# Stiffness-Oriented Structure Topology Optimization for Hinge-Free Compliant Mechanisms Design
## 2 Complaint Mechanism Optimization Problem and Formulation
### Checkerboard and Minimum Member Size Control  
A well-known problem that often happened in continuum topology optimization results is the so-called checkerboard phenomenon. [PERSON] and [PERSON] [31] introduced that the stiffness of the checkerboard pattern is overestimated and stiffer than any real material. As a result, the checkerboard pattern is artificially preferred by the optimization algorithm. Different approaches have been developed to overcome the checkerboard issue, [PERSON] and [PERSON] [32] have given a review of the regularization methods as perimeter constraint, sensitivity filtering, and density gradient constraints.  
Based on the understanding that the reason for the checkerboard pattern is due to bad numerical modeling that overestimates the stiffness of checkerboards, [PERSON] and [PERSON] [33] introduced a constraint on the local gradient of the slope of element densities for guaranteeing the accuracy of the finite element formulation. However, these additional linear constraints make this approach computationally prohibitive for practical applications. Based on the basic concept of a slope constraint, [PERSON] et al. [34] enforced the formula by an adaptive constraint strategy in the optimization algorithm that is similar to adding move limits and does not require any extra computational effort. In the meantime, this algorithm is given an opportunity that can control the minimum member size.  
For achieving a half predetermine minimum member size of the radius \(r_{\text{min}}=D_{\text{min}}/2\), \(D_{\text{min}}\) denotes the minimum member size, the slope constraint can be formulated for a general irregular finite element mesh as follows:  
\[|\rho_{\text{i}}-\rho_{\text{k}}|\leq\frac{(1.0-\rho_{\text{min}})\text{dist }(\text{i},\text{k})}{r_{\text{min}}} \tag{7}\]  
where \(\text{dist}(\text{i},\text{k})\) is the distance between adjacent elements \(\text{i}\) and \(\text{k}\), and \(\text{k}\in\Omega_{\text{i}}\) with \(\Omega_{\text{i}}\) denoting the set of elements adjacent to element \(\text{i}\). However, the above formulation still adds additional linear constraints and result in computationally prohibitive as well. The density slope constraints in (7) can be improved through enforcement of adaptive lower bounds on the density as follows:  
\[\rho_{\text{i}}\geq\text{max}\big{[}\mu,\rho_{\text{j}}-(1.0-\rho_{\text{min }})\text{dist}(\text{i},\text{k})/r_{\text{min}}\big{]} \tag{8}\]  
where \(\mu\) is the lower limit of density (in this paper, all examples \(\mu=0.6\)) and \(\rho_{\text{j}}\) is the density of element \(\text{j}\) at previous iteration that has the highest density among all elements that are adjacent to element \(\text{i}\)  
\[\rho_{\text{j}}=\text{max}(\rho_{\text{k}}|\text{k}\in\Omega_{\text{i}}) \tag{9}\]  
Formula (8) shows that the only works with the values of the box-constraints on the density \(\rho_{\text{j}}\), which at the \((\text{i}+1)\)th iteration step are modified to restrict the variations in the design. It is easy to see that except for the negligible computation associated with the adjustment of the side constraints of the design variables.