## Analytic adjoint solutions for the quasi-onedimensional Euler equations (2001)

Venue: | J. Fluid Mechanics |

Citations: | 15 - 6 self |

### BibTeX

@ARTICLE{Giles01analyticadjoint,

author = {B. Giles and Niles and A. Pierce},

title = {Analytic adjoint solutions for the quasi-onedimensional Euler equations},

journal = {J. Fluid Mechanics},

year = {2001},

pages = {327--345}

}

### Years of Citing Articles

### OpenURL

### Abstract

The analytic properties of adjoint solutions are examined for the quasi-onedimensional 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.

### Citations

207 | Aerodynamic Design Via Control Theory
- Jameson
- 1988
(Show Context)
Citation Context ... effort has been devoted to the development of optimal design methods based on the adjoint approach. Some methods use curvilinear coordinates and the differential adjoint, as outlined above (see e.g. =-=Jameson 1988-=-, 1995, 1999; Reuther et al. 1996, 1999a, b; Jameson, Pierce & Martinelli 1998). Other methods first discretize the nonlinear p.d.e. and then use the adjoint (transpose) of the linear discrete matrix ... |

108 | Optimum Aerodynamic Design Using the Navier-Stokes Equations," AIAA Paper 970101 - 40Jameson, Pierce, et al. - 1997 |

76 | Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers: Part II - Reuther, Jameson, et al. - 1998 |

53 | An Introduction to the Adjoint Approach to Design - Giles |

53 | Adjoint recovery of superconvergent functionals from PDE approximations
- Pierce, Giles
- 2000
(Show Context)
Citation Context ...l Poisson equation and the quasi-one-dimensional Euler equations, this has been shown to lead to corrected values of twice the order of accuracy of the flow-field solution (Giles & Pierce 1998, 1999; =-=Pierce & Giles 1998-=-, 2000). While significant effort has been dedicated to developing methods for calculating adjoint solutions to compressible flow equations, there has been little discussion of the properties of the a... |

48 |
Optimum aerodynamic design using control theory
- Jameson
- 1995
(Show Context)
Citation Context ...ome given computational domain. When calculating the two-dimensional flow around an aerofoil, one technique is to use curvilinear coordinates (ξ,η) in which the aerofoil surface corresponds to η = 0 (=-=Jameson 1995-=-). Using these coordinates, the p.d.e. can be written as R(U) =0, (1.1) where U is the flow solution and R is a nonlinear differential operator which depends on the mapping from (ξ,η) to the Cartesian... |

46 | A Posteriori Finite Element Bounds for Linear-Functional Outputs of Elliptic Partial Differential Equations. Computer methods in applied mechanics and engineering - Paraschivoiu, Peraire, et al. - 1997 |

42 |
Weighted a posteriori error control in finite element methods
- Becker, Rannacher
- 1996
(Show Context)
Citation Context ...a exceeds some threshold, to try to achieve the maximum reduction in the magnitude of the error for a given computational effort (Johnson, Rannacher & Boman 1995; Paraschivoiu, Peraire & Patera 1997; =-=Becker & Rannacher 1998-=-; Süli 1998). Alternatively, this error term can be carefully evaluated and used to correct the value of the objective function given by the calculated flow field. For the two-dimensional Poisson equa... |

33 | Practical 3D aerodynamic design and optimization using unstructured meshes
- Elliot, Peraire
- 1996
(Show Context)
Citation Context ...99; Reuther et al. 1996, 1999a, b; Jameson, Pierce & Martinelli 1998). Other methods first discretize the nonlinear p.d.e. and then use the adjoint (transpose) of the linear discrete matrix operator (=-=Elliott & Peraire 1997-=-; Anderson & Bonhaus 1999). For a more comprehensive introduction to adjoint methods in aerodynamic design and a discussion of the relative advantages of the two main approaches, see Giles & Pierce (2... |

33 | M.: Numerical and hydrodynamic stability: Towards error control in computational fluid dynamics - Johnson, Rannacher, et al. - 1995 |

27 | Improved Lift and Drag Estimates Using Adjoint Euler Equations".Technical Report 99-3293 - Giles, Pierce - 1999 |

27 | 17 of sensitivity analysis and shape optimization for complex aerodynamic configurations - Barnwell, Newman, et al. - 1999 |

23 | Re-engineering the design process through computation. AIAA paper 97-0641, 35th Aerospace Sciences Meeting and Exhibit - Jameson - 1997 |

22 |
Airfoil design on unstructured grids for turbulent flows
- Anderson, Bonhaus
- 1999
(Show Context)
Citation Context ... 1999a, b; Jameson, Pierce & Martinelli 1998). Other methods first discretize the nonlinear p.d.e. and then use the adjoint (transpose) of the linear discrete matrix operator (Elliott & Peraire 1997; =-=Anderson & Bonhaus 1999-=-). For a more comprehensive introduction to adjoint methods in aerodynamic design and a discussion of the relative advantages of the two main approaches, see Giles & Pierce (2001). For a review of the... |

22 | Contribution to the Optimal Shape Design of Two-Dimensional Internal Flows with Embedded Shocks
- Iollo, Salas
- 1995
(Show Context)
Citation Context ... owing to the disparity in the number of adjoint characteristics entering and leaving the shock. However, the conclusions differ from those of previous investigators (see Iollo, Salas & Ta’asan 1993; =-=Iollo & Salas 1996-=-; Cliff, Heinkenschloss & Shenoy 1996, 1998). The analytic adjoint solutions are then derived in closed form for all Mach regimes. This is accomplished by constructing the Green’s functions for the li... |

19 | Shape Optimization Governed by the Euler Equations Using an Adjoint Method - Iollo, Salas, et al. - 1993 |

18 | On the properties of solutions of the adjoint Euler equations
- Giles, Pierce
- 1998
(Show Context)
Citation Context ...eld. For the two-dimensional Poisson equation and the quasi-one-dimensional Euler equations, this has been shown to lead to corrected values of twice the order of accuracy of the flow-field solution (=-=Giles & Pierce 1998-=-, 1999; Pierce & Giles 1998, 2000). While significant effort has been dedicated to developing methods for calculating adjoint solutions to compressible flow equations, there has been little discussion... |

17 | A posteriori error analysis and adaptivity for finite element approximations of hyperbolic problems
- Suli
- 1998
(Show Context)
Citation Context ... to try to achieve the maximum reduction in the magnitude of the error for a given computational effort (Johnson, Rannacher & Boman 1995; Paraschivoiu, Peraire & Patera 1997; Becker & Rannacher 1998; =-=Süli 1998-=-). Alternatively, this error term can be carefully evaluated and used to correct the value of the objective function given by the calculated flow field. For the two-dimensional Poisson equation and th... |

9 | Adjoint-based methods in aerodynamic design-optimization - Cliff, Heinkenschloss, et al. - 1998 |

5 | On the optimality system for a 1-D Euler flow problem, AIAA Paper - Cliff, Heinkenschloss, et al. - 1996 |

1 |
Aerodynamic shape optimization of computer aircraft configurations via an adjoint formulation. AIAA Paper
- Reuther, Jameson, et al.
- 1996
(Show Context)
Citation Context ...o the development of optimal design methods based on the adjoint approach. Some methods use curvilinear coordinates and the differential adjoint, as outlined above (see e.g. Jameson 1988, 1995, 1999; =-=Reuther et al. 1996-=-, 1999a, b; Jameson, Pierce & Martinelli 1998). Other methods first discretize the nonlinear p.d.e. and then use the adjoint (transpose) of the linear discrete matrix operator (Elliott & Peraire 1997;... |