## Discrete adjoint approximations with shocks (2002)

Venue: | CONFERENCE ON HYPERBOLIC PROBLEMS |

Citations: | 7 - 3 self |

### BibTeX

@INPROCEEDINGS{Giles02discreteadjoint,

author = {M. B. Giles},

title = {Discrete adjoint approximations with shocks},

booktitle = {CONFERENCE ON HYPERBOLIC PROBLEMS},

year = {2002},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

In recent years there has been considerable research into the use of adjoint flow equations for design optimisation (e.g. [Jam95]) and error analysis (e.g. [PG00, BR01]). In almost every case, the adjoint equations have been formulated under the assumption that the original nonlinear flow solution is smooth. Since most applications have been for incompressible or subsonic flow, this has been valid, however there is now increasing use of such techniques in transonic design applications for which there are shocks. It is therefore of interest to investigate the formulation and discretisation of adjoint equations when in the presence of shocks. The reason that shocks present a problem is that the adjoint equations are defined to be adjoint to the equations obtained by linearising the original nonlinear flow equations. Therefore, this raises the whole issue of linearised perturbations to the shock. The validity of linearised shock capturing for harmonically oscillating shocks in flutter analysis was investigated by Lindquist and Giles [LG94] who showed that the shock capturing produces the correct prediction of integral quantities such as unsteady lift and moment provided the shock is smeared over a number of grid points. As a result, linearised shock capturing is now the standard method of turbomachinery aeroelastic analysis [HCL94], benefitting from the computational advantages of the linearised approach, without the many drawbacks of shock fitting. There has been very little prior research into adjoint equations for flows with shocks. Giles and Pierce [GP01] have shown that the analytic derivation of the adjoint equations for the steady quasi-one-dimensional Euler equations requires the specification of an internal adjoint boundary condition at the shock. However, the numerical evidence [GP98] is that the correct adjoint solution is obtained using either the "fully discrete" approach (in which one linearises the discrete equations and uses the transpose) or the "continuous" approach (in which one discretises the analytic adjoint equations). It is not clear though that this will remain true in two dimensions, for which there is a similar adjoint boundary condition along a shock. In this paper, we consider unsteady one-dimensional hyperbolic equations with a convex scalar flux, and in particular obtain numerical results for Burgers equation. Tadmor [Tad91] developed a Lip' topology for the formulation of adjoint equations for this problem, with application to linear post-processing functionals. Building on this and the work of Bouchut and James [BJ98], Ulbrich has very recently introduced the concept of shiftdifferentiability [Ulb02a, Ulb02b] to handle nonlinear functionals of the type considered in this paper. This supplies the analytic adjoint solution against which the numerical solutions in this paper will be compared. An alternative derivation of this analytic solution is presented in an expanded version of this paper [Gil02].

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(Show Context)
Citation Context ...dary condition along a shock. In this paper, we consider unsteady one-dimensional hyperbolic equations with a convex scalarsux, and in particular obtain numerical results for Burgers equation. Tadmor =-=[Tad91]-=- developed a Lip' topology for the formulation of adjoint equations for this problem, with application to linear post-processing functionals. Building on this and the work of Bouchut 2 M.B. Giles and ... |

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(Show Context)
Citation Context ...oped a Lip' topology for the formulation of adjoint equations for this problem, with application to linear post-processing functionals. Building on this and the work of Bouchut 2 M.B. Giles and James =-=[BJ98-=-], Ulbrich has very recently introduced the concept of shiftdi erentiability [Ulb02a, Ulb02b] to handle nonlinear functionals of the type considered in this paper. This supplies the analytic adjoint s... |

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(Show Context)
Citation Context ...versity Computing Laboratory, Oxford, U.K. giles@comlab.ox.ac.uk 1 Introduction In recent years there has been considerable research into the use of adjoint ow equations for design optimisation (e.g. =-=[Jam95]-=-) and error analysis (e.g. [PG00, BR01]). In almost every case, the adjoint equations have been formulated under the assumption that the original nonlinearsow solution is smooth. Since most applicatio... |

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(Show Context)
Citation Context ...ivation of the adjoint equations for the steady quasi-one-dimensional Euler equations requires the specication of an internal adjoint boundary condition at the shock. However, the numerical evidence [=-=GP98] is that t-=-he correct adjoint solution is obtained using either the \fully discrete" approach (in which one linearises the discrete equations and uses the transpose) or the \continuous" approach (in wh... |

18 | A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms - Ulbrich |

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(Show Context)
Citation Context ... computational advantages of the linearised approach, without the many drawbacks of shockstting. There has been very little prior research into adjoint equations forsows with shocks. Giles and Pierce =-=[GP01-=-] have shown that the analytic derivation of the adjoint equations for the steady quasi-one-dimensional Euler equations requires the specication of an internal adjoint boundary condition at the shock.... |

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(Show Context)
Citation Context ... such as unsteady lift and moment provided the shock is smeared over a number of grid points. As a result, linearised shock capturing is now the standard method of turbomachinery aeroelastic analysis =-=[HCL94-=-], benetting from the computational advantages of the linearised approach, without the many drawbacks of shockstting. There has been very little prior research into adjoint equations forsows with shoc... |

7 | The adaptive computation of far patterns by a posteriori error estimates of linear functionals - Monk, Suli - 1998 |

5 |
B.: On the Validity of Linearized Unsteady Euler Equations With Shock Capturing
- Lindquist, Giles
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(Show Context)
Citation Context ...ises the whole issue of linearised perturbations to the shock. The validity of linearised shock capturing for harmonically oscillating shocks insutter analysis was investigated by Lindquist and Giles =-=[LG94]-=- who showed that the shock capturing produces the correct prediction of integral quantities such as unsteady lift and moment provided the shock is smeared over a number of grid points. As a result, li... |

3 | Time- and Frequency-Domain Numerical Simulation of Linearized Euler - Sreenivas, Whitfield - 1998 |

1 |
Adjoint equations and discrete approximations in the presence of shocks
- Giles
- 2002
(Show Context)
Citation Context ...e analytic adjoint solution against which the numerical solutions in this paper will be compared. An alternative derivation of this analytic solution is presented in an expanded version of this paper =-=[Gil02]-=-. 2 Analytic adjoint solution Let u(x; t) be the solution of the scalar equation @u @t + @f(u) @x = 0; 0 0 subject to initial conditions u(x; 0) = u 0 (x). Numerical results will be presented later fo... |