Optimization techniques applied to single point ... - Springer Link

Int J Adv Manuf Technol (2018) 95:1789–1804 https://doi.org/10.1007/s00170-017-1305-y

ORIGINAL ARTICLE

Optimization techniques applied to single point incremental forming process for biomedical application M. Sbayti 1 & R. Bahloul 1 & H. BelHadjSalah 1 & F. Zemzemi 2

Received: 4 April 2017 / Accepted: 1 November 2017 / Published online: 15 November 2017 # Springer-Verlag London Ltd., part of Springer Nature 2017

Abstract Single point incremental forming (SPIF) process has the potential to replace conventional sheet forming process in industrial applications. For this, its major defects, especially poor geometrical accuracy, should be overcome. This process is influenced by many factors such as step size, tool diameter, and friction coefficient. The optimum selection of these process parameters plays a significant role to ensure the quality of the product. This paper presents the optimization aspects of SPIF parameters for titanium denture plate. The optimization strategy is determined by numerical simulation based on Box–Behnken design of experiments and response surface methodology. The Multi-Objective Genetic Algorithm and the Global Optimum Determination by Linking and Interchanging Kindred Evaluators algorithm have been proposed for application to find the optimum solutions. Minimizing the sheet thickness, the final achieved depth and the maximum forming force were considered as objectives. For results evaluation, the denture plate was manufactured using SPIF with the optimum process parameters. The comparison of the final geometry with the target geometry was conducted using an optical measurement system. It is shown that the applied method provides a robust way for the selection of optimum parameters in SPIF. Keywords Denture plate . SPIF . CAD-CAM . Simulation . Design of experiments . Optimization . MOGA . GODLIKE . Optical measurement * M. Sbayti [email protected]

1

Mechanical Engineering Laboratory, National Engineering School of Monastir, University of Monastir, 5019 Monastir, Tunisia

2

Laboratory of Mechanics of Sousse, National School of Engineers of Sousse, University of Sousse, 4023 Sousse, Tunisia

1 Introduction Single point incremental forming (SPIF) is a variant of incremental sheet forming (ISF) especially suitable for prototyping, customized components, and small-batch production due to its flexibility and low tool effort, low production cost, and lead time [1, 2]. The beneficial aspects of this technique are mainly ascribed to its essence; that is progression of forming process by transmission of localized plastic deformation near the tool end [3]. The process is capable to form metallic as well as polymer sheet components [4, 5] and was primarily applied in the automobile and aerospace industry [6, 7]. However, there are other branches with an important potential for the technology, such as the biomedical field [8, 9]. The ISF process suffers from poor geometric accuracy of the formed parts. The sheet springback and sheet bending [10] account for the geometric inaccuracies reported in the most areas of the formed parts. In addition, the “pillow effect,” an unwanted curved surface, typically occurs on the flat base of the part formed in SPIF [11]. The ISF toolpath is one of the main parameters influencing the geometric accuracy of the part since a simple tool follows the predefined toolpath to form the desired part [12]. Recently, many studies have focused on the toolpath optimization. In order to reduce the manufacturing time and homogenize thickness distribution of an asymmetric part, Azaouzi et al. [13] developed a parameterized forming strategy for the tool trajectory optimization in SPIF. Lu et al. [14] used model predictive control (MPC) to enhance ISF accuracy. The incremental step size of the toolpath was optimized based on shape state feedback during the forming process. The results showed that the geometric errors were improved in the base areas of the formed part, but the errors in the part wall areas were still relatively large since the proposed control algorithm only dealt with toolpath correction in the vertical direction. Additionally, obvious “pillow effect” was

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Material properties of CP-Ti Gr.1 pure titanium [8]

Table 1 Young’s modulus E (MPa)

Density ρ (kg/m3)

Poisson’s ratio υ

Yield strength σS (MPa)

Elongation δ (%)

Tensile stress σb (MPa)

105,000

4510

0.37

205

36

370

the two last works is the absence of the experimental validation of the process parameters optimization. The present paper attempts to evaluate the feasibility of producing customized Commercially Pure Titanium Grade 1 (CP-Ti Gr.1) denture plate with an acceptable geometrical accuracy by means of SPIF process. This work aims to control some process parameters and improve the final product quality by means of geometrical accuracy (final achieved depth, sheet thinning) and low energy (reduced forming forces). For this end, the study provides a detailed development and comparison of multi-objective optimization methods of single point incremental forming based on numerical simulation and experimental validation.

