**3.1 Single-objective optimization**

Regarding NiTi recovery stress, there would appear to be an optimal thickness range for which recovery stress reaches max values as depicted in **Figure 3**, below which there is a sharp drop off. Therefore the tendency for increased deflections for thinner materials reaches a point of diminishing returns due to the effect of decreasing recovery stress. Below 100–150 nm, shape memory properties have been

**Figure 3.** *Relation between recovery stress (MPa) and NiTi film thickness (nm).*

*Optimization of MEMS Actuator Driven by Shape Memory Alloy Thin Film Phase Change DOI: http://dx.doi.org/10.5772/intechopen.92393*

shown to drop off completely, so we impose constraints for NiTi thickness to vary between 150 and 1300 nm.

Based on the curve fitting equation (*third-order polynomial*), the single-objective optimization problem can be written as following:

#### *maximize* :

$$d = \frac{3E\_{\rm{NIT}}\sigma\_{\rm{nr}}t\_{\rm{NIT}}t\_{\rm{SU-8}}(t\_{\rm{NIT}} + t\_{\rm{SU-8}})l^2}{E\_{\rm{NIT}}^2t\_{\rm{NIT}}^4 + E\_{\rm{SU-8}}E\_{\rm{NIT}}(4t\_{\rm{NIT}}^3t\_{\rm{SU-8}} + 6t\_{\rm{NIT}}^2t\_{\rm{SU-8}}^2 + 4t\_{\rm{NIT}}t\_{\rm{SU-8}}^3) + E\_{\rm{SU-8}}^2t\_{\rm{SU-8}}^4} \tag{2}$$

*subjected to*::

$$100\ \mu m \le l \le \text{300 } \mu m \tag{3}$$

$$150\ mm \le t\_{NiTi} \le 1000\ mm \tag{4}$$

$$200\ mm \le t\_{SU \cdot 8} \le 2000\ nm \tag{5}$$

$$
\sigma\_{\text{nc}} \ge 0 \tag{6}
$$

$$
\sigma\_{\rm rec} = \textbf{5.3} \textbf{6E} \textbf{26t}\_{\rm NiTi}^{-3} - 2.1 \textbf{5E} \textbf{21t}\_{\rm NiTi}^{-2} + 2.4 \textbf{5E} \textbf{15t}\_{\rm NiTi} - 2.5 \textbf{8E} \textbf{08} \text{ (SI units)}\tag{7}
$$

Covert to standard form in SI units:

$$\begin{array}{c} \text{minimize}:\\ f = \frac{-3E\_{\text{NTi}}\sigma\_{\text{re}}t\_{\text{NTi}}t\_{\text{SU-8}}(t\_{\text{NTi}} + t\_{\text{SU-8}})l^2}{E\_{\text{NTi}}^2t\_{\text{NTi}}^4 + E\_{\text{SU-8}}E\_{\text{NTi}}\left(4t\_{\text{NTi}}^3t\_{\text{SU-8}} + 6t\_{\text{NTi}}^2t\_{\text{SU-8}}^2 + 4t\_{\text{NTi}}t\_{\text{SU-8}}^3\right) + E\_{\text{SU-8}}^2t\_{\text{SU-8}}^4} \end{array} \tag{8}$$

*subjected to*::

**3. Methods, results, and discussions**

*SMA MEMS fabrication process for SU-8 on NiTi bimorph.*

Regarding NiTi recovery stress, there would appear to be an optimal thickness range for which recovery stress reaches max values as depicted in **Figure 3**, below which there is a sharp drop off. Therefore the tendency for increased deflections for

decreasing recovery stress. Below 100–150 nm, shape memory properties have been

thinner materials reaches a point of diminishing returns due to the effect of

**3.1 Single-objective optimization**

*Advanced Functional Materials*

**Figure 2.**

**Figure 3.**

**28**

*Relation between recovery stress (MPa) and NiTi film thickness (nm).*

$$g\_1: 100 \times 10^{-6} - l \le 0;\tag{9}$$

$$\text{g}\_2: l - \text{300} \times 10^{-6} \le 0;\tag{10}$$

$$\mathbf{g}\_3: \mathbf{150} \times \mathbf{10^{-9}} - t\_{\mathrm{NiTi}} \le \mathbf{0};\tag{11}$$

