A Lagrangian shape and topology optimization framework based on semi-discrete optimal transport

This article revolves around shape and topology optimization, in the applicative context where the objective and constraint functionals depend on the solution to a physical boundary value problem posed on the optimized domain. We introduce a novel framework based on modern concepts from computational geometry, optimal transport and numerical analysis. Its pivotal feature is a representation of the optimized shape by the cells of an adapted version of a Laguerre diagram. Although such objects are originally described by a collection of seed points and weights, recent results from optimal transport theory suggest a more intuitive parametrization in terms of the seed points and measures of the associated cells. The polygonal mesh of the shape induced by this diagram serves as support for the deployment of the Virtual Element Method for the numerical solution of the physical boundary value problem at play and the calculation of the objective and constraint functionals. The sensitivities of the latter are derived next; at first, we calculate their derivatives with respect to the positions of the vertices of the Laguerre diagram by shape calculus techniques; a suitable adjoint methodology is then developed to express them in terms of the seed points and cell measures of the diagram. The evolution of the shape is realized by first updating the design variables according to these sensitivities and then reconstructing the diagram with efficient algorithms from computational geometry. Our shape optimization strategy is versatile: it can be applied to a wide gammut of physical situations. It is Lagrangian by essence, and it thereby benefits from all the assets of a consistently meshed representation of the shape. Yet, it naturally handles dramatic motions, including topological changes, in a very robust fashion. These features, among others, are illustrated by a series of 2d numerical examples.