Shape Design Using The Euler Equations

Our next example is a similar minimization problem but this time subject to the Euler equation. Namely,

$$\min_{\Gamma} \frac{1}{2} \int_{\Gamma} (p - p^*) ^2 ds$$ (82)

where $p = p(U)$, and $U$ is the solution of the Euler equation. Here $U$ stands for the variables $(\rho, \rho {\bf u} , E )$ and ${\bf u} = (u,v,w)$. The Euler equations in conservation form are written as

$$f_x + g_y + h_z = 0$$ (83)

where

$$f = A U \qquad g = B U \qquad h = C U,$$ (84)

and where the matrices $A,B,C$ can be found, for example, in Hirsch [12]. An important property of the equation that we use here is

$$f_x = A U_x \qquad g_y = B U_y \qquad h = C u_z.$$ (85)

The change $\tilde f$ in the flux vector $f$ satisfies,

$$f( U + \epsilon \tilde U) = f(U) + \epsilon \frac{\partial f}{\pa... ...silon ^2) = f(U) + \epsilon A \tilde U + O(\epsilon ^2) = f + \epsilon \tilde f$$ (86)

and similar expressions for $\tilde g, \tilde h$. The equation for the perturbation quantities reads

$$\tilde f _x + \tilde g _y + \tilde h _z = 0$$ (87)

or equivalently,

$$(A \tilde U )_x + (B \tilde U) _y + (C \tilde U) _z = 0.$$ (88)

Now consider the following identity which follows from integration by parts,

$$\begin{array}{ll} \int _\Omega \Lambda \cdot (A \tilde U)_x dV & ... ...\int _{\partial\Omega} A^T \Lambda \cdot \tilde U {\bf n\cdot i} ds \end{array}$$ (89)

and similar integrals for the $g$ and $h$ terms. Combining these identities we arrive at

$$\begin{array}{ll} -\int _\Omega ( A^T \Lambda _x + B^T \Lambda _y... ...n \cdot j} + C^T {\bf n \cdot k} ) \Lambda \cdot \tilde U ds & = 0, \end{array}$$ (90)

for an arbitrary $\Lambda$. We will use the notation

$$D = A {\bf n \cdot i } + B {\bf n \cdot j} + C {\bf n \cdot k}$$ (91)

and note that $DU$ is the normal flux at the boundary which has the form, see Hirsch [12],

$$D U = \left( \begin{array}{c} \rho {\bf u} \cdot {\bf n} \\ \rho... ... n}) {\bf u} + p {\bf n} \\ (E + p ) {\bf u} \cdot {\bf n} \end{array}\right),$$ (92)

and at a wall where ${\bf u} \cdot {\bf n} = 0$, it reduces to

$$D U _{\vert _\Gamma} = \left( \begin{array}{c} 0 \\ p {\bf n} \\ 0 \end{array}\right).$$ (93)

We have $D \tilde U = \tilde{DU}$ following (84),(86) and its analog for the $\tilde g, \tilde h$ terms, and

$$\tilde {( p {\bf n} )} = \tilde p {\bf n} + p \tilde {\bf n}.$$ (94)

Combining the last equalities and $\tilde{\bf n} = - \sum _{j=1}^{d-1} \frac{\partial \alpha}{\partial t_j} {\bf t}_j$ from (75), we get

$$\int _\Gamma \Lambda \cdot D \tilde U ds = \int _\Gamma \Lambda \... ..._{j=1}^{d-1} \frac{\partial \alpha }{\partial t_j} (\lambda \cdot {\bf t}_j )ds$$ (95)

where we used the notation $\Lambda = ( \lambda _1, \lambda , \lambda _5 )$, and $\lambda = (\lambda _2, \lambda _3, \lambda _4 )$. The wall boundary condition

$${\bf u \cdot n}_{\vert _\Gamma} = 0$$ (96)

becomes upon perturbation

$${\bf u}^\epsilon \cdot {\bf n^\epsilon} _{\vert _\Gamma ^\epsilon} = 0,$$ (97)

and as before we transfer this boundary condition to the original boundary $\Gamma$,

$$({\bf u + \epsilon \tilde u}) \cdot {\bf n}^\epsilon _{\vert _{\G... ...partial \alpha}{\partial t_j} {\bf t_j} ) _{\vert _{\Gamma}} + O (\epsilon ^2).$$ (98)

