# Stiffness-Oriented Structure Topology Optimization for Hinge-Free Compliant Mechanisms Design
## 1 Introduction  
Distributed or hinge-free compliant mechanisms are continuous, monolithic (one-piece) and flexible mechanisms, which can transfer displacement and force from input port to output port in other directions by elastic deformation and have enough stiffness to withstand the external loads as well [1]. Compared with traditional rigid multi-part mechanisms, compliant mechanisms efficiently reduce the size, material usage, displacement loss, backlash loss, noise, and vibration [2, 3]. Therefore, it shows considerable promise application in micro-scale and nano-scale mechanical systems such as biomedical devices, semiconductor industry, micro-electromechanical systems (MEMS), and many other applications. To obtain the compliant mechanisms, there are mainly two kinds of approaches, namely, the kinematics-based method and the structure topology optimization-based method. Structure topology optimization has been successfully employed to achieve optimal topology design through various topology optimization algorithms, such as the homogenization algorithm [4], solid isotropic material with penalization (SIMP) algorithm [5], evolutionary structural optimization (ESO) algorithm [6, 7], non-uniform rational basis spline (NURBS) based hyper-surfaces theory [8, 9, 10], and level-set [11] and phase-field approaches [12]. Numerical engineering examples for topology optimization validate its advantages in terms of systematism and high efficiency, compared with the kinematics-based approach. Therefore, the application of this technology has been further expanded to other fields over the last decades, such as compliant mechanisms design. Compliant mechanisms are expected to achieve designated flexibility in the specified directions and to satisfy enough structure stiffness requirements to sustain external loads simultaneously. Accordingly, compliant mechanisms optimization can be categorizedto the multi-objectives problem using the topology optimization method. As a multi-objectives problem, one of the main difficulties in compliant mechanisms design is the provided formulation should properly tradeoff between flexibility and stiffness since there're conflicting design objectives with different fundamental units [13; 14]. [PERSON] and [PERSON] [15] were the pioneers and proposed an energy-based formulation to unify these fundamental units by introducing mutual potential energy (MPE) to define flexibility requirement and strain energy (SE) to represent stiffness requirement. To maximize MPE and minimize SE simultaneously, [PERSON] et al. [16] presented a multi-criteria objective using a weighted linear combination of these two objectives. The difficulty is that the order of these two objectives is often not comparable and easy to result in one objective dominated by the other. A later approach by [PERSON] et al. [1] solved this problem using another multi-criteria formulation by maximizing the ratio of MPE and SE. Benefiting from that, the formulation successfully balanced the kinematics and structural stiffness. A further approach is presented by [PERSON] and [PERSON] [17; 18] using a more general objective formulation by the ratio of \(MPE^{m}\) and \(SE^{n}\). Recently, [PERSON] et al. [19] proposed a new multi-objective formulation for topology optimization of compliant mechanisms which considers to maximum the MPE and minimum the mean compliance simultaneously. Besides energy-based formulation, other approaches to design compliant mechanisms using topology optimization are studied to balance flexibility and stiffness requirements. An early study on topology synthesis of compliant mechanisms was carried out by [PERSON] et al. [20] who introduced a formulation to maximize mechanical advantage (MA) with the volume and input displacement constraints. Along similar lines, [PERSON] et al. [21] proposed a functional specification method by the ratio of geometry advantage (GA), MA to optimize the design of compliant mechanisms.