Three-dimensional aerodynamic shape optimization with high-order direct discontinuous Galerkin schemes

In this work, a three-dimensional aerodynamic shape optimization (ASO) framework is established based on the high-order (direct) discontinuous Galerkin (DG/DDG) discretization, which serves as a flow solver for solving compressible Euler/Navier–Stokes equations. The design variables are introduced to represent the shapes via two parameterization approaches, including Hicks–Henne and Free-Form deformation methods. At each loop, the radial basis function mesh deformation technique is employed to redistribute the meshes. Two typical gradient-based optimization methods are employed to update the shapes. One is the Sequential Quadratic Programming method, whose gradients are computed based on the discrete adjoint-based method using finite difference approximation via perturbating the design variables individually. The other is the steepest decent approach, where the gradients are provided by the Simultaneous Perturbation Stochastic Approximation method using finite difference approximation via perturbating the design variables simultaneously and stochastically. These modules work with the DG/DDG flow solver to search for improved shapes in ASO framework. Several airfoil drag minimization experiments involving 2D/3D inviscid/viscous flow are presented to demonstrate the performance of high-order DG/DDG flow solver in ASO, where the drag coefficients can be reduced significantly with the constraints well preserved.