^{1}

^{2}

^{3}

^{1}

^{1}

^{2}

^{3}

This paper proposes a new evolutionary multiobjective optimization technique for a linear compliant mechanism of nanoindentation tester. The mechanism design is inspired by the elastic deformation of flexure hinge. To improve overall static performances, a multiobjective optimization design was carried out. An efficient hybrid optimization approach of central composite design (CDD), finite element method (FEM), artificial neural network (ANN), and multiobjective genetic algorithm (MOGA) is developed to solve the optimization problem. In this procedure, the CDD is used to lay out the experimental data. The FEM is developed to retrieve the quality performances. And then, the ANN is developed as black box to call the pseudo-objective functions. Unlike previous studies on multiobjective evolutionary algorithms, most of which generating only one Pareto-optimal solution, this proposed approach can generate more than three Pareto-optimal solutions. Based on the user’s real-work problem, one of the best optimal solutions is chosen. The results showed that the optimal results were found at the displacement of 330.68

Nanoindentation tester has attracted great interest from researchers from academics and industry. This device is now widely used in areas of material science. It is desired to probe the mechanical properties such as harness, creep, elastic-plastic modulus, and roughness surface. Materials can be tested, including hard and soft types from tissue, biological cell, nanomaterial, optics, material science, semiconductor, biomechanics, micro-electromechanical systems, and electronics [

During the indentation process, multiple microscopes are used to record the image of sample before and after indenting test to characterize the curve of displacement versus load. High precision positioning and stability of the system are dependent not only on the microscope, controller, and imaging technique, but also a mechanical-driven platform. The platform is utilized to bring the material in front of microscope and then a picture is taken. And then, the platform takes the material to the location of indenter so as to do an indentation. At last, the platform brings the material to return back the location of microscope to characterize the curve of displacement versus load. It can be concluded that the mechanical-driven platform is a critical mechanism for the nanoindentation tester. In commercialization, the current instruments are relatively effective but so costly. The reason is that, in the current nanoindentation tester system, a ball-screw driven system by servo motors or linear motors is used to ensure the linear precision. However, such mechanical-driven systems possess some common defects such as assembly gap, wear and aging, and friction error, which may restrict their further applications in the nanoindentation industries where micro or nanoscale positioning accuracy is needed. Hence, it has been a difficult challenge for researchers to make a cheaper system with better accuracy.

To eliminate the cost and enhance the positioning accuracy, a linear compliant mechanism (LCM) is proposed in this study to bring the observed material. The proposed LCM is operated based on ideal of elastic springs and has no backlash, free friction, high precision, and monolithic structure [

To meet the practice requirements of advanced material sciences, the capability of a large positioning space and long working life of the LCM should be improved further. To overcome the limitation of performances, common procedures are, for example, topology and size optimization. Topology represents the connectivity of the domain. Size optimization denotes design parameters such as thickness, cross-sectional area, and length [

For these reasons, this study introduces a new evolutionary multiobjective optimization technique for optimizing the performances of LCM to decrease the modeling errors. The proposed optimization approach is a hybrid intelligent integration of central composite design (CDD), finite element method (FEM), artificial neural network (ANN), and multiobjective genetic algorithm (MOGA). Each of the mentioned methods has been used widely but individually. For example, CDD was investigated in building and optimizing [

The computationally evolutionary optimization approach is developed to satisfy a tradeoff between performance characteristics of LCM. In this hybrid optimization approach, a model is created via using finite element method. Subsequently, based on central composite design (CCD), a number of simulated experiments are generated and the results of performances are automatically generated via FEM. And then, relationship between design parameters and output performances will be generated well by using the ANN technique. ANN is as a black box to map well the inputs and outputs. Finally, MOGA is integrated to seek an optimal solution. This optimization process is totally automatic solution to gain accurate results.

The purposes of this paper are to propose an optimal design strategy for the LCM in terms of good static characteristics. Elastic joints of the LCM are designed based on elastic springs. A lever-displacement amplifier is used to amplify the workspace. To improve the static performances, a hybrid optimization algorithm is developed. A validation is performed to evaluate predicted results, efficiency, and robustness of the proposed approach.

Although the piezoelectric stack actuator (PSA) was a common actuator for precision positioning platform and manipulator, it is still limited by its working travel. To overcome this limitation, the lever-displacement amplification is often used to amplify the travel of PSA.

A traditional lever-displacement amplifier is depicted, as seen in Figure _{1} and output distance_{o}. Figure

Traditional lever-displacement amplification: (a) model, (b) kinematics.

The traditional amplification ratio can be calculated as follows:

in which,_{1} and_{2} are the input displacement and output displacement, respectively.

In the field of precision positioning system, compliant mechanisms (CMs) are operated relying on the elastic deformation of thin cross section links [

Linear compliant mechanism: (a) 2D diagram, (b) 3D model.

The basic mathematical model for the CDA was computed as follows:

in which,

As seen in (

As depicted in Figure _{1} and length_{1} served as flexure hinge and helped to eliminate the parasitic motion along the x-axis. Other links with thicknesses_{2} and_{3} and lengths_{2} and_{3} were geometrical parameters of CDA. The LCM had width of

Properties of material Al 7075.

Material properties | |||

| |||

Density | Poison’s ratio | Young’s modulus | Yield strength |

| |||

2810 kg/m^{3} | 0.33 | 71.70 GPa | 503 MPa |

| |||

Parameters | Unit | Dimension | |

_{ 1 }, _{2}, _{3},_{1}, _{2}, _{3} | mm | Variables |

In the material field, inspecting mechanical properties of a soft and thin material sample during nanoindentation test is critical work. Figure

2D model of a nanoindentation tester.

