## Practical 3D Aerodynamic Design and Optimization Using Unstructured Meshes (1997)

Venue: | AIAA J |

Citations: | 35 - 4 self |

### BibTeX

@ARTICLE{Elliott97practical3d,

author = {Jonathan Elliott and Jaime Peraire},

title = {Practical 3D Aerodynamic Design and Optimization Using Unstructured Meshes},

journal = {AIAA J},

year = {1997},

volume = {35},

pages = {35--9}

}

### Years of Citing Articles

### OpenURL

### Abstract

A methodology for performing optimization on 3D unstructured grids based on the Euler equations is presented. The same, low-memory-cost 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 Local-ExtremumDiminishing (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...

### Citations

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Citation Context ...BFGS method. The BFGS method is a quasi-Newton method in which an increasingly accurate estimate of the Hessian matrix, H , is made based on changes in the gradient vector along the optimization path =-=[7]-=-. Resulting geometry and surface pressure evolutions are shown in Figures 4 and 5 while the evolution of design variables and cost function is shown in Figure 6. It should be noted that after the eigh... |

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Citation Context ...een the given node and the freestream state vector U1 . ~ F b i \Delta ~ S b = ~ F i + ~ F1 2 \Delta ~ S b \Gamma S b 2 jAj(U i \Gamma U1 ) where A = @( ~ F \Deltasn b )=@U is the standard Roe matrix =-=[19]-=- resolved in the direction of the boundary normalsn b = ~ S b =S b and can be formed from the Roe-averaging of U i and U1 . These expressions for the fluxes at the boundaries ensure that correct bound... |

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Citation Context ...ctical 3D optimization on unstructured grids in which the cost of the gradient calculation is practically independent of the number of design variables. This is done using the discrete adjoint method =-=[12, 10, 9, 8, 20]-=- and solving the resulting system with the same explicit multistage timestepping algorithm used to solve the flow equations. For structured grids, generation of analytic sensitivities based on the Eul... |

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Citation Context ... used to solve the flow equations. For structured grids, generation of analytic sensitivities based on the Euler equations in 2D and 3D has recently been performed using the analytic adjoint approach =-=[12, 17, 18]-=-, in which flow and adjoint equations are discretized separately. The same relaxation algorithm was used to solve flow and adjoint equations. In [12] an alternating direction method with upwind biasin... |

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Citation Context ...discretized on 2D and 3D unstructured grids. LED properties are obtained by introducing just enough dissipation to upwind each characteristic component through the use of a Roe-averaged flux jacobian =-=[11]-=-. Minmod-limited antidiffusion is added to improve the accuracy of the scheme to 2nd order in smooth regions of the solution. Furthermore the mesh movement algorithm can handle arbitrary geometries an... |

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Citation Context ...ctical 3D optimization on unstructured grids in which the cost of the gradient calculation is practically independent of the number of design variables. This is done using the discrete adjoint method =-=[12, 10, 9, 8, 20]-=- and solving the resulting system with the same explicit multistage timestepping algorithm used to solve the flow equations. For structured grids, generation of analytic sensitivities based on the Eul... |

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Citation Context ...ower required core memory (due to the smaller stencil) and the second order Jacobian on the right hand side. A 3D unstructured sensitivity capability based on the Euler equations was also reported in =-=[5]-=- by the authors. The extension of this capability to the optimization of realistic, complete aircraft geometries is discussed in this paper. This extension has been made possible (for practical design... |

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Citation Context ...ctical 3D optimization on unstructured grids in which the cost of the gradient calculation is practically independent of the number of design variables. This is done using the discrete adjoint method =-=[12, 10, 9, 8, 20]-=- and solving the resulting system with the same explicit multistage timestepping algorithm used to solve the flow equations. For structured grids, generation of analytic sensitivities based on the Eul... |

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Citation Context ...r convergence acceleration, A i is the area of the control volume and oe represents the local CFL number. The residual at a point can be calculated as a sum over all the edges that contain that point =-=[16]-=-. R i = K X ik=1 [R] ik (4) where ik represents the edge connecting nodes i and k and the residual increment for an interior edge is, [R] ik = \Gamma ~ F i + ~ F k 2 \Delta ~ S ik (5) where ~ S ik is ... |

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Citation Context ... are, by construction, always positive. The addition of antidiffusion causes the scheme to now resemble a high order MUSCL scheme. For further details of similar algorithms, the reader is referred to =-=[16, 15, 4]-=-. Timestep calculation It was found that use of a linearized Fourier stability analysis to determine the allowable timestep [6], allows convergence to machine accuracy only for very low CFL numbers. F... |

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Citation Context ... multistage relaxation scheme was used in a scheme incorporating scalar dissipation with pressure switching to allow shock capturing. Newman, Taylor and Burgreen presented a capability to generate 2D =-=[14]-=-, and recently 3D [13], unstructured sensitivities for a 2nd order scheme using Van Leer flux vector splitting. This was done using an incremental iterative implicit time-marching scheme. They used th... |

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Citation Context ... scheme was used in a scheme incorporating scalar dissipation with pressure switching to allow shock capturing. Newman, Taylor and Burgreen presented a capability to generate 2D [14], and recently 3D =-=[13]-=-, unstructured sensitivities for a 2nd order scheme using Van Leer flux vector splitting. This was done using an incremental iterative implicit time-marching scheme. They used the discrete sensitivity... |

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Citation Context ... are, by construction, always positive. The addition of antidiffusion causes the scheme to now resemble a high order MUSCL scheme. For further details of similar algorithms, the reader is referred to =-=[16, 15, 4]-=-. Timestep calculation It was found that use of a linearized Fourier stability analysis to determine the allowable timestep [6], allows convergence to machine accuracy only for very low CFL numbers. F... |

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Supersonic Wing and Wing-Body Shape Optimization using an Adjoint Formulation
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Citation Context ... used to solve the flow equations. For structured grids, generation of analytic sensitivities based on the Euler equations in 2D and 3D has recently been performed using the analytic adjoint approach =-=[12, 17, 18]-=-, in which flow and adjoint equations are discretized separately. The same relaxation algorithm was used to solve flow and adjoint equations. In [12] an alternating direction method with upwind biasin... |

4 |
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Citation Context ...been the high cost associated with the calculation of sensitivities. However, in recent years impressive efforts have been made using structured grids that allow for efficient sensitivity calculation =-=[1, 2]-=-. For the solution of flowfields over complex domains, unstructured grid methods have several advantages over their alternative structured counterparts. For example, the time associated with grid gene... |

2 |
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