Shape optimization of an electrostatic capacitive sensor

Abstract

This paper describes the shape optimization of an electrostatic capacitive sensor used to detect fingers. We consider two state determination problems. The first is a basic electrostatic field problem consisting of sensing electrodes, an earth electrode, and air. The second is an electrostatic field problem in which fingers are added to the basic electrostatic field problem. An objective cost function is defined using the negative-signed squared $H^{1}$-norm of the difference between the solutions of the two state determination problems. The volume of the sensing electrode is used as the cost function. Using the solutions of the two state determination problems and the two adjoint problems, we present a method for evaluating the shape derivative of the objective cost function. To solve the shape optimization problem and minimize the negative-signed difference norm under the volume constraint, we use an iterative algorithm based on the $H^{1}$ gradient method. An algorithm for the shape optimization problem is developed to solve the boundary value problems. Numerical examples show that reasonable shapes are obtained using the present approach.

Appendix A: Formula of shape derivative

In Sect. 6, the following formula is used [2, p. 96 Proposition 4.4].

Proposition 1

(Shape derivative of domain integral) Let $\varvec{\phi } \in \mathcal {D}$, $u \in \mathcal {U} = C^1 \left( \mathcal {D} ; H^1 \left( \mathbb {R}^d ; \mathbb {R} \right) \right)$, $\varvec{\nabla } u \in \mathcal {V} = C^1 \left( \mathcal {D} ; L^2 \left( \mathbb {R}^d ; \mathbb {R}^d \right) \right)$, and $h \left( u, \varvec{\nabla } u \right) \in C^1 \left( \mathcal {U} \times \mathcal {V} ; L^2 \left( \mathbb {R}^d ; \mathbb {R} \right) \right)$. Writing $\varvec{z} = \varvec{x} + \varvec{\varphi } \left( \varvec{x} \right)$, let

$$\begin{aligned} f \left( \varvec{\phi } + \varvec{\varphi }, u \left( \varvec{\phi } + \varvec{\varphi } \right) , \varvec{\nabla }_z u \left( \varvec{\phi } + \varvec{\varphi } \right) \right) = \int _{\varOmega \left( \varvec{\phi } + \varvec{\varphi } \right) } h \left( u \left( \varvec{\phi } + \varvec{\varphi } \right) , \varvec{\nabla }_z u \left( \varvec{\phi } + \varvec{\varphi } \right) \right) \mathrm {d}z \end{aligned}$$

for an arbitrary $\varvec{\varphi } \in {\mathcal {D}}$. Then, the shape derivative (Fréchet derivative with respect to domain variation) of f is given by

$$\begin{aligned} f^\prime \left( \varvec{\phi }, u \left( \varvec{\phi } \right) , \varvec{\nabla } u \left( \varvec{\phi } \right) \right) \left[ \varvec{\varphi } \right]&= \int _{\varOmega \left( \varvec{\phi } \right) } \bigl \{ h_u \left( u \left( \varvec{\phi } \right) , \varvec{\nabla } u \left( \varvec{\phi } \right) \right) \left[ u^\prime \left( \varvec{\phi } \right) \left[ \varvec{\varphi } \right] \right] \nonumber \\&\quad +\;h_{\varvec{\nabla } u} \left( u \left( \varvec{\phi } \right) , \varvec{\nabla } u \left( \varvec{\phi } \right) \right) \left[ \varvec{\nabla } u^\prime \left( \varvec{\phi } \right) \left[ \varvec{\varphi } \right] - \varvec{\nabla } \varvec{\varphi }^\mathrm {T} \varvec{\nabla } u \left( \varvec{\phi } \right) \right] \nonumber \\&\quad +\;h \left( u \left( \varvec{\phi } \right) , \varvec{\nabla } u \left( \varvec{\phi } \right) \right) \varvec{\nabla } \cdot \varvec{\varphi } \bigr \} \mathrm {d}x, \end{aligned}$$

(A.1)

where $u^\prime \left( \varvec{\phi } \right) \left[ \varvec{\varphi } \right]$ is the shape derivative of $u \left( \varvec{\phi } \right)$ with respect to $\varvec{\varphi } \in {\mathcal {D}}$.