Jacobian-Based Topology Optimization Method Using an Improved Stiffness Evaluation

Compliant mechanisms are elastic structures that can transmit force or motion from input to output. Due to the combined characteristics of mechanism and structure, the analysis and design of compliant mechanisms are more challenging than that of rigid-body mechanisms. There are two main synthesis approaches for compliant mechanisms, i.e., rigid-body replacement approach [1–3] and topology optimization approach [4–7].

The rigid-body replacement approach has a wide design scope including all kinds of degree-of-freedom (DOF) compliant mechanisms. This approach synthesizes most of compliant parallel mechanisms (CPMs), whose synthesis method is the focus of this paper, by replacing the kinematic joints of existing rigid-body mechanisms with flexure hinges. Thus, the compliant mechanisms designed by this approach rely on the topologies of rigid-body mechanisms. While this approach is successful in designing multi-DOF CPMs for precision applications, it is limited by the fact that a compliant mechanism may still be unable to fully reproduce the motion of its rigid-body counterpart even using rigorous analysis and optimization techniques [8]. Moreover, this approach cannot select the best topology for a specific problem currently, which is quite important for developing mechanisms with high performance.

The topology optimization approach regards the synthesis of compliant mechanism as finding out the optimal material distribution within a given design domain, by maximizing the motion or force transmission between specific input and output ports. Due to this advantage, the topology optimization approach has been successfully applied to the synthesis problems of multiple input and output ports [5,9–13], multiple materials or physics [9–11], three-dimensional simple compliant mechanisms [14–16], etc. The compliant mechanisms designed by this approach possess structural type of topology, i.e., no flexure hinges, which is different from the flexure hinge-based mechanisms obtained by the rigid-body replacement approach.

To introduce the idea of topology optimization into the design scope of the rigid-body replacement approach, our previous works [17–19] tried to synthesize the flexure-based compliant mechanisms with simple motion based on the idea of topology optimization. Lum et al. [20–22] presented a hybrid topological and structural optimization method. This method first synthesizes the compliant joints with the optimal stiffness characteristics by topology optimization. The resulting compliant joints are then assembled into a CPM based on existing rigid-body mechanism topology.

Recently, we proposed a Jacobian-based topology optimization method [23] for the optimal synthesis of planar CPMs. Traditional topology optimization methods realize multiple outputs by predefining specific output displacements at output ports. The premise of this realization is that the position and direction of output ports are known. However, the output motion of multi-DOF CPMs is unknown. The traditional way of predefining specific output ports cannot be applied to multi-DOF CPMs topology optimization directly. To solve this problem, Jacobian matrix [24,25] is introduced into the field of topology optimization by the proposed method. The Jacobian matrix describes all the freedoms of CPM's mobile platform in a unified and concise form, and contains the information of CPM's DOF and direct kinematics simultaneously. By optimizing the Jacobian matrix, one can synthesize a compliant mechanism with desired DOF (mechanism's function) and optimized direct kinematics (mechanism's performance). In addition to the above kinematic realization, the mechanism's input and output characteristic stiffness [26,27] (C-stiffness for short) are also optimized to achieve enough stiffness to bear the external loads.

Lately, we found that the C-stiffness fails to ensure a valid topology result in some particular problems. In this paper, an improved stiffness evaluation based on its definition is incorporated into the problem formulation, and will be compared with the C-stiffness formulation by using several design problems.

The rest of this paper is organized as follows: Section 2 describes the problem formulations of Jacobian-based topology optimization method. Section 3 illustrates the topology analysis of CPMs. The sensitivity analysis and optimization algorithm are described in Sec. 4. Section 5 gives numerical examples to verify the stiffness formulation. The conclusions are presented in Sec. 6.

The topology of a CPM is determined by the number, arrangement, and topology structures of its constituent compliant limbs. The proposed method regards the problem of CPMs topology optimization as finding the best topology of compliant limbs within several given design domains.

A CPM composed of n compliant limbs is shown in Fig. 1. The mechanism's compliant limbs are assumed to be synthesized within n predefined design domains, $Γ={Ω1,Ω2,…,Ωn}$⁠. The corresponding design variables to these design domains are illustrated by $X={X1,X2,…,Xn}$⁠. The number of compliant limbs, n, is determined by the number of CPM's DOF. The position of input points a_i (⁠$i=1,2,…,n$⁠) and output point o is defined by designers.

where $Uo=[ux,uy,θz]T$ contains the three freedoms' displacement of output point o at the mobile platform, and $Ua=[u1,u2,…,un]T$ is the vector of input displacements at input points a_i. Since the compliant mechanisms are usually driven by linear motion actuators, we limit the displacements in U_a to the translational freedoms of input points.

