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28
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 32 (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 22 (6 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
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 14 (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 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 11 (3 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.
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
 Institute for Computer
, 1996
"... . 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 5 (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 di#erent flow speeds and achieve an order of magnitude reduction in the cost functional at a computational e#ort of a full solution of the analys...
Mesh Movement for a DiscreteAdjoint NewtonKrylov Algorithm for Aerodynamic Optimization
"... A grid movement algorithm based on the linear elasticity method with multiple increments is presented. The method is computationally expensive, but is exceptionally robust, producing high quality elements even for large shape changes. It is integrated with an aerodynamic shape optimization algorithm ..."
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Cited by 3 (2 self)
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A grid movement algorithm based on the linear elasticity method with multiple increments is presented. The method is computationally expensive, but is exceptionally robust, producing high quality elements even for large shape changes. It is integrated with an aerodynamic shape optimization algorithm that uses an augmented adjoint method for gradient calculation. The discrete adjoint equations are augmented to explicitly include the sensitivities of the mesh movement, resulting in an increase in efficiency and numerical accuracy. This gradient computation method requires less computational time than a function evaluation, and leads to significant computational savings as dimensionality is increased. The results from application of these techniques to several large deformation and optimization cases are presented. I.