Abstract
In this work, we present a novel inverse approach to characterize the nonhomogeneous mechanical behavior of linear elastic solids. In this approach, we optimize the geometric parameters and shear modulus values of the predefined moving morphable inclusions (MMIs) to solve the inverse problem. Thereby, the total number of the optimization parameters is remarkably reduced compared with the conventional iterative inverse algorithms to identify the nonhomogeneous shear modulus distribution of solids. The proposed inverse approach is tested by multiple numerical examples, and we observe that this approach is capable of preserving the shape and the shear moduli of the inclusions well. In particular, this inverse approach performs well even without any regularization when the noise level is not very high. Overall, the proposed approach provides a new paradigm to solve the inverse problem in elasticity and has potential of addressing the issue of computational inefficacy existing in the conventional inverse approaches.
Issue Section:
Research Papers
References
1.
Ophir
, J.
, Cespedes
, I.
, Ponnekanti
, H.
, Yazdi
, Y.
, and Li
, X.
, 1991
, “Elastography: A Quantitative Method for Imaging the Elasticity of Biological Tissues
,” Ultrason. Imaging.
, 13
(2
), pp. 111
–134
. 10.1177/0161734691013002012.
Ophir
, J.
, Alam
, S. K.
, Garra
, B.
, Kallel
, F.
, Konofagou
, E.
, Krouskop
, T.
, and Varghese
, T.
, 1999
, “Elastography: Ultrasonic Estimation and Imaging of the Elastic Properties of Tissues
,” Proc. Inst. Mech. Eng. Part H: J. Eng. Med.
, 213
(3
), pp. 203
–233
. 10.1243/09544119915349333.
Sack
, I.
, Beierbach
, B.
, Hamhaber
, U.
, Klatt
, D.
, and Braun
, A.
, 2008
, “Non-Invasive Measurement of Brain Viscoelasticity Using Magnetic Resonance Elastography
,” NMR Biomed.
, 21
(3
), pp. 265
–271
. 10.1002/nbm.11894.
Trabelsi
, O.
, Gutierrez
, M.
, Farzaneh
, S.
, Duprey
, A.
, and Avril
, S.
, 2017
, “A Non-Invasive Methodology for ATAA Rupture Risk Estimation
,” J. Biomech.
, 66
, pp. 119
–126
. 10.1016/j.jbiomech.2017.11.0125.
Shore
, S. W.
, Barbone
, P. E.
, Oberai
, A. A.
, and Morgan
, E. F.
, 2011
, “Transversely Isotropic Elasticity Imaging of Cancellous Bone
,” ASME J. Biomech. Eng.
, 133
(6
), pp. 1
–11
. 10.1115/1.40042316.
Zhu
, Y.
, Hall
, T. J.
, and Jiang
, J.
, 2003
, “A Finite-Element Approach for Young’s Modulus Reconstruction
,” IEEE Trans. Med. Imaging.
, 22
(7
), pp. 890
–901
. 10.1109/TMI.2003.8150657.
Pan
, X.
, Liu
, K.
, Bai
, J.
, and Luo
, J. W.
, 2014
, “A Regularization-Free Elasticity Reconstruction Method for Ultrasound Elastography With Freehand Scan
,” Biomed. Eng. Online.
, 13
(1
), p. 132
. 10.1186/1475-925X-13-1328.
Oberai
, A. A.
, Gokhale
, N. H.
, and Feijoo
, G. R.
, 2003
, “Solution of Inverse Problems in Elasticity Imaging Using the Adjoint Method
,” Inverse Probl.
, 19
(2
), pp. 297
–313
. 10.1088/0266-5611/19/2/3049.
Simón-allué
, R.
, Calvo
, B.
, Oberai
, A. A.
, and Barbone
, P. E.
, 2017
, “Towards the Mechanical Characterization of Abdominal Wall by Inverse Analysis
,” J. Mech. Behav. Biomed. Mater.
, 66
, pp. 127
–137
. 10.1016/j.jmbbm.2016.11.00710.
Liu
, Z.
, Sun
, Y. L.
, Deng
, J. W.
, Zhao
, D. M.
, Mei
, Y.
, and Luo
, J. W.
, 2020
, “A Comparative Study of Direct and Iterative Inversion Approaches to Determine the Spatial Shear Modulus Distribution of Elastic Solids
,” Int. J. Appl. Mech.
, 11
(10
), p. 1950097
. 10.1142/S175882511950097211.
Lalitha Sridhar
, S.
, Mei
, Y.
, and Goenezen
, S.
, 2018
, “Improving the Sensitivity to Map Nonlinear Parameters for Hyperelastic Problems
,” Comput. Methods Appl. Mech. Eng.
, 331
, pp. 474
–491
. 10.1016/j.cma.2017.11.02812.
Goenezen
, S.
, Barbone
, P.
, and Oberai
, A. A.
, 2011
, “Solution of the Nonlinear Elasticity Imaging Inverse Problem: The Incompressible Case
,” Comput. Methods Appl. Mech. Eng.
, 200
(13–16
), pp. 1406
–1420
. 10.1016/j.cma.2010.12.01813.
Gokhale
, N. H.
, Barbone
, P. E.
, and Oberai
, A. A.
, 2008
, “Solution of the Nonlinear Elasticity Imaging Inverse Problem: The Compressible Case
,” Inverse Probl.
, 24
(4
), p. 045010
. 10.1088/0266-5611/24/4/04501014.
Bendsøe
, M. P.
, and Sigmund
, O.
, 2013
, Topology Optimization: Theory, Methods, and Applications
, Springer Science & Business Media
, New York
.15.
Guo
, X.
, Zhang
, W. S.
, and Zhong
, W. L.
