Topological and Shape Optimization of Flexure Hinges for Designing Compliant Mechanisms Using the Level Set Method

A flexure hinge is a major component in designing compliant mechanisms that offers unique possibilities in a wide range of application fields in which high positioning accuracy is required. Although various flexure hinges with different configurations have been successively proposed, they are often designed based on designers’ experiences and inspirations. This study presents a systematic method for topological optimization of flexure hinges by using the level set method. Optimization formulations are developed by considering the functional requirements and geometrical constraints of flexure hinges. The functional requirements are first constructed by maximizing the compliance in the desired direction while minimizing the compliances in the other directions. The weighting sum method is used to construct an objective function in which a self-adjust method is used to set the weighting factors. A constraint on the symmetry of the obtained configuration is developed. Several numerical examples are presented to demonstrate the validity of the proposed method. The obtained results reveal that the design of a flexure hinge starting from the topology level can yield more choices for compliant mechanism design and obtain better designs that achieve higher performance.


Introduction
Compliant mechanisms have been widely used in precision engineering applications in which precision of motion, compactness, and feasibility are required. Examples can be easily found in, for example, micro-and nano-manipulation and manufacturing [1][2][3] and optical alignment [4,5]. Compliant mechanisms often refer to monolithic or jointless structures that transfer an input force or displacement to another point through elastic body deformation [6,7]. For example, in a microelectro-mechanical system (MEMS), movement between adjacent members is nearly entirely achieved through flexure hinges. Therefore, to design compliant mechanisms, a flexure hinge is regarded as a key component [8,9]. Unlike the rigid mechanical connections that allow different kinematic degrees of freedom (DOFs), a flexure hinge can only provide a limited relative motion between two rigid parts, as illustrated in Figure 1 Since Paros and Weisbord [10] proposed right circular flexure hinges and introduced an analytical approach to derive exact and approximate compliance equations, substantial research has been spent on creating flexure hinges having new configurations that achieve high performance [11][12][13][14]. For instance, a hybrid flexure hinge, which consists of half a hyperbolic flexure hinge and half a corner-filleted flexure hinge, was developed in Ref. [15]. Comparison studies with five kinds of common notch flexure hinges have also been conducted quantitatively with respect to compliance, precision, compliance precision ratios, and maximum stress. In addition, right-elliptical, corner-filleted, V-shaped and cycloidal, and polynomial-type flexure hinges, as well as those with conic sections, have also been developed [12,16,17].
However, although many new configurations of flexure hinges have been proposed, the design procedures for (2019) 32:13 these configurations are often intuitive and thus depend to a large extent on designers' individual intuition, experience, and inspiration. For conceptually designing flexible joints, screw theory-based methods have been developed [18][19][20][21]. However, these types of methods are often adopted when designing compound-type flexure hinges. Recently, structural topology optimization methods have also been adopted for designing configurations of flexure hinges. Examples of using these methods can be found in Refs. [22][23][24][25][26]. The unique feature of the topology optimization method is that the optimal solution can be generated automatically even though the topology of the mechanism is not known in advance. Since Bendsoe and Kikuchi developed the homogenization method [27], topology optimization has been exhaustively explored in both theoretical studies and practical applications [28][29][30][31]. Recently developed topology optimization methods include density-based approaches, such as the solid isotropic material with penalization (SIMP) [31][32][33] and evolutionary structural optimization (ESO) [34,35] as well as the level set method [36,37]. In contrast to density-based methods, the level set method can avoid ambiguities of intermediate material phases (i.e., provide a smooth boundary of the designed structures [38]). In using topology optimization methods to design flexure hinges, the first task it to devise an optimization formulation. Several formulations have been established as in Refs. [22,39]. The objective is to maximize the compliance in the desired direction while minimizing the compliances in the other directions. In addition, a constraint to obtain the accuracy of the revolution movement is also used. However, the obtained flexure hinges seem asymmetric with respect to the longitudinal axis [12]. This increases the difficulty of real-world applications of the obtained flexure hinges.
Therefore, in this study, we present a systematic method for designing revolute flexure hinges by considering the symmetric constraint. An optimization model is developed by considering the functional requirements of the flexure hinge (i.e., maximizing the compliance in the desired direction while minimizing the compliances in the other directions). The weighting sum method is used to construct an objective function in which a selfadjust method is used to set the weighting factors. A constraint on the symmetry of the obtained configuration is developed.
The remainder of the paper is organized as follows. In Section 2, a level set method that does not require re-initialization is introduced. In Section 3, optimization formulations are developed for the topology optimization of flexure hinges by considering an asymmetric constraint. In Section 4, sensitivity analysis and an optimization algorithm are introduced. In Section 5, several numerical examples are presented to demonstrate the validity of the proposed method. A conclusion is provided in Section 6.

