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317
Aerodynamic Design Optimization On Unstructured Meshes Using the NavierStokes Equations
 AIAA J
, 1999
"... A continuous adjoint approach for obtaining sensitivity derivatives on unstructured grids is developed and analyzed. The derivation of the costate equations is presented, and a secondorder accurate discretization method is described. The relationship between the continuous formulation and a discret ..."
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Cited by 145 (4 self)
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A continuous adjoint approach for obtaining sensitivity derivatives on unstructured grids is developed and analyzed. The derivation of the costate equations is presented, and a secondorder accurate discretization method is described. The relationship between the continuous formulation and a discrete formulation is explored for inviscid, as well as for viscous flow. Several limitations in a strict adherence to the continuous approach are uncovered, and an approach that circumvents these diculties is presented. The issue of grid sensitivities, which do not arise naturally in the continuous formulation, is investigated and is observed to be of importance when dealing with geometric singularities. A method is described for modifying inviscid and viscous meshes during the design cycle to accommodate changes in the surface shape. The accuracy of the sensitivity derivatives is established by comparing with finite
PROBLEM FORMULATION FOR MULTIDISCIPLINARY OPTIMIZATION
, 1994
"... This paper is about multidisciplinary (design) optimization, or MDO, the coupling of two or more analysis disciplines with numerical optimization. The paper has three goals. First, it is an expository introduction to MDO aimed at those who do research on optimization algorithms, since the optimizati ..."
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Cited by 129 (8 self)
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This paper is about multidisciplinary (design) optimization, or MDO, the coupling of two or more analysis disciplines with numerical optimization. The paper has three goals. First, it is an expository introduction to MDO aimed at those who do research on optimization algorithms, since the optimization community has much to contribute to this important class of computational engineering problems. Second, this paper presents to the MDO research community a new abstraction for multidisciplinary analysis and design problems as well as new decomposition formulations for these problems. Third, the "individual discipline feasible " (IDF) approaches introduced here make use of existing specialized analysis codes, and they introduce significant opportunities for coarsegrained computational parallelism particularly well suited to heterogeneous computing environments. The key distinguishing characteristic of the three fundamental approaches to MDO formulation discussed here is the kind of disciplinary feasibility that must be maintained at each optimization iteration. Other formulation issues, such as the sensitivities required, are also considered. This discussion highlights the tradeoffs between reuse of existing software, computational requirements, and probability of success.
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 86 (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
Optimum aerodynamic design using CFD and control theory
 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 70 (23 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.
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 57 (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 52 (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 49 (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...
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 45 (5 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...