2 Modeling and statistical analysis 2.1 Finite element (FE) model The finite element method was used to investigate the sheet deformation in ISF processing for the denture plate. The sheet material used in the numerical simulation is CP-Ti Gr.1. It was modeled as an elasto-plastic material with isotropic hardening. Table 1 shows the material properties of CP-Ti Gr.1 pure titanium. Lu et al. [8] had examined these properties by using tensile tests on an Instron tensile testing machine. Figure 1 shows the flow stress of pure titanium. As can be seen from the figure, the initial yield stress is about 200 MPa while the stress at fracture is about 350 MPa. 400 350

True Stress (MPa)

observed at the flat bases of the formed parts. Based on this work, an improved MPC algorithm was developed by the same authors [15] for two point incremental forming (TPIF) with a partial die. The toolpath correction was performed through properly adjusting the toolpath in two directions based on the optimized toolpath parameters at each step. Despite good improvement in geometric accuracy that was achieved, this method is not able to perfectly compensate the springback in the partial fillet areas of the test shape. Other researches were performed to reduce the main drawbacks of ISF with new approaches for optimization problems by studying the effect of ISF process parameters on the sheet formability. Ham et al. [16] determined the effect of part shape, material type, sheet thickness, vertical step size, and tool size on the maximum forming angle using a Box– Behnken experimental design. It is noted that the material type has the greatest effect on formability, followed by the shape. Hussain et al. [17] attempted to present the trends between ISF process parameters and the formability of commercially pure titanium sheet in ISF process. They concluded that increasing the feed rate, pitch size, or tool size has a negative effect on the sheet formability. In another study, Hussain et al. [18] used the design of experiments as a reliable methodology to evaluate the maximum achievable forming angle of aluminum sheet AA-2024. All experiments were planned according to a central composite design (CCD) of experiments. It was demonstrated that the interaction between the punch radius and the vertical step size has the most important effect on the maximum achievable forming angle. Furthermore, they showed that forming speed of annealed sheet can be increased highly in favor of process productivity without affecting the formability of AA-2024 sheets. However, inconsistent result was observed in the case of pre-aged AA-2024 sheet. Despite such noteworthy efforts, no comprehensive study exists to date scrutinizing how formability is affected by individual and interactional effects of all ISF process parameters. Moreover, far less attention has been devoted so far to the optimization of the process. Liu et al. [19] attempted to evaluate the surface roughness. Response surface methodology (RSM) with Box–Behnken design and multi-objective function was employed to establish the prediction model and examine the impact of four forming parameters (step down size, sheet thickness, tool diameter, and feed rate) on the overall surface roughness. The most influencing parameters on the overall surface finish are the sheet thickness followed by the step down. In contrast, feed rate and tool diameter have little influence on the overall surface roughness. Bahloul et al. [20] examined and minimized the sheet thinning rate and the punch loads for conical shape formed with SPIF process by response surface methods (RSMs) and genetic algorithms. Arfa et al. [21] developed a multi-objective optimization strategy based on numerical simulation to optimize the thinning rate and the springback in SPIF. The main of

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300 250 200 150 100 50 0

0

0.1

0.2 True Strain

Fig. 1 Flow stress of CP-Ti Gr.1 [8]

0.3

0.4

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∆x, ∆y, ∆z

(b)

(a)

Fig. 2 a Schematic of the setup in FEA simulations: initial configuration of the blank shape during the SPIF operation, b FE meshing configuration and tool position for the SPIF simulation

(a)

(b) Rigid clamp Rigid tool Limit line Titanium sheet

Rigid rig Part

Fig. 3 a Three-dimensional exploded view of numerical model used in SPIF process, b punch trajectory