The element J_ji in J represents the geometry advantage (GA) between the jth freedom of U_o and the ith input. If all the elements of the jth row in J are equal or close to zero, all the inputs in U_a will not produce displacement in the jth freedom of mobile platform, i.e., this freedom is suppressed. While the other freedoms corresponding to nonzero row vectors are considered as the CPM's DOF. By maximizing the absolute value of the elements in the nonzero row vectors, the forward kinematics and motion transmission performance of CPM can be optimized. Thus, the Jacobian matrix contains the information of CPM's function and performance simultaneously. By optimizing the Jacobian matrix, we can synthesize a compliant mechanism with desired DOF to realize the mechanism's function and optimized direct kinematics to achieve higher performance.

The objective function of CPMs topology optimization used in Ref. [23] is developed on the basis of Chen and Wang's formulation [26,27] that utilizes the C-stiffness. This paper tries to modify the formulation using a new stiffness evaluation that calculates the stiffness based on its definition.

where GA is the geometry advantage of the mechanism, and GA* is the desired geometry advantage, f and S represent the kinematic and stiffness requirement, respectively.

According to Eq. (2), the physical meanings of k₁₁ and k₂₁ are the forces that should be acted on input and output ports, if u_in = 1 and u_out = 0 are expected. The input C-stiffness k₁₁ just describes part of the stiffness relationship between f_in and u_in, and so does the output C-stiffness k₂₂.

By comparing k₁₁ and k₂₂ with k_in and k_out, respectively, C-stiffness is part of the stiffness. While the C-stiffness has successfully evaluated the mechanism's stiffness property in many design problems, some problems show that the stiffness is more reliable than C-stiffness. Thus, this paper will modify the formulations of Jacobian-based topology optimization method by replacing the C-stiffness with stiffness.

where ω (0 < ω < 1) is the weight indicating the relative significance of kinematic requirement f. V(X) is the volume fraction of topology candidate, and V_o is the allowed volume fraction.

The calculation of Jacobian matrix J, input and output stiffness will be illustrated in Secs. 3.3 and 3.4, respectively.

This section shows how the Jacobian matrix, and input and output stiffnesses of CPMs can be obtained by using the finite element analysis and matrix methods [31].

where N_i is the number of the elements in design domain Ω_i, $Npi$ is the number of the elements in $Ωpi$⁠, K_e is the element stiffness matrix in the global level, x^e is the material density (design variable) of each element in Ω_i with value between the lower limit $xmine$ (void) and 1 (solid), ρ is the penalty factor, the elements in $Ωpi$ are solid.

On the basis of sensitivity analysis, the topology optimization problem is solved by modifying the 99 line matlab code proposed by Sigmund [34]. The optimality criteria-based optimizer and filtering technique [7] of the 99 line Matlab code are used to update the design variables of each domain and ensure existence of solutions, respectively. For each numerical example in Sec. 5, the initial design X is defined by setting the material density of each element to be the value of the allowed volume fraction, i.e., x^e = V_o. The convergence criterion is the change in design variables, which is set to 0.005 in this paper. The move limit in the heuristic updating scheme is 0.1. The filter radius r_min is set to 1.2, i.e., the filter length scale d_min (d_min = 2r_min) is 2.4. It should be pointed out that the volume constraint is active during the whole optimization process. For more detail about the optimality criteria-based optimizer and filtering technique, readers can refer to Ref. [34].

This section will compare several topology optimization results obtained by using the C-stiffness and stiffness formulations, respectively. The artificial material properties for these examples are described as: Young's modulus is E = 1 GPa and Poisson's ratio is υ = 0.3. The characteristic length l_c is set to 10 mm in this study. All the numerical examples are carried out on a computer with Intel Core i7 – 6700 (3.40 GHz) CPU, 8.00GB RAM, and MatlabR2009a. Note that the filter length scale d_min = 2.4 is represented by a red bar in each figure showing the final topology.

In this section, a 2DOF CPM will be synthesized using two asymmetrical compliant limbs within the design domain shown in Fig. 4. As can be seen, the two compliant limbs have the same size and boundary conditions. Each compliant limb is discretized by 100 × 100 finite elements for elastic analysis. The allowable amount of material is 20%.