Existing nanoindentation tester has attracted a great interest in material engineering. It is used for various materials such as biological tissues and soft gels. Commercialized instruments can monitor easily the mechanical properties of a sample but its operating costs are rather high. A high cost may come from positioning system where motors and actuators are utilized simultaneously. To fulfill higher precise probe requirements in several tens of micrometers to hundreds of millimeters, the proposed LCM was used to decrease the motors and actuators. Hence, the considerable challenges have been faced when developing the nanoindentation devices; the instruments should have the following specifications: (i) able to give a large displacement; (ii) able to speed up the indentation process; (iii) able to ensure a long working time; (iv) able to gain an excellent stability; and (v) able to record the displacement and force signal itself before and after indentation process.

Specifically, the LCM could give a large displacement and a long working time for nanoindentation tester. In this system, the LCM should fulfill the following important requirements: to satisfy faced challenges as

a large working stroke to allow suitability for various materials;

a high safety factor to enhance the fatigue life.

In general, these requirements conflicted together. A tradeoff multiple responses could be gained by developing a new hybrid optimization approach.

It was noted that the proposed LCM was very sensitive to geometric dimensions such as thicknesses, lengths, and width. Therefore, these parameters were considered as the design variables. The vector of design variables was set as

The lower and upper bounds for the design variables were assigned as follows:_{i} represents length of

The multiple quality performances of LCM were as follows: (i) the displacement,

The resulting stress of the LCM must be under the yield strength of the material, which was described as follows:

Before conducting an engineering optimization problem, mathematical models are traditionally established, and then a suitable optimization algorithm is utilized. The fact is that there would be an error between mathematical models and simulations or experiments. This is because those models may not be right and depend mainly on capability and knowledge of researchers about engineering and mathematical theory. Therefore, an evolution optimization algorithm can then generate inaccurate results. To overcome this limitation, a hybrid approach of RSM, FEM, ANN, and MOGA was proposed in this study so as to improve the quality performances of the LCM. The robustness and efficiency of proposed approach could be validated through simulated experiments. Figure

The flowchart of hybrid optimization procedure.

Nowadays, with a fast development of computer, a high performance computer (HPC) can perform a thousands of calculations per a second, which is considered as a workstation. Based on these advantages, machine learning and artificial intelligent algorithms can be programed well to meet a research and development in engineering. It can be called computer aid engineering application and design (CAD A&D). The first stage to optimize the CDA, a CAD A&D process for the CDA, was conducted by following steps.

The CDA was deigned to serve as a displacement amplifier for applications in precision engineering micro- or nanopositioning systems. In these devices, a piezo stack actuator (PSA) serves as an actuator but its travel is limited. Therefore, the CDA was intended to amplify the work travel for PSA, a product from Physik Instrumente Co., Ltd. Multiobjective optimization problem for the CDA was conducted to improve its quality performances.

A mechanical structure was proposed based on designer’s experience and perception. Initially, many drafts were drawn to illustrate the highlights of CDA. A final model was chosen but its specifications cannot meet the requirements of positioning system.

The lengths and thicknesses of flexure hinge and the width of the CDA were determined as design variables. The reason was that these parameters affect the performances of the CDA. These variables can be seen as Figure

To meet the practical requirements of customers, a precision positioning system must have a large workspace, a high speed, a high safety factor, and a minimum stress. All these performances can be fulfilled by optimizing the CDA because the mechanical structure and geometries of CDA contribute to the system. Hence, the mentioned quality characteristics were taken into account as four objective functions.

A 3D model was drawn by using FEM. The FEM is a computational technique used to obtain approximate solutions of problems in engineering. Postprocessor contains sophisticated routines used for sorting, printing, and plotting selected results from a finite element solution. Because the CDA was an elastic structure, during the analysis some of the following relationships were used.

According to the linear Hooke’s law, the equation describing a relation between the force and displacement was computed as

Also by Hooke’s law, the relationship between stress and strain was calculated as follows:

The 3D model was drawn in Step

Prior to optimization process, virtually pseudo-mathematical equations could be treated as fitness functions. To do this, a number of experiments were generated and simulated data were collected. At last, the regression models were established to map design variables and outcome performances.

By using the RSM-integrated FEM, a numbers of simulation experiments were planned via using the central composite design integrated with RSM. The number of necessary experiments was determined by the following equation:_{c} = 1 was the number of replicates at the center point of the design space.

The simulated experimentations were collected based on an integration of RSM and FEM. First, they were based on the 3D model designed in the FEM in the Step

There were various regression models such as full 2nd-order polynomials, artificial neural network, and nonparametric regression. In this study, ANN was a suitable candidate for the retrieved data. It was used as a regression approach for the estimated database. It was considered as a black box to approximate the complex nonlinear relationship between design parameters and the qualities. ANN model can find a pseudo-objective functions. These objective functions were used for the MOGA algorithm.

After the pseudo-objective functions were determined in Step

Flowchart of MOGA algorithm.

The multiobjective genetic algorithm (MOGA) was a variant of the nondominated sorted genetic algorithm-II (NSGA-II) that relied on controlled elitism concepts. It can solve multiple objectives and constraints. At last, the MOGA can find the global optimum solution. The controllable parameters of MOGA in this study were given in Table

Controllable parameters for MOGA algorithm.