Due to large displacements and large located strains in incremental forming, the explicit solution scheme was adopted. In this work, ABAQUS Explicit software package was employed. Based on the scanned shape of an existing custom-made denture base, the tool path has been generated to produce simultaneously two identical denture base parts. The shape of the titanium sheet has been considered square with a size of 120 mm × 120 mm. The sheet was clamped into a fixed rectangular blank holder and a backing plate is placed underneath it to support the part (Fig. 2a). The sides of the sheet blank are fixed in all directions. In the FE model, quadrilateral shell elements with four nodes and six degrees of freedom per node (S4R) and seven Gaussian reduced integration points through the thickness direction were used to model the sheet blank. The mesh size is 1 mm with approximately 14,400 elements and 14,641 nodes. The finite

element meshing subdivision of the initial blank is depicted in Fig. 2b. The punch, the clamp, and the rig are supposed to be analytical rigid surfaces (Fig. 3a). The sheet blank is deformed progressively by the punch along a sequence of small increments, which cause a large number of increments to be calculated. Proceeding in an Table 2

Variation of CPU time with trajectory parameters

Trajectory parameters Diameter (mm) 5 5 5

Step size (mm) 0.02 0.11 0.2

CPU time (h) 31:32:54 16:37:42 10:11:07

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Fig. 4 Optimization strategy for SPIF process of denture plate

incremental way, the tool is moved along contours which follow the desired final geometrical shape of the part as described by CAD-CAM CATIA software. In the FE model, the boundary conditions followed by the forming tool during the process are given by the path shown in Fig. 3b. The rigid tool is allowed to move in all directions simultaneously. A Coulomb model was considered for frictional actions. The parameters of the finite element model (meshing, element type, and contact algorithm) are selected after several numerical simulations to evaluate their influence on the computational time and to achieve good results. The simulations were

Table 3

ANOVA table for STH

Source

DF

Adj SS

Adj MS

F value

P value

Model Linear D Δz μ Square D×D Δz × Δz μ×μ 2-way interaction D × Δz D×μ Δz × μ Error Lack of fit Pure error Total

9 3 1 1 1 3 1 1 1 3 1 1 1 5 3 2 14

0.002349 0.000699 0.000006 0.000694 0 0.001515 0.000916 0.000703 0.000001 0.000135 0.000132 0 0.000003 0.000119 0.000119 0.000000 0.002469

0.000261 0.000233 0.000006 0.000694 0 0.000505 0.000916 0.000703 0.000001 0.000045 0.000132 0 0.000003 0.000024 0.00004 0.000000

10.92 9.75 0.23 29.03 0 21.12 38.32 29.42 0.03 1.89 5.53 0 0.13

0.008 0.016 0.648 0.003 1 0.003 0.002 0.003 0.859 0.25 0.065 0.992 0.735

–

–

Model summary: R2 = 95.16%, R2 (adj) = 86.45%

performed using four cores under “RWTH Compute Cluster” running under the Linux operating system. Table 2 reports the required hours to implement a FE approach to simulate the SPIF process of titanium denture plate. It can be noted that this time is related to the part trajectory parameters especially the step size. Design of experiments (DOE) is a cost-effective tool for controlling the influence of parameters in manufacturing processes. Its usage decreases the number of experiments and material resources. Furthermore, the analysis performed on the results is easily realized and experimental errors

Table 4

ANOVA table for depth

Source

DF

Adj SS

Adj MS

F value

P value

Model Linear D Δz μ Square D×D Δz × Δz μ×μ 2-way interaction D × Δz D×μ Δz × μ Error Lack of fit Pure error Total

9 3 1 1 1 3 1 1 1 3 1 1 1 5 3 2 14

0.842358 0.663225 0.221112 0.4418 0.000313 0.144883 0.0351 0.100523 0.028269 0.03425 0.034225 0 0.000025 0.009775 0.009775 0 0.852133

0.093595 0.221075 0.221112 0.4418 0.000313 0.048294 0.0351 0.100523 0.028269 0.011417 0.034225 0 0.000025 0.001955 0.003258 0