, 2014
, “Doing Topology Optimization Explicitly and Geometrically-A New Moving Morphable Components Based Framework
,” ASME J. Appl. Mech.
, 81
(8
), p. 081009
. 10.1115/1.402760916.
Guo
, X.
, Zhang
, W. S.
, Zhang
, J.
, and Yuan
, J.
, 2016
, “Explicit Structural Topology Optimization Based on Moving Morphable Components (MMC) With Curved Skeletons
,” Comput. Methods Appl. Mech. Eng.
, 310
, pp. 711
–748
. 10.1016/j.cma.2016.07.01817.
Zhang
, W. S.
, Yang
, W. Y.
, Zhou
, J. H.
, Li
, D.
, and Guo
, X.
, 2017
, “Structural Topology Optimization Through Explicit Boundary Evolution
,” ASME J. Appl. Mech.
, 84
(1
), p. 011011
. 10.1115/1.403497218.
Zhang
, W. S.
, Chen
, J. S.
, Zhu
, X. F.
, Zhou
, J. H.
, Xue
, D. C.
, Lei
, X.
, and Guo
, X.
, 2017
, “Explicit Three Dimensional Topology Optimization via Moving Morphable Void (MMV) Approach
,” Comput. Methods Appl. Mech. Eng.
, 322
, pp. 590
–614
. 10.1016/j.cma.2017.05.00219.
Xue
, R. Y.
, Liu
, C.
, Zhang
, W. S.
, Zhu
, Y. C.
, Tang
, S.
, Du
, Z. L.
, and Guo
, X.
, 2019
, “Explicit Structural Topology Optimization Under Finite Deformation via Moving Morphable Void (MMV) Approach
,” Comput. Methods Appl. Mech. Engrg.
, 344
, pp. 798
–818
. 10.1016/j.cma.2018.10.01120.
Liu
, C.
, Du
, Z. L.
, Zhang
, W. S.
, Zhu
, Y. C.
, and Guo
, X.
, 2017
, “Additive Manufacturing-Oriented Design of Graded Lattice Structures Through Explicit Topology Optimization
,” ASME J. Appl. Mech.
, 84
(8
), p. 081008
. 10.1115/1.403694121.
Liu
, C.
, Zhu
, Y. C.
, Sun
, Z.
, Li
, D. D.
, Du
, Z. L.
, Zhang
, W. S.
, and Guo
, X.
, 2018
, “An Efficient Moving Morphable Component (MMC)-Based Approach for Multi-Resolution Topology Optimization
,” Struct. Multidiscip. Optim.
, 58
(6
), pp. 2455
–2479
. 10.1007/s00158-018-2114-022.
Liu
, D.
, and Du
, J. F.
, 2019
, “A Moving Morphable Components Based Shape Reconstruction Framework for Electrical Impedance Tomography
,” IEEE Trans. Med. Imaging.
, 38
(12
), pp. 2937
–2948
. 10.1109/TMI.2019.291856623.
Liu
, D.
, Gu
, D. P.
, Smyl
, D.
, Deng
, J. S.
, and Du
, J. F.
, 2020
, “Shape Reconstruction Using Boolean Operations in Electrical Impedance Tomography
,” IEEE Trans. Med. Imaging.
, 39
(9
), pp. 2954
–2964
. 10.1109/TMI.2020.298305524.
Reddy
, J. N.
, 1993
, An Introduction to the Finite Element Method
, McGraw-Hill
, New York
.25.
Hughes
, T. J.
, 2012
, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis
, Courier Corporation
, New York
.26.
Zienkiewicz
, O. C.
, Taylor
, R. L.
, Nithiarasu
, P.
, and Zhu
, J. Z.
, The Finite Element Method
, McGraw-Hill
, London
.27.
Mei
, Y.
, Stover
, B.
, Afsar Kazerooni
, N.
, Srinivasa
, A.
, Hajhashemkhani
, M.
, Hematiyan
, M. R.
, and Goenezen
, S.
, 2018
, “A Comparative Study of Two Constitutive Models Within an Inverse Approach to Determine the Spatial Stiffness Distribution in Soft Materials
,” Int. J. Mech. Sci.
, 140
, pp. 446
–454
. 10.1016/j.ijmecsci.2018.03.00428.
Dong
, L.
, Wijesinghe
, P.
, Dantuono
, J. T.
, Sampson
, D. D.
, Munro
, P. R. T.
, Kennedy
, B. F.
, and Kennedy
, A. A.
, 2016
, “Quantitative Compression Optical Coherence Elastography as an Inverse Elasticity Problem
,” IEEE J. Sel. Top. QUANTUM Electron.
, 22
(3
), p. 6802211
. 10.1109/jstqe.2015.251259729.
Byrd
, R. H.
, Lu
, P.
, Nocedal
, J.
, and Zhu
, C.
, 1995
, “A Limited Memory Algorithm for Bound Constrained Optimization
,” SIAM J. Sci. Comput.
, 16
(5
), pp. 1190
–1208
. 10.1137/091606930.
Mei
, Y.
, Kuznetsov
, S.
, and Goenezen
, S.
, 2016
, “Reduced Boundary Sensitivity and Improved Contrast of the Regularized Inverse Problem Solution in Elasticity
,” Trans. ASME J. Appl. Mech.
, 83
(3
), p. 031001
. 10.1115/1.403193731.
Mei
, Y.
, Tajderi
, M.
, and Goenezen
, S.
, 2017
, “Regularizing Biomechanical Maps for Partially Known Material Properties
,” Int. J. Appl. Mech.
, 9
(2
), p. 1750020
. 10.1142/S175882511750020XCopyright © 2020 by ASME
You do not currently have access to this content.
