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Approaches to accommodate noisy data in the direct solution of inverse problems in incompressible plane strain elasticity. (English) Zbl 1321.74049

Summary: We apply the adjoint-weighted equation method (AWE) to the direct solution of inverse problems of incompressible plane strain elasticity. We show that based on untreated noisy displacements, the reconstruction of the shear modulus can be very poor. We link this poor performance to loss of coercivity of the weak form when treating problems with discontinuous coefficients. We demonstrate that by smoothing the displacements and appending a regularization term to the AWE formulation, a dramatic improvement in the reconstruction can be achieved. With these improvements, the advantages of the AWE method as a direct solution approach can be extended to a wider range of problems.

MSC:

74L15 Biomechanical solid mechanics
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
74G75 Inverse problems in equilibrium solid mechanics
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[1] DOI: 10.1016/0161-7346(91)90079-W · doi:10.1016/0161-7346(91)90079-W
[2] DOI: 10.1177/016173469301500201 · doi:10.1177/016173469301500201
[3] DOI: 10.1016/0301-5629(94)90048-5 · doi:10.1016/0301-5629(94)90048-5
[4] DOI: 10.1016/0929-8266(95)00134-4 · doi:10.1016/0929-8266(95)00134-4
[5] DOI: 10.1148/radiology.202.1.8988195 · doi:10.1148/radiology.202.1.8988195
[6] DOI: 10.1088/0031-9155/45/6/311 · doi:10.1088/0031-9155/45/6/311
[7] DOI: 10.1016/S0301-5629(02)00488-X · doi:10.1016/S0301-5629(02)00488-X
[8] DOI: 10.1016/j.ultrasmedbio.2005.01.017 · doi:10.1016/j.ultrasmedbio.2005.01.017
[9] DOI: 10.1088/0031-9155/52/21/013 · doi:10.1088/0031-9155/52/21/013
[10] DOI: 10.1186/bcr2787 · doi:10.1186/bcr2787
[11] DOI: 10.1002/(SICI)1522-2594(199910)42:4<779::AID-MRM21>3.0.CO;2-Z · doi:10.1002/(SICI)1522-2594(199910)42:4<779::AID-MRM21>3.0.CO;2-Z
[12] DOI: 10.1088/0031-9155/45/6/309 · doi:10.1088/0031-9155/45/6/309
[13] DOI: 10.1088/0266-5611/19/2/304 · Zbl 1171.35490 · doi:10.1088/0266-5611/19/2/304
[14] DOI: 10.1088/0266-5611/23/6/003 · Zbl 1189.35368 · doi:10.1088/0266-5611/23/6/003
[15] DOI: 10.1016/j.cma.2009.02.034 · Zbl 1229.74047 · doi:10.1016/j.cma.2009.02.034
[16] Barbone PE, Int. J. Numer. Meth. Eng 81 pp 1713– (2010)
[17] DOI: 10.1002/nme.4372 · Zbl 1352.74072 · doi:10.1002/nme.4372
[18] DOI: 10.1088/0266-5611/20/1/017 · Zbl 1056.74020 · doi:10.1088/0266-5611/20/1/017
[19] DOI: 10.1016/0045-7825(92)90143-8 · Zbl 0759.76040 · doi:10.1016/0045-7825(92)90143-8
[20] DOI: 10.1016/0045-7825(95)00844-9 · Zbl 0866.76044 · doi:10.1016/0045-7825(95)00844-9
[21] DOI: 10.1016/S0045-7825(01)00191-8 · Zbl 0985.65145 · doi:10.1016/S0045-7825(01)00191-8
[22] DOI: 10.1016/0045-7825(82)90071-8 · Zbl 0497.76041 · doi:10.1016/0045-7825(82)90071-8
[23] Hughes TJR, Finite element methods for convection dominated flows, AMD 34 pp 19– (1979)
[24] DOI: 10.1007/978-1-4612-4026-6 · doi:10.1007/978-1-4612-4026-6
[25] DOI: 10.1063/1.1529180 · Zbl 1185.76144 · doi:10.1063/1.1529180
[26] DOI: 10.1016/0167-2789(92)90242-F · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F
[27] DOI: 10.1109/83.679423 · Zbl 0993.94519 · doi:10.1109/83.679423
[28] DOI: 10.1016/0045-7825(86)90153-2 · Zbl 0587.76120 · doi:10.1016/0045-7825(86)90153-2
[29] DOI: 10.1016/0045-7825(86)90003-4 · Zbl 0593.76096 · doi:10.1016/0045-7825(86)90003-4
[30] DOI: 10.1016/0045-7825(88)90108-9 · Zbl 0626.76091 · doi:10.1016/0045-7825(88)90108-9
[31] DOI: 10.1016/0045-7825(93)90213-H · Zbl 0844.76048 · doi:10.1016/0045-7825(93)90213-H
[32] DOI: 10.1137/1.9780898717570 · Zbl 1008.65103 · doi:10.1137/1.9780898717570
[33] Hughes TJR, The finite element method (2000)
[34] Richards MS. Quantitative three dimensional elasticity imaging [PhD thesis]. Boston (MA): Boston University; 2007.
[35] DOI: 10.1088/0031-9155/54/3/019 · doi:10.1088/0031-9155/54/3/019
[36] DOI: 10.1016/j.ultrasmedbio.2012.09.023 · doi:10.1016/j.ultrasmedbio.2012.09.023
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