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20
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.
Multipoint and Multiobjective Aerodynamic
 Shape Optimization,” AIAA Journal
"... A gradientbased Newton–Krylov algorithm is presented for the aerodynamic shape optimization of single and multielement airfoil configurations. The flow is governed by the compressible Navier–Stokes equations in conjunction with a oneequation transport turbulence model. The preconditioned general ..."
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Cited by 22 (15 self)
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A gradientbased Newton–Krylov algorithm is presented for the aerodynamic shape optimization of single and multielement airfoil configurations. The flow is governed by the compressible Navier–Stokes equations in conjunction with a oneequation transport turbulence model. The preconditioned generalized minimal residual method is applied to solve the discreteadjoint equation, which leads to a fast computation of accurate objective function gradients. Optimization constraints are enforced through a penalty formulation, and the resulting unconstrained problem is solved via a quasiNewton method. The new algorithm is evaluated for several design examples, including the lift enhancement of a takeoff configuration and a liftconstrained drag minimization at multiple transonic operating points. Furthermore, the new algorithm is used to compute a Pareto front based on competing objectives, and the results are validated using a genetic algorithm. Overall, the new algorithm provides an efficient approach for addressing the issues of complex aerodynamic design.
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
Recent Improvements in Aerodynamic Design Optimization On Unstructured Meshes
 AIAA JOURNAL
, 2002
"... Recent improvements in an unstructuredgrid method for largescale aerodynamic design are presented. Previous work had shown such computations to be prohibitively long in a sequential processing environment. Also, robust adjoint solutions and mesh movement procedures were difficult to realize, par ..."
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Cited by 18 (1 self)
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Recent improvements in an unstructuredgrid method for largescale aerodynamic design are presented. Previous work had shown such computations to be prohibitively long in a sequential processing environment. Also, robust adjoint solutions and mesh movement procedures were difficult to realize, particularly for viscous flows. To overcome these limiting factors, a set of design codes based on a discrete adjoint method is extended to a multiprocessor environment using a shared memory approach. A nearly linear speedup is demonstrated, and the consistency of the linearizations is shown to remain valid. The full linearization of the residual is used to precondition the adjoint system, and a significantly improved convergence rate is obtained. A new mesh movement algorithm is implemented and several advantages over an existing technique are presented. Several design cases are shown for turbulent flows in two and three dimensions.
Analytic adjoint solutions for the quasionedimensional Euler equations
 J. Fluid Mechanics
, 2001
"... The analytic properties of adjoint solutions are examined for the quasionedimensional Euler equations. For shocked flow, the derivation of the adjoint problem reveals that the adjoint variables are continuous with zero gradient at the shock, and that an internal adjoint boundary condition is requir ..."
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Cited by 15 (6 self)
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The analytic properties of adjoint solutions are examined for the quasionedimensional Euler equations. For shocked flow, the derivation of the adjoint problem reveals that the adjoint variables are continuous with zero gradient at the shock, and that an internal adjoint boundary condition is required at the shock. A Green’s function approach is used to derive the analytic adjoint solutions corresponding to supersonic, subsonic, isentropic and shocked transonic flows in a converging–diverging duct of arbitrary shape. This analysis reveals a logarithmic singularity at the sonic throat and confirms the expected properties at the shock. 1.
On the use of RungeKutta timemarching and multigrid for the solution of steady adjoint equations
, 2000
"... This paper considers the solution of steady adjoint equations using a class of iterative methods which includes preconditioned RungeKutta timemarching with multigrid. It is shown that, if formulated correctly, equal numbers of iterations of the direct and adjoint iterative solvers will result in t ..."
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Cited by 14 (5 self)
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This paper considers the solution of steady adjoint equations using a class of iterative methods which includes preconditioned RungeKutta timemarching with multigrid. It is shown that, if formulated correctly, equal numbers of iterations of the direct and adjoint iterative solvers will result in the same value for the linear functional being sought. The precise details of the adjoint iteration are formulated for the case of RungeKutta timemarching with partial updates, which is commonly used in CFD computations. The theory is supported by numerical results from a MATLAB program for two model problems, and from programs for the solution of the linear and adjoint 3D NavierStokes equations.
Adjoint Error Correction for Integral Outputs
"... Introduction 1.1 Output functionals Why do engineers perform CFD calculations? In the case of a transport aircraft at cruise conditions, a calculation might be performed to investigate whether there is an adverse pressure gradient near the leading edge of the wing, causing boundary layer separatio ..."
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Cited by 14 (2 self)
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Introduction 1.1 Output functionals Why do engineers perform CFD calculations? In the case of a transport aircraft at cruise conditions, a calculation might be performed to investigate whether there is an adverse pressure gradient near the leading edge of the wing, causing boundary layer separation and premature transition. Alternatively, one might be concerned about wing/pylon/nacelle integration, in which case one might be looking to see if there are any shocks on the pylon, leading to unacceptable integration losses. In both of these examples, qualitative information is being obtained from the computed ow eld to understand and interpret the impact of the phenomena on the quantitative outputs of most concern to the aeronautical engineer, the lift and drag on the aircraft. The quality of the CFD calculation is judged, rst and foremost, by the accuracy of the lift and drag predictions. The details of the ow eld are much less important, and are used in a more qualitative manner t
Adjoint Code Developments Using the Exact Discrete Approach
 AIAA Paper
, 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. With a new iterative procedure for solving the adjoint equaitons, exact numeri ..."
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Cited by 10 (4 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. With a new iterative procedure for solving the adjoint equaitons, exact numerical equivalence is maintained between the linear and adjoint discretisations. The incorporation of strong boundary conditions within the discrete approach is discussed, as well as a new application of adjoint methods to linear unsteady ow in turbomachinery
An Exact Dual Adjoint Solution Method for Turbulent Flows on Unstructured Grids
 Computers & Fluids
, 2003
"... this report is for accurate reporting and does not constitute an official endorsement, either expressed or implied, of such products or manufacturers by the National Aeronautics and Space Administration ..."
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Cited by 9 (4 self)
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this report is for accurate reporting and does not constitute an official endorsement, either expressed or implied, of such products or manufacturers by the National Aeronautics and Space Administration