47.87 113.08 113.1 225.98 0.16 24.7 17.95 51.42 14.46 5.84 17.51 0 0.01

0 0 0 0 0.706 0.002 0.008 0.001 0.013 0.043 0.009 1 0.914

–

–

Model summary: R= 98.85%2 , R2 (adj) = 96.79%

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Table 5

ANOVA table for force

Source

DF

Adj SS

Adj MS

F value

P value

Model

9

209,373

23,264

14.76

0.004

Linear

3

158,352

52,784

33.48

0.001

D Δz

1 1

21,070 135,083

21,070 135,083

13.36 85.68

0.015 0.000

μ Square

1 3

2198 47,596

2198 15,865

1.39 10.06

0.291 0.015

D×D

1

37,310

37,310

23.66

0.005

Δz × Δz μ×μ

1 1

4308 4921

4308 4921

2.73 3.12

0.159 0.138

2-way interaction D × Δz

3 1

3425 456

1142 456

0.72 0.29

0.580 0.614

D×μ

1

2

2

0.00

0.975

Δz × μ Error

1 5

2967 7883

2967 1577

1.88

0.228

Lack of fit Pure error

3 2

7883 0

2628 0

–

–

14

217,256

Total

Model summary: R = 96.37%2 , R2 (adj) = 89.84%

are minimized. Statistical methods measure the effects of change on the operating variables and their mutual interactions on process through experimental design way [22]. Normal Probability Plot

1. Scan of the denture base; 2. Development of the tool trajectory under CAD/CAM software; 3. Identify the important factors, which influence the SPIF of titanium denture plate; 4. Develop the experimental design matrix using design of experiments;

(b)

(response is Depth)

Normal Probability Plot

95 90

80 70 60 50 40 30 20

80 70 60 50 40 30 20

10 5 1 -0,075

(response is STH)

99

95 90

Percent

Percent

(a) 99

Box–Behnken designs (BBDs) [23] are a class of rotatable or nearly rotatable second-order designs based on three-level incomplete factorial designs. In this work, the Box–Behnken experimental design was chosen for finding out the relationship between the process response functions (final sheet thickness, forming force, and final achieved depth) and the variables (tool diameter, vertical step size, and friction coefficient). The design for three factors consists of 15 numerical simulation runs that can be split into three blocks with one center point at each block. The considered parameters and levels are tool diameter (D = 5–7.5– 10 mm), step down (Δz = 0.02–0.11–0.2 mm), and friction coefficient (μ = 0.05–0.15–0.25). The main objective of this procedure is to define the most critical process parameters and their effect on the considered responses for denture plate manufacturing. As summarized in Fig. 4, the present investigation has been planned in the following steps:

10 5 -0,050

-0,025

0,000

0,025

1 -0,008

0,050

-0,006

-0,004

-0,002

Residual

(c)

Normal Probability Plot (response is Force)

99 95 90

Percent

0,000

Residual

80 70 60 50 40 30 20 10 5 1

-50

-25

0

25

Residual

Fig. 5 Normal probability plots for the residuals for the responses: a depth, b STH, and c force

50

0,002

0,004

0,006

0,008

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parameters used for evaluating model fitness are R2 (multiple correlation coefficient) and adjusted R2 (R2 adj). Generally speaking, the larger the values of R2 and R2 adj are, the better the fitness is [24]. The important coefficient R2 measures the percentage of data variation that is explained by the regression equation. The adjusted R2 value is particularly useful when comparing models with different number of terms. When R2 approaches to unity, the response model fits the actual data effectively. The value of R 2 obtained was 0.9516, indicating that 95.16% of the total variation is explained by the model. The large value for the coefficient of multiple determination (R2 = 0.9516/R2 = 0.9885/R2 = 0.9637) implied that the quadratic model adequately represented the simulation results. However, the adequacy of the developed model was further examined by analyzing the normal distribution of residual, which is the difference between an observed response and the predicted response. Normal probability plot of residuals is presented in Fig. 5 for the three process responses. The points on the normal probability plots of the residual should form a straight line to get an adequate model. Residuals for the studied responses are verified to follow a normal distribution assumption,

5. Conduct the numerical simulation as per the design matrix; 6. Analyze the results using ANOVA; 7. Empirical modeling to approximate the relationship (i.e., the response surface) between responses and factors; 8. Optimize the chosen factor levels to minimize process responses (final sheet thickness, forming force, and final achieved depth) using two multi-objective optimization methods: MOGA and GODLIKE; 9. Confirm experiments and verify the predicted performance characteristics.