Parameters | Range of value |
---|---|

Population size | 20-100 |

Number of generations | 50-150 |

Crossover probability | 0.2-0.9 |

Mutation probability | 0.01-0.02 |

Maximum number candidates | 3 |

This step was critical to change the fitness so as to gain the fitness values of the individuals.

The usual selection operator, such as Monte Carlo, was used and combined with the elitist model.

Crossover operation was utilized to produce parent generation.

Mutation operation was used to generate new generation.

If the gross generation was gained and the best individual was found through predetermined iterations, the MOGA was stopped. Otherwise, Step

The best candidates were found by using MOGA and then the final phase was overall evaluations of these candidates.

Multiobjective optimization problem was solved by using MOGA, and then the tradeoff the objective optimizations was treated as Pareto-optimal set. At last, the optimal results were found.

If the optimal results were satisfying, the extra validations were conducted to evaluate the robustness and efficiency of the proposed hybrid optimization approach. The optimization process was ended herein. If they were not well refined, the optimization process would be further enhanced by Step 17 or Step

In performing the optimization process, a refinement was the most important phase to seek the optimal solutions. After optimal candidates were generated and based on the initial requirements of quality characteristics, the researchers would evaluate the candidates. If there was no candidate that was satisfying, the ranges of quality characteristics or the range of design variables must be controlled or refined again from Step

If Step

A 3D model was created in FEM design modeler, and then FEA simulations were conducted to test the initial performance of LCM. As shown in Table

Initial parameters and performances.

Design parameter | Initial design | Unit |
---|---|---|

_{ 1 } | 0.5 | mm |

_{ 2 } | 0.4 | mm |

_{ 3 } | 0.5 | mm |

_{ 1 } | 30 | mm |

_{ 2 } | 20 | mm |

_{ 3 } | 6 | mm |

| 10 | mm |

Displacement | 295.59 | |

Safety factor | 2.7331 |

As proposed hybrid optimization approach, a 3D model of the LCM was designed via using FEM. With six design variables, the numbers of experiments were calculated using the central composite design in (

And then, based on Kriging regression model, each pseudo-objective function was made for the displacement, resulting stress, and safety factor. Finally, by integrating FEM, RSM, and ANN regression model, the real values of three quality responses were automatically computed, as given in Table

Design of experiments and training data using CCD.

No. | _{ 1 } | _{ 1 } | _{ 2 } | _{ 2 } | _{ 3 } | _{ 3 } | | Displacement | Safety factor |
---|---|---|---|---|---|---|---|---|---|

