of scatterers are provided. These experiments verify the overall approach, and demonstrate that it is robust, accurate, and effective even at high levels of measurement noise.
Citation: Jaedal Jung, Ertugrul Taciroglu. A divide-alternate-and-conquer approach for localization and shape identification of multiple scatterers in heterogeneous media using dynamic XFEM. Inverse Problems and Imaging, 2016, 10 (1) : 165-193. doi: 10.3934/ipi.2016.10.165
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H. Ammari, H. Kang, E. Kim, M. Lim and K. Louati, A direct algorithm for ultrasound imaging of internal corrosion, SIAM J. Numer. Anal., 49 (2011), 1177-1193. doi: 10.1137/100784710.

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B. A. Benowitz and H. Waisman, A spline-based enrichment function for arbitrary inclusions in extended finite element method with application to finite deformations, Int. J. Numer. Methods Eng., 95 (2013), 361-386. doi: 10.1002/nme.4508.

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M. Bonnet and B. B. Guzina, Elastic-wave identification of penetrable obstacles using shape-material sensitivity framework, J. Comput. Phys., 228 (2009), 294-311. doi: 10.1016/j.jcp.2008.09.009.

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[29]

J. Jung, C. Jeong and E. Taciroglu, Identification of a scatterer embedded in elastic heterogeneous media using dynamic XFEM, Comput. Methods Appl. Mech. Eng., 259 (2013), 50-63. doi: 10.1016/j.cma.2013.03.001.

[30]

J. Jung and E. Taciroglu, Modeling and identification of an arbitrarily shaped scatterer using dynamic XFEM with cubic splines, Comp. Methods Appl. Mech. Eng., 278 (2014), 101-118. doi: 10.1016/j.cma.2014.05.001.

[31]

L. F. Kallivokas, A. Fathi, S. Kucukcoban, K. H. Stokoe II, J. Bielak and O. Ghattas, Site characterization using full waveform inversion, Soil Dyn. Earthq. Eng., 47 (2013), 62-82. doi: 10.1016/j.soildyn.2012.12.012.

[32]

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C. W. Liu and E. Taciroglu, Enriched reproducing kernel particle method for piezoelectric structures with arbitrary interfaces, Int. J. Numer. Methods Eng., 67 (2006), 1565-1586. doi: 10.1002/nme.1684.

[37]

C. W. Liu and E. Taciroglu, Shape optimization of piezoelectric devices using an enriched meshfree method, Int. J. Numer. Methods Eng., 78 (2009), 151-171. doi: 10.1002/nme.2479.

[38]

M. Marija and K. Kaspars, Application of Ultrasonic Imaging Technique as Structural Health Monitoring Tool for Assessment of Defects in Glass Fiber Composite Structures, Proceeding of the International Conference on Civil Engineering, 4 (2013), 180-184.

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J. M. Melenk and I. Babuška, The partition of unity finite element method: Basic theory and applications, Comput. Methods Appl. Mech. Eng., 139 (1996), 289-314. doi: 10.1016/S0045-7825(96)01087-0.

[40]

N. Moës, M. Cloirec, P. Cartraud and J. F. Remacle, A computational approach to handle complex microstructure geometries, Comput. Methods Appl. Mech. Eng., 192 (2003), 3163-3177.

[41]

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[42]

R. Potthast, A survey on sampling and probe methods for inverse problems, Inverse Problems, 22 (2006), 1-47. doi: 10.1088/0266-5611/22/2/R01.

[43]

D. Rabinovich, D. Givoli and S. Vigdergauz, XFEM-based crack detection scheme using a genetic algorithm, Int. J. Numer. Methods Eng., 71 (2007), 1051-1080. doi: 10.1002/nme.1975.

[44]

D. Rabinovich, D. Givoli and S. Vigdergauz, Crack identification by arrival time using XFEM and a genetic algorithm, Int. J. Numer. Methods Eng., 77 (2009), 337-359. doi: 10.1002/nme.2416.

[45]

C. L. Richardson, J. Hegemann, E. Sifakis, J. Hellrung and J. M. Teran, An XFEM method for modeling geometrically elaborate crack propagation in brittle materials, Int. J. Numer. Methods Eng., 88 (2011), 1042-1065. doi: 10.1002/nme.3211.

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J. H. Rose, Elastic wave inverse scattering in nondestructive evaluation, Pure Appl. Geophys., 131 (1989), 715-739. doi: 10.1007/978-3-0348-6363-6_7.

