Adjoint-Based Aerodynamic Shape and Mesh Optimization with High-order Discontinuous Galerkin Methods

Abstract

The aerodynamic shape of an aircraft plays a pivotal role in determining its overall performance. Aerodynamic Shape Optimization (ASO) involves systematically modifying the geometry to optimize the aerodynamic objective functions such as drag and lift, and improve the aircraft's aerodynamic efficiency. ASO combines numerical optimization algorithms with Computational Fluid Dynamics (CFD) simulations, leveraging the latter to evaluate the objective functions. Gradient-based numerical optimization techniques are extensively utilized in ASO due to their computational efficiency and scalability. The efficiency of gradient-based methods comes from using an adjoint solution for computing gradients of the objective functions with respect to the shape parameters. Using adjoint solution makes gradient computation independent of the number of shape parameters. Since the CFD simulations, using which the objective functions are evaluated, contain discretization errors, they can affect the reliability of the optimization process. High-order numerical methods, such as the Discontinuous Galerkin (DG) method, offer enhanced accuracy for solving compressible flow problems while maintaining computational cost comparable to traditional Finite Volume methods, making them well-suited for ASO applications.

Towards this, the present work performs ASO using high-order DG methods. The shape is defined using smooth splines, and the Free Form Deformation (FFD) method controls shape changes. With changes in the geometry, the mesh needs to move to be consistent with the modified shape. We use a mesh deformation strategy to ensure the mesh evolves smoothly with changes in the geometry. A gradient-based method employing the Sequential Quadratic Programming (SQP) algorithm is used for optimization. The discrete adjoint solution is used to compute the gradients of the objective function and passes them to the optimization algorithm. Optimization for 2D drag minimization problems, including the benchmark Aerodynamic Design Optimization Discussion Group (ADODG) test case 1 and inverse design problems, is performed. A systematic study was conducted to examine the influence of mesh resolution on the resulting optimal geometry, highlighting its dependency on the mesh used during the optimization process. The findings underscore the need for an approach to determine appropriate mesh resolutions dynamically within the optimization cycles. Adjoint-based mesh adaptation strategy presents a suitable approach as the meshes can be adapted to minimize errors in the same objective function. The adjoint solution, already computed for gradient evaluations, can be leveraged for this purpose. An important contribution of the present work is developing and implementing a strategy to integrate adjoint-based mesh adaptation within the ASO framework. The strategy ensures that, within the shape optimization cycles, optimum meshes for given error tolerances are used. The strategy was found to improve the optimization process's reliability and computational efficiency.