Adjoint-based airfoil optimization with adaptive isogeometric discontinuous Galerkin method

In this work, an adjoint-based airfoil shape optimization algorithm is developed based on the adaptive isogeometric discontinuous Galerkin method for compressible Euler equations to investigate the significance of each design variable of airfoil B-spline parameterization. We first parameterize the airfoil by B-spline curve approximation with some control points viewed as design variables, and build the B-spline representation of the flow field with the curve to apply the goal-oriented h-adaptive isogeometric DG method for flow solution. Then we compute and employ the discrete adjoint solutions for both multi-target error estimation in adaptive mesh refinement. With the isogeometric nature, not only all the geometrical cells but also the numerical basis functions can be analytically expressed by the design variables, indicating that the numerical solutions and objective could be differentiable with respect to those variables. Consequently, the gradient is totally computed in an accurate approach, and the sensitivity analysis is thus improved, by reducing the spatial discretization error and introducing the analytical expression of derivative, to reveal the key parameters for optimization in an intuitive and efficient manner. Although the SQP optimization algorithm is adopted in the paper, the given accurate gradient can be applied to any gradient-based optimization algorithm. The proposed algorithm is demonstrated on RAE2822 airfoil with inviscid transonic flow, where the shape is optimized to minimize the drag coefficient at a constrained lift and airfoil area. The numerical results show that the drag is much more sensitive to the design variables near the tailing edge at the beginning but sensitivity is reduced when optimal.