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M. Safdari, A. R. Najafi, N. R. Sottos and P. H. Geubelle, A NURBS-based interface-enriched generalized finite element method for problems with complex discontinuous gradient fields, Int. J. Num. Meth. Engng., 101 (2015), 950-964. doi: 10.1002/nme.4852.

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H. Sauerland and T. P. Fries, A stable XFEM for two-phase flows, Comput. Fluids, 87 (2013), 41-49. doi: 10.1016/j.compfluid.2012.10.017.

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B. G. Smith, B. L. Vaughan, Jr and D. L. Chopp, The extended finite element method for boundary layer problems in biofilm growth, Comm. App. Math. and Comp. Sci., 2 (2007), 35-56. doi: 10.2140/camcos.2007.2.35.

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H. Sun, H. Waisman and R. Betti, Nondestructive identification of multiple flaws using XFEM and a topologically adapting artificial bee colony algorithm, Int. J. Numer. Methods Eng., 95 (2013), 871-900. doi: 10.1002/nme.4529.

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H. Theodoros, B. Efstratios and T. N. Georgios, Application of Ultrasonic C-Scan Techniques for Tracing Defects in Laminated Composite Materials, J. Mech. Eng., 57 (2011), 192-203.

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show all references

References:
[1]

H. Ammari, E. Iakovleva and D. Lesselier, A MUSIC algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a fixed frequency, Multiscale Model. Simul., 3 (2005), 597-628. doi: 10.1137/040610854.

[2]

H. Ammari, H. Kang, E. Kim, M. Lim and K. Louati, A direct algorithm for ultrasound imaging of internal corrosion, SIAM J. Numer. Anal., 49 (2011), 1177-1193. doi: 10.1137/100784710.

[3]

H. T. Banks, Y. Wang and K. Ito., Well-posedness for damped second order systems with unbounded input operators, Differential and Integral Eqs., 8 (1995), 587-606.

[4]

T. Belytschko and T. Black, Elastic crack growth in finite elements with minimal remeshing, Int. J. Numer. Methods Eng., 45 (1999), 601-620. doi: 10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S.

[5]

T. Belytschko and R. Gracie, On XFEM applications to dislocations and interfaces, Int. J. Plast., 23 (2007), 1721-1738. doi: 10.1016/j.ijplas.2007.03.003.

[6]

B. A. Benowitz and H. Waisman, A spline-based enrichment function for arbitrary inclusions in extended finite element method with application to finite deformations, Int. J. Numer. Methods Eng., 95 (2013), 361-386. doi: 10.1002/nme.4508.

[7]

M. Bonnet and A. Constantinescu, Inverse problems in elasticity, Inverse Probl., 21 (2005), 1-50. doi: 10.1088/0266-5611/21/2/R01.

[8]

M. Bonnet and B. B. Guzina, Elastic-wave identification of penetrable obstacles using shape-material sensitivity framework, J. Comput. Phys., 228 (2009), 294-311. doi: 10.1016/j.jcp.2008.09.009.

[9]

J. C. Brigham, W. Aquino, F. G. Mitri, J. F. Greenleaf and M. Fatemi, Inverse estimation of viscoelastic material properties for solids immersed in fluids using vibroacoustic techniques, J. Appl. Phys., 101 (2007), 023509. doi: 10.1063/1.2423227.

[10]

F. Cakoni, D. Colton and P. Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering, SIAM, Philadelphia, 2011. doi: 10.1137/1.9780898719406.

[11]

F. Cakoni, M. Fares and H. Haddar, Analysis of two linear sampling methods applied to electromagnetic imaging of buried objects, Inverse Problems, 22 (2006), 845-867. doi: 10.1088/0266-5611/22/3/007.

[12]

E. N. Chatzi, B. Hiriyur, H. Waisman and A. W. Smyth, Experimental application and enhancement of the XFEM-GA algorithm for the detection of flaws in structures, Comput. Struct., 89 (2011), 556-570. doi: 10.1016/j.compstruc.2010.12.014.

[13]

X. Chen and Y. Zhong, MUSIC electromagnetic imaging with enhanced resolution for small inclusions, Inverse Problems, 25 (2009), 015008, (12pp). doi: 10.1088/0266-5611/25/1/015008.

