Driving force profile design in comb drive electrostatic actuators using a level set-based shape optimization method

Abstract

Electrostatic actuators, actuators actuated by electrostatic forces, are now widely used as sensors and switches, especially in Micro-Electro-Mechanical Systems (MEMS). Among different kinds of electrostatic actuators, the comb drive type is one of the most popular because it has a relatively large range of displacement. In design problems for electrostatic actuators, the driving force profile is of primary engineering importance. In this paper, we develop a structural optimization method for comb drive electrostatic actuators that achieves prescribed driving force profiles, based on a level set-based shape optimization method that provides optimal configurations with clear boundaries, solutions that are valid in an engineering sense. Accurate calculation of the electrostatic forces that occur on the structural boundaries during optimization is important for developing actuators that operate with prescribed driving forces. In the conventional level set-based shape optimization methods, inaccuracies in the calculation of these electrostatic forces occur because the structural boundaries are seldom aligned with the finite element method (FEM) nodes. To precisely calculate the electrostatic forces, we developed a mesh adaptation scheme by which the finite element nodes are brought into alignment with the structural boundaries at every iteration of the optimization procedure. In the following, we explain the details of the proposed level set-based shape optimization method, in which a multi-objective optimization problem is formulated to achieve a prescribed driving force profile. The sensitivity is derived using the adjoint variable method. Four numerical examples are provided, to examine the suitability of the proposed optimization method.

Appendix: Sensitivity Analysis

The details of the derivation of the sensitivity are now provided. First, the Lagrangian $\overline {F}$ is rewritten from (29) as follows:

$$\begin{array}{@{}rcl@{}} \overline{F} &=&\! \log\! \left[ \sum\limits_{j=1}^{m} \exp \left\{ \mathit{w}^{j}\, \left( {\int}_{D} \boldsymbol{\mathit{t}}(\boldsymbol{\nabla} V^{j} ) \overline{\boldsymbol\delta}(\boldsymbol\phi) \mathrm{d}D -\boldsymbol{\mathit{T}}^{j*} \right)^{2} \right\} \right]\\ &+&\sum\limits_{j=1}^{m} \left[ -{\int}_{D} (\boldsymbol{\nabla} \hat{V}^{j} )\cdot\{ \upepsilon (\boldsymbol\phi)\boldsymbol{\nabla} V^{j} \}~ \mathrm{d}D +{\int}_{\Gamma} \hat{V}^{j} q^{j} \mathrm{d}{\Gamma} \right]\\ \end{array} $$

(33)

The Lagrangian $\overline {F}$ changes when the design variable ϕ slightly changes, so we have

$$\begin{array}{rll}\overline{F}&+& \delta \overline{F} \notag\\&=&\!\log\! \left[ \! \sum\limits_{j=1}^{m} \exp\! \left\{ \mathit{w}^{j}\!\left\{\! \int_{D}\! \left( \!\boldsymbol{\mathit{t}}(\boldsymbol{\nabla} V^{j})\!+\!\frac{\partial \boldsymbol{\mathit{t}}(\boldsymbol{\nabla} V^{j}) }{\partial (\boldsymbol{\nabla} V^{j})}{\boldsymbol\delta}(\boldsymbol{\nabla} V^{j})\! \right)\right.\right.\right.\notag\\&&\left.\left.\left.\left(\overline{\boldsymbol\delta}(\boldsymbol{\phi})+\frac{\partial \overline{\boldsymbol\delta}(\boldsymbol{\phi})}{\partial \boldsymbol{\phi}}{\boldsymbol\delta} \boldsymbol{\phi} \right) \mathrm{d}D -\boldsymbol{\mathit{T}}^{j*} \right\}^{2} \, \right\} \right] \notag\\&+&\sum\limits_{j=1}^{m} \Biggl[ -\int_{D} (\boldsymbol{\nabla} \hat{V}^{j} )\cdot\left\{ \left(\upepsilon(\boldsymbol{\phi})+\frac{\partial \upepsilon (\boldsymbol{\phi})}{\partial \boldsymbol{\phi}} \delta \boldsymbol{\phi} \right)\right. \notag\end{array} $$

