Adjoint variable-based shape optimization with bounding surface constraints

This article presents an algorithm for constraining shape deformations in adjoint-based aerodynamic shape optimization. The algorithm considers known bounding surfaces and constrains the shape undergoing optimization not to intersect with them. For each and every node on the shape, its signed distance to the bounding surfaces is computed. The signed distance function returns a positive value, in case a node lies outside the bounds, or a negative one otherwise. It can, therefore, serve as an inequality constraint, the satisfaction of which ensures that this node does not violate the no-intersection criteria. To increase the efficiency and make this imposition of constraints usable with any parameterization scheme, the resulting inequality constraints are transformed to a single-equality constraint by means of a penalty function which is, then, summed over the shape to be optimized. The shape parameterization used in this article is spline-based but this is not restrictive and any other parameterization type (node-based, RBF, etc) can be used instead. The gradient of the objective function of the aerodynamic problem with respect to the design variables defined by the parameterization is computed using the continuous adjoint method. The method is demonstrated in the optimization of a 2D manifold case and two industrial-like cases including the shape optimization of a U-bend cooling duct, constrained to stay within a box, and the side mirror of a passenger car constrained to retain an almost undeformed reflector glass.