An energy gap functional: Cavity identification in linear elasticity

References

[1] S. Amstutz, I. Horchani and M. Masmoudi, Crack detection by the topological gradient method, Control Cybernet. 34 (2005), no. 1, 81–101. Search in Google Scholar

[2] S. Andrieux, T. N. Baranger and A. Ben Abda, Solving Cauchy problems by minimizing an energy-like functional, Inverse Problems 22 (2006), no. 1, 115–133. 10.1088/0266-5611/22/1/007Search in Google Scholar

[3] J. B. Bacani and G. Peichl, On the first-order shape derivative of Kohn–Vogelius cost functional of the Bernoulli problem, Abstr. Appl. Anal. 2013 (2013), Article ID 384320. 10.1155/2013/384320Search in Google Scholar

[4] J. B. Bacani and G. Peichl, The second-order shape derivative of Kohn–Vogelius-type cost functional using the boundary differentiation approach, Mathematics 2 (2014), no. 4, 196–217. 10.3390/math2040196Search in Google Scholar

[5] A. Ben Abda, F. Bouchon, G. H. Peichl, M. Sayeh and R. Touzani, A Dirichlet–Neumann cost functional approach for the Bernoulli problem, J. Engrg. Math. 81 (2013), no. 1, 157–176. 10.1007/s10665-012-9608-3Search in Google Scholar

[6] A. Ben Abda, M. Hassine, M. Jaoua and M. Masmoudi, Topological sensitivity analysis for the location of small cavities in Stokes flow, SIAM J. Control Optim. 48 (2009), no. 5, 2871–2900. 10.1137/070704332Search in Google Scholar

[7] H. Ben Ameur, M. Burger and B. Hackl, Level set methods for geometric inverse problems in linear elasticity, Inverse Problems 20 (2004), no. 3, 673–696. 10.1088/0266-5611/20/3/003Search in Google Scholar

[8] H. Ben Ameur, M. Burger and B. Hackl, Cavity identification in linear elasticity and thermoelasticity, Math. Methods Appl. Sci. 30 (2007), no. 6, 625–647. 10.1002/mma.772Search in Google Scholar

[9] S. Chaabane, C. Elhechmi and M. Jaoua, A stable recovery method for the Robin inverse problem, Math. Comput. Simulation 66 (2004), no. 4–5, 367–383. 10.1016/j.matcom.2004.02.016Search in Google Scholar

[10] M. C. Delfour and J. P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus and Optimization, SIAM, Philadelphia, 2001. Search in Google Scholar

[11] P. Destuynder and M. Djaoua, Sur une interprétation mathématique de l’intégrale de Rice en théorie de la rupture fragile, Math. Methods Appl. Sci. 3 (1981), no. 1, 70–87. 10.1002/mma.1670030106Search in Google Scholar

[12] K. Eppler and H. Harbrecht, On a Kohn–Vogelius like formulation of free boundary problems, Comput. Optim. Appl. 52 (2012), no. 1, 69–85. 10.1007/s10589-010-9345-3Search in Google Scholar

[13] S. Gottlieb, C. W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001), no. 1, 89–112. 10.1137/S003614450036757XSearch in Google Scholar

[14] K. Ito, K. Kunisch and Z. Li, Level-set function approach to an inverse interface problem, Inverse Problems 17 (2001), no. 5, 1225–1242. 10.1088/0266-5611/17/5/301Search in Google Scholar

[15] E. Jaïem and S. Khalfallah 2016, An energy-gap cost functional for cavities identification, preprint (2016); to appear in Nonlinear Stud. Search in Google Scholar

[16] G. S. Jiang and D. Peng, Weighted ENO schemes for Hamilton–Jacobi equations, SIAM J. Sci. Comput. 21 (2000), no. 6, 2126–2143. 10.1137/S106482759732455XSearch in Google Scholar

[17] A. Karageorghis, D. Lesnic and L. Marin, The method of fundamental solutions for the detection of rigid inclusions and cavities in plane linear elastic bodies, Comput. Struct. 106–107 (2012), 176–188. 10.1016/j.compstruc.2012.05.001Search in Google Scholar

[18] A. Karageorghis, D. Lesnic and L. Marin, The method of fundamental solutions for three-dimensional inverse geometric elasticity problems, Comput. Struct. 166 (2016), 51–59. 10.1016/j.compstruc.2016.01.010Search in Google Scholar

[19] R. V. Kohn and M. Vogelius, Relaxation of a variational method for impedance computed tomography, Comm. Pure Appl. Math. 40 (1987), no. 6, 745–777. 10.1002/cpa.3160400605Search in Google Scholar

[20] J. Lemaitre, A Course on Damage Mechanics, Springer, Berlin, 1996. 10.1007/978-3-642-18255-6Search in Google Scholar

[21] H. Meftahi and J. P. Zolésio, Sensitivity analysis for some inverse problems in linear elasticity via minimax differentiability, Appl. Math. Model. 39 (2015), no. 5–6, 1554–1576. 10.1016/j.apm.2014.09.026Search in Google Scholar

[22] L. Mishnaevsky, Computational Mesomechanics of Composites: Numerical Analysis of the Effect of Microstructures of Composites on Their Strength and Damage Resistance, John Wiley & Sons, New York, 2007. 10.1002/9780470513170Search in Google Scholar

[23] N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Springer, Berlin, 1977. 10.1007/978-94-017-3034-1Search in Google Scholar

[24] S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Appl. Math. Sci. 153, Springer, New York, 2002. 10.1007/b98879Search in Google Scholar

[25] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton–Jacobi formulations, J. Comput. Phys. 79 (1988), no. 1, 12–49. 10.1016/0021-9991(88)90002-2Search in Google Scholar

[26] J. Sokolowski and J. P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer, Berlin, 1992. 10.1007/978-3-642-58106-9Search in Google Scholar

[27] R. Valid, La mécanique des milieux continus et le calcul des structures, Eyrolles, Paris, 1977. Search in Google Scholar

[28] A. Wexler, B. Fry and M. R. Neuman, Impedance-computed tomography algorithm and system, Appl. Optics 24 (1985), no. 23, 3985–3992. 10.1364/AO.24.003985Search in Google Scholar