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A NewtonKrylov Algorithm for the Euler Equations Using Unstructured Grids,” AIAA Paper
, 2003
"... A NewtonKrylov flow solver is presented for the Euler equations on unstructured grids. The algorithm uses a preconditioned matrixfree GMRES method to solve the linear system that arises at each Newton iteration. The preconditioner is an incomplete lowerupper factorization of an approximation to t ..."
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Cited by 4 (2 self)
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A NewtonKrylov flow solver is presented for the Euler equations on unstructured grids. The algorithm uses a preconditioned matrixfree GMRES method to solve the linear system that arises at each Newton iteration. The preconditioner is an incomplete lowerupper factorization of an approximation to the Jacobian matrix after applying the reverse CuthillMcKee reordering. The algorithm successfully converges for a wide range of steady two and threedimensional aerodynamic flows. A tenorder reduction of the density residual is obtained in a computing time equivalent to fewer than 520 and 1, 800 residual evaluations for the twodimensional and threedimensional cases, respectively.
Toward practical aerodynamic design through numerical optimization,” AIAA paper 20073950
, 2007
"... A NewtonKrylov algorithm for aerodynamic optimization is applied to the multipoint design of an airfoil for eighteen different operating conditions. The operating conditions include four cruise conditions and four longrange cruise conditions at maximum and minimum cruise weights and altitudes. In ..."
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A NewtonKrylov algorithm for aerodynamic optimization is applied to the multipoint design of an airfoil for eighteen different operating conditions. The operating conditions include four cruise conditions and four longrange cruise conditions at maximum and minimum cruise weights and altitudes. In addition, eight operating points are included in order to provide adequate maneuvering capabilities under dive conditions at the same maximum and minimum weights and altitudes with two different load factors. Finally, two lowspeed operating conditions are included at the maximum and minimum weights. The problem is posed as a multipoint optimization problem with a composite objective function that is formed by a weighted sum of the individual objective functions. The NewtonKrylov algorithm, which employs the discreteadjoint method, has been extended to include the lift constraint among the governing equations, leading to an improved liftconstrained drag minimization capability. The optimized airfoil performs well throughout the flight envelope. This example demonstrates how numerical optimization can be applied to practical aerodynamic design. Beginning with the work of Hicks et al. 1 I.
Optimized NaturalLaminarFlow Airfoils
"... A twodimensional NewtonKrylov aerodynamic shape optimization algorithm has been modified to incorporate the prediction of laminarturbulent transition. Modifications to the discreteadjoint gradient computation were required to allow the optimization algorithm to manipulate the transition point th ..."
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Cited by 3 (2 self)
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A twodimensional NewtonKrylov aerodynamic shape optimization algorithm has been modified to incorporate the prediction of laminarturbulent transition. Modifications to the discreteadjoint gradient computation were required to allow the optimization algorithm to manipulate the transition point through shape changes. The coupled Euler and boundarylayer solver, MSES, is used to obtain transition locations, which are then used in Optima2D, a NewtonKrylov discreteadjoint optimization algorithm based on the compressible Reynoldsaveraged NavierStokes equations. The algorithm is applied to the design of airfoils with maximum liftdrag ratio, endurance factor, and lift coefficient. The design examples demonstrate that the optimizer is able to control the transition point locations to provide optimum performance, in effect designing optimized naturallaminarflow airfoils. In particular, the optimization algorithm is able to design an airfoil that is very similar, in terms of both shape and performance, to one of the highlift airfoils designed by Liebeck (J. of Aircraft, 10:610617, 1973) in the 1970’s. I.
On Aerodynamic Optimization Under a Range of Operating Conditions
 AIAA Journal
"... In aerodynamic design, good performance is generally required under a range of operating conditions, including offdesign conditions. This can be achieved through multipoint optimization. The desired performance objective and operating conditions must be specified, and the resulting optimization pro ..."
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In aerodynamic design, good performance is generally required under a range of operating conditions, including offdesign conditions. This can be achieved through multipoint optimization. The desired performance objective and operating conditions must be specified, and the resulting optimization problem must be solved in such a manner that the desired performance is achieved. Issues involved in formulating multipoint optimization problems are discussed. A technique is proposed for automatically choosing sampling points within the operating range and their weights to obtain the desired performance over the range of operating conditions. Examples are given involving liftconstrained drag minimization over a range of Mach numbers. Tradeoffs and their implications for the formulation of multipoint problems are presented and discussed. I.
A HighOrder Parallel NewtonKrylov flow solver for the Euler Equations
, 2009
"... This work presents a parallel NewtonKrylov flow solver employing third and fourthorder spatial discretizations to solve the threedimensional Euler equations on structured multiblock meshes. The fluxes are discretized using summationbyparts operators; boundary and interface conditions are implem ..."
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This work presents a parallel NewtonKrylov flow solver employing third and fourthorder spatial discretizations to solve the threedimensional Euler equations on structured multiblock meshes. The fluxes are discretized using summationbyparts operators; boundary and interface conditions are implemented using simultaneous approximation terms. Functionals, drag and lift, are calculated using Simpson’s rule. The solver is verified using the method of manufactured solutions and Ringleb flow and validated using the ONERA M6 wing. The results demonstrate that the combination of highorder finitedifference operators with a parallel NewtonKrylov solution technique is an excellent option for efficient computation of aerodynamic flows. I.
