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 proposes 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 filtered using the level set method. The interface is fit with non-uniform rational B-spline (NURBS) curves through application of sensitivities garnered 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.

In fluid mechanics, two of the most widely used optimization approaches are shape and topology optimization, which are generally treated as mutually-exclusive. This paper presents a general continuous adjoint formulation for both shape and topology optimization, focusing on (a) crucial aspects of computing accurate shape sensitivity derivatives such as the differentiation of the turbulence model PDEs and the proper treatment of grid sensitivities and (b) a synergistic, sequential application of topology and shape optimization, in which topology is used to define a preliminary solution from which shape optimization can be initiated. To achieve this, a transition process is used to accurately represent and parameterize the topological solution with NURBS surfaces. The flow cases presented in this work and the in-house code pertaining to the optimization and transition process are implemented within OpenFOAM. Results from purely aerodynamic as well as multidisciplinary applications including Conjugate Heat Transfer (CHT) are presented.

Abstract. The processes of topology and shape optimization are well known methods in the field of fluid mechanics. Although successful in their own rights, it is conceivable that the two methods will find choicest solutions in tandem: i.e. if shape optimization were able to improve a topological solution. Conjoining the two methods in this manner is not straightforward, however, since there is no existing process to connect one to the other. Toward this goal, a novel transitional process is proposed to process level set topology solutions obtained using the continuous adjoint method such that a shape optimization loop using the continuous adjoint can be initialized, run and ultimately produce a refined, parameterized solution. First, the topology optimization process is enhanced using the level set method to both maintain an explicit description of the interface between the solid and fluid topological domains and prevent the formation of fluid or solid islands which would not be viable for manufacturing. The interface is then fitted with Non-Uniform Rational B-Splines (NURBS) through application of sensitivities garnered from the solution of an auxiliary optimization problem which aims at reducing the difference between the signed distance fields generated about each NURBS curve and its corresponding interface section. A body-fitted mesh is generated for the geometry defined by the fitted NURBS, allowing a shape optimization loop to be initiated. The parameterized result of the topology to shape transition process will be compared to that of shape optimization in two 2D cases with internal, incompressible fluid flows.

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