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44
NewtonKrylov Algorithm for Aerodynamic Design Using the NavierStokes Equations
 AIAA JOURNAL
, 2002
"... A Newton–Krylov algorithm is presented for twodimensional Navier–Stokes aerodynamic shape optimization problems. The algorithm is applied to both the discreteadjoint and the discrete flowsensitivity methods for calculating the gradient of the objective function. The adjoint and flowsensitivity e ..."
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Cited by 35 (24 self)
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A Newton–Krylov algorithm is presented for twodimensional Navier–Stokes aerodynamic shape optimization problems. The algorithm is applied to both the discreteadjoint and the discrete flowsensitivity methods for calculating the gradient of the objective function. The adjoint and flowsensitivity equations are solved using a novel preconditioned generalized minimum residual (GMRES) strategy. Together with a complete linearization of the discretized Navier–Stokes and turbulence model equations, this results in an accurate and efficient evaluation of the gradient. Furthermore, fast flow solutions are obtained using the same preconditioned GMRES strategy in conjunction with an inexact Newton approach. The performance of the new algorithm is demonstrated for several design examples, including inverse design, liftconstrained drag minimization,lift enhancement, and maximization of lifttodrag ratio. In all examples, the norm of the gradient is reduced by several orders of magnitude, indicating that a local minimum has been obtained. By the use of the adjoint method, the gradient is obtained in from onefifth to onehalf of the time required to converge a flow solution.
Algorithm Developments for Discrete Adjoint Methods
, 2001
"... This paper presents a number of algorithm developments for adjoint methods using the `discrete' approach in which the discretisation of the nonlinear equations is linearised and the resulting matrix is then transposed ..."
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Cited by 25 (7 self)
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This paper presents a number of algorithm developments for adjoint methods using the `discrete' approach in which the discretisation of the nonlinear equations is linearised and the resulting matrix is then transposed
Aerodynamic Design on Unstructured Grids for Turbulent Flows
 NASA TM
, 1997
"... An aerodynamic design algorithm for turbulent flows using unstructured grids is described. The current approach uses adjoint (costate) variables to obtain derivatives of the cost function. The solution of the adjoint equations is obtained by using an implicit formulation in which the turbulence mo ..."
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Cited by 16 (4 self)
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An aerodynamic design algorithm for turbulent flows using unstructured grids is described. The current approach uses adjoint (costate) variables to obtain derivatives of the cost function. The solution of the adjoint equations is obtained by using an implicit formulation in which the turbulence model is fully coupled with the flow equations when solving for the costate variables. The accuracy of the derivatives is demonstrated by comparison with finitedifference gradients and a few sample computations are shown. In addition, a user interface is described that significantly reduces the time required to set up the design problems. Recommendations on directions of further research into the NavierStokes design process are made.
PseudoTime Method for Optimal Shape Design Using the Euler Equations
, 1995
"... In this paper we exploit a novel idea for the optimization of flows governed by the Euler equations. The algorithm consists of marching on the design hypersurface while improving the distance to the state and costate hypersurfaces. We consider the problem of matching the pressure distribution to ade ..."
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Cited by 15 (1 self)
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In this paper we exploit a novel idea for the optimization of flows governed by the Euler equations. The algorithm consists of marching on the design hypersurface while improving the distance to the state and costate hypersurfaces. We consider the problem of matching the pressure distribution to adesired one, subject to the Euler equations, both for subsonic and supersonic flows. The rate of convergence to the minimum for the cases considered is 3 to 4 times slower than that of the analysis problem. Results are given for Ringleb flow and a shockless recompression case.
Aerodynamic Design Sensitivities on an Unstructured Mesh Using the NavierStokes Equations and a Discrete Adjoint Formulation
, 1998
"... A discrete adjoint method is developed and demonstrated for aerodynamic design optimization on unstructured grids. The governing equations are the threedimensional Reynoldsaveraged NavierStokes equations coupled with a oneequation turbulence model. A discussion of the numerical implementation of ..."
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Cited by 10 (3 self)
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A discrete adjoint method is developed and demonstrated for aerodynamic design optimization on unstructured grids. The governing equations are the threedimensional Reynoldsaveraged NavierStokes equations coupled with a oneequation turbulence model. A discussion of the numerical implementation of the flow and adjoint equations is presented. Both compressible and incompressible solvers are differentiated, and the accuracy of the sensitivity derivatives is verified by comparing with gradients obtained using finite differences and a complexvariable approach. Several simplifying approximations to the complete linearization of the residual are also presented. A firstorder approximation to the dependent variables is implemented in the adjoint and design equations, and the effect of a “frozen ” eddy viscosity and neglecting mesh sensitivity terms is also examined. The resulting derivatives from these approximations are all shown to be inaccurate and often of incorrect sign. However, a partiallyconverged adjoint solution is shown to be sufficient for computing accurate sensitivity derivatives, yielding a potentially large cost savings in the design process. The convergence rate of the adjoint solver is compared to that of the flow solver. For inviscid adjoint solutions, the cost is roughly one to four times that of a flow solution, whereas for turbulent computations, this ratio can reach as high as ten. Sample optimizations are
Optimum Transonic Airfoils based on the Euler Equations
 ICASE Rep
, 1996
"... We solve theproblem of determining airfoils that approximate, in a least square sense, given surface pressure distributions in transonic ight regimes. The owis modeled by means of the Euler equations and thesolution procedure is an adjointbased minimization algorithm that makesuseoftheinverse Theodo ..."
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Cited by 7 (1 self)
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We solve theproblem of determining airfoils that approximate, in a least square sense, given surface pressure distributions in transonic ight regimes. The owis modeled by means of the Euler equations and thesolution procedure is an adjointbased minimization algorithm that makesuseoftheinverse Theodorsen transform in order to parameterize the airfoil. Fast convergence to theoptimal solution is obtained by means of the pseudotime method. Results are obtained using three di erent pressure distributions for several free stream conditions. The airfoils obtained have given a trailing edge angle. 1
A Preconditioning Method for Shape Optimization Governed by the Euler Equations
, 1998
"... We consider a classical aerodynamic shape optimization problem subject to the compressible Euler flow equations. The gradient of the cost functional with respect to the shape variables is derived with the adjoint method at the continuous level. The Hessian (second order derivative of the cost func ..."
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Cited by 7 (4 self)
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We consider a classical aerodynamic shape optimization problem subject to the compressible Euler flow equations. The gradient of the cost functional with respect to the shape variables is derived with the adjoint method at the continuous level. The Hessian (second order derivative of the cost functional with respect to the shape variables) is approximated also at the continuous level, as first introduced by Arian and Ta’asan (1996). The approximation of the Hessian is used to approximate the Newton step which is essential to accelerate the numerical solution of the optimization problem. The design space is discretized in the maximum dimension, i.e., the location of each point on the intersection of the computational mesh with the airfoil is taken to be an independent design variable. We give numerical examples for 86 design variables in two different flow speeds and achieve an order of magnitude reduction in the cost functional at a computational effort of a full solution of the analysis partial differential equation (PDE).