## 3 Numerical Examples
### Numerical Example--Maximum Displacement Mechanism Design  
We define the optimization problem for the maximization of a mechanism's output displacement \(u_{\text{out}}\) (problem type I) as  
\[ \begin{split}&\min_{\underline{x}}\{f_{u}\}\\ \text{where}& f_{u}(\underline{x})=-L^{\text{T}} \underline{u}(\underline{x})\\ \text{subject to}& g_{V}(\underline{x})=\frac{\varphi( \underline{x})}{\varphi_{\text{allow}}}-1\leq 0\\ & g_{e}(\underline{x})=\frac{\hat{\sigma}_{p,\underline{x}}( \underline{x})}{\sigma_{\text{allow}}}-1\leq 0\\ & 0\leq x_{i}\leq 1\\ \text{governed by}&\underline{R}=\underline{P}_{\text{ int}}-\underline{P}_{\text{ext}}=\underline{0}\end{split} , \tag{64}\]  
where the output displacement function is defined by Equation (16). The example for this optimization problem is a gripping mechanism illustrated in Figure 2. Within a rectangular domain, a topology is sought that transfers an input displacement caused by the input force \(P_{\text{in}}\) into a maximum output displacement \(u_{\text{out}}\) at the right-hand side of the domain. Springs \(k_{\text{in}}\) and \(k_{\text{out}}\) are added to simulate the stiffness of the input mechanism and output workpiece and hence to pose a minimum load transferring requirement on the mechanism. The gray area represents the designable domain, whereas black and white areas represent non-designable solid and void, respectively. All non-designable solid regions have a height of 0.05 h. The corresponding parameter values for the optimization problem defined by Equation (64) and Figure 2 are listed in Tables 1 and 2. In order to solve the optimization problem, the design space is discretized with bilinear quadrilateral finite elements and the symmetry of the design problem is exploited. The integrals for the residuum and stiffness are approximated by Gauss integration using four Gauss points per element. Results for linear and nonlinear FEA, both with and without stress constraints are shown in Figure 3. Four different topologies can be observed. It should be noted that the results shown are the results of the topology optimization without smoothing. To the left, solutions with volume constraints are illustrated and to the right, stress constraints are additionally applied. For all the results, the volume constraint is satisfied. For the problem formulation with only a volume constraint, the maximum stresses exceed the limit stress, which would lead to failure of this mechanisms at its first actuation. This, as well as the occurrence of nonphysical one node connected hinges in those solutions, excludes practical realization of the mechanisms. The stress constraints successfully alleviate both of the mentioned problems, by eliminating high stress localized hinge regions from the topologies and substituting them with so-called distributed compliance hinges. Hence, the maximum occurring stress level can be lowered to the prescribed limit. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Symbol** & **Value** & **Units** \\ \hline Young’s modulus & \(E\) & 4232 & MPa \\ Poisson’s ratio & \(\
u\) & 0.36 & – \\ Limit stress & \(\sigma_{\text{limit}}\) & 100 & MPa \\ Max allowable stress & \(\sigma_{\text{allow}}\) & 50 & MPa \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material parameters for a generic polyether ether ketone (PEEK). Comparing the linear to the nonlinear results, also notable differences in the topology can be observed. Commonality is seen though in the large rigid regions, which are dosed based on the value of the volume fraction constraint. The differences become especially evident by looking at the mechanisms performance in terms of output displacement and maximum stress level. Those result values are listed for the different solution topologies in Table 3, where also a nonlinear analysis is performed on the linear optimization results. As shown by [14] it can be seen that, due to the neglected nonlinear effects the performance of the linearly obtained mechanism topologies is drastically overestimated.