Abstract
This work focuses on the identification of heterogeneous linear elastic moduli in the context of frequency-domain, coupled acoustic-structure interaction (ASI), using either solid displacement or fluid pressure measurement data. The approach postulates the inverse problem as an optimization problem where the solution is obtained by minimizing a modified error in constitutive equation (MECE) functional. The latter measures the discrepancy in the constitutive equations that connect kinematically admissible strains and dynamically admissible stresses, while incorporating the measurement data as additional quadratic error terms. We demonstrate two strategies for selecting the MECE weighting coefficient to produce regularized solutions to the ill-posed identification problem: 1) the discrepancy principle of Morozov, and 2) an error-balance approach that selects the weight parameter as the minimizer of another functional involving the ECE and the data misfit. Numerical results demonstrate that the proposed methodology can successfully recover elastic parameters in 2D and 3D ASI systems from response measurements taken in either the solid or fluid subdomains. Furthermore, both regularization strategies are shown to produce accurate reconstructions when the measurement data is polluted with noise. The discrepancy principle is shown to produce nearly optimal solutions, while the error-balance approach, although not optimal, remains effective and does not need a priori information on the noise level.
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Oberai AA, Gokhale NH, Doyley MM, Bamber JC (2004) Evaluation of the adjoint equation based algorithm for elasticity imaging. Phys Med Biol 49:2955–2974
Levasseur S, Malécot Y, Boulon M, Flavigny E (2007) Soil parameter identification using a genetic algorithm. Int J Numer Anal Metho Geomech 32:189–213
Stull CJ, Earls CJ, Koutsourelakis PS (2011) Model-based structural health monitoring of naval ship hulls. Comp Meth Appl Mech Eng 200:1137–1149
Banks HT, Joyner ML, Wincheski B, Winfree WP (2002) Real time computational algorithms for eddy-current-based damage detection. Inverse Probl 18:795–823
Greenleaf JF, Fatemi M, Insana M (2003) Selected methods for imaging elastic properties in biological tissues. Ann Rev Biomed Eng 5:57–78
Bernal M, Nenadic I, Urban MW, Greenleaf JF (2011) Material property estimation for tubes and arteries using ultrasound radiation force and analysis of propagating modes. J Acoust Soc Am 129(3):1344–1354
Plewes DB, Bishop J, Samani A, Sciarretta J (2000) Visualization and quantification of breast cancer biomechanical properties with magnetic resonance elastography. Phys Med Biol 45:1591–1610
Kingwell BA, Gatzka CD (2002) Arterial stiffness and prediction of cardiovascular risk. J Hypertens 20:2337–2340
Chen S, Fatemi M, Greenleaf JF (2004) Quantifying elasticity and viscosity from measurement of shear wave speed dispersion. J Acoust Soc Am 115(6):2781–2785
Muthupillai R, Lomas DJ, Rossman PJ, Greenleaf JF, Manduca A, Ehman RL (1995) Magnetic resonance elastography by direct visualization of propogating acoustic strain waves. Science 269(5232):1854–1857
Fatemi M, Greenleaf JF (1998) Ultrasound-stimulated vibro-acoustic spectrography. Science 280:82–85
Fatemi M, Greenleaf JF (1999) Vibro-acoustography: An imaging modality based on ultrasound-stimulated acoustic emission. Proc Natl Acad Sci 96:6603–6608
Brigham JC, Aquino W, Mitri FG, Greenleaf JF, Fatemi M (2007) Inverse estimation of viscoelastic material properties for solids immersed in fluids using vibroacoustic techniques. J Appl Phys 101(23509):1–14
Brigham JC, Aquino W (2009) Inverse viscoelastic material characterization using pod reduced-order modeling in acoustic-structure interaction. Comp Meth Appl Mech Eng 198:893–903
