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 numerical 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 discrete polygonal mesh of the shape induced by this diagram is a convenient support for the deployment of the Virtual Element Method for the numerical solution of the boundary value problem characterizing the physics at play and for the evaluation of the functionals of the domain involved in the optimization problem. 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 through the optimization process is realized by first updating the design variables according to these sensitivities and then reconstructing the diagram thanks to efficient algorithms from computational geometry. Our shape and topology optimization strategy is versatile: it can be applied to a whole gammut of physical situations (such as thermal and structural mechanics) and optimal design settings (including single- and multi-phase problems). It is Lagrangian and body-fitted by essence, and it thereby benefits from all the assets of an explicit, meshed representation of the shape at each stage of the process. Yet, it naturally handles dramatic motions of the optimized shape, including topological changes, in a very robust fashion. These features, among others, are illustrated by a series of 2d numerical examples.