2.2 Analysis of variance (ANOVA) The analysis of variance (ANOVA) is used to check the fitness of response surface approximation model. This statistical analysis technique was carried out using the commercial software Minitab17. Tables 3, 4, and 5 show ANOVA results, respectively, for the quadratic model of the final sheet thickness, the final achieved depth, and the maximum forming force of titanium denture plate part forming. These tables summarize the analysis of variance for each response and show the significant model terms. The major statistical

Main Effects Plot for Depth

(a) D

Δz

μ

Data Means

D

0,310

-11,6

Δz

μ

0,305

Mean

-11,7

Mean

Main Effects Plot for STH

(b)

Data Means

-11,8 -11,9

0,300 0,295 0,290

-12,0

0,285

-12,1 5.0

10.0

0.02

0.20

0.05

0.25

5.0

10.0

0.02

0.20

0.05

Main Effects Plot for Force Data Means

(c)

D

Δz

μ

800

Mean

700

600

500

400 5,0

7,5

10,0

0,02

0,11

0,20

0,05

0,15

0,25

Fig. 6 Graphs of the main effects of three studied factors for a final achieved depth, b final sheet thickness (STH), and c forming force

0.25

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considering an approximately straight line distribution of the data point. This confirmed that the model is correct and can be effectively used to navigate the design space. The proposed model can be helpful to predict the formability of CP Ti Gr.1 sheet in the investigated range of parameters. Consequently, the second-order model developed herein can be used to determine optimal values.

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2.3 Mathematical model For quadratic polynomial regression model, the predicted response value y could be calculated by: k

k

i¼1

i¼1

y ¼ β 0 þ ∑ β i X i þ ∑ βii X 2i þ ∑kj ∑β ij X i X j þ ε

ð1Þ

Fig. 7 Results given in form of second-order response surfaces and contours plots of two variables with μ = 0.15 for final sheet thickness (a1, b1), final achieved depth (a2, b2), and maximum punch load (a3, b3)

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where k is the number of design variables, Xi is the set of design variables, βi are the polynomial coefficients, and ε is minor error.

In our case, the regression equations defining the SPIF process responses in terms of final sheet thickness, final achieved depth, and maximum forming force are as follows:

STH ¼ 0:1213 þ 0:04026 D þ 0:685 Δz þ 0:024 μ−0:002520 D  D−1:704 Δz  Δz −0:048 μ  μ−0:0256 D  Δz þ 0:00010 D  μ−0:097 Δz  μ

ð2Þ

Depth ¼ −12:326 þ 0:1223 D þ 3:97 Δz þ 2:53 μ−0:01560 D  D −20:37 Δz  Δz−8:75 μ  μ þ 0:4111 D  Δz þ 0:0000 D  μ þ 0:28 Δz  μ

ð3Þ

Force ¼ 1035−226:3 D þ 1562 Δz−1614 μ þ 16:08 D  D−4217 Δz  Δz þ3651 μ  μ þ 47:5 D  Δz þ 2:6 D  μ þ 3026 Δz  μ

ð4Þ

It means that the response surface prediction model has high precision, can accurately reflect the mathematical relationship between objective function and design parameters, and can replace the actual physical models for multiobjective optimization.

2.4 Process parameters effect Figure 6 presents the main effects plots for three process responses: forming force, final achieved depth, and final sheet thickness (STH). Comparing the impacts of the process factors: tool diameter (D), increment step size (Δz), and friction coefficient (μ) on these responses, it can be noted that changes carried out between the low and high levels of different parameters affect, more or less, the variation of the response. As

can be seen in Fig. 6c, a variation of Δz from 0.02 to 0.2 mm leads to an amplification of the forming force from 410 to 700 N. The graphs of the main effects emphasize immediately the significant factors. According to Fig. 4, the vertical increment (Δz) seems to be the most influencing factor on the three process responses while the friction coefficient (μ) is the factor with much less important effects on the responses. Based on the preceding results, the RSM can be used to construct the global approximation of the responses at various sampled points of design space. Figure 7 shows the threedimensional representation of the relative variation of the predicted thickness (STH), depth, and force given in form of surface and contour plots. Due to its low effect, the friction coefficient (μ) can be fixed to its medium value. The results show that these responses evolve in a nonlinear way according to the considered parameters (D, Δz). Also, it can be noted that these responses are more sensitive to the vertical step size than to the punch diameter.