Rosario E, Brigham JC, Aquino W (2008) Identification of material properties of orthotropic elastic cylinders immersed in fluid using vibroacoustic techniques. Ultrasonics 48:547–552
Aguilo M, Aquino W, Brigham JC, Fatemi M (2010) An inverse problem approach for elasticity imaging through vibroacoustics. IEEE Trans Med Imaging 29(4):1012–1021
Ladeveze P, Leguillon D (1983) Error estimate procedure in the finite element method and applications. SIAM J Numer Anal 20:485–509
Ladevèze P, Nedjar D, Reynier M (1994) Updating of finite element models using vibration tests. AIAA J 32:1485–1491
Barthe D, Deraemaeker A, Ladevèze P, Le Loch S (2004) Validation and updating of industrial models based on the constitutive relation error. Am Inst Aeronaut Astronaut 42:1427–1434
Allix O, Feissel P, Nguyen HM (2005) Identification strategy in the presence of corrupted measurements. Eng Comput 22:487–504
Feissel P, Allix O (2007) Modified constitutive relation error identification strategy for transient dynamics with corrupted data. Comp Meth Appl Mech Eng 196:1968–1983
Banerjee B, Walsh TF, Aquino W, Bonnet M (2013) Large scale parameter estimation problems in frequency-domain elastodynamics using an error in constitutive equation functional. Comp Meth Appl Mech Eng 253:60–72
Isakov V (1998) Inverse Problems for Partial Differential Equations. Springer, New York
Colton D, Kress R (1998) Inverse Acoustic and Electromagnetic Scattering Theory. Springer, Berlin
Hasegawa T, Kido T, Iizuka T, Matsuoka C (2000) A general theory of rayleigh and langevin radiation pressures. Journal-Acoustical Society of Japan, 21(3)
Zhang X, Greenleaf JF (2007) Estimation of tissues elasticity with surface wave speed. J Acoustl Soc Am 122(5):2522–2525
Berenger JP (1994) A perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys 114:185–200
Hinze M, Pinnau R, Ulbrich M, Ulbrich S (2009) Optimization with PDE Constraints. Springer, New York
Bathe KJ (2006) Finite Element Procedures. Prentice Hall, Pearson Education Inc, USA
Schenk O, Gärtner K (2004) Solving unsymmetric sparse systems of linear equations with pardiso. J Future Gener Comput Syst 20(3):475–487
Schenk O, Gärtner K (2006) On fast factorization pivoting methods for symmetric indefinite systems. Elec Trans Numer Anal 23:158–179
Epanomeritakis I, Akcelik V, Ghattas O, Bielak J (2008) A newton-cg method for large-scale three-dimensional elastic full-waveform seismic inversion. Inverse Prob, 24:034015 (26p)
Calvetti D, Morigi S, Reichel L, Sgallari F (2000) Tikhonov regularization and the l-curve for large discrete ill-posed problems. J Comput Appl Math 123:423–446
Hughes TJR (2000) The finite element method: linear static and dynamic finite element anaysis. Dover Publications Inc, Mineola, New York, USA
Sinkus R, Tanter M, Catheline S, Lorenzen J, Kuhl C, Sondermann E, Fink M (2005) Imaging anisotropic and viscous properties of breast tissue by magnetic resonance-elastography. Magn Reson Med 53(2):372–387
Kwon OI, Park C, Nam HS, Woo EJ, Seo JK, Glaser KJ, Manduca A, Ehman RL (2009) Shear modulus decomposition algorithm in magnetic resonance elastography. IEEE Trans. Med. Imaging 28(10):1526–1533
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This work was partially supported by NIH Grants \(\#\)EB002640 and \(\#\)EB002167.
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Warner, J.E., Diaz, M.I., Aquino, W. et al. Inverse material identification in coupled acoustic-structure interaction using a modified error in constitutive equation functional. Comput Mech 54, 645–659 (2014). https://doi.org/10.1007/s00466-014-1018-0
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DOI: https://doi.org/10.1007/s00466-014-1018-0