## 2 Topology Optimization for the Synthesis of Compliant Mechanisms
### Formulation of Topology Optimization
#### 2.2.6 Multiresolution Topology Optimization  
The multiresolution topology optimization (MTOP) formulation was introduced by [PERSON] et al. [25] and reduces the computational cost necessary for the calculation of high resolution topologies. Since the most significant part of the computational cost of topology optimization is finite-element analysis, [25] applies different discretization levels for the displacements (FE analysis), the densities (material representation) and the design variables (optimization). In general, the FE analysis is performed on a coarser mesh, whereas the densities and the design variables are mapped on to finer meshes. By means of this uncoupling of FE displacement elements, densities and design variables, higher resolution results can be obtained with a computational effort comparable to the original solution. Based on the work of [PERSON] et al. [25, 26], a new multiresolution topology optimization scheme is proposed, extending to the consideration of stress constraints. In this work the design variable mesh is chosen to be coincident to the density mesh. The key to multiresolution topology optimization is to find a functional relationship between the displacement element's internal force vector and stiffness matrix for structural analysis and the single densities of the density elements. This relationship can be found in the summation terms of the Gauss integration approximation by assigning a density element to each Gauss integration point. By this means, e.g., the internal force vector is approximated as  
\[\underline{P}_{\text{int,i}}=\int_{\Omega_{i}}\underline{B}_{\text{int,}}^{ \intercal}\underline{\underline{F}}\underline{\underline{C}}_{\text{VK}} \varepsilon_{\text{GL,}i}dV\approx\sum_{j=1}^{n_{\text{SF}}}\underline{P}_{ \text{int,i},j} \tag{28}\]  
where \(\underline{P}_{\text{int,i},j}\) is the integrand for the internal force vector (see Equation (2)) of the displacement element \(i\) at the Gauss point \(j\) assigned to the corresponding density element. For the derivation of \(\underline{P}_{\text{int,i},j}\), the constitutive matrix \(\underline{\underline{C}}_{\text{VK},j}\) of each summation term uses the specific [PERSON]'s modulus \(E\) for the corresponding density element, obtained by the SIMP approach. In Figure 1, an exemplary subdivision of a displacement element (a) in four density elements and its corresponding Gauss integration points (b) are given. The isoparametric element is divided in four equally sized density subelements, each of which influences the summation term of the internal force vector and the tangent stiffness matrix corresponding to the Gauss point within the density elements domain. The number of subelements must be equal to the number of integration points used. For material representation, all the density elements with density values \(\tilde{\pi}_{i,j}\) are assembled to a global density mesh. The global density vector \(\underline{\tilde{\pi}}\) and hence the design variable vector \(\underline{x}\) in this approach are of the size \(n_{\text{eff}}\times n_{\text{SF}}\). The definition of the displacement elements and the global displacement vector \(\underline{u}\) for multiresolution topology optimization does not change compared to standard topology optimization. Therefore, the objective function definitions in Equations (16) and (17) remain unchanged also for the multiresolution topology optimization approach. The utilized volume \(V_{\text{util}}\) for the calculation of the volume fraction \(\varphi\) defining the volume constraint in Equation (19) is changed for multiresolution topology optimization. It can be expressed by the following summation term:  
\[V_{\text{util}}=\sum_{i=1}\sum_{j=1}\tilde{\pi}_{i,j}V_{i,j}, \tag{29}\]  
where \(\tilde{\pi}_{i,j}\) and \(V_{i,j}\) respectively are the density and the volume of the subelement \(j\) of the displacement element \(i\).