Start

Generate initial population

3 Multi-objective optimization problems

Calculate vector objective function

Once the response surface is constructed for the considered process outputs, the multi-objective optimization technique can be used to search for the optimal solutions. A generic formulation of the Multi-Objective Optimization Problems (MOOP) can be defined by the following equations:

Calculate ranking

Reproduction Genetic operation (selection, crossover, mutation)

Table 6 Parameters setting value used in multi-objective genetic algorithm

Select new population

Terminal condition

No

Population size Npop

Crossover probability Pc

Mutation probability Pm

Selection rule

Generation size

0.8

0.007

Roulette wheel

400

Yes Stop

Fig. 8 Flowchart of the multi-objective genetic algorithm

200

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!
! Minimize ! y ¼ F X h
!
!
!iT ¼ f 1 X ; f 2 X ; …; f N X
! Subject to gi X ≤ 0; i ¼ 1; 2; …; m
! h j X ≤ 0; j ¼ 1; 2; …; l T !  Where X ¼ X 1 ; X 2 ; …; X p ∈Ω

3.1 Multi-objective optimization based on genetic algorithm ð5Þ ð6Þ ð7Þ ð8Þ

! y is the objective vector, a vector with the values of
scalar
!  ! objective functions to be minimized. The gi X and h j X represent, respectively, the inequality and equality constraints ! functions and X is a P-dimensional vector representing the decision variables within a parameter space Ω. Two methods will be studied to optimize incremental forming responses:

Fig. 9 Flowchart of GODLIKE algorithm

Figure 8 presents the flowchart of the Multi-Objective Genetic Algorithm (MOGA). In the MOGA method, first the population is initialized within the specified variable ranges. After evaluation of this population, based on a non-dominated sorting approach, the generated alternatives are classified into different fronts. The population members are ranked according to their fitness values (frank) and are selected for genetic operation. To compare candidate solutions to the multi-objective problems, the concepts of Pareto dominance and Pareto optimality are commonly used [25]. The evolutionary process of Pareto multi-objective optimization is accomplished by using critical parameters given in Table 6:

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3.2 Multi-objective optimization based on GODLIKE algorithm Optimal.D (mm)

The optimization problem is also solved using Global Optimum Determination by Linking and Interchanging Kindred Evaluators (GODLIKE) [26]. Figure 9 shows graphically the basic operations of GODLIKE solver. Thus, this algorithm solves optimization problems using relatively basic implementations of genetic algorithm (GA), differential evolution, particle swarm optimization, and adaptive simulated annealing. GODLIKE is primarily intended to improve the robustness of the optimization method by exploiting the simultaneous running of the involved algorithms to decrease the chance of premature convergence to a local solution and to diminish the influence of the fine-tuning of each algorithm on the final solution. By using multiple optimizers simultaneously, it is essentially equal to performing four (or more) consecutive optimizations all at once, which already improves the probability of exploration and searching for the global optima. The weak points related to each of these cited algorithms are neglected by the robustness and the performance of another; at the same time, the strength and the efficiency of these algorithmic techniques simply add up simultaneously. The exchange of individuals in successive generations will lead to immigrant introduction by GODLIKE algorithm into the new population, which could be conducted to better alternative

Thickness-Force

Tickness-Depth

Force-Depth

10 9 8 7 6 5 4 0.01

0.04

0.07

0.1

0.13

0.16

0.19

0.22

Optimal.Δz (mm)

Fig. 11 Optimal factors of three bi-objective optimization problems

solutions compared to the ones already explored by one of the algorithms. These immigrants allow to alter some individuals of the population, and they can move quickly in the unexplored areas of the search space and thus diversify the research. This step allows to quickly browse the interesting region of the search space for explore it in detail, while increasing the chances to determine the global minimum. The interchange operator is very useful for MOOP; when one population is completely non-dominated, interchanging individuals between populations will usually lead to a dominated population, which continues the search for the Pareto front, instead of reporting convergence. GODLIKE does not aim to make either of the algorithms more efficient in terms of function evaluations.