(34)

Therefore, we have

$$\begin{array}{@{}rcl@{}} \delta \overline{F} &=& \log \left[ \sum\limits_{j=1}^{m} \exp \left\{ \mathit{w}^{j}\, \left\{ {\int}_{D} \left( \boldsymbol{\mathit{t}}+\frac{\partial \boldsymbol{\mathit{t}}(\boldsymbol{\nabla} V^{j})} {\partial (\boldsymbol{\nabla} V^{j})} {\boldsymbol\delta}(\boldsymbol{\nabla} V^{j}) \right) \right.\right.\right.\\ &&\left.\left.\left.\overline{\boldsymbol\delta}(\boldsymbol\phi)~ \mathrm{d}D + {\int}_{D} \left( \boldsymbol{\mathit{t}}(\boldsymbol{\nabla} V^{j}) \frac{\partial \overline{\boldsymbol\delta}(\boldsymbol\phi)}{\partial \boldsymbol{\phi}}\delta \boldsymbol{\phi} \right) \mathrm{d}D -\boldsymbol{\mathit{T}}^{j*} \right\}^{2} \right\} \right]\\ &-&\!\log \!\left[ \! \sum\limits_{j=1}^{m} \exp\! \left\{ \mathit{w}^{j}\, \left( {\int}_{D} \boldsymbol{\mathit{t}} (\boldsymbol{\nabla} V^{j}) \overline{\boldsymbol\delta}(\boldsymbol{\phi})~ \mathrm{d}D -\boldsymbol{\mathit{T}}^{j*} \right)^{2} \right\} \!\right]\\ &+&\sum\limits_{j=1}^{m} \left[ -{\int}_{D} (\boldsymbol{\nabla} \hat{V}^{j} )\cdot\left\{{\vphantom{\frac{\partial \upepsilon(\boldsymbol{\phi})}{\partial \boldsymbol{\phi}}}} \upepsilon(\boldsymbol{\phi}){\boldsymbol\delta}(\boldsymbol{\nabla} V^{j})\right.\right. \\ &+&\left.\left.\frac{\partial \upepsilon(\boldsymbol{\phi})}{\partial \boldsymbol{\phi}} \delta \boldsymbol{\phi} \boldsymbol{\nabla} V^{j} \right\} \mathrm{d}D+{\int}_{\Gamma} \hat{V}^{j} \delta q^{j} \mathrm{d}{\Gamma} \right], \end{array} $$

(35)

where a second order variation is neglected. Here, following notations are introduced.

$$\boldsymbol{\mathit{T}}^{j} :={\int}_{D} \boldsymbol{\mathit{t}}(\boldsymbol{\nabla} V^{j}) \cdot \overline{\boldsymbol\delta}(\boldsymbol\phi) \mathrm{d}D$$

(36)

$$\begin{array}{@{}rcl@{}} {\boldsymbol\delta}\boldsymbol{\mathit{T}}^{j}&:=& {\int}_{D} \left( \frac{\partial \boldsymbol{\mathit{t}}(\boldsymbol{\nabla} V^{j}) }{\partial (\boldsymbol{\nabla} V^{j})} {\boldsymbol\delta}(\boldsymbol{\nabla} V^{j}) \right) {\boldsymbol\delta}(\boldsymbol{\phi})~ \mathrm{d}D\\ &+& {\int}_{D} \left( \boldsymbol{\mathit{t}}(\boldsymbol{\nabla} V^{j}) \frac{\partial \overline{\boldsymbol\delta}(\boldsymbol{\phi})}{\partial \boldsymbol{\phi}}\delta \boldsymbol{\phi} \right) \mathrm{d}D\end{array} $$