[14]

M. Cheney, The linear sampling method and the music algorithm, Inverse Problems, 17 (2001), 591-595. doi: 10.1088/0266-5611/17/4/301.

[15]

D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003.

[16]

S. W. Doebling, C. R. Farrar and M. B. Prime, A summary review of vibration-based damage identification methods, Shock Vib. Dig., 20 (1998), 91-105.

[17]

H. Edelsbrunner, Geometry and Topology for Mesh Generation, Cambridge University Press, 2001. doi: 10.1017/CBO9780511530067.

[18]

Y. Fan, T. Jiang and D. J. Evans, The parallel genetic algorithm for electromagnetic inverse scattering of a conductor, Int. J. Computer Math., 79 (2002), 573-586. doi: 10.1080/00207160210955.

[19]

M. Fatemi and J. F. Greenleaf, Vibro-acoustography: An imaging modality based on ultrasound-stimulated acoustic emission, Proc. Natl. Acad. Sci. USA, 96 (1999), 6603-6608. doi: 10.1073/pnas.96.12.6603.

[20]

E. M. Feericka, X. C. Liub and P. McGarrya, Anisotropic mode-dependent damage of cortical bone using the extended finite element method (XFEM), J. Mech. Behav. Biomed. Mater., 20 (2013), 77-89. doi: 10.1016/j.jmbbm.2012.12.004.

[21]

M. Fleming, Y. A. Chu, B. Moran and T. Belytschko, Enriched element-free Galerkin methods for crack tip fields, Int. J. Numer. Methods Eng., 40 (1997), 1483-1504. doi: 10.1002/(SICI)1097-0207(19970430)40:8<1483::AID-NME123>3.0.CO;2-6.

[22]

T. P. Fries and T. Belytschko, The extended/generalized finite element method: An overview of the method and its applications, Int. J. Numer. Methods Eng., 84 (2010), 253-304. doi: 10.1002/nme.2914.

[23]

D. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, 1989.

[24]

C. J. Hellier, Handbook of Nondestructive Evaluation, McGraw-Hill, NY, 2003.

[25]

K. Ito, B. Jin and J. Zou, A direct sampling method to an inverse medium scattering problem, Inverse Problems, 28 (2012), 025003, 11pp. doi: 10.1088/0266-5611/28/2/025003.

[26]

K. Ito, B. Jin and J. Zou, A direct sampling method for inverse electromagnetic medium scattering, Inverse Problems, 29 (2013), 095018, 19pp. doi: 10.1088/0266-5611/29/9/095018.

[27]

K. Ito, B. Jin and J. Zou, A direct sampling method for electrical impedance tomography, Inverse Problems, 30 (2014), 095003, 25pp. doi: 10.1088/0266-5611/30/9/095003.

[28]

H. Jia, T. Takenaka and T. Tanaka, Time-domain inverse scattering method for cross-borehole radar imaging, IEEE Trans. on Geoscience and Remote Sensing, 40 (2002), 1640-1647. doi: 10.1109/TGRS.2002.800440.

[29]

J. Jung, C. Jeong and E. Taciroglu, Identification of a scatterer embedded in elastic heterogeneous media using dynamic XFEM, Comput. Methods Appl. Mech. Eng., 259 (2013), 50-63. doi: 10.1016/j.cma.2013.03.001.

[30]

J. Jung and E. Taciroglu, Modeling and identification of an arbitrarily shaped scatterer using dynamic XFEM with cubic splines, Comp. Methods Appl. Mech. Eng., 278 (2014), 101-118. doi: 10.1016/j.cma.2014.05.001.

[31]

L. F. Kallivokas, A. Fathi, S. Kucukcoban, K. H. Stokoe II, J. Bielak and O. Ghattas, Site characterization using full waveform inversion, Soil Dyn. Earthq. Eng., 47 (2013), 62-82. doi: 10.1016/j.soildyn.2012.12.012.

[32]

D. Karaboga, An Idea Based On Honey Bee Swarm for Numerical Optimization, Tech. Report, TR06, Erciyes University, Computer Engineering Department.

[33]

J. Krautkrämer and H. Krautkrämer, Ultrasonic Testing of Materials, Springer-Verlag, Berlin, New York, 1990.

[34]

J. Li and J. Zou, A direct sampling method for inverse scattering using far-field data, Inverse Problems and Imaging, 7 (2013), 757-775. doi: 10.3934/ipi.2013.7.757.

