Discrete adjoint aerodynamic shape optimization using symbolic analysis with OpenFEMflow

Abstract

The combination of gradient-based optimization with the adjoint method for sensitivity analysis is a very powerful and popular approach for aerodynamic shape optimization. However, differentiating CFD codes is a time consuming and sometimes a challenging task. Although there are a few open-source adjoint CFD codes available, due to the complexity of the code, they might not be very suitable to be used for educational purposes. An adjoint CFD code is developed to support students for learning adjoint aerodynamic shape optimization as well as developing differentiated CFD codes. To achieve this goal, we used symbolic analysis to develop a discrete adjoint CFD code. The least-squares finite element method is used to solve the compressible Euler equations around airfoils in the transonic regime. The symbolic analysis method is used for exact integration to generate the element stiffness and force matrices. The symbolic analysis is also used to compute the exact derivatives of the residuals with respect to both design variables (e.g., the airfoil geometry) and the state variables (e.g., the flow velocity). Besides, the symbolic analysis allows us to compute the exact Jacobian of the governing equations in a computationally efficient way, which is used for Newton iteration. The code includes a build-in gradient-based optimization algorithm and is released as open-source to be available freely for educational purposes.