(37)

$$\mathit{w}^{j} := \exp \left( \mathit{w}^{j}\, \left( \boldsymbol{\mathit{T}}^{j} -\boldsymbol{\mathit{T}}^{j*} \right)^{2} \right) $$

(38)

Using the above notations, the first and second terms of the variation δ F can be evaluated as follows:

$$\begin{array}{@{}rcl@{}}\delta F&=& \log \left[\sum\limits_{j=1}^{m} \exp \left\{ \mathit{w}^{j}\, \left( \boldsymbol{\mathit{T}}^{j} + \delta\boldsymbol{\mathit{T}}^{j} -\boldsymbol{\mathit{T}}^{j*} \right)^{2} \right\} \right]\\ &\qquad -\log \left[\sum\limits_{j=1}^{m} \exp \left\{ \mathit{w}^{j}\, \left( \boldsymbol{\mathit{T}} -\boldsymbol{\mathit{T}}^{j*} \right)^{2} \right\} \right]\end{array} $$

(39)

$$ = \log \left[\frac{\sum\limits_{j=1}^{m} \exp \left\{ \mathit{w}^{j}\, \left( \boldsymbol{\mathit{T}}^{j} +\delta \boldsymbol{\mathit{T}}^{j} -\boldsymbol{\mathit{T}}^{j*} \right)^{2} \right\}}{\sum\limits_{j=1}^{m} \exp \left\{ \mathit{w}^{j}\, \left( \boldsymbol{\mathit{T}}^{j} -\boldsymbol{\mathit{T}}^{j*} \right)^{2} \right\}}\right] $$

(40)

Here, we recall the Maclaurin expansion with respect to exp(x):

$$ \exp(\mathit{x})=\sum\limits_{n=0}^{\infty} \frac{x^{n}}{n!} $$

(41)

Therefore, we have

$$\delta F = \log \left[1+ \frac{ 2\sum\limits_{j=1}^{m} \mathit{w}^{j} \mathit{W}^{j} (\boldsymbol{\mathit{T}}^{j})-\boldsymbol{\mathit{T}}^{j*}) \cdot \boldsymbol{\delta}\boldsymbol{\mathit{T}}^{j} } { \sum\limits_{j=1}^{m} \mathit{W}^{j} } \right] $$

(42)

Furthermore, we recall the following relationship:

$$ e=\lim\limits_{x\rightarrow 0} \left( 1+x \right)^{\frac{1}{x}}. $$

(43)

Therefore, we have

$$ \delta F = \left(\frac{2}{\sum \mathit{W}^{j}}\right) \sum\limits_{j=1}^{m} \mathit{w}^{j}\mathit{W}^{j} (\boldsymbol{\mathit{T}}^{j}-\boldsymbol{\mathit{T}}^{j*}) \cdot \boldsymbol{\delta}\boldsymbol{\mathit{T}}^{j} $$

(44)

Given the boundary condition imposed on Γ _q , we have following relationship:

$$ \delta q^{j} =0 \qquad \text{on} \quad {\Gamma}_{q}\,. $$

(45)

Thus, the boundary integral term can be evaluated as follows:

$$ {\int}_{\Gamma} \hat{V}^{j} \boldsymbol{\delta} q^{j} \mathrm{d}{\Gamma} \rightarrow {\int}_{{\Gamma}_{V}} \hat{V}^{j} \boldsymbol{\delta} q^{j} \mathrm{d}{\Gamma}. $$

(46)

