Gradient-Limiting Shape Control for Efficient Aerodynamic Optimisation

Abstract

Local shape control methods, such as B-Spline surfaces, are well-conditioned such that they allow high-fidelity design optimisation, however this comes at the cost of degraded optimisation convergence rate as control fidelity is refined due to the resulting exponential increase in the size of the design space. Moreover, optimisations in higher-fidelity design spaces become illposed due to high frequency shape components being insufficiently bounded; this can lead to non-smooth and oscillatory geometries that are invalid both in physicality (shape) and discretisation (mesh). This issue is addressed here by developing a geometrically-meaningful constraint to reduce the effective degrees of freedom and improve the design space thereby improving optimisation convergence rate and final result. A new approach to shape control is presented using coordinate control (x, z) to recover shape-relevant displacements and surface gradient constraints to ensure smooth and valid iterates. The new formulation transforms constraints directly onto design variables, and these bound the out-of-plane variations to ensure smooth shapes as well as the in-plane variations for mesh validity. Shape gradient constraints approximating a C 2 continuity condition are derived and demonstrated on a challenging test case: inviscid transonic drag minimisation of a symmetric NACA0012 aerofoil. Significantly the regularised shape problem is shown to have an optimisation convergence rate independent of both shape control fidelity and numerical mesh resolution, while still making use of increased control fidelity to achieve improved results. Consequently, a value of 1.6 drag counts is achieved on the test case, the lowest value achieved by any method.

Gradient-Limiting Shape Optimisation Applied to AIAA ADODG Test Cases

“…The objective of the gradient-limiting approach is to generalise the regularisation effects of the methods discussed above and remove dependence of the optimisation procedure on the shape control method. In [4] shape gradient constraints are derived and it is demonstrated that a sufficiently constrained problem space is all that is required for a well-posed and consistent shape optimisation problem without limitation on the numerical method used for solution. In the remainder of this paper, motivation for the gradient-limiting approach is first presented followed by the mathematical formulation and details on numerical implementation.…”

“…Recent work by the authors [4] has developed a new approach to shape control which, like existing methods, regularises the shape problem to maintain surface smoothness but without limitation on the resolution of shape control. The new approach uses two-dimensional control (x,z) to recovery shape-relevant displacements and surface gradient constraints to ensure smooth and valid iterates.…”

“…Several benchmark optimisation cases are laid out by the AIAA Aerodynamic Design Optimisation Discussion Group (ADODG) to aid in the collaborative evaluation and comparison among shape optimisation practitioners. In this paper, an extension to work previously presented [4], two such test cases are considered here.…”

Structural Topology Optimisation with R-Snakes Volume of Solid

“…Deformative methods specify changes away from a known geometry, a non-exhaustive list includes: Hicks-Henne bump functions [38], Free Form Deformation (FFD) methods [39], data base singular value decomposition (SVD) [40] and subdivision curves [41]. Constrained smooth mesh-point movement can also be used to control the evolution of the geometry [42]. Many of these parameterisation are directly compared inside a single ASO framework, in geometric and aerodynamic terms, in [43] and [44] respectively.…”

Objectives and Constraints for Transonic Wing Optimization

“…[1][2][3][4][5] The authors have presented work in this area, with developments such as the domain element method, 6 SVD modes, 7 subdivision surfaces 8 and curvature constraints. 9 Optimizing for minimum drag is a commonly studied problem. For transonic flow, a substantial source of drag is due to the shock; this causes wave drag and also affects the boundary layer.…”