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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 144 (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.
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 50 (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 48 (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...
Contribution to the Optimal Shape Design of TwoDimensional Internal Flows with Embedded Shocks
, 1995
"... We explore the praticability of optimal shape design for ows modeled by the Euler equations. We de ne a functional whose minimum represents the optimality condition. The gradient of the functional with respect to the geometry is calculated with the Lagrange multipliers, which are determined by solvi ..."
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Cited by 28 (5 self)
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We explore the praticability of optimal shape design for ows modeled by the Euler equations. We de ne a functional whose minimum represents the optimality condition. The gradient of the functional with respect to the geometry is calculated with the Lagrange multipliers, which are determined by solving a costate equation. The optimization problem is then examined by comparing the performance of several gradientbased optimization algorithms. In this formulation, the ow eld can be computed to an arbitrary order of accuracy. Finally, some results for internal ows with embedded shocks are presented, including a case for which the solution to the inverse problem does not belong to the design space.
Control theory based airfoil design for potential flow . . .
 AIAA PAPER 944272, 5TH AIAA/USAF/NASA/ISSMO SYMPOSIUM ON MULTIDISCIPLINARY ANALYSIS AND OPTIMIZATION, PANAMA CITY BEACH, FL
, 1994
"... This paper describes the implementation of optimization techniques based on control theory for airfoil design. In previous studies [6, 71 it was shown that control theory could be used to devise an effective optimization procedure for twodimensional profiles in which the shape is determined by a co ..."
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Cited by 27 (8 self)
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This paper describes the implementation of optimization techniques based on control theory for airfoil design. In previous studies [6, 71 it was shown that control theory could be used to devise an effective optimization procedure for twodimensional profiles in which the shape is determined by a conformal transformation from a unit circle, and the control is the mapping function. The goal of our present work is to develop a method which does not depend on conformal mapping, so that it can be extended to treat threedimensional problems. Therefore, we have developed a method which can address arbitrary geometric shapes through the use of a finite volume method to discretize the potential flow equation. Here the control law serves to provide computationally inexpensive gradient information to a standard numerical optimization method. Results are presented, where both target speed distributions and minimum drag are used as objective functions.
Aerodynamic Shape Optimization Using the Adjoint Method
 VKI Lecture Series on Aerodynamic Drag Prediction and Reduction, von Karman Institute of Fluid Dynamics, Rhode St Genese
, 2003
"... These Lecture Notes 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 sh ..."
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Cited by 26 (9 self)
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These Lecture Notes 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.
On the Properties of Solutions of the Adjoint Euler Equations
 Numerical Methods for Fluid Dynamics VI. ICFD
, 1998
"... The behavior of analytic and numerical adjoint solutions is examined for the quasi1D 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 ..."
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Cited by 25 (10 self)
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The behavior of analytic and numerical adjoint solutions is examined for the quasi1D 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 isentropic and shocked transonic flow, revealing a logarithmic singularity at the sonic throat and confirming the expected properties at the shock. Numerical solutions obtained using both discrete and continuous adjoint formulations reveal that there is no need to explicitly enforce the adjoint shock boundary condition. Adjoint methods are demonstrated to play an important role in the error estimation of integrated quantities such as lift and drag. 1 Introduction Adjoint problems arise naturally in the formulation of methods for optimal aerodynamic design and optimal error control. F...