# Stiffness-Oriented Structure Topology Optimization for Hinge-Free Compliant Mechanisms Design
## 1 Introduction
To address the convex issue which may occur when regarding the maximum MA or GA as the objective function, [PERSON] et al. [22] proposed a non-convex objective formulation and verified it with numerical examples. [PERSON] et al. [23] introduced a material path-generating method and formulated an objective function for the synthesis design of large-displacement compliant mechanisms which was built by a global Lagrangian finite element formulation. In addition, [PERSON] et al. [24] introduced the ESO method to compliant mechanisms design by incorporating the elastic SE and MA into the objective function. Programing in this line of work, [PERSON] et al. [25] extended this method to the design of a 3D-compliant mechanism with a finite element addition scheme and introduced the elastic strain energy of the mechanism into the proposed objective function. Apart from the difficulty to balance the flexibility and stiffness requirement in compliant mechanisms optimization formulation, the single-node hinge feature is another difficulty which frequently encounters in the design of compliant mechanisms using topology optimization, often leading to the optimized result close to a rigid-body mechanism [26]. The undesirable hinge feature is unfeasible in most practical application since it is very difficult to fabricate and easy to break due to its high-stress concentration; in other words, the single-node hinge feature often result in potential reliability issue. The optimization algorithm and formulation naturally lead to a single-node hinge feature, because it can generate large displacement under a given force while not increasing strain energy [27], therefore, worse local optima or non-optimal may occur in the topology optimization process. To achieve the design of hinge-free compliant mechanisms, [PERSON] [28] cited a minimum length scale constraint in topology optimization and successfully generated distributed flexible compliant mechanisms. Furthermore, [PERSON] et al. [29] developed a continuum structural topology optimization with hinge-free compliant mechanisms by attaching artificial springs to the input and output point of the optimization model and the single-node hinge zone was eliminated with an increase in the relative stiffness of the artificial springs. Recently, [PERSON] et al. [2] introduced a new BESO algorithm to maximize the ratio of GA/SE, and the hinge zone was precluded by introducing the total strain energy of the structure. In addition, the stiffness objective in general optimization formation is always an overall stiffness of the structure which is often achieved by the introduction of minimizing the SE into the formulation, but the actual applications are often required to focus on the stiffness in an interesting direction for optimizing the usage of material. Inspired by daily work on compliance mechanisms design, the method presented in this paper focuses on the stiffness in an interesting direction using eigenmodes by weighting its corresponding eigen-frequency. Because eigenmode is always related to specifically oriented stiffness and each eigenmode has its corresponding eigen-frequency, the optimization process will focus on the specified eigen-frequency when properly added into the objective formulation. Therefore, this paper develops a stiffness-oriented structure topological synthesis method to design a hinge-free compliant mechanism in which flexibility and stiffness requirements are considered. Based on the topology optimization and sensitivity analysis, a systematic formulation is deduced to tradeoffs between flexibility controlled by mutual potential and stiffness represented by eigen-frequency. Moreover, it will be shown that the unwanted single-node hinge connection issue has been successfully addressed by introducing the eigen-frequency constraints into the optimization formulation. This approach can provide an alternative method for hinge-free compliant design.