Mesh Movement for a DiscreteAdjoint NewtonKrylov Algorithm for Aerodynamic Optimization
"... A grid movement algorithm based on the linear elasticity method with multiple increments is presented. The method is computationally expensive, but is exceptionally robust, producing high quality elements even for large shape changes. It is integrated with an aerodynamic shape optimization algorithm ..."
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A grid movement algorithm based on the linear elasticity method with multiple increments is presented. The method is computationally expensive, but is exceptionally robust, producing high quality elements even for large shape changes. It is integrated with an aerodynamic shape optimization algorithm that uses an augmented adjoint method for gradient calculation. The discrete adjoint equations are augmented to explicitly include the sensitivities of the mesh movement, resulting in an increase in efficiency and numerical accuracy. This gradient computation method requires less computational time than a function evaluation, and leads to significant computational savings as dimensionality is increased. The results from application of these techniques to several large deformation and optimization cases are presented. I.
Aerodynamic Shape Optimization Using a Cartesian Adjoint Method and CAD Geometry
"... We present a new approach for the computation of shape sensitivities using the discrete adjoint and flowsensitivity methods on Cartesian meshes with general polyhedral cells (cutcells) at the wall boundaries. By directly linearizing the cutcell geometric constructors of the mesh generator, an effi ..."
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We present a new approach for the computation of shape sensitivities using the discrete adjoint and flowsensitivity methods on Cartesian meshes with general polyhedral cells (cutcells) at the wall boundaries. By directly linearizing the cutcell geometric constructors of the mesh generator, an efficient and robust computation of shape sensitivities is achieved. We show that the error convergence rate of the flow solution and its sensitivity, as well as the objective function and its gradient is consistent with the secondorder spatial discretization of the threedimensional Euler equations. The performance of the approach is demonstrated for an airfoil optimization problem in transonic flow, and a CADbased shape optimization of a reentry capsule in hypersonic flow. The approach is wellsuited for conceptual design studies where fast turnaround time is required. I.
Adaptive Airfoils for Drag Reduction at Transonic Speeds
"... Adaptive airfoils and wings can provide superior performance at the expense of increased cost and complexity. In this paper, an aerodynamic optimization algorithm is used to assess an adaptive airfoil concept for drag reduction at transonic speeds. The objective is to quantify both the improvements ..."
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Adaptive airfoils and wings can provide superior performance at the expense of increased cost and complexity. In this paper, an aerodynamic optimization algorithm is used to assess an adaptive airfoil concept for drag reduction at transonic speeds. The objective is to quantify both the improvements in drag that can be achieved and the magnitude of the shape changes needed. In an initial study, a baseline airfoil is designed to produce low drag at a fixed lift coefficient over a range of Mach numbers. This airfoil is compared with a sequence of nine airfoils, each designed to be optimal at a single operating point in the Mach number range. Shape changes of less than 2 % chord lead to drag reductions of 46% over a range of Mach numbers from 0.68 to 0.76. If the shape changes are restricted to the upper surface only, then changes of less than 1 % chord lead to drag reduction of 35%. In a second study, a baseline airfoil is designed based on a multipoint optimization over eighteen operating points, including dive and lowspeed offdesign requirements. Adaptive airfoils are designed through singlepoint optimization for the operating points corresponding to cruise conditions, producing drag reductions ranging from 9.7 to 16.7 % with shape changes on the order of a few percent chord. I.
A CADFree and a CADBased Geometry Control System for Aerodynamic Shape Optimization,” AIAA Paper 2005–0451
, 2005
"... The performance of an aerodynamic shape optimization routine is greatly dependent on its geometry control system. This system must accurately parameterize the initial geometry and generate a flexible set of design variables for the optimization cycle. It must also generate new instances of the geome ..."
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The performance of an aerodynamic shape optimization routine is greatly dependent on its geometry control system. This system must accurately parameterize the initial geometry and generate a flexible set of design variables for the optimization cycle. It must also generate new instances of the geometry based on the changes to the design variables dictated by the optimization routine. In response to changes in the geometry, it is also desirable to generate a new surface grid with the same topology as the original grid. This new surface grid can be used to perturb the associated volume grid. This paper presents two geometry control systems, a CADfree system, and a CATIA V5 CADbased system. The two systems provide practical tools for aerodynamic optimization. They also provide a basis for comparing CADfree and CADbased systems and understanding additional issues that need to be addressed in order to develop reliable optimization systems.
Towards Aircraft Design Using an Automatic Discrete Adjoint Solver
"... The ADjoint method is applied to a threedimensional Computational Fluid Dynamics (CFD) solver to generate the sensitivities required for aerodynamic shape optimization. The ADjoint approach selectively uses Automatic Differentiation (AD) to generate the partial derivative terms in the discrete adjo ..."
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The ADjoint method is applied to a threedimensional Computational Fluid Dynamics (CFD) solver to generate the sensitivities required for aerodynamic shape optimization. The ADjoint approach selectively uses Automatic Differentiation (AD) to generate the partial derivative terms in the discrete adjoint equations. By selectively applying AD techniques, the computational cost and memory overhead incurred by using AD are significantly reduced, while still allowing for the the accurate treatment of arbitrarily complex governing equations and boundary conditions. Once formulated, the discrete adjoint equations are solved using the Portable, Extensible Toolkit for Scientific computation (PETSc). With this approach, the computed adjoint vector can be used to calculate the total sensitivities required for aerodynamic shape optimization of a complete aircraft configuration. The resulting sensitivities are compared with complexstep derivatives to establish their accuracy. The tools developed are applied to an infinite wing test case to demonstrate the accuracy and efficiency of the method. I.