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174
Optimum aerodynamic design using the NavierStokes equations
 Theoretical and Computational Fluid Dynamics
, 1998
"... The ultimate success of an aircraft design depends on the resolution of complex multidisciplinary tradeo s between factors such as aerodynamic eciency, structural weight, stability and control, and ..."
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Cited by 106 (45 self)
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The ultimate success of an aircraft design depends on the resolution of complex multidisciplinary tradeo s between factors such as aerodynamic eciency, structural weight, stability and control, and
Problem Formulation for Multidisciplinary Optimization
 SIAM Journal on Optimization
, 1993
"... This paper is concerned with the optimization of systems of coupled simulations. In computational engineering, this frequently is called multidisciplinary (design) optimization, or MDO. We present an expository introduction to MDO for optimization researchers. We believe the optimization community h ..."
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Cited by 73 (9 self)
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This paper is concerned with the optimization of systems of coupled simulations. In computational engineering, this frequently is called multidisciplinary (design) optimization, or MDO. We present an expository introduction to MDO for optimization researchers. We believe the optimization community has much to contribute to this important class of computational engineering problems. In addition, this paper presents a new abstraction for multidisciplinary analysis and design problems as well as new decomposition formulations for these problems. Furthermore, the "individual discipline feasible" (IDF) approaches suggested here make use of existing specialized simulation analysis codes, and they introduce significant opportunities for coarsegrained computational parallelism particularly wellsuited to heterogeneous computing environments. The key issue in the three fundamental approaches to MDO formulation discussed here is the kind of feasibility that must be maintained at each optimizati...
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 55 (9 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
Optimum aerodynamic design using CFD and control theory. AIAA paper 951729
 AIAA 12th Computational Fluid Dynamics Conference
, 1995
"... This paper describes the implementation of optimization techniques based on control theory for airfoil and wing design. The theory is applied to a system defined by the partial differential equations of the flow, with control by the boundary as a free surface. The Frechet derivative of the cost func ..."
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Cited by 43 (24 self)
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This paper describes the implementation of optimization techniques based on control theory for airfoil and wing design. The theory is applied to a system defined by the partial differential equations of the flow, with control by the boundary as a free surface. 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. The cost is kept low by using multigrid techniques, which yield a sufficiently accurate solution in about 15 iterations. Satisfactory designs are usually obtained with 1020 design cycles. 1 The Design Problem as a Control Problem Aerodynamic design has traditionally been carried out on a cut and try basis, with the aerodynamic expertise of the designer guiding the selection of each shape modification. Although considerable gains in aerodynamic performance have been achieved by this approach, continued improvement will most probably be much more difficult to attain. The subtlety and complexity of fluid flow is such that it is unlikely that repeated trials in an interactive analysis and design procedure can lead to a truly optimum design. Automatic design techniques are therefore needed in order to fully realize the potential improvements in aerodynamic efficiency. The simplest approach to optimization is to define the geometry through a set of design parameters, which may, for example, be the weights ai applied to a set of shape functions bi(x) so that the shape is represented as Then a cost function I is selected which might, for example, be the drag coefficient or the lift to drag ratio, and I is regarded as a function of the parameters ai. The sensitivities 5 may now be estimated by making
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 36 (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
Practical 3D Aerodynamic Design and Optimization Using Unstructured Meshes
 AIAA J
, 1997
"... A methodology for performing optimization on 3D unstructured grids based on the Euler equations is presented. The same, lowmemorycost explicit relaxation algorithm is used to resolve the discrete equations which govern the flow, linearized direct and adjoint problems. The analysis scheme is a high ..."
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Cited by 34 (4 self)
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A methodology for performing optimization on 3D unstructured grids based on the Euler equations is presented. The same, lowmemorycost explicit relaxation algorithm is used to resolve the discrete equations which govern the flow, linearized direct and adjoint problems. The analysis scheme is a high resolution LocalExtremumDiminishing (LED) type scheme which uses Roe decomposition for the dissipative fluxes. Mesh movement is performed in such a way that optimization of arbitrary geometries is allowed. The parallelization of the algorithm, which permits its extension to optimization of realistic, complete aircraft geometries is presented. Two sample optimizations are performed. The first is the inverse design of a transonic wing/body configuration using a surface target pressure distribution found by analyzing the geometry for known design variable deflections. The second exercise is the inverse design of a business jet configuration consisting of wing, body, strut, nacelle, horizontal...
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.
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 31 (7 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...
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 30 (12 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...