1 | 30 | 0.5 | 20 | 0.4 | 0.5 | 6 | 10 | 472.59 | 3.63 |

2 | 30 | 0.5 | 20 | 0.4 | 0.5 | 6 | 5 | 476.33 | 3.68 |

3 | 30 | 0.5 | 20 | 0.4 | 0.5 | 6 | 15 | 474.79 | 3.57 |

4 | 30 | 0.5 | 18 | 0.4 | 0.5 | 6 | 10 | 602.70 | 1.87 |

5 | 30 | 0.5 | 22 | 0.4 | 0.5 | 6 | 10 | 392.22 | 2.61 |

6 | 30 | 0.5 | 20 | 0.36 | 0.5 | 6 | 10 | 481.24 | 3.62 |

7 | 30 | 0.5 | 20 | 0.44 | 0.5 | 6 | 10 | 467.73 | 3.77 |

8 | 30 | 0.5 | 20 | 0.4 | 0.45 | 6 | 10 | 468.62 | 3.72 |

9 | 30 | 0.5 | 20 | 0.4 | 0.55 | 6 | 10 | 475.77 | 3.68 |

10 | 30 | 0.5 | 20 | 0.4 | 0.5 | 5.4 | 10 | 473.37 | 3.60 |

11 | 30 | 0.5 | 20 | 0.4 | 0.5 | 6.6 | 10 | 471.30 | 3.67 |

12 | 27 | 0.5 | 20 | 0.4 | 0.5 | 6 | 10 | 465.98 | 3.74 |

13 | 33 | 0.5 | 20 | 0.4 | 0.5 | 6 | 10 | 478.28 | 3.66 |

14 | 30 | 0.45 | 20 | 0.4 | 0.5 | 6 | 10 | 473.23 | 3.64 |

15 | 30 | 0.55 | 20 | 0.4 | 0.5 | 6 | 10 | 470.26 | 3.71 |

16 | 28.93 | 0.51 | 19.29 | 0.38 | 0.48 | 5.78 | 8.23 | 521.26 | 2.94 |

17 | 28.93 | 0.48 | 19.29 | 0.38 | 0.48 | 5.78 | 11.76 | 524.74 | 3.04 |

18 | 28.93 | 0.48 | 20.70 | 0.38 | 0.48 | 5.78 | 8.23 | 431.56 | 3.36 |

19 | 28.93 | 0.51 | 20.70 | 0.38 | 0.48 | 5.78 | 11.76 | 434.51 | 3.78 |

20 | 28.93 | 0.48 | 19.29 | 0.41 | 0.48 | 5.78 | 8.23 | 515.78 | 3.01 |

21 | 28.93 | 0.51 | 19.29 | 0.41 | 0.48 | 5.78 | 11.76 | 514.26 | 3.12 |

22 | 28.93 | 0.51 | 20.70 | 0.41 | 0.48 | 5.78 | 8.23 | 428.35 | 3.35 |

23 | 28.93 | 0.48 | 20.70 | 0.41 | 0.482 | 5.78 | 11.76 | 432.74 | 3.78 |

24 | 28.93 | 0.48 | 19.29 | 0.38 | 0.51 | 5.78 | 8.23 | 527.63 | 2.88 |

25 | 28.93 | 0.51 | 19.29 | 0.38 | 0.51 | 5.78 | 11.76 | 526.40 | 3.03 |

26 | 28.93 | 0.51 | 20.70 | 0.38 | 0.51 | 5.78 | 8.23 | 438.24 | 3.53 |

27 | 28.93 | 0.48 | 20.70 | 0.38 | 0.51 | 5.78 | 11.76 | 436.99 | 3.76 |

28 | 28.93 | 0.51 | 19.29 | 0.41 | 0.51 | 5.78 | 8.23 | 517.02 | 2.97 |

29 | 28.93 | 0.48 | 19.29 | 0.41 | 0.51 | 5.78 | 11.76 | 520.74 | 3.04 |

30 | 28.93 | 0.48 | 20.70 | 0.41 | 0.51 | 5.78 | 8.23 | 434.69 | 3.49 |

31 | 28.93 | 0.51 | 20.70 | 0.41 | 0.51 | 5.78 | 11.76 | 431.84 | 3.71 |

32 | 28.93 | 0.48 | 19.29 | 0.38 | 0.48 | 6.21 | 8.23 | 522.59 | 3.00 |

33 | 28.93 | 0.51 | 19.29 | 0.38 | 0.48 | 6.21 | 11.76 | 524.15 | 3.05 |

34 | 28.93 | 0.51 | 20.70 | 0.38 | 0.48 | 6.21 | 8.23 | 430.91 | 3.42 |

35 | 28.93 | 0.48 | 20.70 | 0.38 | 0.48 | 6.21 | 11.76 | 434.53 | 3.96 |

36 | 28.93 | 0.51 | 19.29 | 0.41 | 0.48 | 6.21 | 8.23 | 512.86 | 3.07 |

37 | 28.93 | 0.48 | 19.29 | 0.41 | 0.48 | 6.21 | 11.76 | 513.27 | 3.12 |

38 | 28.93 | 0.48 | 20.70 | 0.41 | 0.48 | 6.21 | 8.23 | 433.47 | 3.44 |

39 | 28.93 | 0.51 | 20.70 | 0.41 | 0.48 | 6.21 | 11.76 | 430.08 | 3.89 |

40 | 28.93 | 0.51 | 19.29 | 0.38 | 0.51 | 6.21 | 8.23 | 523.21 | 2.93 |

41 | 28.93 | 0.48 | 19.29 | 0.38 | 0.51 | 6.21 | 11.76 | 526.51 | 3.04 |

42 | 28.93 | 0.48 | 20.70 | 0.38 | 0.51 | 6.21 | 8.23 | 438.89 | 3.67 |

43 | 28.93 | 0.51 | 20.70 | 0.38 | 0.51 | 6.21 | 11.76 | 437.81 | 3.79 |

44 | 28.93 | 0.48 | 19.29 | 0.41 | 0.51 | 6.21 | 8.23 | 517.41 | 2.99 |

45 | 28.93 | 0.51 | 19.29 | 0.41 | 0.51 | 6.21 | 11.76 | 514.68 | 3.06 |

46 | 28.93 | 0.51 | 20.70 | 0.41 | 0.51 | 6.21 | 8.23 | 433.78 | 3.33 |

47 | 28.93 | 0.48 | 20.70 | 0.41 | 0.51 | 6.21 | 11.76 | 433.24 | 3.74 |

48 | 31.06 | 0.48 | 19.29 | 0.38 | 0.48 | 5.78 | 8.23 | 529.57 | 2.93 |

49 | 31.06 | 0.51 | 19.29 | 0.38 | 0.48 | 5.78 | 11.76 | 527.76 | 3.06 |

50 | 31.06 | 0.51 | 20.70 | 0.38 | 0.48 | 5.78 | 8.23 | 441.45 | 3.60 |

51 | 31.06 | 0.48 | 20.70 | 0.38 | 0.48 | 5.78 | 11.76 | 440.86 | 3.82 |

52 | 31.06 | 0.51 | 19.29 | 0.41 | 0.48 | 5.78 | 8.23 | 520.39 | 2.99 |

