in a domain , where ( denotes the velocity vector), with the solid wall boundary condition

(64) |

It is assumed that the problem was obtained by a linearization in a vicinity of a boundary point, and that the far field boundary conditions were given in terms of characteristic variables, which are not used explicitly in the derivation of the approximate Hessian. The minimization problem is

(65) |

If a change produces a change in the pressure then, the variation in this functional can be written as

(66) |

We calculate the Hessian in a slightly different way than before to illustrate another approach. If one can express the quadratic term in in terms of one can identify the Hessian. That is,

(67) |

This means that we can calculate the Hessian without going through the adjoint variable. We need to express in terms of , and we do it in the Fourier space. From the boundary condition at the wall

(68) |

The calculation of in terms of is done by solving the system of the linearized Euler equation with the above boundary condition for . We look for solution of the form

(69) |

The term here is the analog of our in the previous example. It is more convenient here due to the form of the symbol of the full equation. The following relation follows by substituting the above expression for into the Linearized Euler equations (63),

(70) |

This is a linear system for
and it has a nontrivial solution
when the determinant is zero,

(71) |

Note that there are five solutions for this equations. Each of them has a corresponding solution for the vector ,

(72) |

where

(73) |

Note that for the subsonic case correspond to the bounded solution, while to the unbounded one. When we have two bounded solution. In that case correspond to the incident wave and therefore its amplitude is zero for the perturbation variables. Thus, we are left with for both subsonic and supersonic cases. The three solution corresponding to are not important for our analysis since they do not affect the changes in pressure (see the corresponding eigenvectors).

To summarize, only contributes to the pressure changes as a
result of changes to the design variables by .
The solution for is given by
for some scalar . The component in this solution is
and this must equal to
form the boundary condition which in the Fourier space is given
by (68).
From that we find
.
Thus the solution is,

(74) |

The last component in this vector gives us the change in the pressure

(75) |

and from this we get

(76) |

Notice that we have taken the complex conjugate of , and since is a complex number its conjugate was taken as well. Since we obtain the symbol of the Hessian in the form,

(77) |

A preconditioner for this problem is done exactly as in the small disturbance equations using (60)-(62). It is also possible to construct the preconditioner based on solution of the linearized Euler equations, but is more complicated and unnecessary. The gradient appearing in (60)-(62) has to be changed to the gradient for this problem, using the adjoint formulation.