# Stiffness-Oriented Structure Topology Optimization for Hinge-Free Compliant Mechanisms Design
## 2 Complaint Mechanism Optimization Problem and Formulation
### Complaint Mechanism Optimization Problem  
Consider a general compliance mechanism design example with design domain \(\mathbf{\Omega}\) under given loads and constraints conditions shown in Figure 1a where applied input force \(\mathrm{F_{A}}\) is the loading at the input port and results in a displacement \(\Delta_{\mathrm{A}}\). To achieve the flexibility requirement of compliant mechanisms, \(\Delta_{\mathrm{B}}\) is the expected output displacement at output port with a workpiece simulated by a spring with constant stiffness. Considering the stiffness requirement, the optimized design should be strong enough to withstand the input force \(\mathrm{F_{A}}\) and output reaction force \(\mathrm{F_{B}}\) from the workpiece. Figure 1: (**a**) Complaint mechanism design; (**b**) mutual potential energy (MPE) calculation. To cater to the flexibility requirement, the compliant mechanism design is expected to be flexible enough to generate the required kinematic motion under the applied loads and reaction force from the workpiece. Generally, the structure flexibility can be represented by output displacement. Maximizing output displacement \(\Delta_{\mathbf{B}}\) at port B can be expressed in terms of maximizing mutual potential energy(MPE), just as shown in Figure 1b, which can be formulated as follows [1]. \[\max\left(\mathbf{f}_{\mathrm{B}}^{\mathrm{T}}\mathbf{u}_{\mathrm{A}}\right)= \max\left(\mathbf{v}_{\mathrm{B}}^{\mathrm{T}}\,\mathbf{K}_{\mathrm{1}}\mathbf{ u}_{\mathrm{A}}\right)=\max\left(\mathrm{MPE}\right) \tag{1}\]  
Subject to:  
\[\mathbf{K}_{\mathrm{1}}\mathbf{u}_{\mathrm{A}}=\mathbf{f}_{\mathrm{A}}\]  
\[\mathbf{K}_{\mathrm{1}}\mathbf{v}_{\mathrm{B}}=\mathbf{f}_{\mathrm{B}}\]  
where \(\mathbf{u}_{\mathrm{A}}\) is the nodal displacement vector due to \(\mathbf{f}_{\mathrm{A}}\), \(\mathbf{v}_{\mathrm{B}}\) is the nodal displacement vector due to the dummy load \(\mathbf{f}_{\mathrm{B}}\), and \(\mathbf{K}_{\mathrm{1}}\) is the symmetric global stiffness matrix. On some occasions, optimized design layout often turns out to be connected with some single-node hinges which are very difficult to fabricate and easy to break due to their high-stress concentration and may result in potential reliability issues. The topology optimization algorithms prefer them because they can generate large displacements without increasing total strain energy. To deal with the hinge pivot issue in topology optimization, a general approach is introducing stiffness requirement into optimization formation as objective or constraint. Mostly, the structure stiffness is formulated in terms of the mean compliance or total strain energy [1,8,24]. In other words, maximizing the stiffness is equivalent to minimizing the mean compliance or total strain energy (SE). However, the current optimization formulation cannot specifically focus on the stiffness in an interesting direction or eigenmode, because both mean compliance and total strain energy are scalar and cannot specify the direction. Generally, the mechanical structural stiffness can be characterized in terms of eigenfrequencies and their corresponding eigenmodes. It is a wise choice if the optimization process can focus on the interesting eigen-frequency and its corresponding eigenmodes. Another advantage of this approach is that eigen-frequency can be easily measured by various measurement methods, such as the sine sweeping-frequency vibration method and the hammering method. In addition, the single-node connected hinge can be precluded in compliant mechanism design by introducing the scheme of maximizing the required eigen-frequency. The reason is that material will be added to the hinge region to achieve the stiffness requirement.