[35]

G. R. Liu and X. Han, Computational Inverse Techniques in Nondestructive Evaluation, CRC Press, Boca Raton, FL, 2003. doi: 10.1201/9780203494486.

[36]

C. W. Liu and E. Taciroglu, Enriched reproducing kernel particle method for piezoelectric structures with arbitrary interfaces, Int. J. Numer. Methods Eng., 67 (2006), 1565-1586. doi: 10.1002/nme.1684.

[37]

C. W. Liu and E. Taciroglu, Shape optimization of piezoelectric devices using an enriched meshfree method, Int. J. Numer. Methods Eng., 78 (2009), 151-171. doi: 10.1002/nme.2479.

[38]

M. Marija and K. Kaspars, Application of Ultrasonic Imaging Technique as Structural Health Monitoring Tool for Assessment of Defects in Glass Fiber Composite Structures, Proceeding of the International Conference on Civil Engineering, 4 (2013), 180-184.

[39]

J. M. Melenk and I. Babuška, The partition of unity finite element method: Basic theory and applications, Comput. Methods Appl. Mech. Eng., 139 (1996), 289-314. doi: 10.1016/S0045-7825(96)01087-0.

[40]

N. Moës, M. Cloirec, P. Cartraud and J. F. Remacle, A computational approach to handle complex microstructure geometries, Comput. Methods Appl. Mech. Eng., 192 (2003), 3163-3177.

[41]

N. M. Newmark, A method of computation for structural dynamics, ASCE J. Engng. Mech. Div., 85 (1959), 67-94.

[42]

R. Potthast, A survey on sampling and probe methods for inverse problems, Inverse Problems, 22 (2006), 1-47. doi: 10.1088/0266-5611/22/2/R01.

[43]

D. Rabinovich, D. Givoli and S. Vigdergauz, XFEM-based crack detection scheme using a genetic algorithm, Int. J. Numer. Methods Eng., 71 (2007), 1051-1080. doi: 10.1002/nme.1975.

[44]

D. Rabinovich, D. Givoli and S. Vigdergauz, Crack identification by arrival time using XFEM and a genetic algorithm, Int. J. Numer. Methods Eng., 77 (2009), 337-359. doi: 10.1002/nme.2416.

[45]

C. L. Richardson, J. Hegemann, E. Sifakis, J. Hellrung and J. M. Teran, An XFEM method for modeling geometrically elaborate crack propagation in brittle materials, Int. J. Numer. Methods Eng., 88 (2011), 1042-1065. doi: 10.1002/nme.3211.

[46]

J. H. Rose, Elastic wave inverse scattering in nondestructive evaluation, Pure Appl. Geophys., 131 (1989), 715-739. doi: 10.1007/978-3-0348-6363-6_7.

[47]

M. Safdari, A. R. Najafi, N. R. Sottos and P. H. Geubelle, A NURBS-based interface-enriched generalized finite element method for problems with complex discontinuous gradient fields, Int. J. Num. Meth. Engng., 101 (2015), 950-964. doi: 10.1002/nme.4852.

[48]

H. Sauerland and T. P. Fries, A stable XFEM for two-phase flows, Comput. Fluids, 87 (2013), 41-49. doi: 10.1016/j.compfluid.2012.10.017.

[49]

B. G. Smith, B. L. Vaughan, Jr and D. L. Chopp, The extended finite element method for boundary layer problems in biofilm growth, Comm. App. Math. and Comp. Sci., 2 (2007), 35-56. doi: 10.2140/camcos.2007.2.35.

[50]

N. Sukumar, D. L. Chopp, N. Moës and T. Belytschko, Modeling holes and inclusions by level sets in the extended finite-element method, Comput. Methods Appl. Mech. Eng., 190 (2001), 6183-6200. doi: 10.1016/S0045-7825(01)00215-8.

[51]

H. Sun, H. Waisman and R. Betti, Nondestructive identification of multiple flaws using XFEM and a topologically adapting artificial bee colony algorithm, Int. J. Numer. Methods Eng., 95 (2013), 871-900. doi: 10.1002/nme.4529.

[52]

H. Theodoros, B. Efstratios and T. N. Georgios, Application of Ultrasonic C-Scan Techniques for Tracing Defects in Laminated Composite Materials, J. Mech. Eng., 57 (2011), 192-203.

[53]

P. Turán, A note of welcome, Journal of Graph Theory, 1 (1977), 7-9.

[54]

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