53 | 31.06 | 0.48 | 19.29 | 0.41 | 0.48 | 5.78 | 11.76 | 522.99 | 3.09 |

54 | 31.06 | 0.48 | 20.70 | 0.41 | 0.48 | 5.78 | 8.23 | 438.41 | 3.54 |

55 | 31.06 | 0.51 | 20.70 | 0.41 | 0.48 | 5.78 | 11.76 | 436.12 | 3.65 |

56 | 31.06 | 0.51 | 19.29 | 0.38 | 0.51 | 5.78 | 8.23 | 530.31 | 2.94 |

57 | 31.06 | 0.48 | 19.29 | 0.38 | 0.51 | 5.78 | 11.76 | 533.15 | 3.01 |

58 | 31.06 | 0.48 | 20.70 | 0.38 | 0.51 | 5.78 | 8.23 | 439.53 | 3.32 |

59 | 31.06 | 0.51 | 20.70 | 0.38 | 0.51 | 5.78 | 11.76 | 444.13 | 3.78 |

60 | 31.06 | 0.48 | 19.29 | 0.41 | 0.51 | 5.78 | 8.23 | 524.97 | 2.94 |

61 | 31.06 | 0.517 | 19.29 | 0.41 | 0.51 | 5.78 | 11.76 | 524.10 | 3.05 |

62 | 31.06 | 0.51 | 20.70 | 0.41 | 0.51 | 5.78 | 8.23 | 435.03 | 3.32 |

63 | 31.06 | 0.48 | 20.70 | 0.41 | 0.51 | 5.78 | 11.76 | 439.34 | 3.72 |

64 | 31.06 | 0.51 | 19.29 | 0.38 | 0.48 | 6.21 | 8.23 | 526.08 | 3.01 |

65 | 31.06 | 0.48 | 19.29 | 0.38 | 0.48 | 6.21 | 11.76 | 529.12 | 3.05 |

66 | 31.06 | 0.48 | 20.70 | 0.38 | 0.48 | 6.21 | 8.23 | 436.64 | 3.44 |

67 | 31.06 | 0.51 | 20.70 | 0.38 | 0.48 | 6.21 | 11.76 | 440.67 | 3.84 |

68 | 31.06 | 0.48 | 19.29 | 0.41 | 0.48 | 6.21 | 8.23 | 520.62 | 2.99 |

69 | 31.06 | 0.51 | 19.29 | 0.41 | 0.48 | 6.21 | 11.76 | 520.96 | 3.13 |

70 | 31.06 | 0.51 | 20.70 | 0.41 | 0.48 | 6.21 | 8.23 | 436.90 | 3.41 |

71 | 31.06 | 0.48 | 20.70 | 0.41 | 0.48 | 6.21 | 11.76 | 438.74 | 3.86 |

72 | 31.06 | 0.48 | 19.29 | 0.38 | 0.51 | 6.21 | 8.23 | 530.51 | 2.92 |

73 | 31.06 | 0.51 | 19.29 | 0.38 | 0.51 | 6.21 | 11.76 | 527.16 | 3.03 |

74 | 31.06 | 0.51 | 20.70 | 0.38 | 0.51 | 6.21 | 8.23 | 442.88 | 3.63 |

75 | 31.06 | 0.48 | 20.70 | 0.38 | 0.51 | 6.21 | 11.76 | 442.26 | 3.88 |

76 | 31.06 | 0.51 | 19.29 | 0.41 | 0.51 | 6.21 | 8.23 | 521.42 | 3.04 |

77 | 31.06 | 0.48 | 19.29 | 0.41 | 0.51 | 6.21 | 11.76 | 522.38 | 3.07 |

78 | 31.06 | 0.48 | 20.70 | 0.41 | 0.51 | 6.21 | 8.23 | 434.91 | 3.32 |

79 | 31.06 | 0.51 | 20.70 | 0.41 | 0.51 | 6.21 | 11.76 | 439.50 | 3.56 |

Compared with other regression models such as full 2nd-order polynomials, nonparametric regression, and Kriging model, artificial neural network gave relatively correct results. The coefficient of determination of ANN model was approximately 1, as given in Table

Goodness of fit of regression models.

Artificial neural network model | ||
---|---|---|

Performance criteria | Displacement | Safety factor |

| ||

Coefficient of determination | 1 | 1 |

| ||

Mean square error | 0 | 0 |

The structure of an ANN was determined by the number of layers, the number of nodes in each layer, and the nature of the transfer functions. This paper used three-layered feed forward back propagation neural network (7:10:1), as shown in Figure

Structure of the developed ANN.

In this study, the Bayesian Regularization was chosen as the training method for the safety factor data because it permitted a coefficient of determination (^{2}) of approximately 1. Meanwhile, to gain the^{2} of 1, the Levenberg-Marquardt was suitable for the displacement data. Training, validation, and testing all had a mean square error of about zero, as shown in Figure

Regression plots: (a) the displacement and (b) safety factor.

The contribution of each parameter and interaction affecting the displacement was analyzed in Table _{2} has a largest contribution of 97.78% with respect to the displacement and statistical significance with p-value of 0.000 (less than 0.05%).

Response surface regression of displacement versus design parameters.

Source | DF | Seq SS | Contribution | Adj SS | Adj MS | F-Value | P-Value |
---|---|---|---|---|---|---|---|