Level Set Method
The underlying idea behind the level-set-based topology optimization is that the structural layout is defined by a level set function φ as: where D is the design domain that completely contains the material domain Ω, D\Ω is the void area, Γ is the structural boundary, x is a point in the design domain, and t is the pseudo-time.
The optimization process is described by the evolution of the level set function governed by [37]: where V n denotes the velocity field that determines the motion of the interface. During the optimization process, the level set function gradients at the design interface must be controlled to ensure stability. This is often achieved by periodically re-initializing the level set function to a signed distance function, which is known as the re-initialization procedure [40]. However, (2) ∂φ ∂t + V n |∇φ| = 0, re-initialization can slightly move the zero level set away from the expected position and introduce some fundamental problems (e.g., when and how to perform the re-initialization) that have yet to be solved [36]. To overcome these problems, a generalized Hamilton-Jacobi equation was developed in Ref. [41]. The idea is to replace Eq.
(2) to update the level set function by using the following equation: where ∇ · (d r (|∇φ|)∇φ) is the diffusion part and ω is a predefined parameter [41]. The diffusive rate d r (|∇φ|) is defined as: The added diffusion can maintain the level set function at a location close to a signed distance function near the structural boundaries while also forcing the level set function to be a constant at locations far removed from the structural boundaries. For additional details, please refer to Ref. [41].

Design Formulations
We developed a practical formulation for the topological synthesis of flexure hinges. A general design domain for topology optimization of flexure hinges is illustrated in Figure 2, where D is the design domain and R is the adjacent rigid member. The designed configuration is for planar applications. This means that the two adjacent rigid members are expected to experience limited relative rotation about the sensitive axis, which is an axis that is perpendicular to the xy plane ( Figure 1). Figure 2(a) indicates the boundary conditions for a generic flexure hinge. The design domain D is fixed at the boundary Γ d . Note that this boundary condition is valid because the relative movement of two members can be replicated by considering that one is fixed. The width L of the rigid member R is set to be at least equal to or greater than the width l of the design domain D. The study in Ref. [42] suggests that forces should be applied at a point where at least L ≥ 3l. Therefore, in this study, L = 3l is chosen.

Objectives
The capacity of rotation is the critical factor in characterizing a flexure hinge. The rotation requirement means that a structure must be found by using the given limit of material that can fulfill a finite rotation around the axis that is perpendicular to the xy plane when the rigid body R is applied on a force F y at point 1, as shown in Figure 2(a). Suppose that the displacement at point 1 due to F y is indicated as u 1y . The design goal can be accomplished by maximizing u 1y .
Note that the problem of maximizing u 1y has a trivial solution. One can, in principle, obtain an infinitely large displacement by removing the whole material in the design domain D. In this study, to overcome this problem, we attach an artificial spring with stiffness of k s to point 1 to limit the magnitude of u 1y . In addition, a stiffness requirement should be imposed on the designed flexure hinge. As mentioned in Ref. [22], an ideal revolute flexure hinge should prevent movement in other directions. This can be accomplished by minimizing the horizontal displacement u 1x of point 1 resulting from vertical load F x , as shown in Figure 2(a).
Therefore, a design problem with two objectives is considered. The easiest means of dealing with this problem is to use the weighting sum method to transfer it into an optimization problem with a single objective. Therefore, the objective for topology optimization of revolute flexure hinges can be expressed as: where α is the weighting factor.
One major issue of the weighting sum method is that the value of the weighting factor must be preset. Normally, no uniform means exists to determine its value, and a different design problem may require different values. To overcome this problem, we adopt the self-adjust scheme proposed in Ref. [43]. The underlying idea is to update the values of the weighting factors during each iteration step by using the following equation: where α k is the weighting factor that changes with each iteration k of the optimization algorithm, and u k 1x and u k 1y denote the values of u 1x and u 1y in the kth iteration, respectively. For the first iteration, one can simply set α to 0. We set the weighting factors in this manner because in iteration k + 1, if u k 1x becomes larger than that of the last iteration, whereas u k 1y becomes smaller, meaning that the obtained structure becomes unsteady (i.e., the tendency of generating a disconnected structure), then α k+1 will increase. This in turn will increase the weight of u 1x , causing the created structure to become much stiffer.

