The topological derivative of stress-based cost functionals in anisotropic elasticity
The topological derivative of cost functionals J J that depend on the stress (through the displacement gradient, assuming a linearly elastic material behavior) is considered in a quite general 3D setting where both the background and the inhomogeneity may have arbitrary anisotropic elastic properties. The topological derivative D J ( z ) of J J quantifies the asymptotic behavior of J J induced by the nucleation in the background elastic medium of a small anisotropic inhomogeneity of characteristic radius a a at a specified location z . The fact that the strain perturbation inside an elastic inhomogeneity remains finite for arbitrarily small a a makes the small-inhomogeneity asymptotics of stress-based cost functionals quite different than that of the more usual displacement-based functionals. The asymptotic perturbation of J J is shown to be of order O(a3) O ( a 3 ) for a wide class of stress-based cost functionals having smooth densities. The topological derivative of J J , i.e. the coefficient of the O(a3) O ( a 3 ) perturbation, is established, and computational procedures then discussed. The resulting small-inhomogeneity expansion of J J is mathematically justified (i.e. its remainder is proved to be of order o(a3) o ( a 3 ) ). Several 2D and 3D numerical examples are presented, in particular demonstrating the proposed formulation of D J on cases involving anisotropic elasticity and non-quadratic cost functionals.
topological derivative stress-based criteria topology optimization