The variation of $\overline {F}$ can therefore be evaluated as

$$\begin{array}{rll}\delta \overline{F} &=&\sum_{j=1}^{m}\Biggl[\int_{D} \Bigl[ -2 \upepsilon_{0} \left\{ \left\{(\boldsymbol{\nabla} V^{j})\otimes \overline{\boldsymbol\delta}(\boldsymbol\phi) \right\}^{T} \boldsymbol{\mathit{a}}_{T}^{j} \right\}\notag\\&+& \upepsilon_{0} \left\{ \boldsymbol{\mathit{a}}_{T}^{j} \cdot \left\{ \boldsymbol{\mathit{I}} \overline{\boldsymbol\delta}(\boldsymbol\phi) \right\} (\boldsymbol{\nabla} V^{j}) \right\}- \upepsilon (\boldsymbol\phi)\boldsymbol{\nabla} \hat{V}^{j} \Bigl] \cdot {\boldsymbol\delta}(\boldsymbol{\nabla} V^{j}) \mathrm{d}D\Biggl]\notag\\&+&\sum_{j=1}^{m}\Biggl[\int_{D} \Biggl\{ \left( \dfrac{2\mathit{w}^{j} \mathit{W}^{j} (\boldsymbol{\mathit{T}}^{j}-\boldsymbol{\mathit{T}}^{j*})}{\sum \mathit{W}^{j} }\right) \cdot \left( \boldsymbol{\mathit{t}}(\boldsymbol{\nabla} V^{j})\dfrac{\partial \overline{\boldsymbol\delta}(\boldsymbol\phi)}{\partial \boldsymbol{\phi}} \right)\notag\\&-&(\boldsymbol{\nabla} \hat{V}^{j} )\cdot\left( \dfrac{\partial \upepsilon(\boldsymbol\phi)}{\partial \boldsymbol{\phi}} \boldsymbol{\nabla} V^{j} \right)\Biggl\} \delta \boldsymbol{\phi} \mathrm{d}D\Biggl]\notag\\&+&\sum_{j=1}^{m} \left[ \int_{\Gamma_{V}} \hat{V}^{j} \boldsymbol{\delta} q^{j} \mathrm{d}\Gamma \right]. \end{array} $$

(47)

To make the terms pertaining to $\overline {\boldsymbol \delta }(\boldsymbol {\nabla } V^{j})$ and δ q ^j equivalent to zero, we define the adjoint system as follows:

$$ \left\{\begin{array}{rll} \boldsymbol{\nabla} \hat{V}^{j} &=& -2 \frac{{\upepsilon}_{0}}{\upepsilon (\boldsymbol{\phi})} \left[ \left\{(\boldsymbol{\nabla} V^{j})\otimes \boldsymbol{{(\boldsymbol\varphi)}} \right\}^{T} \boldsymbol{\mathit{a}}_{T}^{j} \right]\\ && +\frac{{\upepsilon}_{0}}{\upepsilon (\boldsymbol{\phi})} \left[ \boldsymbol{\mathit{a}}_{T}^{j} \cdot \left\{ \boldsymbol{\mathit{I}} \overline{\boldsymbol\delta}(\boldsymbol{\phi}) \right\} (\boldsymbol{\nabla} V^{j}) \right] \qquad \text{in} \quad D \\ &&\qquad \hat{V}^{j} =0 \qquad \qquad \text{on} \quad {\Gamma}_{V} \end{array}\right. $$

(48)

Then, $\delta \overline {F}$ can be expressed as

$$\begin{array}{@{}rcl@{}} \delta \overline{F} &=\sum\limits_{j=1}^{m} \int_{D} \Biggl\{ \left( \frac{2\mathit{w}^{j} \mathit{W}^{j} (\boldsymbol{\mathit{T}}^{j}-\boldsymbol{\mathit{T}}^{j*})}{\sum \mathit{W}^{j} }\right)\! \cdot \! \left( \boldsymbol{\mathit{t}}(\boldsymbol{\nabla} V^{j})\frac{\partial \overline{\boldsymbol\delta}(\boldsymbol{\phi})}{\partial \boldsymbol{\phi}} \right)\notag\\&\qquad-(\boldsymbol{\nabla} \hat{V}^{j} )\cdot\left( \frac{\partial \upepsilon (\boldsymbol{\phi})}{\partial \boldsymbol{\phi}} \boldsymbol{\nabla} V^{j} \right)\Biggl\} \delta \boldsymbol{\phi}~ \mathrm{d}D \end{array} $$

(49)