Shape Design Using The Full Potential Equation

Consider the Full Potential (FP) equation

$$\begin{array}{ll} \nabla \rho \nabla \phi = 0 & \Omega \\ \vspac... ...ga - \Gamma \\ \vspace{2mm} \nabla \phi \cdot {\bf n} = 0 & \Gamma \end{array}$$ (63)

where $\rho = f (q)$ with $q = \frac{1}{2} \vert \nabla \phi \vert ^2$ and the following shape optimization problem,

$$\min_{\Gamma} \frac{1}{2} \int_{\Gamma} (p - p^*) ^2 ds$$ (64)

where $p = g(q)$. We derive the necessary conditions for this problem, and obtain a formula for the gradient for this functional subject to the FP equation (63).

As a result of changes in the shape $\Gamma$, the potential changes to $\phi + \epsilon \tilde \phi + O(\epsilon ^2)$ and $\rho$ into $\rho + \epsilon \tilde \rho + O( \epsilon ^2)$. Moreover,

$$\tilde \rho = \frac{\partial f}{\partial q}\nabla \phi \cdot \nabla \tilde \phi$$ (65)

and the equation governing $\tilde \phi$ is

$$\nabla \tilde \rho \nabla \phi + \nabla \rho \nabla \tilde \phi = 0.$$ (66)

The functional variation with respect to $\Gamma$ (see equation (62)) can be written as

$$\delta J = \epsilon \int _{\Gamma} (p - p^*) \tilde p ds + \epsil... ...c{\partial p}{\partial n}(p-p^*) - \frac{(p-p^*)^2}{2R} ] ds + O( \epsilon ^2).$$ (67)

From the boundary condition $\frac{\partial \phi}{\partial n}= 0$ some terms are simplified, in particular $\frac{\partial p}{\partial n}= \frac{1}{2} \frac{\partial g}{\partial q}\frac{\... ...{\partial n}) ^2 + \sum_{j=1}^{d-1} (\frac{\partial \phi}{\partial t_j})^2] = 0$. We also have the relation

$$\tilde p = \frac{\partial p}{\partial q}( \frac{\partial \phi}{\p... ...rac{\partial g}{\partial q}\nabla _\Gamma \phi \cdot \nabla _\Gamma \tilde \phi$$ (68)

where $\nabla _\Gamma$ stands for the tangential gradient. Substituting these into $\delta J$ and using integration by parts for the $\nabla _\Gamma \tilde \phi$ terms gives

$$\delta J = - \int _\Gamma [ \tilde \phi \sum _{j=1}^{d-1} \frac{\... ...) \frac{\partial \phi}{\partial t_j}\right) + \alpha \frac{ (p-p^*)^2}{2R} ] ds$$ (69)

This expression depends on $\tilde \phi$ which is to be eliminated using the same idea as before. To this end we use the identity

$$\int _\Omega \lambda ( \nabla \tilde \rho \nabla \phi + \nabla \rho \nabla \tilde \phi ) dx = 0$$ (70)

which follows from (66) and holds for an arbitrary $\lambda$. The relation $\tilde \rho = \frac{\partial f}{\partial q}\nabla \tilde \phi \cdot \nabla \phi$ and integration by parts of each of the terms in the above integral give

$$\int _\Omega \lambda \nabla \frac{\partial f}{\partial q}[\nabla ... ...\frac{\partial f}{\partial q}[\nabla \lambda \cdot \nabla \phi ] \nabla \phi dx$$ (71)

$$\int _\Omega \lambda \nabla \rho \nabla \tilde \phi dx = \int _\O... ...al \tilde\phi}{\partial n}- \tilde \phi \frac{\partial \lambda}{\partial n}) ds$$ (72)

where in the first integral we used the relation $\nabla \phi \cdot {\bf n} = 0$ on $\Gamma$ as well as $\tilde \phi = 0$ and $\nabla _\Gamma \tilde \phi = 0$ on $\partial \Omega - \Gamma$. Thus,

$$\int _\Omega \tilde \phi ( \nabla \rho \nabla \lambda + \nabla \f... ...tilde\phi}{\partial n}- \tilde \phi \frac{\partial \lambda}{\partial n}) ds = 0$$ (73)

for an arbitrary smooth function $\lambda$. This is the analog of equation (10) of section (2.2).

Before we add this term to the functional we need to express certain terms in the boundary. The wall boundary condition for $\phi$ on $\Gamma _\epsilon$

$$\nabla \phi ^\epsilon \cdot {\bf n ^\epsilon} _{\vert _{\Gamma ^\... ...\epsilon \tilde \phi ) \cdot {\bf n ^\epsilon} _{\vert _{\Gamma ^\epsilon}} = 0$$ (74)

will be transfered to $\Gamma$. It is easy to see that

$${\bf n^\epsilon} = {\bf n} - \epsilon \sum_{j=1}^{d-1} \frac{\partial \alpha}{\partial t_j} {\bf t_j} + O ( \epsilon ^2)$$ (75)

by considering one dimension at a time. Therefore

$$\nabla ( \phi + \epsilon \tilde \phi + \epsilon \alpha \frac{\par... ...partial \alpha}{\partial t_j} {\bf t_j} + O(\epsilon ^2) ) _{\vert _\Gamma} = 0$$ (76)

Using the boundary condition $\nabla \phi \cdot {\bf n} _{\vert _\Gamma} = 0$ we get

$$\nabla \tilde \phi \cdot {\bf n} + \alpha \dphi2dn - \sum _{j=1}^... ...tial \alpha}{\partial t_j} \nabla \phi \cdot {\bf t_j} = 0 \qquad \qquad \Gamma$$ (77)

The expression for the change in the functional as given in (69) depends on $\alpha$ as well as on $\tilde \phi$. To eliminate the dependence on $\tilde \phi$ we add the left hand side of (73). We then collect terms involving $\tilde \phi$ separately from terms involving $\frac{\partial \tilde\phi}{\partial n}$, and use the boundary condition for $\frac{\partial \tilde\phi}{\partial n}$ on $\Gamma$, giving

$$\begin{array}{ll} \delta J = & -\int _\Gamma \tilde \phi [\sum _{... ... f}{\partial q}[\nabla \lambda \cdot \nabla \phi ] \nabla \phi ) dx \end{array}$$ (78)

Now we choose $\lambda$ such that it satisfies

$$\mbox{\tt Adjoint Equation: } \begin{array}{ll} \nabla \rho \nabl... ... 0 & \Gamma \\ \vspace{2mm} \lambda = 0 & \partial \Omega - \Gamma \end{array}$$ (79)

and then the variation of the functional simplifies to

$$\delta J = - \int _\Gamma \alpha [ \rho \lambda \dphi2dn + \frac{... ...partial}{\partial t_j} ( \lambda \rho \frac{\partial \phi}{\partial t_j}) ] ds.$$ (80)

The gradient of the functional is given by

$$\nabla J = - \rho \lambda \dphi2dn - \frac{(p-p^*)^2}{2R} + \sum ... ... t_j} ( \lambda \rho \frac{\partial \phi}{\partial t_j} ) \qquad \qquad \Gamma.$$ (81)