# Topology Optimization of Multi-Materials Compliant Mechanisms
## 3 Results and Discussion
### Inverter  
The first benchmark example is an inverter with single input and output which is illustrated in Figure 4. The input and output stiffness has an equal value of 0.1. This mechanism is considered to be designed with four material properties. To investigate the effect of multi-material, three different cases for volume fraction are presented. The design parameters are presented in Table 1, that consist of three solid material phases, including polyetherimide, polyurethane and polyvinylidinefluoride (Material 1, Material 2 and Material 3, respectively).  
In case 1, the multi-material method is degenerated into the traditional single material method which only contains real material and empty phase, so as to verify the correctness of the multi-material compliant mechanisms design method proposed in this paper.  
To verify the result, after using the equal load conditions, the topology of compliant inverter of traditional SIMP method [55] and degenerate solution of multi-material method was obtained, as showed in Figure 5a,b the distribution form of material is basically the same, and differences only in very small localities. In Figure 5c, the output displacement of the optimized result after the degradation of multi-material is \(-5.3\) mm, which is only \(3.7\%\) error compared with \(-5.1\) mm of the traditional single material (see Table 2). The results indicate a slight difference in the maximum displacement of compliant inverter, this shows that the principle and method of compliant mechanism design for multi-material proposed in this paper are correct and effective.  
By observing Figures 6a, 7a and 8a, it can be found that the high modulus material phase is mainly distributed near the fixed end of the mechanism to maintain the shape, the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Material 1** & **Material 2** & **Material 3** & **Material 4** \\ \hline Young’s modulus (MPa) & 10 & 5 & 2.5 & \(1.0\times 10^{-9}\) \\ Input force(N) & 3 & 3 & 3 & 3 \\ Poisson ratio & 0.3 & 0.3 & 0.3 & 0.3 \\ Volume fraction Case 1 & 0.3 & 0.01 & 0.01 & 0.68 \\ Volume fraction Case 2 & 0.16 & 0.08 & 0.08 & 0.68 \\ Volume fraction Case 3 & 0.08 & 0.16 & 0.08 & 0.68 \\ Volume fraction Case 4 & 0.08 & 0.08 & 0.16 & 0.68 \\ Color & Red & Black & Blue & White \\ \hline \hline \end{tabular}
\end{table}
Table 1: Design parameters for the inverter mechanism.  
Figure 4: The design domain of a compliant inverter mechanism.  
medium modulus material is mainly distributed in the deformation guiding and execution area of the compliant mechanism, while the low modulus material is more concentrated in the local deformation area, which can avoid the problem of high local stress. The output displacements in Figures 6b, 7b and 8b are \(-8.4\) mm, \(-9.4\) mm and \(-10.5\) mm, respectively, with 13% increase in deformation capacity when the dominant material in the four phases is changed from high to medium modulus. When the dominant material in the four phases changed from high to low modulus, the deformation ability of the mechanism increased by 25%.  
Figure 5: Optimization results: (**a**) Traditional SIMP method topologies; (**b**) degenerate solution of multi-material method (case 1); (**c**) convergence plot.  
Figure 6: Optimization results of case 2: (**a**) Identified topologies; (**b**) convergence plot.