The GA^* is first set to −3 and ω is set to 0.5. The corresponding topology optimization problem is solved by using the C-stiffness and stiffness formulations, respectively. The optimizations were run for 200 iterations. The resulting topologies of the two formulations are shown in Fig. 5, which shows that both of C-stiffness and stiffness formulations can obtain valid topologies in this case. Figures 6 and 7 show the iteration history of objective value, kinematic requirement f, and stiffness requirement S in the optimization process of the two formulations, respectively. It can be seen that oscillations exist in the iteration curves. The oscillations may be caused by the material distribution at some specific elements. Fortunately, the topologies at later period of iteration are stable, which can be regarded as the optimal topology. The corresponding J of the two final topologies are listed in the first two rows of Table 1. It should be noted that only the elements in the first column vector of J are displayed for brevity, since J₁₁ ≃ J₂₂, J₂₁ ≃ J₁₂, and J₃₁ ≃ J₃₂. One can see that both of C-stiffness and stiffness formulations can force J of CPMs to be close to J^*, i.e., the kinematics requirement f is realized. In addition, the C-stiffness (Ck_a1 and Ck_ox) and stiffness (k_a1 and k_ox) of the input and output ports in x-axis of the two final topologies are given in the first two rows of Table 2. The results show that both of C-stiffness and stiffness formulations realize the stiffness requirement S effectively. The resulting output stiffness is smaller than the input stiffness to achieve GA, whereas the value of C-stiffness is larger than stiffness for the same mechanism.

However, when GA^* is set to be positive, e.g., GA^* = 2.5, it is found that the C-stiffness formulation fails to ensure the stiffness requirement S and results in invalid topologies. Figure 8(a) shows the final topology obtained by using the C-stiffness in the case of GA^* = 2.5. Although the input and output points of the CPM are connected by solid material successfully, there is no material connection between the compliant limbs and fixed ports. Consequently, the displacements of the input and output ports in one direction are equal, i.e., J₁₁ = 1 (in the third row of Table 1). As shown in the third row of Table 2, the C-stiffness of the input and output ports in x-axis is 101.4 N/mm, whereas the corresponding stiffness k_a1 and k_ox of this invalid topology are approximate to zero. Obviously, the C-stiffness fails to evaluate the stiffness of mechanism in this case.

Then, this problem is solved by using the stiffness formulation. The final topology is shown in Fig. 8(b), where ω is set to 0.7. One can see that there are valid material connections between the input, output and fixed ports of the CPM. As shown in the last rows of Tables 1 and 2, the resulting J is close to J^*, e.g., J₁₁ = 2.27. The values of its C-stiffness and stiffness are reasonable. Thus, the problem of C-stiffness is avoided and a valid final topology can be ensured by the stiffness formulation.

The second design problem of CPMs is to synthesize a 3DOF CPM with three symmetrically arranged compliant limbs. The positions of the CPM's input, output, and fixed points are shown in Fig. 10. Each design domain Ω_i is discretized by 50 × 80 finite elements for elastic analysis under the same boundary condition. The allowable amount of material is 20%. Since the kinematics of 3DOF CPM is complex, it is hard to predefine a desired Jacobian matrix for the optimization. The second form of kinematic requirement f₂ shown in Eq. (9) will be used in the objective function. For the planar 3DOF CPMs, no freedom should be suppressed, i.e., the design objective is to maximize the workspace of these three freedoms.

Although the Jacobian-based topology optimization method is developed for the CPMs, this method is applicable to the typical compliant mechanisms designed by the current topology optimization methods, e.g., displacement inverter, amplifier, and redirector.

When GA^* is negative, e.g., GA^* = −3, the design problem is to synthesize a displacement inverter. The corresponding topology optimization problem is also solved by using the C-stiffness and stiffness formulations, respectively. Figure 14(a) shows the resulting topology obtained by using the C-stiffness formulation and setting the weight as ω = 0.5. Its corresponding Jacobian matrix is J = [−2.7, 0, 0]^T, i.e., the optimized GA of displacement inverter is −2.7. The C-stiffness at the input and output ports are 40.9 and 4.7 N/mm, while the stiffness at input and output ports are 6.4 and 0.7 N/mm, respectively. Figure 14(b) shows the resulting topology obtained by using the stiffness formulation and setting ω to 0.5. Its corresponding Jacobian matrix is J = [−2.8, 0, 0]^T. The C-stiffness at the input and output ports are 57.2 and 2.7 N/mm, while the stiffness at input and output ports are 41.3 and 2.0 N/mm, respectively. One can see that both of the C-stiffness and stiffness formulations can obtain valid topologies in this case.