$$\text{g}\_4: t\_{\text{NTi}} - 1300 \times 10^{-9} \le 0;\tag{12}$$

$$\text{g}\_{\text{5}} : 200 \times 10^{-9} - t\_{SU \cdot 8} \le 0; \tag{13}$$

$$\mathbf{g}\_{\mathbf{6}} : \mathbf{t}\_{\mathrm{SU}\cdot\mathbf{8}} - 2000 \times 10^{-9} \le \mathbf{0};\tag{14}$$

$$\mathbf{g}\_{\mathcal{T}} : -\sigma\_{\rm rec} \le \mathbf{0} \tag{15}$$

*<sup>h</sup>*1: *<sup>σ</sup>rec* � <sup>5</sup>*:*<sup>36</sup> � <sup>10</sup><sup>26</sup>*tNiTi* <sup>3</sup> <sup>þ</sup> <sup>2</sup>*:*<sup>15</sup> � <sup>10</sup><sup>21</sup>*tNiTi* <sup>2</sup> � <sup>2</sup>*:*<sup>45</sup> � 1015*tNiTi* <sup>þ</sup> <sup>2</sup>*:*<sup>58</sup> � <sup>10</sup><sup>8</sup> <sup>¼</sup> <sup>0</sup> (16)

#### **3.2 MATLAB optimization toolbox (fmincon)**

According to the toolbox (and as shown in **Figure 4**), optimal solution is: *tNiTi* <sup>¼</sup> <sup>359</sup> *nm*, *tSU*‐<sup>8</sup> <sup>¼</sup> <sup>824</sup> *nm*, *<sup>l</sup>* <sup>¼</sup> <sup>300</sup> *<sup>μ</sup>m*.

#### **3.3 Multi-objective optimization**

The curvature of a bilayer elastic material [46] is given as

$$K = \frac{-E\_{SU \cdot \\$}' t\_{SU \cdot \\$} E\_{NT\overline{\text{t}}}' t\_{N\overline{\text{t}}\overline{\text{t}}} (t\_{N\overline{\text{t}}\overline{\text{t}}} + t\_{SU \cdot \\$})}{G \left( E\_{SU \cdot \\$}' t\_{SU \cdot \\$} + E\_{N\overline{\text{t}}\overline{\text{T}}}' t\_{N\overline{\text{t}}\overline{\text{t}}} \right)} \Delta \varepsilon \tag{17}$$

#### **Figure 4.**

*Optimization contours for the case where SU-8 elastic modulus is 2 GPa. Variables considered are individual layer thicknesses: NiTi (x-axis) and SU-8 (y-axis).*

$$G = E'\_{\text{SU-S}}t\_{\text{SU-S}}^2 \left(\frac{\text{t}\_{\text{NTT}}}{2} - \frac{\text{t}\_{\text{SU-S}}}{6} - \theta\right) \\ - E'\_{\text{MTT}}\mathbf{t}\_{\text{NTT}} \left[\mathbf{t}\_{\text{SU-S}} \left(\mathbf{t}\_{\text{SU-S}} + \frac{\mathbf{t}\_{\text{NTT}}}{2}\right) + \frac{\mathbf{t}\_{\text{NTT}}^2}{6} + \theta \left(\mathbf{2}\_{\text{SU-S}} + \mathbf{t}\_{\text{NTT}}\right)\right] \tag{18}$$

$$\theta = \frac{t\_{\text{NiTi}} t\_{\text{SU-8}} \left( E\_{\text{NiTi}}' - E\_{\text{SU-8}}' \right)}{2 \left( E\_{\text{SU-8}}' t\_{\text{SU-8}} + E\_{\text{NiTi}}' t\_{\text{NiTi}} \right)} \tag{19}$$

Maximize:

**Figure 5.**

Subjected to:

*h*<sup>1</sup> : *G* � *E*<sup>0</sup>

� *E*<sup>0</sup>

*maximize:*

**31**

*SU*‐<sup>8</sup>*<sup>t</sup>* 2 *SU*‐<sup>8</sup>

tion problem can be stated as follows:

*<sup>ρ</sup>* <sup>¼</sup> <sup>1</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.92393*

*<sup>K</sup>* ¼ � *G E*<sup>0</sup>

*and lower bound on SU-8 thickness are active constraints for objective function 2.*

