## 5 Application Discussion
### A Micro-Force Sensor  
A piezoelectric micro-force sensor [43] has several advantages, including a reliable structure, fast response, and simple driving circuits. It has been extensively applied in the fields of precision positioning, ultrasonic devices, micro-force measurement, etc. Given that uncertainties are inevitable in structural sizes and material parameters, robust optimization is essential to ensure the performance of the sensor. As shown in Figure 4, the core part of the micro-force sensor is a piezoelectric cantilever beam, which consists of a piezoelectric film, a silicon-based layer, and two electrodes. The force at the free end causes bending deformation on the beam, which drives the piezoelectric film to output polarization charges through the piezoelectric effect. The charge is transmitted to the circuit by the electrodes and converted into a voltage signal. According to the theoretical model proposed by [PERSON] et al. [43], this voltage can be formulated as:  
\[U=\frac{3\cdot d_{31}^{p}\cdot S_{11}^{Si}\cdot S_{11}^{p}\cdot h\cdot h^{p} \cdot\left(h+h^{p}\right)\cdot L\cdot F}{K\cdot e_{33}^{p}\cdot w} \tag{18}\]  
Figure 3: The flowchart of the proposed method. EBRO: evidence-theory-based robust optimization; BPA: basic probability assignment. where  
\[ \begin{array}{l}K=4\cdot S_{11}^{Si}\cdot S_{11}^{P}\cdot h\cdot\left(h^{P} \right)^{3}+4\cdot S_{11}^{Si}\cdot S_{11}^{P}\cdot h^{3}\cdot h^{P}\\ +\left(S_{11}^{P}\right)^{2}\cdot h^{4}+\left(S_{11}^{Si}\right)^{2}\cdot\left( h^{P}\right)^{4}+4\cdot S_{11}^{Si}\cdot S_{11}^{P}\cdot\left(S_{11}^{Si} \right)^{2}\cdot\left(h^{P}\right)^{2}\end{array}  \tag{19}\]  
where \(F\) is the concentration force; \(L\), \(w\) represent the length and width of the beam; \(h\), \(h^{P}\) denote the thickness of the silicon-base layer and piezoelectric film; \(S_{11}^{Si}\), \(S_{11}^{P}\) are the compliance coefficient of the silicon-based layer and piezoelectric film; and \(d_{31}^{P}\), \(e_{33}^{P}\) is the piezoelectric coefficient and dielectric constant of the piezoelectric film. The constants in Equation (18) include \(h^{P}=5\times 10^{-4}\) mm, \(S_{11}^{P}=18.97\times 10^{-12}\)m\({}^{2}\)/N, and \(S_{11}^{Si}=7.70\times 10^{-12}\)m\({}^{2}\)/N, The structural sizes of \(L\), \(w\), \(h\) and the material parameter of \(d_{31}^{P}\), \(e_{33}^{P}\) are viewed as evidence variables. The marginal BPAs of the variables are shown in Figure 5, and the nominal values of \(d_{31}^{P}\), \(e_{33}^{P}\) are, respectively, 1.8 C/N, 1.6 F/m. Figure 4: A piezoelectric cantilever beam. Figure 5: Marginal BPAs of variables in the micro-force sensor problem. BPA: basic probability assignment; FD: frame of discernment. In engineering, the greater the output voltage, the higher the theoretical accuracy of the sensor. Thus, \(U\) is regarded as the objective function. The design variables are \(L\), \(w\) and \(h\). The constraints of shape, stiffness and strength are considered, which are expressed as \(\eta\geq 0.83\), \(\delta\leq 2.5\,\upmu\)m, and \(\sigma\leq 32.0\,\)MPa, where \(\eta\) is the ratio of \(w\) to \(h\), \(\delta\) denotes the displacement at the free end of the beam, and \(\sigma\) denotes the maximum stress of the beam. \(\delta\) and \(\sigma\) can be written as [43]. \[ \begin{array}{l}\sigma=\frac{6\cdot F\cdot L\cdot S_{11}^{\mathrm{S}_{11} ^{\mathrm{S}_{11}^{\mathrm{P}},h+S_{11}^{\mathrm{S}_{11}^{\mathrm{P}}}}}\cdot \left(h+h^{\mathrm{P}}\right)}{K\cdot w}\\ \delta=\frac{4\cdot F\cdot L^{3}\cdot S_{11}^{\mathrm{S}_{11}^{\mathrm{P}}, \left(S_{11}^{\mathrm{P}},h+S_{11}^{\mathrm{S}_{11}^{\mathrm{S}_{11}^{\mathrm{P }}}}\right)}}{K\cdot w}\end{array}  \tag{20}\]  
Due to uncertainties in the structure, \(\eta\), \(\delta\) and \(\sigma\) are also uncertain. Theoretically, the three constraints should be modeled as reliability constraints. To focus on the topic of robust optimization, the constraints are considered as deterministic in this example.