The topological gradient in anisotropic elasticity with an eye towards lightweight design
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 reinforced materials. If the materials are transversely isotropic and the perturbations are spheriodal, we even obtain an analytical 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 comp osites or reinforcing analysis of brittle materials.