Research Article Received 18 March 2013 Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/mma.2918 MOS subject classification: 74B05; 74P15 The topological gradient in anisotropic elasticity with an eye towards lightweight design Matti Schneider a * † and Heiko Andrä b Communicated by H. Ammari We derive a representation formula for the topological gradient with respect to arbitrary quadratic yield functionals and anisotropic elastic materials, thus laying the theoretical foundations for topological sensitivity analysis in lightweight design. For compliance, minimization involving general anisotropic materials and ellipsoidal perturbations, we give a closed formula for the topological gradient, enabling topology optimization of integrated designs involving several rein- forced materials. If the materials are transversely isotropic and the perturbations are spheriodal, we even obtain an ana- lytical formula. For general anisotropy, recent advances in the computation of Eshelby’s tensor enable rapid numerical computation of the topological gradient. Restricting to isotropic materials and spheroidal inclusions, we obtain an analytical formula for minimizing isotropic yield functionals with applications to microscale-scale sensitivity analysis of fiber reinforced composites or reinforcing analysis of brittle materials. Copyright © 2013 John Wiley & Sons, Ltd. Keywords: topological gradient; topology optimization; asymptotic analysis; elastic moment tensor; Eshelby’s tensor; anisotropic elasticity; lightweight design 1. Introduction The topological gradient On a fixed domain, the topological gradient, introduced in [1], measures the sensitivity of a given shape functional with respect to the creation of a small hole or inhomogeneity. During the last decade, the topological gradient has been successfully applied to problems in topology optimization [2–4], the detection and simulation of damage [5, 6], medical imaging [7, 8], anomaly detection [9], and image processing [10–13]. Topology optimization Topology optimization on classes of sets is ill-posed in general, see [3, 14, 15]. One possible remedy is to enlarge the class of admissible sets, for instance, taking into account densities. This leads to power-law- density or homogenization-based algorithms [16–21]. These methods, however, tend to develop microstructures as optima, that is, densities that are largely between 0 and 1. To obtain classic sets in the end, a penalization projection is applied. Instead of enlarging the class of admissible sets, the level-set method restricts to sets that can be written as the level-set of a suffi- ciently regular function. The dynamics of sets are translated into the dynamics of a level-set function that are more in line with classical mathematical understanding. In addition to avoiding problems with microstructure, level-set algorithms can often be performed on a fixed mesh, rendering costly remeshing superfluous. Coupled with the classical shape derivative [3, 22] or employing gradient flow techniques [4], the topological gradient leads to fast and robust topology optimization algorithms. a Faculty of Mechanical Engineering, Department of Lightweight Structures and Polymer Technology, Chemnitz University of Technology, 09107 Chemnitz, Germany b Flow and Material Simulation, Fraunhofer Institute for Industrial Mathematics ITWM, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany *Correspondence to: Matti Schneider, Faculty of Mechanical Engineering, Department of Lightweight Structures and Polymer Technology, Chemnitz University of Technology, 09107 Chemnitz, Germany. † E-mail: matti.schneider@mb.tu-chemnitz.de Copyright © 2013 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2013