Model | 35 | 143821 | 99.61% | 143821 | 4109 | 316.88 | 0.000 |

Linear | 7 | 142517 | 98.71% | 110807 | 15830 | 1220.71 | 0.000 |

_{ 1 } | 1 | 548 | 0.38% | 436 | 436 | 33.63 | 0.000 |

_{ 1 } | 1 | 17 | 0.01% | 39 | 39 | 2.99 | 0.091 |

_{ 2 } | 1 | 141172 | 97.78% | 109802 | 109802 | 8467.46 | 0.000 |

_{ 2 } | 1 | 657 | 0.46% | 454 | 454 | 34.99 | 0.000 |

_{ 3 } | 1 | 102 | 0.07% | 56 | 56 | 4.35 | 0.043 |

_{ 3 } | 1 | 14 | 0.01% | 12 | 12 | 0.94 | 0.339 |

| 1 | 7 | 0.01% | 3 | 3 | 0.23 | 0.635 |

Square | 7 | 1064 | 0.74% | 1061 | 152 | 11.69 | 0.000 |

_{ 1 } _{ 1 } | 1 | 42 | 0.03% | 6 | 6 | 0.44 | 0.511 |

_{ 1 } _{ 1 } | 1 | 74 | 0.05% | 9 | 9 | 0.70 | 0.409 |

_{ 2 } _{ 2 } | 1 | 924 | 0.64% | 421 | 421 | 32.44 | 0.000 |

_{ 2 } _{ 2 } | 1 | 2 | 0.00% | 0 | 0 | 0.00 | 0.997 |

_{ 3 } _{ 3 } | 1 | 11 | 0.01% | 8 | 8 | 0.65 | 0.424 |

_{ 3 } _{ 3 } | 1 | 11 | 0.01% | 5 | 5 | 0.36 | 0.551 |

| 1 | 0 | 0.00% | 0 | 0 | 0.04 | 0.849 |

2-Way Interaction | 21 | 240 | 0.17% | 240 | 11 | 0.88 | 0.614 |

_{ 1 } | 1 | 4 | 0.00% | 5 | 5 | 0.35 | 0.555 |

_{ 1 } _{ 2 } | 1 | 0 | 0.00% | 0 | 0 | 0.02 | 0.902 |

_{ 1 } _{ 2 } | 1 | 1 | 0.00% | 1 | 1 | 0.11 | 0.745 |

_{ 1 } _{ 3 } | 1 | 16 | 0.01% | 16 | 16 | 1.21 | 0.277 |

_{ 1 } _{ 3 } | 1 | 1 | 0.00% | 1 | 1 | 0.10 | 0.748 |

_{ 1 } | 1 | 2 | 0.00% | 2 | 2 | 0.17 | 0.682 |

_{ 1 } _{ 2 } | 1 | 38 | 0.03% | 33 | 33 | 2.54 | 0.118 |

_{ 1 } _{ 2 } | 1 | 3 | 0.00% | 3 | 3 | 0.23 | 0.635 |

_{ 1 } _{ 3 } | 1 | 0 | 0.00% | 1 | 1 | 0.05 | 0.820 |

_{ 1 } _{ 3 } | 1 | 1 | 0.00% | 1 | 1 | 0.07 | 0.795 |

_{ 1 } | 1 | 0 | 0.00% | 0 | 0 | 0.00 | 0.990 |

_{ 2 } _{ 2 } | 1 | 139 | 0.10% | 138 | 138 | 10.63 | 0.002 |

_{ 2 } _{ 3 } | 1 | 5 | 0.00% | 5 | 5 | 0.39 | 0.534 |

_{ 2 } _{ 3 } | 1 | 16 | 0.01% | 16 | 16 | 1.26 | 0.269 |

_{ 2 } | 1 | 1 | 0.00% | 1 | 1 | 0.04 | 0.837 |

_{ 2 } _{ 3 } | 1 | 5 | 0.00% | 5 | 5 | 0.42 | 0.520 |

_{ 2 } _{ 3 } | 1 | 0 | 0.00% | 0 | 0 | 0.00 | 0.964 |

_{ 2 } | 1 | 3 | 0.00% | 2 | 2 | 0.19 | 0.663 |

_{ 3 } _{ 3 } | 1 | 1 | 0.00% | 1 | 1 | 0.11 | 0.743 |

_{ 3 } | 1 | 2 | 0.00% | 2 | 2 | 0.13 | 0.722 |

_{ 3 } | 1 | 0 | 0.00% | 0 | 0 | 0.03 | 0.855 |

Error | 43 | 558 | 0.39% | 558 | 13 | ||

Total | 78 | 144379 | 100.00% |

As shown in Table _{2} also had a largest contribution on the safety factor with 50.14% and a good statistical significance with p-value of 0.000 (less than 0.05%).

Response surface regression of safety factor versus design parameters.

