Aerodynamic Shape Optimization Design Based On Continuous Adjoint Equation And Free-form Deformation

Based on unstructured grids,a stable and efficient dynamic shape optimization method is developed by using Euler equation and combining continuous adjoint method,free-form deformation strategy and sequential quadratic programming optimization algorithm.Firstly,the method of calculating flow is introduced in detail,including the Euler governing equations,spatial and temporal discretization,boundary conditions and etc.Roe scheme is adopted for the convection flux calculation,while the left and right states are obtained by piecewise linear reconstruction.An implicit LU-SGS algorithm is employed as well for the sake of better efficiency.The methods mentioned above are verified by the simulation of flows over the ONERA M6 wing.Then the derivation of continuous adjoint equation is studied.According to control theory,the adjoint equation and boundary conditions can be obtained.Afterwards,the calculation methods of adjoint equation are studied,including spatial and temporal discretization.The convective fluxes of adjoint equation is evaluated using Roe scheme,and the piecewise linear reconstruction method is also adopted here,while temporal discretization is implemented by LU-SGS implicit algorithm as well.Besides,aerodynamic shape parametric methods and dynamic grid methods are introduced.Among all the methods,FFD parametric method and spring dynamic grid approach are expounded in detail,and the ONERA M6 deformation is showed to verify the reliability of the methods.Finally,the Sequential quadratic programming optimization algorithm is introduced.The developed approach is used to optimize transonic wings,transonic flying wing configuration and transonic airliner to minimize the drag coefficient.The optimization results show that the drag coefficients of optimized wings are decreased by 21.81%,15.40%,14.11% and 10.30%,respectively,while the lift coefficient and volume that used as constraint conditions remain the same generally.It’s proved that the developed approach has obvious effect on the transonic aerodynamic shape optimization,while the continuity and smoothness of geometry surface are ensured.Besides,the approach is of high efficiency.