Transition from Adjoint Level Set Topology to Shape Optimization for 2D Fluid Mechanics

Most optimization problems in the field of fluid mechanics can be classified as either topology or shape optimization. Although topology and shape have been considered mutually exclusive optimization methods since their inception, it is conceivable that they will find choicest solutions in tandem, with shape optimization refining a solution found by topology. However, linking the topology optimization problem to that of shape is not trivial and, to the authors' knowledge, has yet to be formally attempted. This paper pro-
poses a novel transitional procedure that post-processes 2D adjoint topology solutions,fitting the interface between the solid and 
fluid topological domains to create a parameterized solution which can be used as either a CAD-compatible representation of the interface or a source for grid generation from which a shape optimization loop can be initialized. The interface to be fit can be extracted from any topological field with distinct fluid and solid domains, meaning that the proposed transition process is independent of the topology approach utilized. To conveniently describe the interface between the solid and fluid topological
domains, the topology optimization process employed in this paper is ltered using the level set method. The interface is fit with non-uniform rational B-spline (NURBS) curves through application of sensitivitiesgarnered from the solution of an auxiliary inverse design problem which aims at reducing the difference between signed-distance fields generated about both the NURBS curve being optimized and the section of interface being fit. The geometry defined by the fit NURBS curves is then (optionally) used to build a
boundary-fitted grid on which a shape optimization loop is performed. The parameterized result of the topology to shape transition process is compared to that of shape optimization in 2D cases with internal, incompressible fluid flows.