Higher order topological derivatives for three-dimensional anisotropic elasticity

Bonnet, Marc; Cornaggia, Rémi

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 6, pp. 2069-2092.

This article concerns an extension of the topological derivative concept for 3D elasticity problems involving elastic inhomogeneities, whereby an objective function 𝕁 is expanded in powers of the characteristic size $a$ of a single small inhomogeneity. The $O\left(a^6\right)$ approximation of 𝕁 is derived and justified for an inhomogeneity of given location, shape and elastic properties embedded in a 3D solid of arbitrary shape and elastic properties; the background and the inhomogeneity materials may both be anisotropic. The generalization to multiple small inhomogeneities is concisely described. Computational issues, and examples of objective functions commonly used in solid mechanics, are discussed.

Classification : 35C20, 45F15, 74B05
Mots clés : Topological derivative, asymptotic expansion, volume integral equation, elastostatics

G. Allaire, F. De Gournay, F. Jouve and A.M. Toader, Structural optimization using topological and shape sensitivity via a level-set method. Control and Cybernetics 34 (2005)59–80. | MR | Zbl

F. Alouges and M. Aussal, The sparse cardinal sine decomposition and its application for fast numerical convolution. Numer. Algorithms (2015) 1–22. | MR

H. Ammari, J. Garnier, V. Jugnon and H. Kang, Stability and resolution analysis for a topological derivative based imaging functional. SIAM J. Control Opt. 50 (2012) 48–76. | DOI | MR | Zbl

H. Ammari and H. Kang, Polarization and moment tensors: with applications to inverse problems and effective medium theory. Vol. 162. Springer (2007). | MR | Zbl

H. Ammari, H. Kang, G. Nakamura and K. Tanuma, Complete asymptotic expansions of solutions of the system of elastostatics in the presence of an inclusion of small diameter and detection of an inclusion. J. Elast. 67 (2002) 97–129. | DOI | MR | Zbl

S. Amstutz and H. Andrä, A new algorithm for topology optimization using a level-set method. J. Comput. Phys. 216 (2006) 573–588. | DOI | MR | Zbl

S. Andrieux and A. Ben Abda, The reciprocity gap: a general concept for flaws identification problems. Mech. Res. Commun. 20 (1993) 415–420. | DOI | Zbl

R.J. Asaro and D.M. Barnett, The non-uniform transformation strain problem for an anisotropic ellipsoidal inclusion. J. Mech. Phys. Solids 23 (1975) 77–83. | DOI | Zbl

C. Bellis and M. Bonnet, A FEM-based topological sensitivity approach for fast qualitative identification of buried cavities from elastodynamic overdetermined boundary data. Int. J. Solids Struct. 47 (2010) 1221–1242. | DOI | Zbl

C. Bellis, M. Bonnet and F. Cakoni, Acoustic inverse scattering using topological derivative of far-field measurements-based $L^2$ cost functionals. Inverse Problems 29 (2013) 075012. | DOI | MR | Zbl

M. Bonnet, Inverse acoustic scattering by small-obstacle expansion of a misfit function. Inverse Problems 24 (2008) 035022. | DOI | MR | Zbl

M. Bonnet, Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems. “Special issue on the advances in mesh reduction methods - In honor of Professor Subrata Mukherjee on the occasion of his 65th birthday”. Eng. Anal. Boundary Elements 35 (2011) 223–235. | DOI | MR | Zbl

M. Bonnet and G. Delgado, The topological derivative in anisotropic elasticity. Quarterly J. Mech. Appl. Math. 66 (2013) 557–586. | DOI | MR | Zbl

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. 0172-5939. Springer Verlag New York (2011). | MR | Zbl

P.G.Ciarlet, Linear and nonlinear functional analysis with applications. SIAM (2013). | MR | Zbl

R. Dautray and J.L. Lions, Analyse mathematique et calcul numerique pour les sciences et les techniques. Masson (1984-85). | MR | Zbl

J. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. Roy. Soc. 241 (1957) 376–396. | MR | Zbl

J. Eshelby, Elastic inclusions and inhomogeneities. Progress in solid Mechanics 2 (1961) 89–140. | MR

A.B. Freidin and V.A. Kucher, Solvability of the equivalent inclusion problem for an ellipsoidal inhomogeneity. Math. Mech. Solids 21 (2016) 255–262. | DOI | MR | Zbl

Y. Fu, K.J. Klimkowski, G.J. Rodin, E. Berger, J.C. Browne, J.R. Singer, R.A. van der Greijn and K.S.Vemaganti, A fast solution method for the three-dimensional many-particle problems of linear elasticity. Int. J. Num. Meth. Eng. 42 (1998) 1215–1229. | DOI | Zbl

S. Garreau, P. Guillaume and M. Masmoudi, The topological asymptotic for PDE systems: the elasticity case. SIAM J. Contr. Opt. 39 (2001) 1756–1778. | DOI | MR | Zbl

D. Gintides and K. Kiriaki, Solvability of the integrodifferential equation of Eshelby’s equivalent inclusion method. The Quarterly J. Mech. Appl. Math. 68 (2015) 85–96. | DOI | MR | Zbl

M. Grédiac and F. Hild, Full-field measurements and identification in solid mechanics. John Wiley and Sons (2012). | Zbl

G.C. Hsiaoa and W.L. Wendland, Boundary integral equations. Springer (2008). | MR | Zbl

R. Kohn and A. Mckenney, Numerical implementation of a variational method for electrical impedance tomography. Inverse Problems 6 (1990) 389–414. | DOI | MR | Zbl

W.P. Kuykendall, W.D. Cash, D.M. Barnett and W. Cai, On the existence of Eshelby’s equivalent ellipsoidal inclusion solution. Math. Mech. Solids 17 (2012) 840–847. | DOI | MR | Zbl

P.A. Martin, Acoustic scattering by inhomogeneous obstacles. SIAM J. Appl. Math. 64 (2003) 297–308. | DOI | MR | Zbl

M. Masmoudi, J. Pommier and B. Samet, The topological asymptotic expansion for the Maxwell equations and some applications. Inverse Problems 21 (2005) 547–564. | DOI | MR | Zbl

W. McLean, Strongly elliptic systems and boundary integral equations. Cambridge (2000). | MR | Zbl

S.G. Mikhlin and S. Prössdorf, Singular integral operators. Vol. 1986. Springer Verlag Berlin (1986). | MR | Zbl

T. Mura, Micromechanics of defects in solids. M. Nijhoff, The Hague and Boston and Higham, Mass (1982). | Zbl

A.G. Ramm, A simple proof of the Fredholm alternative and a characterization of the Fredholm operators. Amer. Math. Monthly. 108 (2001) 855–860. | DOI | MR | Zbl

B. Samet, S. Amstutz and M. Masmoudi, The topological asymptotic for the Helmholtz equation. SIAM J. Contr. Opt. 42 (2004) 1523–1544. | DOI | MR | Zbl

M. Silva, M. Matalon and D.A. Tortorelli, Higher order topological derivatives in elasticity. Inter. J. Solids Structures 47 (2010) 3053–3066. | DOI | Zbl

J. Sokolowski and A. Zochowski, Topological derivatives of shape functionals for elasticity systems. Mech. Based Design Struct. Machines 21 (2001) 331–349. | DOI | MR

T. Touhei, A fast volume integral equation method for elastic wave propagation in a half space. Int. J. Solids Struct. 48 (2011) 3194–3208. | DOI

S. Xinchun and R.S. Lakes, Stability of elastic material with negative stiffness and negative Poisson’s ratio. Phys. Stat. Sol. 244 (2007) 1008–1026. | DOI