Aerodynamic Shape Optimization Based on Free-Form Deformation

This paper presents a free-form deformation technique suitable for aerodynamic shape optimization. Because the proposed technique is independent of grid topology, we can treat structured and unstructured computational fluid dynamics grids in the same manner. The proposed technique is an alternative shape parameterization technique to a trivariate volume technique. It retains the flexibility and freedom of trivariate volumes for CFD shape optimization, but it uses a bivariate surface representation. This reduces the number of design variables by an order of magnitude, and it provides much better control for surface shape changes. The proposed technique is simple, compact, and efficient. The analytical sensitivity derivatives are independent of the design variables and are easily computed for use in a gradient-based optimization. The paper includes the complete formulation and aerodynamics shape optimization results. Nomenclature B = B-spline basis functions C = product of B-spline basis functions I = maximum number of points in u-direction J = maximum number of points in v-direction M = number of surface points V = control points W = weights b = baseline grid m,n = grid point number r = coordinates (x, y, z) v = design variable vector ξ,η = surface bivariate coordinates I.