Source | DF | Seq SS | Contribution | Adj SS | Adj MS | F-Value | P-Value |

| |||||||

Model | 35 | 10.3266 | 91.71% | 10.3266 | 0.29505 | 13.59 | 0.000 |

Linear | 7 | 6.2102 | 55.15% | 4.2000 | 0.60000 | 27.63 | 0.000 |

_{ 1 } | 1 | 0.0040 | 0.04% | 0.0070 | 0.00698 | 0.32 | 0.574 |

_{ 1 } | 1 | 0.0033 | 0.03% | 0.0198 | 0.01980 | 0.91 | 0.345 |

_{ 2 } | 1 | 5.6464 | 50.14% | 3.8575 | 3.85746 | 177.66 | 0.000 |

_{ 2 } | 1 | 0.0019 | 0.02% | 0.0195 | 0.01955 | 0.90 | 0.348 |

_{ 3 } | 1 | 0.0062 | 0.05% | 0.0002 | 0.00015 | 0.01 | 0.934 |

_{ 3 } | 1 | 0.0237 | 0.21% | 0.0132 | 0.01323 | 0.61 | 0.439 |

| 1 | 0.5248 | 4.66% | 0.3058 | 0.30581 | 14.08 | 0.001 |

Square | 7 | 3.6100 | 32.06% | 3.6212 | 0.51731 | 23.82 | 0.000 |

_{ 1 } _{ 1 } | 1 | 0.1209 | 1.07% | 0.0723 | 0.07234 | 3.33 | 0.075 |

_{ 1 } _{ 1 } | 1 | 0.1764 | 1.57% | 0.0812 | 0.08121 | 3.74 | 0.060 |

_{ 2 } _{ 2 } | 1 | 3.2129 | 28.53% | 1.1001 | 1.10007 | 50.66 | 0.000 |

_{ 2 } _{ 2 } | 1 | 0.0127 | 0.11% | 0.0783 | 0.07827 | 3.60 | 0.064 |

_{ 3 } _{ 3 } | 1 | 0.0418 | 0.37% | 0.0967 | 0.09670 | 4.45 | 0.041 |

_{ 3 } _{ 3 } | 1 | 0.0105 | 0.09% | 0.0434 | 0.04343 | 2.00 | 0.164 |

| 1 | 0.0348 | 0.31% | 0.0408 | 0.04083 | 1.88 | 0.177 |

2-Way Interaction | 21 | 0.5064 | 4.50% | 0.5064 | 0.02411 | 1.11 | 0.374 |

_{ 1 } | 1 | 0.0023 | 0.02% | 0.0029 | 0.00294 | 0.14 | 0.715 |

_{ 1 } _{ 2 } | 1 | 0.0010 | 0.01% | 0.0010 | 0.00104 | 0.05 | 0.828 |

_{ 1 } _{ 2 } | 1 | 0.0054 | 0.05% | 0.0053 | 0.00527 | 0.24 | 0.625 |

_{ 1 } _{ 3 } | 1 | 0.0051 | 0.05% | 0.0047 | 0.00475 | 0.22 | 0.642 |

_{ 1 } _{ 3 } | 1 | 0.0010 | 0.01% | 0.0009 | 0.00092 | 0.04 | 0.837 |

_{ 1 } | 1 | 0.0016 | 0.01% | 0.0016 | 0.00165 | 0.08 | 0.784 |

_{ 1 } _{ 2 } | 1 | 0.0320 | 0.28% | 0.0306 | 0.03061 | 1.41 | 0.242 |

_{ 1 } _{ 2 } | 1 | 0.0079 | 0.07% | 0.0073 | 0.00729 | 0.34 | 0.565 |

_{ 1 } _{ 3 } | 1 | 0.0000 | 0.00% | 0.0001 | 0.00008 | 0.00 | 0.953 |

_{ 1 } _{ 3 } | 1 | 0.0015 | 0.01% | 0.0016 | 0.00163 | 0.08 | 0.785 |

_{ 1 } | 1 | 0.0265 | 0.24% | 0.0246 | 0.02464 | 1.13 | 0.293 |

_{ 2 } _{ 2 } | 1 | 0.1314 | 1.17% | 0.1296 | 0.12961 | 5.97 | 0.019 |

_{ 2 } _{ 3 } | 1 | 0.0041 | 0.04% | 0.0043 | 0.00433 | 0.20 | 0.657 |

_{ 2 } _{ 3 } | 1 | 0.0003 | 0.00% | 0.0002 | 0.00022 | 0.01 | 0.920 |

_{ 2 } | 1 | 0.2419 | 2.15% | 0.2428 | 0.24277 | 11.18 | 0.002 |

_{ 2 } _{ 3 } | 1 | 0.0105 | 0.09% | 0.0107 | 0.01069 | 0.49 | 0.487 |

_{ 2 } _{ 3 } | 1 | 0.0058 | 0.05% | 0.0058 | 0.00580 | 0.27 | 0.608 |

_{ 2 } | 1 | 0.0045 | 0.04% | 0.0040 | 0.00402 | 0.18 | 0.669 |

_{ 3 } _{ 3 } | 1 | 0.0001 | 0.00% | 0.0001 | 0.00006 | 0.00 | 0.959 |

_{ 3 } | 1 | 0.0233 | 0.21% | 0.0233 | 0.02333 | 1.07 | 0.306 |

_{ 3 } | 1 | 0.0002 | 0.00% | 0.0002 | 0.00022 | 0.01 | 0.921 |

Error | 43 | 0.9337 | 8.29% | 0.9337 | 0.02171 | ||

Total | 78 | 11.2603 | 100.00% |

By using Phase

The larger-the better objective function: the range of this function should not exceed an allowable value. For example, the maximum displacement of LCM was desired to be about 400

A minimum high safety was also required, and this function was constrained to be 3, as seen in Table

Bounds of the quality responses.

Characteristics | Constraint type | Lower | Upper | Unit |
---|---|---|---|---|

Maximum displacement | Lower <= Values <= Upper | N/A | 400 | |

Minimum safety factor | Values >= Lower | N/A | 3 |

The history charts of displacement, stress, and safety factor were retrieved from the results of the proposed hybrid optimization algorithm. As seen in Figure

History chart of the displacement.

History chart of the safety factor.

The pseudo- or virtual mathematical models were found by using ANN approach. To achieve the optimal results for the LCM, the constraints for two objective functions were set up, as given in Table

Potentially optimal design parameters.

Candidates | _{ 1 } (mm) | _{ 1 } (mm) | _{ 2 } (mm) | _{ 2 } (mm) | _{ 3 } (mm) | _{ 3 } (mm) | |
---|---|---|---|---|---|---|---|

Candidate 1 | 27.04 | 0.46 | 21.45 | 0.38 | 5.73 | 0.45 | 14.23 |

Candidate 2 | 28.48 | 0.45 | 21.79 | 0.36 | 6.57 | 0.52 | 14.32 |

Candidate 3 | 29.94 | 0.46 | 21.81 | 0.37 | 6.56 | 0.53 | 14.39 |

Before selecting the best optimal solution for the LCM, the best tuning parameters of MOGA algorithm were selected. As given in Table

A shown in Table

Comparison among potentially optimal performances.

