Elasticity equations with random domains—the shape derivative approach

In this work, we discuss elasticity equations on a two-dimensional domain with random boundaries and we apply these equations to modelling human corneas. References R. C. Augustyn, D. Nankivil, A. Mohamed, B. Maceo, F. Pierre, and J.-M. Parel. Human ocular biometry. Exp. Eye Res. 102 (2012), pp. 70–75. doi: 10.1016/j.exer.2012.06.009. F. Ballarin, A. Manzoni, G. Rozza, and S. Salsa. Shape optimization by free-form deformation: Existence results and numerical solution for Stokes flows. J. Sci. Comput. 60.3 (2014), pp. 537–563. doi: 10.1007/s10915-013-9807-8. S. C. Brenner and L.-Y. Sung. Linear finite element methods for planar linear elasticity. Math. Comp. 59 (1992), pp. 321–338. doi: 10.2307/2153060. M. C. Delfour and J.-P. Zolesio. Shapes and geometries. Advances in Design and Control. SIAM, Philadelphia, 2001. doi: 10.1137/1.9780898719826. J. Dick. Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands. Ann. Statist. 39.3 (2011), pp. 1372–1398. doi: 10.1214/11-AOS880. J. Dick, F. Y. Kuo, Q. T. Le Gia, and Ch. Schwab. Multilevel higher order QMC Petrov–Galerkin discretization for affine parametric operator equations. SIAM J. Num. Anal. 4.54 (2015), pp. 2541–2568. doi: 10.1137/16M1078690 A. Eilaghi, J. G. Flanagan, I. Tertinegg, C. A. Simmons, G. W. Brodland, and C. R. Ethier. Biaxial testing of human sclera. J. Biomech. 43 (2010), pp. 1696–1701. doi: 10.1016/j.jbiomech.2010.02.031. U. Fares, A. M Otri, M. A. Al-Aqaba, and H. S. Dua. Correlation of central and peripheral corneal thickness in healthy corneas. Cont. Lens Anterior Eye 35 (2012), pp. 39–45. doi: 10.1016/j.clae.2011.07.004. R. N. Gantner and Ch. Schwab. Computational higher order quasi-Monte Carlo integration. Monte Carlo and quasi-Monte Carlo methods. Vol. 163. Springer Proc. Math. Stat. Springer, 2016, pp. 271–288. doi: 10.1007/978-3-319-33507-0_12. on p. C129). H. Harbrecht. Second moment analysis for Robin boundary value problems on random domains. Singular Phenomena and Scaling in Mathematical Models. Ed. by M. Griebel. Springer, 2014, pp. 361–381. doi: 10.1007/978-3-319-00786-1_16. H. Harbrecht, M. Peters, and M. Siebenmorgen. Analysis of the domain mapping method for elliptic diffusion problems on random domains. Numer. Math. 134.4 (2016), pp. 823–856. doi: 10.1007/s00211-016-0791-4. H. Harbrecht, R. Schneider, and Ch. Schwab. Sparse second moment analysis for elliptic problems in stochastic domains. Numer. Math. 109.3 (2008), pp. 385–414. doi: 10.1007/s00211-008-0147-9. R. Hiptmair and J. Li. Shape derivatives in differential forms I: an intrinsic perspective. Ann. Matematica Pura Appl. 192 (2013), pp. 1077–1098. doi: 10.1007/s10231-012-0259-9. C. R. de Lima, L. A. Mello, R. G. Lima, and E. C. N. Silva. Electrical impedance tomography through constrained sequential linear programming: a topology optimization approach. Meas. Sci. Tech. 18.9 (2007), pp. 2847–2858. doi: 10.1088/0957-0233/18/9/014. M. Loeve. Probability theory I. Graduate Texts in Mathematics. Springer-Verlag, 1978. doi: 10.1007/978-1-4684-9464-8. R. Martin, S. Jonuscheit, A. Rio-Cristobal, and M. J. Doughty. Repeatability of Pentacam peripheral corneal thickness measurements. Cont. Lens Anterior Eye 38 (2015), pp. 424–429. doi: 10.1016/j.clae.2015.05.001. R. von Mises. Mechanik der festen Körper im plastisch-deformablen Zustand. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse 1 (1913), pp. 562–592. url: https://eudml.org/doc/58894 F. Orucoglu and E. Toker. Comparative analysis of anterior segment parameters in normal and keratoconic eyes generated by Scheimpflug tomography. J. Ophthalmol., 925414 (2015), pp. 1–8. doi: 10.1155/2015/925414. D. C. Pye. A clinical method for estimating the modulus of elasticity of the human cornea in vivo. PLOS One 15 (2020), pp. 1–19. doi: 10.1371/journal.pone.0224824.