Determination of rigid inclusions immersed in an isotropic elastic body from boundary measurement. (English) Zbl 1527.35189
Summary: We study the determination of some rigid inclusions immersed in an isotropic elastic medium from overdetermined boundary data. We propose an accurate approach based on the topological sensitivity technique and the reciprocity gap concept. We derive a higher-order asymptotic formula, connecting the known boundary data and the unknown inclusion parameters. The obtained formula is interesting and useful tool for developing accurate and robust numerical algorithms in geometric inverse problems.
MSC:
| 35N25 | Overdetermined boundary value problems for PDEs and systems of PDEs |
| 35J57 | Boundary value problems for second-order elliptic systems |
| 35R30 | Inverse problems for PDEs |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 49K40 | Sensitivity, stability, well-posedness |
Keywords:
linear elasticity; reciprocity gap functional; calculus of variations; topological sensitivityReferences:
| [1] | Pasquale, C.; Leszek, G.; Roberto, L.; Junior Joao, S., Multiplicity of positive solutions for a degenerate nonlocal problem with p-laplacian, Adv. Nonlinear Anal., 11, 1, 357-368 (2022) · Zbl 1475.35135 |
| [2] | Blatt, S.; Hopper, C.; Vorderobermeier, N., A regularized gradient flow for the p-elastic energy, Adv. Nonlinear Anal., 11, 1383-1411 (2022) · Zbl 1503.53174 · doi:10.1515/anona-2022-0244 |
| [3] | Muvasharkhan, J.; Murat, R.; Madi, Y., On the numerical solution of one inverse problem for a linearized two-dimensional system of navier-stokes equations, Opusc. Math., 42, 5, 709-725 (2022) · Zbl 1498.35390 · doi:10.7494/OpMath.2022.42.5.709 |
| [4] | Garcke, H.; Huttl, P.; Knopf, P., Shape and topology optimization involving the eigenvalues of an elastic structure: a multi-phase-field approach, Adv. Nonlinear Anal., 11, 159-197 (2022) · Zbl 1475.35217 · doi:10.1515/anona-2020-0183 |
| [5] | Abdelwahed, M.; Chorfi, N.; Hassine, M., Asymptotic formulas for the identification of small inhonogeneities in a fluid midium, Electron. J. Differ. Equ., 186 (2015) · Zbl 1322.35170 |
| [6] | Andrieux, S.; Ben Abda, A., Identification of planar cracks by complete overdetermined data: inversion formulae, Inverse Probl., 12, 553-563 (1996) · Zbl 0858.35131 · doi:10.1088/0266-5611/12/5/002 |
| [7] | Alves, C.; Silvestre, A. L., On the determination of point-forces on a stokes system, Math. Comput. Simul., 66, 385-397 (2004) · Zbl 1075.76020 · doi:10.1016/j.matcom.2004.02.007 |
| [8] | Abdelwahed, M.; Hassine, M., Topological optimization method for a geometric control problem in stokes flow, Appl. Numer. Math., 59, 1823-1838 (2009) · Zbl 1165.76011 · doi:10.1016/j.apnum.2009.01.008 |
| [9] | Abdelwahed, M.; Hassine, M.; Masmoudi, M., Optimal shape design for fluid flow using topological perturbation technique, J. Math. Anal. Appl., 356, 548-563 (2009) · Zbl 1172.35049 · doi:10.1016/j.jmaa.2009.02.045 |
| [10] | Badra, M.; Caubet, F.; Dambrine, M., Detecting an obstacle immersed in a fluid by shape optimization methods, Math. Models Methods Appl. Sci., 21, 2069-2101 (2011) · Zbl 1239.35182 · doi:10.1142/S0218202511005660 |
| [11] | Garreau, S.; Guillaume, P.; Masmoudi, M., The topological asymptotic for pde systems: The elastic case, SIAM J. Control Optim., 39, 1756-1778 (2011) · Zbl 0990.49028 · doi:10.1137/S0363012900369538 |
| [12] | Guillaume, P.; Sid Idris, K., Topological sensitivity and shape optimization for the stokes equations, SIAM J. Control Optim., 43, 1-31 (2004) · Zbl 1093.49029 · doi:10.1137/S0363012902411210 |
| [13] | Hassine, M.; Khelif, K., On the high-order topological asymptotic expansion for shape functions, Electron. J. Differ. Equ., 110 (2016) · Zbl 1342.35014 |
| [14] | Hassine, M.; Masmoudi, M., The topological asymptotic expansion for the quasi-stokes problem, ESAIM Control Optim. Calc. Var., 10, 478-504 (2004) · Zbl 1072.49027 · doi:10.1051/cocv:2004016 |
| [15] | Sokolowski, J.; Zochowski, A., On the topological derivative in shape optimization, SIAM J. Control Optim., 37, 1251-1272 (1999) · Zbl 0940.49026 · doi:10.1137/S0363012997323230 |
| [16] | Astashova, I.; Bartusek, M.; Dosla, Z.; Marini, M., Asymptotic proximity to higher order nonlinear differential equations, Adv. Nonlinear Anal., 11, 1598-1613 (2022) · Zbl 1503.34083 · doi:10.1515/anona-2022-0254 |
| [17] | Wang, Y.; Wu, W., Initial boundary value problems for the three-dimensional compressible elastic navier-stokes-poisson equations, Adv. Nonlinear Anal., 10, 1356-1383 (2021) · Zbl 1472.76003 · doi:10.1515/anona-2020-0184 |
| [18] | Dautray, R.; Lions, J., Analyse Mathémathique et Calcul Numérique Pour les Sciences et les Techniques (1987) · Zbl 0708.35003 |
| [19] | Watanabe, K., Stability of stationary solutions to the three-dimensional navier-stokes equations with surface tension, Adv. Nonlinear Anal., 12, 279-284 (2023) · Zbl 1506.35147 |
| [20] | Simsen, J.; Simsen, M.; Wittbold, P., Reaction-diffusion coupled inclusions with variable exponents and large diffusion, Opusc. Math., 41, 539-570 (2021) · Zbl 1476.35058 · doi:10.7494/OpMath.2021.41.4.539 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
