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116
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
The complexstep derivative approximation
 ACM Transactions on Mathematical Software
"... The complexstep derivative approximation and its application to numerical algorithms are presented. Improvements to the basic method are suggested that further increase its accuracy and robustness and unveil the connection to algorithmic differentiation theory. A general procedure for the implement ..."
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Cited by 68 (15 self)
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The complexstep derivative approximation and its application to numerical algorithms are presented. Improvements to the basic method are suggested that further increase its accuracy and robustness and unveil the connection to algorithmic differentiation theory. A general procedure for the implementation of the complexstep method is described in detail and a script is developed that automates its implementation. Automatic implementations of the complexstep method for Fortran and C/C++ are presented and compared to existing algorithmic differentiation tools. The complexstep method is tested in two large multidisciplinary solvers and the resulting sensitivities are compared to results given by finite differences. The resulting sensitivities are shown to be as accurate as the analyses. Accuracy, robustness, ease of implementation and maintainability make these complexstep derivative approximation tools very attractive options for sensitivity analysis.
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
HighFidelity AeroStructural Design Optimization of a Supersonic Business Jet
 Journal of Aircraft
, 2002
"... This paper focuses on the demonstration of an integrated aerostructural method for the design of aerospace vehicles. Both aerodynamics and structures are represented using highfidelity models such as the Euler equations for the aerodynamics and a detailed finite element model for the primary struc ..."
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Cited by 46 (22 self)
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This paper focuses on the demonstration of an integrated aerostructural method for the design of aerospace vehicles. Both aerodynamics and structures are represented using highfidelity models such as the Euler equations for the aerodynamics and a detailed finite element model for the primary structure. The aerodynamic outer mold line (OML) and a structure of fixed topology are parameterized using a large number of design variables
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 40 (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 36 (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.
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 35 (9 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.
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 32 (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.
A CoupledAdjoint Sensitivity Analysis Method for HighFidelity AeroStructural Design
 Optimization and Engineering
, 2005
"... Abstract. This paper presents an adjoint method for sensitivity analysis that is used in an aerostructural aircraft design framework. The aerostructural analysis uses highfidelity models of both the aerodynamics and the structures. Aerostructural sensitivities are computed using a coupledadjoin ..."
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Cited by 32 (15 self)
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Abstract. This paper presents an adjoint method for sensitivity analysis that is used in an aerostructural aircraft design framework. The aerostructural analysis uses highfidelity models of both the aerodynamics and the structures. Aerostructural sensitivities are computed using a coupledadjoint approach that is based on previously developed single discipline sensitivity analysis. Alternative strategies for coupled sensitivity analysis are also discussed. The aircraft geometry and a structure of fixed topology are parameterized using a large number of design variables. The aerostructural sensitivities of aerodynamic and structural functions with respect to these design variables are computed and compared with results given by the complexstep derivative approximation. The coupledadjoint procedure is shown to yield very accurate sensitivities and to be computationally efficient, making highfidelity aerostructural design feasible for problems with thousands of design variables.