Gradient-Limiting Shape Optimisation Applied to AIAA ADODG Test Cases

Abstract

Without addressing shape smoothness, gradient-based optimisation methods naturally amplify high-frequency shape components which can lead to poor convergence of the optimisation problem and a convergence rate exhibiting dependency on the fidelity of shape-control. Recent work by the authors demonstrated that this problem arises due to the discrete shape problem being ill-posed by naturally including geometries that are invalid both in physicality (shape) and discretisation (mesh), and a new shape control methodology has been developed which addresses this explicitly. The new approach uses two-dimensional control to recover shaperelevant displacements and surface gradient constraints to ensure smooth and valid iterates. The shape gradient constraints, approximating a C 2 continuity condition, are derived for two local shape control methods: discrete grid point control and cubic B-Splines. The local control methods provide high-fidelity control whereas the surface constraints exclude non-physical shapes from the design space making the shape problem well-posed. As a result, high-fidelity shape optimisation is possible at a reasonable computational cost. In this paper, further results are presented for the aerodynamic optimisation of two standard test cases defined by the AIAA aerodynamic design optimisation discussion group. A value of 7 drag counts is achieved on the inviscid case and 66 drag counts on the viscous case; to the authors' knowledge these are the lowest results published for either case.

Generic Modal Design Variables for Efficient Aerodynamic Optimization

“…Recent work by the authors [7,8] addresses both shape smoothness and design space conditioning explicitly and demonstrates that a sufficiently constrained and well-conditioned design space is all that is required for a well-posed shape optimisation problem without limitation on the attainable design fidelity. In the previous work by the authors, shape gradient constraints approximating a C2 continuity condition for discrete mesh points and cubic B-Spline curves were derived, and demonstrated on a standard test case.…”

“…A gradient-limiting methodology has been developed by the authors [7,8] to control smoothness during shape optimisation; this is done by applying a linear constraint to the design variables which bounds the third derivative as approximated by undivided differences or spline derivatives. Given a shape control basis 𝝓, derived from a local shape control method such as grid-point control or cubic B-Spine, the discretised geometry x for analysis is given by:…”

“…Smaller values of 𝜇 correspond to enforcing higher shape smoothness with a smaller overall design space, and vice versa for larger values. Previous work [1,7,8] has shown this approximate 𝐶 2 condition ensures surface smoothness and mesh validity during shape optimisation which reduce the effective degrees of freedom in a geometrically meaningful manner and thereby resulting in better optimisation performance.…”

“…Previous work by the authors [1,7,8] has demonstrated that the problem arises due to the discrete shape problem being ill-posed since it naturally includes geometries that are invalid both in physicality (shape) and discretisation (mesh).…”