*E*0

*tNiTi* <sup>2</sup> � *tSU*‐<sup>8</sup>

*NiTitNiTi tSU*‐<sup>8</sup> *tSU*‐<sup>8</sup> <sup>þ</sup>

*SU*‐<sup>8</sup>*tSU*‐<sup>8</sup>*E*<sup>0</sup>

*MATLAB-generated contour plot of curvature radius (m), against the primary design variables (i.e., tNiTi and tSU-8). Curvature radius is maximized for the thickest values of NiTi and thinnest values of SU-8. The result is intuitive because this is the stiffest beam (from the perspective of thickest NiTi with much larger Young's modulus compared to SU-8). Thinner SU-8 means the effect from strain differential and CTE mismatch is minimized and contributes less to curvature radius. Overall, this means that upper bound on NiTi thickness*

*Optimization of MEMS Actuator Driven by Shape Memory Alloy Thin Film Phase Change*

<sup>6</sup> � *<sup>θ</sup>*

*tNiTi* 2 <sup>þ</sup>

*SU*‐<sup>8</sup>*tSU*‐<sup>8</sup> <sup>þ</sup> *<sup>E</sup>*<sup>0</sup>

For multi-objective optimization, the deflection and curvature radius of *SMA* bimorph actuator are maximized simultaneously. So, the multi-objective optimiza-

> *SU*‐<sup>8</sup>*tSU*‐<sup>8</sup> <sup>þ</sup> *<sup>E</sup>*<sup>0</sup> *NiTitNiTi*

*t* 2 *NiTi* <sup>6</sup> <sup>þ</sup> *<sup>θ</sup>*ð Þ <sup>2</sup>*tSU*‐<sup>8</sup> <sup>þ</sup> *tNiTi*

¼ 0 (27)

*NiTi* � *E*<sup>0</sup>

*SU*‐<sup>8</sup>

*NiTitNiTi* <sup>¼</sup> <sup>0</sup> (28)

*NiTitNiTi*ð Þ *tNiTi* <sup>þ</sup> *tSU*‐<sup>8</sup> *Δε* (29)

*<sup>h</sup>*<sup>2</sup> : *<sup>θ</sup>* � *tNiTitSU*‐<sup>8</sup> *<sup>E</sup>*<sup>0</sup>

2 *E*<sup>0</sup>

*<sup>f</sup>* <sup>1</sup> : *<sup>ρ</sup>* ¼ � *G E*<sup>0</sup>

*E*0

*SU*‐<sup>8</sup>*tSU*‐<sup>8</sup>*E*<sup>0</sup>

*SU*‐<sup>8</sup>*tSU*‐<sup>8</sup> <sup>þ</sup> *<sup>E</sup>*<sup>0</sup> *NiTitNiTi*

*NiTitNiTi*ð Þ *tNiTi* <sup>þ</sup> *tSU*‐<sup>8</sup> *Δε* (22)

*<sup>g</sup>*<sup>1</sup> : <sup>150</sup> � <sup>10</sup>�<sup>9</sup> � *tNiTi* <sup>≤</sup> 0; (23) *<sup>g</sup>*<sup>2</sup> : *tNiTi* � <sup>1300</sup> � <sup>10</sup>�<sup>9</sup> <sup>≤</sup> 0; (24)

*<sup>g</sup>*<sup>3</sup> : <sup>200</sup> � <sup>10</sup>�<sup>9</sup> � *tSU*‐<sup>8</sup> <sup>≤</sup>0; (25) *<sup>g</sup>*<sup>4</sup> : *tSU*‐<sup>8</sup> � <sup>2000</sup> � <sup>10</sup>�<sup>9</sup> <sup>≤</sup>0; (26)

$$
\Delta \varepsilon = (a\_{\text{SU}:\\$} - a\_{\text{NTi}}) \Delta T \tag{20}
$$

ρ is the curvature radius generally expressed in units of μm. Δε is a strain differential term resulting from CTE mismatch and temperature difference experienced during the processing. θ is a correction factor used in the placement of neutral plane. E<sup>0</sup> is the biaxial modulus defined as E/(1 � v) where v is Poisson ratio and E is Young's modulus. Poisson ratios are assumed to be 0.22 for SU-8 and 0.33 for NiTi. <sup>α</sup>\_SU-8 is reported to be 52 � <sup>10</sup>–<sup>6</sup> /°C. α\_NiTi (depending on austenite or martensite phase) is reported to be 6.6 or 11 � <sup>10</sup>–<sup>6</sup> /°C. For simplicity sake, we assume an intermediate value of <sup>α</sup>\_NiTi = 9 � <sup>10</sup>–<sup>6</sup> /°C. Units for theta term is nm or m. Units for G term is Pa � nm3 or Pa � <sup>m</sup><sup>3</sup> . Therefore units for curvature is in nm or m. Δε term is unit less.

