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150
Grid adaptation for functional outputs: application to twodimensional inviscid flows
 J. Comput. Phys
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Adjoint Recovery of Superconvergent Functionals from Approximate Solutions of Partial Differential Equations
, 1998
"... Abstract. Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that ..."
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Cited by 85 (12 self)
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Abstract. Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that uses an adjoint PDE to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest. Numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy. Numerical results are presented for the Poisson equation in one and two dimensions and for the nonlinear quasionedimensional Euler equations. The theory is equally applicable to nonlinear equations in complex multidimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations. Key words. PDEs, adjoint equations, error analysis, superconvergence AMS subject classifications. 65G99, 76N15 PII. S0036144598349423
A Perspective on Computational Algorithms for Aerodynamic Analysis and Design
 Progress in Aerospace Sciences
, 2001
"... This paper exam nes the use of computational fluid dynamics as a tool for aircraft design. It addresses the requirements for effective industrial use, and tradeoffs between modeling accuracy and computational costs. Essential elements of algorithm design are discussed in detail, together with a uni ..."
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Cited by 58 (19 self)
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This paper exam nes the use of computational fluid dynamics as a tool for aircraft design. It addresses the requirements for effective industrial use, and tradeoffs between modeling accuracy and computational costs. Essential elements of algorithm design are discussed in detail, together with a unified approach to the design of shock capturing schemes. Finally, the paper discusses the use of techniques drawn from control theory to determine optimal aerodynamic shapes. In the future multidisciplinary analysis and optimization should be combined to take account of the tradeoffs in the overall performance of the complete system
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 53 (13 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
Adjoint equations in CFD: duality, boundary conditions and solution behaviour
, 1997
"... The first half of this paper derives the adjoint equations for inviscid and viscous compressible flow, with the emphasis being on the correct formulation of the adjoint boundary conditions and restrictions on the permissible choice of operators in the linearised functional. It is also shown that the ..."
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Cited by 47 (13 self)
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The first half of this paper derives the adjoint equations for inviscid and viscous compressible flow, with the emphasis being on the correct formulation of the adjoint boundary conditions and restrictions on the permissible choice of operators in the linearised functional. It is also shown that the boundary conditions for the adjoint problem can be simplified through the use of a linearised perturbation to generalised coordinates. The second half of the paper constructs the Green's functions for the quasi1D and 2D Euler equations. These are used to show that the adjoint variables have a logarithmic singularity at the sonic line in the quasi1D case, and a weak inverse squareroot singularity at the upstream stagnation streamline in the 2D case, but are continuous at shocks in both cases. 1 Introduction The last few years have seen considerable progress in the use of adjoint equations in CFD for optimal design [19]. In all of the methods, the heart of the algorithm is an optimisati...
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 39 (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.
Aerodynamic Shape Optimization Techniques Based On Control Theory
 CONTROL THEORY, CIME (INTERNATIONAL MATHEMATICAL SUMMER
, 1998
"... This paper review the formulation and application of optimization techniques based on control theory for aerodynamic shape design in both inviscid and viscous compressible flow . The theory is applied to a system defined by the partial differential equations of the flow, with the boundary shape acti ..."
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Cited by 35 (25 self)
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This paper review the formulation and application of optimization techniques based on control theory for aerodynamic shape design in both inviscid and viscous compressible flow . The theory is applied to a system defined by the partial differential equations of the flow, with the boundary shape acting as the control. The Frechet derivative of the cost function is determined via the solution of an adjoint partial differential equation, and the boundary shape is then modified in a direction of descent. This process is repeated until an optimum solution is approached. Each design cycle requires the numerical solution of both the flow and the adjoint equations, leading to a computational cost roughly equal to the cost of two flow solutions. Representative results are presented for viscous optimization of transonic wingbody combinations and inviscid optimization of complex configurations.
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 35 (16 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.
Improved lift and drag estimates using adjoint Euler equations
 AIAA Paper
, 1999
"... This paper demonstrates the use of adjoint error analysis to improve the order of accuracy of integral functionals obtained from CFD calculations. Using second order accurate finite element solutions of the Poisson equation, fourth order accuracy is achieved for two different categories of functiona ..."
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Cited by 33 (8 self)
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This paper demonstrates the use of adjoint error analysis to improve the order of accuracy of integral functionals obtained from CFD calculations. Using second order accurate finite element solutions of the Poisson equation, fourth order accuracy is achieved for two different categories of functional in the presence of both curved boundaries and singularities. Similarly, numerical results for the Euler equations obtained using standard second order accurate approximations demonstrate fourth order accuracy for the integrated pressure in two quasi1D test cases, and a significant improvement in accuracy in a twodimensional case. This additional accuracy is achieved at the cost of an adjoint calculation similar to those performed for design optimization. 1 Introduction In aeronautical CFD, engineers desire very accurate prediction of the lift and drag on aircraft, but they are less concerned with the precise details of the flow field in general, although there is a clear need to underst...