Collecting only the $\epsilon$ terms we get

$$\tilde {\bf u} \cdot {\bf n} - \sum_{j=1}^{d-1} \frac{\partial \a... ...j} + \alpha \frac{\partial {\bf u}}{\partial n}\cdot {\bf n} = 0 \qquad \Gamma.$$ (99)

The variation of the functional

$$\delta J = \int _{\Gamma} [ (p - p^*) \tilde p + \alpha (p - p^*) \frac{\partial p}{\partial n}- \alpha \frac{(p-p^*)^2}{2R} ] ds$$ (100)

will be simplified by adding (90) to it, but with a choice of $\Lambda$ which makes the volume integral vanish. Thus, we assume that

$$A^T \Lambda _x + B^T \Lambda _y + C^T \Lambda _z = 0 \qquad \qquad \Omega$$ (101)

Using (90),(95) it leads to

$$\begin{array}{ll} \delta J & = \int _{\Gamma} [ p - p^* + {\bf\la... ...}& + \int _{\partial \Omega - \Gamma} D^T \Lambda \cdot \tilde U ds \end{array}$$ (102)

Now we come to use the boundary conditions for $\tilde U$. We begin with the far field $\partial \Omega - \Gamma$. We assume that the boundary conditions there are given in terms of characteristic variables and assume that $T$ is the matrix such that $T \tilde U$ are the characteristic variables. We write the far field term as

$$\int _{\partial \Omega - \Gamma } D^T \Lambda \cdot \tilde U ds =... ...U ds = \int _{\partial \Omega - \Gamma } T^{-T} D^T \Lambda \cdot T \tilde U ds$$ (103)

We distinguish the following cases. Supersonic inflow: all variables are specified at inflow, and thus $\tilde U = 0$. Thus, no boundary conditions are imposed on $\Lambda$. Supersonic outflow: No boundary conditions are specified for $U$, hence $\tilde U$ is arbitrary there and therefore we are led to the choice $\Lambda = 0$ at supersonic outflow. Subsonic inflow: 4 conditions are specified (3 in 2D), and those are $(T \tilde U)_{1,2,3,4} = 0$, thus $(T \tilde U)_5$ is arbitrary, leading to $(T^{-T} D^T \Lambda )_5 = 0$. Subsonic outflow: one condition is given for $U$ which implies $(T \tilde U)_5 = 0$ and therefore $(T^{-T} D \Lambda )_{1,2,3,4} = 0$. On the wall $\Gamma$ we choose

$${\bf\lambda } \cdot {\bf n} + p - p^* = 0.$$ (104)

In summary, the boundary conditions for $\Lambda$ are

$$\begin{array}{lr} \mbox{\tt supersonic inflow} & \mbox{\rm none }... ...,3,4} = 0 \\ \mbox{\tt Wall} & \lambda \cdot {\bf n} + p - p^* = 0 \end{array}$$ (105)

With this choice for $\Lambda$ together with the interior equation (101) we get that $\delta J$ involves integrals depending on $\alpha$ and $\Lambda$ and not on $\tilde U$ terms. Rearrangement by using integration by parts gives,

$$\delta J = \int _\Gamma \alpha [- \frac{(p-p^*)^2}{2R} + (p-p^*) \frac{\partial p}{\partial n}-div (p \lambda )]ds.$$ (106)

The gradient of the functional in this case is therefore given by

$$\nabla _\Gamma J = - \frac{(p-p^*)^2}{2R} + (p-p^*) \frac{\partial p}{\partial n}-div (p \lambda ).$$ (107)