The objective number 2 is to maximize curvature radius. We determine the pareto frontier and strong pareto points using the epsilon constrained method. In this epsilon constrained method, we minimize f1 while keeping f2 less than or equal to different values of epsilon. As a first step for objective function 2 (curvature of bimorph) we coded MATLAB script to generate contour plots as a function of the two main design variables (i.e., thickness of NiTi and SU-8). The problem formulation for objective function 2 is as follows (and contour plot is shown in **Figure 5**).

Curvature is:

$$K = -\frac{E\_{SU\cdot\-8}' t\_{SU\cdot\-8} E\_{NiTi}' t\_{NiTi} (t\_{NiTi} + t\_{SU\cdot\-8})}{G \left( E\_{SU\cdot\-8}' t\_{SU\cdot\-8} + E\_{NiTi}' t\_{NiTi} \right)} \Delta \varepsilon \tag{21}$$

*Optimization of MEMS Actuator Driven by Shape Memory Alloy Thin Film Phase Change DOI: http://dx.doi.org/10.5772/intechopen.92393*

#### **Figure 5.**

*G* ¼ *E*<sup>0</sup> *SU*‐<sup>8</sup>*<sup>t</sup>* 2 *SU*‐<sup>8</sup>

**Figure 4.**

*tNiTi* <sup>2</sup> � *tSU*‐<sup>8</sup> <sup>6</sup> � *<sup>θ</sup>* 

*Advanced Functional Materials*

*layer thicknesses: NiTi (x-axis) and SU-8 (y-axis).*

<sup>α</sup>\_SU-8 is reported to be 52 � <sup>10</sup>–<sup>6</sup>

for G term is Pa � nm3 or Pa � <sup>m</sup><sup>3</sup>

term is unit less.

Curvature is:

**30**

ite phase) is reported to be 6.6 or 11 � <sup>10</sup>–<sup>6</sup>

*<sup>K</sup>* ¼ � *<sup>E</sup>*<sup>0</sup>

*SU*‐<sup>8</sup>*tSU*‐<sup>8</sup>*E*<sup>0</sup>

*G E*<sup>0</sup>

intermediate value of <sup>α</sup>\_NiTi = 9 � <sup>10</sup>–<sup>6</sup>

� *E*<sup>0</sup>

*NiTitNiTi tSU*‐<sup>8</sup> *tSU*‐<sup>8</sup> <sup>þ</sup>

*Optimization contours for the case where SU-8 elastic modulus is 2 GPa. Variables considered are individual*

*SU*‐<sup>8</sup>*tSU*‐<sup>8</sup> <sup>þ</sup> *<sup>E</sup>*<sup>0</sup>

ρ is the curvature radius generally expressed in units of μm. Δε is a strain differential term resulting from CTE mismatch and temperature difference experienced during the processing. θ is a correction factor used in the placement of neutral plane. E<sup>0</sup> is the biaxial modulus defined as E/(1 � v) where v is Poisson ratio and E is Young's modulus. Poisson ratios are assumed to be 0.22 for SU-8 and 0.33 for NiTi.

The objective number 2 is to maximize curvature radius. We determine the pareto frontier and strong pareto points using the epsilon constrained method. In this epsilon constrained method, we minimize f1 while keeping f2 less than or equal to different values of epsilon. As a first step for objective function 2 (curvature of bimorph) we coded MATLAB script to generate contour plots as a function of the two main design variables (i.e., thickness of NiTi and SU-8). The problem formulation for objective function 2 is as follows (and contour plot is shown in **Figure 5**).

*SU*‐<sup>8</sup>*tSU*‐<sup>8</sup> <sup>þ</sup> *<sup>E</sup>*<sup>0</sup>

*NiTitNiTi*ð Þ *tNiTi* <sup>þ</sup> *tSU*‐<sup>8</sup>

*NiTitNiTi*

*Δε* (21)

*<sup>θ</sup>* <sup>¼</sup> *tNiTitSU*‐<sup>8</sup> *<sup>E</sup>*<sup>0</sup>

2 *E*<sup>0</sup>

*tNiTi* 2 

*SU*‐<sup>8</sup>

*NiTitNiTi*

*NiTi* � *E*<sup>0</sup>

þ *t* 2 *NiTi*

(19)