Fig. 10 Graphical representation of Pareto front for bi-objective optimization: a relationship between final sheet thickness and final achieved depth, b relationship between final sheet thickness and forming force, c relationship between forming force and final achieved depth

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3.3 Bi-objective optimization problem

forming force (Fmax)), and (Final sheet thickness (STH), final achieved depth (Depth)) to be simultaneously optimized with respect to the design variables. The bi-objective optimization problems can be formulated in the following form:

The couples of two conflicting objectives in this section are (Final sheet thickness (STH), maximum forming force (Fmax)), (final achieved depth (Depth), maximum



  Minimize f 1 ¼ STH ðΔz; DÞ Minimize f 3 ¼ Fmax ðΔz; DÞ Minimize f 1 ¼ STHðΔz; DÞ Minimize f 2 ¼ Depth ðΔz; DÞ Minimize f 3 ¼ FmaxðΔz; DÞ Minimize f 2 ¼ Depth ðΔz; DÞ Subject to the following inequality constraints 5 < D < 10 μ ¼ 0:15 0:02 < Δz < 0:2

Fig. 12 Pareto front for bi-objective optimization with GODLIKE: a1, a2, a3 relationship between forming force and final achieved depth; b1, b2, b3 relationship between final sheet thickness and final achieved depth; c1, c2, c3 relationship between final sheet thickness and forming force

1800

3.4 Optimization results For this work, the strength Pareto approach (SPA) for MOOP has been used. So, the trade-off between the objective functions was indicated by the shape of the Pareto surface and the intention of the optimization is to minimize both objective functions. However, both objectives cannot be simultaneously minimized since there is a conflicting relationship between

Int J Adv Manuf Technol (2018) 95:1789–1804

them. Being optimal for one objective implies being suboptimal for the other. The Pareto optimal solutions are some compromise solutions in the feasible region solutions [27]. The optimal Pareto fronts are plotted in Fig. 10a–c. It can be concluded that MOGA provides a variety of alternative design solutions. It can be observed from Fig. 8a that since the forming force is equal to 327 N, the final achieved depth is about 11.46 mm. While a depth of 12.3 mm can be achieved with a forming force of 768 N. Another observation from the graphs is that MOGA detected a large number of solutions on the Pareto optimal front in a single run. The optimal factors for each bi-objective optimization have been extracted from the results of the optimized responses. As shown in Fig. 11, we have a large number of solutions. It can be noted that the Pareto-optimal solutions found by MOGA are feasible in practice. Comparing these results, it can be concluded that is no common interval of factors variation between the three bi-objective optimizations. However, it can be noted some common solutions (D, Δz) between two biobjective optimizations: In our simulation experiments, the SPIF process responses were also optimized using the MATLAB-based solver GODLIKE. The default settings of algorithm parameters were assumed in the run of GODLIKE solver. Figure 12 shows the final Pareto fronts of the three optimization problems at three steps of the GODLIKE algorithm evolution. The green dots in the final Pareto plots (Fig. 12a3, b3, c3) are the corresponding most efficient points of the MOOP. As can be seen for both optimization methods, the same trend of Pareto plots was obtained for the studied process responses. However, the number of solutions increased with GODLIKE algorithm. Table 7 presents the optimization results of studied problems in terms of their most efficient solutions. With this method, there is a common solution only between two optimization problems. Therefore, tri-objective optimization will be done. 3.5 Tri-objective optimization problem The tri-objective optimization design of SPIF can offer more choices for a designer. Moreover, it helps subsume all those two-objective optimization results presented in Table 7 Bi-objective optimization efficient solution given by GODLIKE algorithm

Fig. 12 (continued)

Bi-objective function

Optimal solution

Optimal factors

STH–depth STH–force Force–depth

(− 12.3253, 0.2822) (0.3169, 326.9246) (− 11.4666, 326.9512)

(9.9987, 0.0200) (6.6856, 0.1552) (6.7772, 0.1552)

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Fig. 13 Three multi-objective resolution: a Pareto of optimal responses, b optimal factors

