Abstract. In this paper a new shape optimization algorithm is presented. As a model application we consider state problems related to fluid mechanics, namely the Navier-Stokes equations for viscous incompressible fluids. The general approach to the problem is described. Next, transformations to classical optimal control problems are presented. Then, the dynamic programming approach is used and sufficient conditions for the shape optimization problem are given. A new numerical method to find the approximate value function is developed.
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.
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