/°C. α\_NiTi (depending on austenite or martens-

. Therefore units for curvature is in nm or m. Δε

/°C. For simplicity sake, we assume an

/°C. Units for theta term is nm or m. Units

*Δε* <sup>¼</sup> ð Þ *<sup>α</sup>SU*‐<sup>8</sup> � *<sup>α</sup>NiTi <sup>Δ</sup><sup>T</sup>* (20)

<sup>6</sup> <sup>þ</sup> *<sup>θ</sup>*ð Þ <sup>2</sup>*tSU*‐<sup>8</sup> <sup>þ</sup> *tNiTi*

(18)

*MATLAB-generated contour plot of curvature radius (m), against the primary design variables (i.e., tNiTi and tSU-8). Curvature radius is maximized for the thickest values of NiTi and thinnest values of SU-8. The result is intuitive because this is the stiffest beam (from the perspective of thickest NiTi with much larger Young's modulus compared to SU-8). Thinner SU-8 means the effect from strain differential and CTE mismatch is minimized and contributes less to curvature radius. Overall, this means that upper bound on NiTi thickness and lower bound on SU-8 thickness are active constraints for objective function 2.*

#### Maximize:

$$\rho = \frac{1}{K} = -\frac{G\left(E\_{SU\cdot\-8}'t\_{SU\cdot\-8} + E\_{N\dot{T}\dot{T}}'t\_{N\dot{T}\dot{T}}\right)}{E\_{SU\cdot\-8}'t\_{SU\cdot\-8}E\_{N\dot{T}\dot{T}}'t\_{N\dot{T}\dot{T}}(t\_{N\dot{T}\dot{T}} + t\_{SU\cdot\-8})\Delta\varepsilon} \tag{22}$$

Subjected to:

$$g\_1: \mathbf{150} \times \mathbf{10^{-9}} - t\_{\text{NiT}} \le \mathbf{0};\tag{23}$$

$$\mathbf{g}\_2: \mathbf{t}\_{\mathrm{NiT}} - \mathbf{1300} \times \mathbf{10^{-9}} \le \mathbf{0};\tag{24}$$

$$\text{g}\_3: 200 \times 10^{-9} - t\_{SU \cdot 8} \le 0;\tag{25}$$

$$\text{g}\_4: t\_{\text{SU}\cdot\text{8}} - 2000 \times 10^{-9} \le 0;\tag{26}$$

$$\begin{aligned} \left[h\_1: G - E'\_{SU \cdot \mathbb{8}} t\_{SU \cdot \mathbb{8}}^2 \left(\frac{t\_{NT\overline{\text{v}}}}{2} - \frac{t\_{SU \cdot \mathbb{8}}}{6} - \theta\right) \\ -E'\_{N\overline{\text{v}}\overline{\text{T}}} t\_{N\overline{\text{T}}\overline{\text{i}}} \left[t\_{SU \cdot \mathbb{8}} \left(t\_{SU \cdot \mathbb{8}} + \frac{t\_{N\overline{\text{T}}\overline{\text{v}}}}{2}\right) + \frac{t\_{N\overline{\text{T}}\overline{\text{v}}}^2}{6} + \theta (2t\_{SU \cdot \mathbb{8}} + t\_{N\overline{\text{T}}\overline{\text{v}}})\right] \\ = \mathbf{0} \end{aligned} \tag{27}$$

$$h\_2: \theta - \frac{t\_{\text{NTi}\uparrow} t\_{\text{SU}\cdot\ $} \left( E'\_{\text{NTi}\uparrow} - E'\_{\text{SU}\cdot\$ } \right)}{2 \left( E'\_{\text{SU}\cdot\ $} t\_{\text{SU}\cdot\$ } + E'\_{\text{NTi}\uparrow} t\_{\text{NTi}\uparrow} \right)} = \mathbf{0} \tag{28}$$

For multi-objective optimization, the deflection and curvature radius of *SMA* bimorph actuator are maximized simultaneously. So, the multi-objective optimization problem can be stated as follows:

*maximize:*

$$f\_1: \rho = -\frac{G\left(E\_{SU \cdot \\$}' t\_{SU \cdot \\$} + E\_{N \dot{\text{T}} \dot{\text{T}}}' t\_{N \dot{\text{T}} \dot{\text{T}}}\right)}{E\_{SU \cdot \\$}' t\_{SU \cdot \\$} E\_{N \dot{\text{T}} \dot{\text{T}}}' t\_{N \dot{\text{T}} \dot{\text{T}}} (t\_{N \dot{\text{T}} \dot{\text{T}}} + t\_{SU \cdot \\$}) \Delta \varepsilon} \tag{29}$$

$$f\_{\cdot 2} : d = \frac{3E\_{\rm NITi} \sigma\_{\rm rec} t\_{\rm NITi} t\_{\rm SU-8} (t\_{\rm NITi} + t\_{\rm SU-8}) l^2}{E\_{\rm NITi}^2 t\_{\rm NITi}^4 + E\_{\rm SU-8} E\_{\rm NITi} (4t\_{\rm NITi}^3 t\_{\rm SU-8} + 6t\_{\rm NITi}^2 t\_{\rm SU-8}^2 + 4t\_{\rm NITi} t\_{\rm SU-8}^3) + E\_{\rm SU-8}^2 t\_{\rm SU-8}^4} \tag{30}$$

Subjected to:

$$\lg\_1: \mathbf{100} \times \mathbf{10^{-6}} - l \le \mathbf{0};\tag{31}$$

$$\text{g}\_2: l - \text{300} \times \text{10}^{-6} \le 0;\tag{32}$$

$$\mathbf{g}\_3: \mathbf{150} \times \mathbf{10}^{-9} - t\_{\text{NiTi}} \le \mathbf{0};\tag{33}$$

$$\mathbf{g}\_4: t\_{\rm{NTi}} - \mathbf{1300} \times \mathbf{10^{-9}} \le \mathbf{0};\tag{34}$$

$$\mathbf{g}\_{\ $}: 200 \times 10^{-9} - t\_{SU \cdot \$ } \le \mathbf{0};\tag{35}$$

$$\mathbf{g}\_{\mathbf{6}} : \mathbf{t}\_{\mathrm{SU}\cdot\mathbf{8}} - 2000 \times 10^{-9} \le \mathbf{0};\tag{36}$$

$$\mathfrak{g}\_{7} \colon -\mathfrak{o}\_{\text{rec}} \le \mathbf{0};\tag{37}$$

deflection. The optimal solution of multi-objective function as shown in **Figure 6**

*Optimization of MEMS Actuator Driven by Shape Memory Alloy Thin Film Phase Change*

*DOI: http://dx.doi.org/10.5772/intechopen.92393*

*Maximum bimorph deflection with a variation of Young's modulus of NiTi and SU-8 layer.*

recovery stress, and plot the deflection over a range of SU-8 thicknesses.

In conclusion, an interesting optimization problem was identified whereby the deflection of shape memory MEMS bimorph actuator was maximized. Original calculations showed that reductions in the thickness of the bimorph layers would yield maximized deflections (for the simplest case assuming constant values of recovery stress in NiTi layer). In the literature, a more complex relationship among recovery stress and the NiTi thickness was identified. A curve fit to this data yielded a much more interesting optimization problem, which was solved graphically (contour plots) and using the Optimization Toolbox in MATLAB. Optimal NiTi and SU-8 thickness were determined to be for the case where SU-8 modulus was 2 GPa to be tNiTi = 359 nm, tSU-8 = 824 nm. After solving the single-objective optimization problem using fmincon, Excel solver, and a hand-coded algorithm, we formulated a second objective function to maximize curvature radius (i.e., to maximize the

Once we have established the optimal objective values for deflection and curvature, we perform a sensitivity analysis regarding the following variables, for which experimentally could be varied with relative ease. These thickness values x1 and x2, corresponding the NiTi and SU-8 thicknesses, can be changed by varying the spin speed for SU-8 coating: faster spins corresponding to thinner films of SU-8 and vice versa. For NiTi, longer sputter time would be used for thicker films and vice versa. Young's modulus can be varied by deposition conditions for NiTi and curing/baking temperatures and conditions for SU-8. To perform the sensitivity analysis for Objective 1, we keep fixed the optimal thickness for SU-8 and vary the NiTi thickness to see how it changes, and plot a function (as shown in **Figure 7**) and generate a table of values. Similarly, we keep fixed the optimal value of NiTi thickness and

has a larger tNiTi.

