Reduction of the Adjoint Gradient Formula in the Continuous Limit

We present a new continuous adjoint method for aerodynamic shape optimization using the Euler equations, which reduces the computational cost of the gradients by reducing the volume integral part of the adjoint gradient formula to a surface integral. The savings are particularly significant for three-dimensional aerodynamic shape optimization problems on general unstructured and overset meshes. In order to validate the concept, the new gradient equations have been tested for various aerodynamic shape optimization problems, including an inverse problem for threedimensional wing configurations, and drag minimization problems of a single-element airfoil and a three-dimensional wing-fuselage configuration. In order to assess their accuracy, the results are compared with finite-difference gradients, complex-step gradients, and gradients calculated by the previous adjoint method which includes a volume integral. ∗Thomas V. Jones Professor of Engineering, Department of Aeronautics and Astronautics, Stanford University, AIAA Fellow †Postdoctoral Fellow, Department of Aeronautics and Astronautics, Stanford University, AIAA Member