In this paper we describe a novel framework for finding numerical solutions to a wide range of shape optimization problems. It is based on classical dynamic programming approach augmented with discretization of the space of trajectories and controls. This allows for straightforward algorithmic implementation. This method has been used to solve a well known problem called the ”dividing tube problem”, a state problem related to fluid mechanics, that requires simultaneous topology and shape optimization in case of elastic contact problems and involves solving the Navier-Stokes equations for viscous incompressible fluids.