**Figure 7.**

**4. Conclusions**

**33**

*<sup>h</sup>*<sup>1</sup> : <sup>σ</sup>rec � <sup>5</sup>*:*<sup>36</sup> � <sup>10</sup>26tNiTi <sup>3</sup> <sup>þ</sup> <sup>2</sup>*:*<sup>15</sup> � 1021tNiTi <sup>2</sup> � <sup>2</sup>*:*<sup>45</sup> � <sup>10</sup>15tNiTi <sup>þ</sup> <sup>2</sup>*:*<sup>58</sup> � <sup>10</sup><sup>8</sup> <sup>¼</sup> 0; (38)

$$\begin{bmatrix} \mathbf{h}\_{2}:\mathbf{G}-\mathbf{E}'\_{\mathrm{SU-3}}\mathbf{t}^{2}\_{\mathrm{SU-3}}\left(\frac{\mathbf{t}\_{\mathrm{NTT}}}{2}-\frac{\mathbf{t}\_{\mathrm{SU-3}}}{6}-\boldsymbol{\Theta}\right)-\mathbf{E}'\_{\mathrm{NTT}}\mathbf{t}\_{\mathrm{NTT}}\left[\mathbf{t}\_{\mathrm{SU-3}}\left(\mathbf{t}\_{\mathrm{SU-3}}+\frac{\mathbf{t}\_{\mathrm{NTT}}}{2}\right)+\frac{\mathbf{t}\_{\mathrm{NTT}}^{2}}{6}+\boldsymbol{\Theta}\left(2\mathbf{t}\_{\mathrm{SU-3}}+\mathbf{t}\_{\mathrm{NTT}}\right)\right]=\mathbf{0};\tag{39}$$

$$h\_3: \theta - \frac{\mathbf{t}\_{\text{NTi}} \mathbf{t}\_{\text{SU-8}} \left(\mathbf{E}'\_{\text{NTi}} - \mathbf{E}'\_{\text{SU-8}}\right)}{2\left(\mathbf{E}'\_{\text{SU-8}} \mathbf{t}\_{\text{SU-8}} + \mathbf{E}'\_{\text{NTi}} \mathbf{t}\_{\text{NTi}}\right)} = \mathbf{0}.\tag{40}$$

Due to the conflicting nature of the two objective functions, the contour plot for the multi-objective function has changed substantially. Maximizing the radius is favored by a larger tNiTi as opposed to a smaller thickness required to maximize

**Figure 6.** *Optimal solution for simultaneous multi-objective optimization of deflection and curvature radius.*

*Optimization of MEMS Actuator Driven by Shape Memory Alloy Thin Film Phase Change DOI: http://dx.doi.org/10.5772/intechopen.92393*

**Figure 7.** *Maximum bimorph deflection with a variation of Young's modulus of NiTi and SU-8 layer.*

deflection. The optimal solution of multi-objective function as shown in **Figure 6** has a larger tNiTi.

Once we have established the optimal objective values for deflection and curvature, we perform a sensitivity analysis regarding the following variables, for which experimentally could be varied with relative ease. These thickness values x1 and x2, corresponding the NiTi and SU-8 thicknesses, can be changed by varying the spin speed for SU-8 coating: faster spins corresponding to thinner films of SU-8 and vice versa. For NiTi, longer sputter time would be used for thicker films and vice versa. Young's modulus can be varied by deposition conditions for NiTi and curing/baking temperatures and conditions for SU-8. To perform the sensitivity analysis for Objective 1, we keep fixed the optimal thickness for SU-8 and vary the NiTi thickness to see how it changes, and plot a function (as shown in **Figure 7**) and generate a table of values. Similarly, we keep fixed the optimal value of NiTi thickness and recovery stress, and plot the deflection over a range of SU-8 thicknesses.

#### **4. Conclusions**

*<sup>f</sup>* <sup>2</sup> : *<sup>d</sup>* <sup>¼</sup> <sup>3</sup>*ENiTiσrectNiTitSU*‐8ð Þ *tNiTi* <sup>þ</sup> *tSU*‐<sup>8</sup> *<sup>l</sup>*

3

<sup>3</sup> <sup>þ</sup> <sup>2</sup>*:*<sup>15</sup> � 1021tNiTi

NiTitNiTi tSU‐<sup>8</sup> tSU‐<sup>8</sup> <sup>þ</sup>

SU‐8tSU‐<sup>8</sup> <sup>þ</sup> <sup>E</sup><sup>0</sup>

*Optimal solution for simultaneous multi-objective optimization of deflection and curvature radius.*

Due to the conflicting nature of the two objective functions, the contour plot for the multi-objective function has changed substantially. Maximizing the radius is favored by a larger tNiTi as opposed to a smaller thickness required to maximize

NiTi � E<sup>0</sup>

� E<sup>0</sup>

*<sup>h</sup>*<sup>3</sup> : <sup>θ</sup> � tNiTitSU‐<sup>8</sup> <sup>E</sup><sup>0</sup>