When GA^* is positive, e.g., GA^* = 3, the design problem is to synthesize a displacement amplifier. It is found that the C-stiffness formulation results in the invalid topology shown in Fig. 15(a). Without material connected to the fixed ports, the Jacobian matrix of the resulting displacement amplifier is J = [1, 0, 0]^T. Both of the input and output C-stiffness are 116.1 N/mm, whereas both of the corresponding input and output stiffness are 4.3 × 10⁻⁷ N/mm. Figure 15(b) shows the resulting topology obtained by using the stiffness formulation and setting ω to 0.6. Its corresponding Jacobian matrix is J = [2.3, 0, 0]^T. The C-stiffness at the input and output ports are 79.5 and 6.2 N/mm, while the stiffness at input and output ports are 47.8 and 3.8 N/mm, respectively. This case shows again that the C-stiffness formulation fails to ensure the stiffness requirement.

The computational expense of the proposed method is compared with the artificial I/O spring formulation by solving the design problem of displacement inverter in Sec. 5.3.1. It takes the artificial I/O spring formulation 47.59 s and 220 iterations to find out the solution, i.e., 216 ms per iteration. The C-stiffness formulation spends 181.5 s and 300 iterations to obtain the topology in Fig. 14(a), i.e., 605 ms per iteration. The stiffness formulation spends 209.37 s and 300 iterations to obtain the topology in Fig. 14(b), i.e., 698 ms per iteration. Obviously, the proposed method is more expensive than the artificial I/O spring formulation in computation. One reason for this is that the finite element with 12 nodal freedoms increases the computational expense of finite element analysis. The other reason is that all the freedoms of the output point o are considered by the proposed method, whereas the spring formulation only concerns the freedom u_x.

The corresponding topology optimization problem is solved by using the C-stiffness and stiffness formulations, respectively. The optimization process was run for 300 iterations. Figure 18(a) shows the resulting topology obtained by using the C-stiffness formulation and setting ω to 0.5. Its corresponding Jacobian matrix is J = [1.7, −1.7]^T. The C-stiffness at the input and output ports are 41.2, 5.9, and 5.9 N/mm, while the stiffness at input and output ports are 10.7, 2.3, and 2.3 N/mm, respectively. Figure 18(b) shows the resulting topology obtained by using the stiffness formulation and setting ω to 0.5. Its corresponding Jacobian matrix is J = [1.5, −1.5]^T. The C-stiffness at the input and output ports are 58.1, 4.1, and 4.1 N/mm, while the stiffness at input and output ports are 40.1, 3.3, and 3.3 N/mm, respectively. This case shows that the proposed method is applicable to the compliant mechanisms with multiple output ports, and both of the C-stiffness and stiffness formulations can obtain valid topologies.

This section is devoted to analyzing the mesh independency of the proposed method. The problem of 2DOF CPM, whose parameter settings except discretization are the same as the case in Fig. 5, is used to illustrate the mesh independency. By using three different element discretizations of 40 × 40, 200 × 200 and 300 × 300, we obtain the corresponding results shown in Fig. 19. The three topologies on the left side of Fig. 19 are solved by the C-stiffness formulation, whereas the other side contains the results of the stiffness formulation. One can see that the results are almost stable under mesh refinement or mesh coarsening. In other words, the proposed method is mesh independent, which is ensured by the filtering technique of the 99 line Matlab code.

This paper presents a new stiffness evaluation based on the definition of stiffness for the Jacobian-based topology optimization method. The proposed stiffness formulation is compared with the C-stiffness formulation by using two synthesis problems of CPMs and three traditional benchmark design problems. The results show that both of the two formulations can achieve valid topologies in most of design cases. In some cases like displacement amplifier, the C-stiffness formulation even the artificial I/O spring formulation cannot obtain valid result, while the stiffness formulation gives an improved stiffness evaluation. Besides, the Jacobian-based topology optimization method shows a general applicability in multi-DOF CPMs and benchmark design problems.

According to the results, the topologies produced by the proposed method are easy to exhibit hinges, especially when the kinematic requirement is much higher than stiffness requirement. Relatively speaking, the stiffness formulation has a better performance than the C-stiffness formulation in avoiding hinges, e.g., the two topologies in Fig. 18. The strategies for alleviating these hinges will be addressed in our future works.

• National Natural Science Foundation of China (Grant Nos. 51275174, 51605166, U1609206, and 51675189).

• Natural Science Foundation of Guangdong Province (Grant No. 2014A030313460).

• Fundamental Research Funds for the Central Universities.