## 2 Topology Optimization for the Synthesis of Compliant Mechanisms
### Formulation of Topology Optimization
#### 2.2.1 Design Variables and Their Stabilization  
In density-based topology optimization, as is the case in this work, the design variables \(x_{i}\) represent the element densities, which are scaled between zero for a void and unity for full material,  
\[x_{i}\in[0,\,1]. \tag{8}\]  
The design variables are mapped to the [PERSON]'s modulus of each element using the value for the base material \(E_{0}\) via the modified SIMP formulation [41],  
\[E_{i}=E_{\text{min}}+(E_{0}-E_{\text{min}})x_{i}^{\theta}. \tag{9}\]  
The inclusion of a minimum value for the [PERSON]'s modulus \(E_{\text{min}}\) avoids numerical singularities in the finite-element analysis that occur when elements have numerically zero stiffness. The value \(\theta\) penalizes the design space of intermediate densities between solid and void is \(\theta=3\) as commonly used in the literature (cf. [PERSON] [12], [PERSON] et al. [14], [PERSON] et al. [28]). To avoid mesh-dependent results and checkerboarding, the design variables are further filtered using the density filtering method introduced by [PERSON] and [PERSON] [42]. This filtering approach is defined according to [PERSON] [41] by  
\[\tilde{x}_{i}=\frac{\sum_{j\in N_{i}}H_{ij}V_{j}x_{j}}{\sum_{j\in N_{i}}H_{ij }V_{j}}, \tag{10}\]  
where \(V_{j}\) is the volume of the \(j\)th element and \(H_{ij}\) is a cone-shaped weighting function of the form  
\[H_{ij}=\max\{0,\,r_{\text{min}}-\Delta(i,j)\}. \tag{11}\]  
The filter radius is denoted by \(r_{\text{min}}\) and \(\Delta(i,j)\) represents the distance between the centroids of the \(i\)th and the \(j\)th element. In addition to the filter, we apply a Heaviside projection scheme according to [PERSON] et al. [43], using the following step approximation:  
\[\tilde{x}_{i}=\frac{\tanh(\beta\eta)+\tanh(\beta(\tilde{x}_{i}-\eta)}{\tanh( \beta\eta)+\tanh(\beta(1-\eta)}. \tag{12}\]  
This is applied in order to avoid so-called intermediate \"gray solutions\", i.e., intermediate density elements in-between solid and void regions. The projection parameters \(\beta\) and \(\eta\) define the degree of discreteness of the step between solid, void and its threshold value. For the results presented below, \(\eta=0.2\) is chosen and the parameter \(\beta\) is sequentially increased over the number of optimization iterations \(j\) from an initial value of \(\beta_{0}=1\) to the final value of \(\beta_{\max}=150\),  
\[\beta_{j}=\beta_{0}2^{\frac{j-1}{30}}. \tag{13}\]  
The resulting modified design variables \(\tilde{x}_{i}\) of Equation (12) replace the original design variables \(x_{i}\) in Equation (9) and, hence, determine the finite-elements stiffness properties. Another difficulty regarding topology optimization using nonlinear finite-element analysis is the numerical instability of low-density elements. With linear FEA, singularity of the stiffness matrix is avoided by introducing \(E_{\min}\) in to the SIMP approach. In nonlinear analysis, this approach only stabilizes the first equilibrium iterations. For the following load steps, large deformations of the low stiffness elements cause negative element volumes and the singularity problems still occur. As a consequence, convergence is heavily disturbed. Several approaches to resolve this issue are reported in literature. [PERSON] and [PERSON] [42] and [PERSON] et al. [30] suggested to use hyperelastic material laws with increased stiffness at high deformations instead of the Saint Venant-Kirchhoff law. [PERSON] et al. [14] reported, that the approach of [PERSON] and [PERSON] was unsuccessful resolving the issues for their numerical examples. They instead excluded low density elements from the convergence checks of the Newton-Raphson iteration. [PERSON] and [PERSON] [44] introduced an element deletion and reintroduction scheme for low density elements. [PERSON] and [PERSON] [45] approached the issues by introducing linear springs, which provide an interconnection between the elements.