2 E<sup>0</sup>

*NiTitSU*‐<sup>8</sup> <sup>þ</sup> <sup>6</sup>*<sup>t</sup>*

2 *NiTit* 2

<sup>þ</sup> *<sup>E</sup>*<sup>2</sup>

*NiTi* <sup>þ</sup> *ESU*‐<sup>8</sup>*ENiTi* <sup>4</sup>*<sup>t</sup>*

*E*2 *NiTit* 4

*Advanced Functional Materials*

*<sup>h</sup>*<sup>1</sup> : <sup>σ</sup>rec � <sup>5</sup>*:*<sup>36</sup> � <sup>10</sup>26tNiTi

tNiTi <sup>2</sup> � tSU‐<sup>8</sup>

<sup>6</sup> � <sup>θ</sup> 

SU‐8t 2 SU‐<sup>8</sup>

*h*<sup>2</sup> : G � E<sup>0</sup>

**Figure 6.**

**32**

Subjected to:

2

3 *SU*‐<sup>8</sup> *SU*‐8*<sup>t</sup>* 4 *SU*‐<sup>8</sup> (30)

(38)

(39)

¼ 0;

*SU*‐<sup>8</sup> <sup>þ</sup> <sup>4</sup>*tNiTit*

*<sup>g</sup>*<sup>1</sup> : <sup>100</sup> � <sup>10</sup>�<sup>6</sup> � *<sup>l</sup>* <sup>≤</sup>0; (31) *<sup>g</sup>*<sup>2</sup> : *<sup>l</sup>* � <sup>300</sup> � <sup>10</sup>�<sup>6</sup> <sup>≤</sup>0; (32) *<sup>g</sup>*<sup>3</sup> : <sup>150</sup> � <sup>10</sup>�<sup>9</sup> � *tNiTi* <sup>≤</sup>0; (33) *<sup>g</sup>*<sup>4</sup> : *tNiTi* � <sup>1300</sup> � <sup>10</sup>�<sup>9</sup> <sup>≤</sup>0; (34)

*<sup>g</sup>*<sup>5</sup> : <sup>200</sup> � <sup>10</sup>�<sup>9</sup> � *tSU*‐<sup>8</sup> <sup>≤</sup>0; (35) *<sup>g</sup>*<sup>6</sup> : *tSU*‐<sup>8</sup> � <sup>2000</sup> � <sup>10</sup>�<sup>9</sup> <sup>≤</sup>0; (36)

> tNiTi 2

> > SU‐<sup>8</sup>

NiTitNiTi

*g*<sup>7</sup> : �σrec ≤0; (37)

þ t 2 NiTi

<sup>¼</sup> <sup>0</sup>*:* (40)

<sup>2</sup> � <sup>2</sup>*:*<sup>45</sup> � <sup>10</sup>15tNiTi <sup>þ</sup> <sup>2</sup>*:*<sup>58</sup> � <sup>10</sup><sup>8</sup> <sup>¼</sup> 0;

<sup>6</sup> <sup>þ</sup> <sup>θ</sup>ð Þ 2tSU‐<sup>8</sup> <sup>þ</sup> tNiTi

In conclusion, an interesting optimization problem was identified whereby the deflection of shape memory MEMS bimorph actuator was maximized. Original calculations showed that reductions in the thickness of the bimorph layers would yield maximized deflections (for the simplest case assuming constant values of recovery stress in NiTi layer). In the literature, a more complex relationship among recovery stress and the NiTi thickness was identified. A curve fit to this data yielded a much more interesting optimization problem, which was solved graphically (contour plots) and using the Optimization Toolbox in MATLAB. Optimal NiTi and SU-8 thickness were determined to be for the case where SU-8 modulus was 2 GPa to be tNiTi = 359 nm, tSU-8 = 824 nm. After solving the single-objective optimization problem using fmincon, Excel solver, and a hand-coded algorithm, we formulated a second objective function to maximize curvature radius (i.e., to maximize the

flatness of the beam because larger curvature radius is a flatter beam). We used fmincon to solve for the optimal values of NiTi and SU-8 to maximize the curvature radius. We determined that the objective functions were conflicting (i.e., there was clearly a tradeoff in order to satisfy both conditions simultaneously), and therefore suitable for multi-objective optimization. We formulated a multi-objective optimization method and solved it using fmincon. Finally, a parametric study or sensitivity analysis was performed pertaining to NiTi and SU-8 Young's modulus.

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