## 2 Topology Optimization for the Synthesis of Compliant Mechanisms
### Nonlinear Finite-Element Analysis  
The central part of topology optimization is structural-mechanical analysis, which is the basis of the values for the optimization functions (objective and constraints) of each optimization iteration. This analysis is carried out discretizing the solution space of the governing equations. The equations describing the nonlinear structural analysis are briefly introduced here. A geometrically nonlinear finite-element method using plane stress and a total Lagrangian formulation [36; 37] is utilized, which was implemented by [PERSON] [38; 39] and further developed by [PERSON] [40].  
Equilibrium is defined as  
\[\underline{R}=\underline{P}_{\text{int}}-\underline{P}_{\text{ext}}=\underline {0}, \tag{1}\]  
where \(\underline{R}\) is the vector of residuals, \(\underline{P}_{\text{ext}}\) is the external force vector and \(\underline{P}_{\text{int}}\) is the internal force vector. The internal force vector is defined element-wise, given here for the \(i\)th element by  
\[\underline{P}_{\text{int,i}}=\int_{\Omega_{i}}\underline{B}_{L,i}^{\text{T}} \overline{\underline{E}}_{i}\underline{\sigma}_{\text{2 PK},i}dV. \tag{2}\]  
The notation here denotes vectors as underlined symbols, matrices as double underlined symbols and scalars as symbols without an underline. It should be noted that we will use Voigt notation unless otherwise denoted.  
For he deformation gradient tensor, we will use both tensor and Voigt notations, the matrix \(\underline{\overline{E}}\) and the vector \(\underline{\underline{E}}\),  
\[\underline{\overline{E}}=\left[\begin{array}{ccc}F_{11}&0&F_{12}\\ 0&F_{22}&F_{21}\\ 0&F_{12}&F_{11}\\ F_{21}&0&F_{22}\end{array}\right], \underline{\underline{E}}=\left[\begin{array}{c}F_{11}\\ F_{22}\\ F_{12}\\ F_{21}\end{array}\right]. \tag{3}\]  
The operator matrix \(\underline{\underline{\underline{\underline{\underline{\underline{\underline{ \underline{\underline{\underline{\underline{\underline{\underline{\underline{ \underline{\underline{\underline{\underline \underline{ \underline{ \underline{ \underline{        }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}\)\)\)\)\) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{\underline{E}}=\left[\begin{array}[\begin{array}{c}F_{11}\\ F_{22}\\ F_{12}\\ F_{21}\\ F_{21}\end{array}\right].\] (4)  
The second Piola-Kirchhoff stresses, which are conjugate to Green-Lagrange strains, are  
\[\underline{\sigma}_{\text{2 PK},i}=\underline{\underline{\underline{\underline {\underline{\underline{\underline{\underline{\underline{\underline{\underline{ \underline{\underline{\underline{\underline{\underline{ \underline{\underline{ \underline{ \underline{ \hatAs the second Piola-Kirchhoff stress lacks physical meaning, it is transformed into Cauchy stress \(\sigma_{C}\) after obtaining the equilibrium deformation state. This transformation is shown for simplicity purposes with the use of matrix notation to describe the stress tensors,  
\[\underline{\sigma}_{C,i}=\frac{1}{J_{i}}\underline{F}_{i}\underline{\sigma}_{ 2\text{PK},i}\underline{F}^{\mathsf{T}}, \tag{7}\]  
where \(J\) is the determinant of the deformation gradient \(\underline{F}\) and is a measure for the volumetric change. It should be noted that due to plane stress, the component \(F_{33}\) governed by the Poisson effect must also be considered for the derivation of the determinant of the deformation gradient. If it is not considered, the plain strain assumption would be applied.  
In this study, bilinear quadrilateral finite elements are used and the integrals are approximated by four Gauss integration points. The stress fields of interest are obtained by evaluation of the element Cauchy stresses \(\underline{\sigma}_{C,i}\) at the element centroid.