Performances | Candidate Point 1 | Candidate Point 2 | Candidate Point 3 |
---|---|---|---|

Equivalent stress (MPa) | 140.6496 | 150.1017 | 155.1464 |

Displacement ( | 398.5 | 399.8 | 399.7 |

Safety factor | 3.5952 | 3.5285 | 3.4986 |

The Pareto-optimal fronts and results obtained from the MOGA algorithm were than compared to those obtained using the Elitist Nondominated Sorting Genetic Algorithm (NSGA-II) [

A basic way to find good parameters for the algorithm is preliminary runs. After thirty times of computational simulations, the best parameters of NSGA-II were chosen as follows: population size of 75, crossover probability of 0.8, mutation probability of 0.2, and number of generation of 110. The comparative results showed that the optimal performances from the MOGA are better than those obtained from the NSGA-II. Specifically, the stress from MOGA was smaller than that from NSGA-II, and the displacement and safety factor from MOGA were higher than those from NSGA-II. Besides, the computational optimization time required for MOGA was less than that for NSGA-II, as given in Table

Comparison of the optimal performances obtained from MOGA an NSGA-II.

Performances | MOGA | NSGA-II |
---|---|---|

Equivalent stress (MPa) | 140.65 | 140.82 |

Displacement ( | 398.5 | 397.4 |

Safety factor | 3.60 | 3.57 |

Computational time (minute) | 9.7 | 10.5 |

In order to evaluate the behavior of the evolutionary algorithms, a statistical analysis is often used. In this study, the Wilcoxon’s rank signed test was applied to describe the behavior of the MOGA. It was a statistically nonparametric technique applied for various fields [

The results of Wilcoxon’s rank signed test.

Pair | R- | R+ | p-value |
---|---|---|---|

MOGA-NSGA-II | -0.2 | 0 | 0.006 |

Along with optimization process, a sensitivity analysis is also a necessary step to determine an influence of each design parameter on each quality response. For the proposed LCM, the sensitivity analysis was mainly focused on three characteristics. Generally, there were many methods that can be applied for calculating the sensitivity such as Nelson method, modal method, matrix perturbation method, differential method, RSM, or statistical analysis [_{i},

In this study, seven variables and two quality characteristics were considered. As seen in Figure _{3} had a highest influence or significant contribution on the displacement while the length_{3} had a lowest contribution. It was noted that a change in_{3} would adjust the displacement as desired. Other parameters were in relatively middle effects. Regarding the safety factor, the lengths_{2} and_{3} had the smallest contributions while the length_{1} had the largest effect on the safety factor. To adjust the safety factor, the_{1} should be changed firstly.

Sensitivity diagram.

Particularly, Figure _{2} on the displacement. The results indicated that the displacement has a linear change corresponding to the width while it has a nonlinear influence with respect to the length. Figure _{2} on the safety factor. The results indicated that the safety factor has a nonlinear effect corresponding to the width and it also has a nonlinear influence with respect to the length. In particular, the length had a more significant contribution to the safety factor compared with the width because this response was changed sharply. An increase in the length made a decrease in the safety factor.

Effect diagram of_{2} on (a) the displacement and (b) safety factor.

The contribution diagrams of thicknesses_{2} and_{3} were plotted in Figures

Effect diagram of_{2} and_{3} on (a) the displacement and (b) safety factor.

As depicted in Figure _{1} and_{3} had a nearly linear influence with the displacement but they had a nonlinear contribution affecting the safety factor.

Effect diagram of_{1} and_{3} on (a) the displacement and (b) safety factor.

As seen in Figure _{1} had a nonlinear influence on the displacement as well as the safety factor.

Effect diagram of_{1} and_{1} on (a) the displacement and (b) safety factor.

Summary, almost the mentioned design parameters had significant contributions to the displacement and safety factor. This would help designers and researchers to make a decision and meet the requirements of a specific system.

To evaluate and validate the optimal performances of the proposed LCM, simulation tests were carried out. Using the optimal design variables of candidate 1 in Table

As given in Table

Competition predicted and experimental results.

Characteristics | Prediction | Experiment | Error (%) |
---|---|---|---|

Displacement ( | 330.68 | 326.0174 | 1.43 |

Stress (MPa) | 140.6496 | 139.4822 | 0.83 |

Safety factor | 3.5952 | 3.5872 | 0.22 |

Comparison of initial design with optimal design.

Responses | Initial design | Optimal design | Improvement (%) |
---|---|---|---|

Displacement ( | 295.59 | 330.68 | 11.87 |

Safety factor | 2.7331 | 3.5952 | 31.54 |

This paper presented a new intelligent evolutionary multiobjective optimization approach for a linear compliant mechanism. The mechanism was designed based on connecting series of the leaf springs. These springs were located in symmetric configuration not only to guarantee a motion linearity but also to increase the working travel. To improve overall static performances, including the large working travel and high safety factor, a hybrid optimization approach was developed. This approach was an integration of FEM, RSM, ANN method, and MOGA. Three optimal candidates were retrieved and then candidate 1 was chosen as the finally optimum solution.

The sensitivity analysis was carried out to determine the significant contribution of each factor. The results revealed that the lengths and thickness are main influencers. The results revealed that the lengths and thickness almost significantly affect both responses. The results showed that the optimal results were found at the displacement of 330.68

The data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this article.

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.01-2016.20.