Inverse design using adjoint approach

Kurzfassung

With the increasing of computational power, the CFD has become a routine tool used for aerodynamic analysis and provides reasonably accurate results. However, the ultimate goal in the design process is to find the optimum shape that maximizes the aerodynamic performance index, which might be the drag coefficient at a fixed lift, the lift-to-drag ratio, or matching the target pressure distribution, with reasonable accuracy and amount of time. The objective of current project is focused on the application of optimization technique based on control theory for airfoil inverse design that was performed using the unstructured Euler and RANS solver developed by DLR (DLR-Tau code). The final goal is to provide an optimization toolbox that is suitable for aerodynamic inverse design, especially for natural laminar flow wing design, for industrial applications. In optimum shape design problems, the true design space is a free surface which has infinite number of design variables and will require N+ 1 flow evaluations for N design variables to calculate the required gradients necessary for gradient-based optimization techniques. To minimize the expensive time required for each flow simulation, we apply the adj oint approach first proposed by Jameson and find the Frechet derivative of the cost function respect to the shape by solving an adjoint equation problem. The total cost, which is independent of number of design parameters, is one flow plus one adjoint solution and this makes this technique very attractive for optimum shape design. The flow solver used in this study is the DLR-Tau code, which solves the Euler and RANS equations on unstructured meshes with multistep time stepping scheme. Rapid convergence to a steady state is achieved via local time stepping, residual averaging, and multi-grid scheme. In contrast to the continuous adjoint approach, the discrete adjoint formulation approach was used to calculate the gradient information of the design variables. In this study, the cost function is defined as the least square of the difference to a given target pressure and this corresponds to an inverse design problem. The shape of airfoil, which is represented by the Class-Shape function Transformation (CST) parameterization, is then modified to match the desired target pressure, Pd. To evaluate the effectiveness of different gradient-based optimization algorithms, three optimization techniques, Steepest descent, Conjugate gradient, and BFGS were tested and the results were presented.