Constraints
In the design of the topology of a flexure hinge, we are interested in the optimal placement of a given elastic material inside D. In other words, we should determine those points that should be material points and those that remain void. The limited given elastic material leads to the following maximum material usage constraint: where the maximal material usage is constrained with an upper limit [Vol]. H(φ) is the Heaviside function defined as: Another important factor in characterizing a flexure hinge is the precision of the rotation (i.e., the center of rotation [16]). For real-world applications, an ideal revolute hinge can cause the center of the rotation to be fixed at point o, as shown in Figure 2(b). We define a parameter e as the offset of the center of rotation. Then, to realize a high precision of rotation, e should be as small as possible. Therefore, for a small rotation angle, the following condition should be satisfied: where u 2y is the vertical displacement at point 2 due to the load F y and ζ is a positive small constant.
The last criterion for topology optimization of flexure hinges concerns their geometric symmetry. As shown in Figure 3, flexure hinges can be identified by their symmetry about the transverse (y) or longitudinal (x) axis. A fully symmetric flexure hinge can be realized if it is symmetric around both the x and y axes. For a single-axis flexure hinge, four cases are possible with respect to the reference domain, as indicated in Figure 3. For the first case, the upper left quarter of the design domain is taken as the reference domain. Then, the level set function in the other three quarters is determined by using the following three equations: where φ(x, y) is the level set function value at point (x, y) in the reference domain and φ x is the level set function value of point x in the domain of B 1 , B 2 or B 3 . We can easily derive the setting rules of the other three cases.

Optimization Model
Incorporating the level set method, a possible formulation for topology optimization of the flexure hinges can be formulated as follows: where u is the state variable, U denotes the space of kinematically admissible displacement fields, and v denotes the arbitrary virtual displacement in the space U. The energy bilinear functional a(u, v, φ) and the load linear functional l(v, φ) are respectively expressed as: where E ijkl and ε ij are the elasticity tensor and strain tensor, respectively. Because the design condition considered in this study does not concern the body force, only the boundary traction f is considered in the previous equations, where δ(φ) is the Dirac delta function defined as:

Sensitivity Analysis and Optimization Algorithm
To use the level set method, finding an appropriate velocity V n is essential. A common technique is to solve the constrained optimization problem by constructing an augmented objective functional as: where λ 1 and λ 2 are Lagrange multipliers. The shape gradient of the general objective function L can be determined effectively as: where G is known as the shape gradient density [44]. To find the optimized structure, we can simply take V n = − G. This is the common strategy for the level set method in structural optimization [36].
For the problem considered in this study, we have: L(u, φ) = J (u, φ) We can see that solving the shape derivative of L can be transferred into solving ∂u 1x ∂t , ∂u 1y ∂t , and ∂u 2y ∂t . By considering the boundary conditions of the considered problem and using the shape sensitivity analysis [45], the shape sensitivities of u 1x , u 1y, and u 2y can be obtained respectively as: where n 0 is the normal unit vector to ∂Ω, u 1 and u 2 denote the displacement fields caused by the load of F x and F y , respectively, and u 3 is the structural displacement field induced by applying only a unit dummy load at point 2 (see Figure 2). By substituting Eqs. (26)- (31) into Eq. (25), we can obtain the shape sensitivity of the Lagrange formulation L. For more details on the sensitivity analysis, please refer to Refs. [37,45].
Note that the geometry constraints B1, B2, and B3 can be easily realized by directly operating the level set function φ. Considering their derivative information is not required. Therefore, a possible algorithm for solving the topology optimization of flexure hinges can be stated as in Algorithm 1. (29)

Results and Discussion
In this section, the numerical results for topology optimization of flexure hinges using the proposed method are presented. For the following numerical cases, the artificial material properties are described as Young's modulus for solid material E is 1 GPa and Poissons ratio υ is 0.3. The input loads F x and F y are both set to 1 mN. The design domain D is discretized by using 50 × 50 bilinear quadrilateral elements, and the rigid member R is discretized by using 50 × 150 bilinear quadrilateral elements for elastic analysis. The size of the element is 1 × 1 mm.

General Case
For the first example, we set k s = 0.02 N/m. The maximum material usage is set to 20% of the total material that can occupy the whole design domain D. The upper limit of the offset e is set to 0.01. To implement the proposed method, new holes cannot be freely created because of the lack of an inherent hole-nucleation scheme. Therefore, a design with a relatively complicated topology (containing numerous holes) is used as the initial configuration, as shown in Figure 4(a). Note that for this studied case, the reference domain is chosen according to Figure 3 Remarkable shape and topological changes occurred at the first 20 iterations of the optimization process. For the last 60 iterations, the overall topology showed little change, although geometrical adjustments still had to be addressed to approach the optimal region. This indicates that the proposed method was computationally effective. The optimized flexure hinge is similar to that of the cartwheel flexure hinge, which is often regarded as a flexure hinge with ultra-high rotation accuracy.
The convergence histories of u 1y , u 1x , volume ratio, and the offset e are shown in Figure 5. The convergence history of the weighting factor α is shown in Figure 6. We can see that by adding a spring with k s = 0.02 N/m, the magnitude of u 1y is restrained within 50 μm. Figure 6 reveals that with the shape and topological change during the optimization process, the weighting factor can adjust its value to ensure a well-posed design. This can overcome the difficulty of setting the value of the weighting factor artificially.

