Optimal topological design through insertion and configuration of finite-sized heterogeneities

In this paper, we develop a procedure for optimal topological design by sequentially inserting finite-sized non-spherical inclusions or holes within a homogeneous domain. We propose a new criterion for topology change that results in a trade-off problem to achieve the greatest/least change in the objective for the least/greatest change in the size of the inclusion/hole respectively. We derive the material derivative of the proposed objective, termed as the configurational derivative, that describes sensitivity of arbitrary functionals to arbitrary motions of the inclusion/hole as well as the domain boundaries. We specifically utilize the sensitivity to position, orientation and scaling of finite-sized heterogeneities to effect topological design. We simplify the configurational derivative to the special case of infinitesimally small spherical inclusions or holes and show that the developed derivative is a generalization of the classical topological derivative. The computational implementation relies on B-spline isogeometric approximations. We demonstrate, through a series of examples, optimal topology achieved through sequential insertion of a heterogeneity of fixed shape and optimization of its configuration (location, orientation and scale).