Fig. 14 Pareto of three objective optimization with GODLIKE: a during algorithm evolution, b final run

the previous section. This will allow us to find trade-off optimum design points from the view point of all three objective functions simultaneously. Therefore, in this section, three objective functions (final sheet thickness, forming force, and final achieved depth) are considered for the multi-objective optimization. The tri-objective optimization problems can be formulated in the following form: 8 < Minimize f 1 ¼ STH ðΔz; DÞ Minimize f 2 ¼ Depth ðΔz; DÞ : Minimize f 3 ¼ Fmax ðΔz; DÞ Subject to the following inequality constraints 5 < D < 10 μ ¼ 0:15 0:02 < Δz < 0:2 Table 8 Three-objective optimization efficient solution given by GODLIKE algorithm Optimal solution (depth, STH, force)

Optimal factors (D, Δz)

(− 11.4668, 0.3171, 326.9502)

(6.7578, 0.1540)

The Pareto of the optimal responses and the optimal factors obtained with the MOGA method are presented in Fig. 13. Figure 14 shows the Pareto funded after using the GODLIKE optimization method. After the evolution of the optimization algorithms (Fig. 14a), the final Pareto was constructed and the most efficient solution appears presented by the green dot. The corresponding optimal solution values are summarized in Table 8.

4 Experimental validation Experiments were performed on Amino DLNC-RB machine (Fig. 15) to manufacture the titanium denture base with initial sheet of 0.5 mm. By considering the geometric accuracy, the most important objective function, one combination of optimal factors has been chosen from the best optimization solutions (Dopt = 10 mm, Δz = 0.02 mm). In order to reduce friction and increase material formability, chlorine-containing forming oils were applied as a lubricant. Figure 16 shows

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Fig. 15 Experimental setup for the SPIF process

Fig. 16 Experimental results: a produced part with SPIF, b geometry comparison with the original part using ATOS system

Fig. 17 Thickness reduction obtained from a ARGUS system, b numerical simulation

Tool trajectory

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the well-formed denture base made by SPIF technology. After forming, the dimensional control of the obtained part was performed by comparing, in GOM ATOS Inspect software environment, with the original CAD model. The thickness reduction after forming of the part was controlled using GOM ARGUS system as shown in Fig. 17a. The results are in agreement with those obtained by numerical simulation (Fig. 17b). The middle area of the plate was not affected; it is still at the initial sheet thickness (0.5 mm). The formed area with SPIF presents additional mass reduction (with a maximum of 40%). This seems to be highly desirable for the comfort of wear to the patient.

1803 Acknowledgements The authors would like to thank the German Research Foundation DFG for the kind support within the Cluster of Excellence “Integrative Production Technology for High-Wage Countries”. Furthermore, the authors thank the Institute of Metal Forming of RWTH Aachen University for their support in the performance and evaluation of the incremental sheet forming experiments.

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5 Conclusion In this study, by evaluating FE simulation of SPIF process for a biomedical component, the feasibility of manufacturing customized titanium denture plates by using the ISF approach is studied based on preliminary optimization procedure. In order to control the precision of incremental forming product, multiobjective optimization methods for SPIF process parameters were proposed based on the combination of numerical optimization techniques and finite element simulation technology validated by experiments. The conclusions of this work may be summarized as follows: 1. The novelty of this study is that two multi-objective optimization methods for the precision SPIF process of titanium denture plate are proposed. This work can cut down the number of experiments by using the response surface approximation model. 2. Mathematical model of the multi-objective optimization was established based on the second-order response surface method. Based on experimental design and finite element simulation, relation between optimal process responses was generated with Pareto-optimal front. For this, MOGA and GODLIKE algorithms were used to obtain the Pareto-optimal solutions and the most efficient ones. It is clearly found that there are collisions between studied process responses. 3. The feasibility of the proposed multi-objective optimization methods was verified by numerical simulation and experiment. First experimental results using a combination of optimal process parameters show promise and the process seems to be satisfactory for the application of Grade 1 titanium denture plate. 4. The optimization methodology based on FEA and multiobjective techniques is a reliable and powerful tool for the optimization of SPIF parameters. 5. Further development is still needed to evaluate the surface quality after forming of the part.

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