Effect of the Reference Domain
We next investigated the effects of the reference domain on the topology results. All parameters were set to the same as those given in Section 5.1. Similarly, an initial configuration like that shown in Figure 4(a) was used for all studied cases. Four cases were studied, including one that did not consider the symmetry constraint and three possible reference domains that are shown in Figure 3(a)-(c). The corresponding optimized flexure hinges are shown in Figure 7. These results are also quantitatively compared with one another in Table 1. Figure 7(a) shows that the optimized flexure hinge is indeed always symmetrical with the longitudinal axis x even though no symmetry constraint is considered. This can also be seen in Figure 4(h) and Figure 7(b)-(d).
Results indicate that choosing the reference domain with respect to the transverse axis y is essential to obtaining a meaningful design. When we chose the reference domain according to Figure 3 Table 1.

Effect of the k s
We further investigated the effects of the output spring k s on the topology results of flexure hinges. Several cases with different output spring stiffness k s were analyzed. The maximum material usage was set to 20% of the total material that could occupy the whole design domain D. The upper limit of the offset e was set to 0.01. Similarly, an initial configuration like that shown in Figure 4 Table 2.
Stiffness can considerably affect the outcome of the optimization process. The spring stiffness must be carefully chosen to obtain a meaningful design.

Effect of the Volume Constraint
We also examined the impact of the volume constraint on the obtained flexure hinges. Two values of the maximum material usage constraint [Vol] were considered that corresponded to 30% and 10% of the whole design domain. The design parameters were set to be the same as those described in Section 5.1. The final designs are shown in Figure 9. The values of u 1x in the two studied cases were 8.3787 μm (Figure 9a) and 12.8315 μm (Figure 9b). The  offset e of the two cases were 0.006 ( Figure 9a) and 0.009 (Figure 9b). These results reveal that the topology of the final designs was affected by changing the maximum material usage.

Effect of the Design Domain Configuration
The aforementioned cases all assumed the design domain to be rectangular. We next examined the optimized configurations of the flexure hinges when a design domain with a different shape or topology was used. Two cases were studied. The design domains are shown in Figure 10.
For the first case, the design domain was actually set to be a right circular flexure hinge. The goal of this design was thus to remove some of the material within a right circular flexure hinge to obtain a new hinge. As indicated in Ref. [24], although right circular flexure hinges are precise in rotation, their compliance and range of motion are small. Therefore, removing some material from the right circular flexure hinge is helpful to improving its performance. The optimization process is shown in Figure 11. The final topology revealed that the four sharp corners of the design domain were removed. The most important change was that two notches were generated on both sides of the design. With the increase in k s , the obtained flexure hinge had a tendency to become two parallel cantilever beams, as indicated in Figure 8(a). This was because when k s is large, u 1x becomes an overwhelming part in  Eq. (5). Therefore, the design problem becomes a problem of minimizing the compliance along the x direction. When the spring stiffness k s is small (e.g., k s = 0.1 or 0.01 N/m), topologies that are similar to that of the cartwheel flexure hinge will be obtained, as shown in Figure 8(b) and (c). When k s is very small, as indicated in Figure 8(d), a notch type flexure hinge will be obtained. In addition, the thickness of the material in the middle of the design domain becomes very thin, which may indicate that this result was not practically applicable. A previous study demonstrated that these types of flexure hinges have better output performance without losing rotational accuracy [24]. For the second case, two holes were inserted inside the rectangular design domain, indicating that these two areas have no material. The optimization process is shown in Figure 12. These results show that the topology of the final designs could be affected by the shape or topology of the design domain.

Conclusions
A method for topology optimization of single-axis flexure hinges based on the level set method was proposed. A capable optimization model was developed by considering a multi-criteria objective function and a symmetry constraint. Several numerical examples were presented to demonstrate the validity of the proposed method. In addition, the obtained results revealed that the design of a flexure hinge starting from the topology level yielded more choices for compliant mechanism design and obtained better designs that achieved higher performance. Future studies will consider flexure hinges that are subjected to large deflections, which means that a nonlinear finite element analysis method will be used for elastic