Optimization of loading curve in tube hydroforming using the multilevel response surface. Mostafa aslani choghiorty

Transcription

1 Optimization of loading curve in tube hydroforming using the multilevel response surface Mostafa aslani choghiorty Department of Mechanical enginnering, Najafabad Branch, Islamic Azad University, Najafabad, Iran Dr Ehsan Baniasadi Universuty of Isfahan Mechanical engineering department Abstract The process optimization and improvement of productivity are among the important activities in which the manufacturing companies invest in order to achieve the quality and assurance and fewer defects in product by the best possible way. Due to the increasingly demand for high quality products with reasonable prices in companies, which work in the field of injection molding, the molder has been forced to control the molding process, shape of parts, the characteristics of molding material and deformation method, and so on. The lack of an analytical correlation between the formability criteria and loading parameters in this process leads to the complexity of optimization process. In this study, the results of applying the load paths on the tube deformation are calculated using the finite element simulation. The optimal mathematical models for estimating the deformation are determined by regression analysis of response surface in statistical software, and then the simulated Annealing algorithm is utilized to design the optimal load paths. On the other hand, taking into account all available parameters increases the complexity in the optimization process of loading paths. In order to access the more precise power curve, a more narrow range of variation is considered around the resulted pressure-displacement curve. Therefore, this curve can be considered as the optimal pressure-displacement curve, and the optimization process can be stopped considering the criteria. We can argue that if this small range of variables was not the neighborhood of optimal response, the response graph would converge to the absolute optimum and place on the boundary points. Like to what happened in the optimization at the first stage, since the reduction of ranges in variables leads to the increased accuracy of model, a broader range of variables is selected at this stage to find the optimum point neighborhood, but a smaller range of variables selected at the next stage for optimal and accurate selection. Keywords: Loading curve, hydroforming process, response surface method (RSM), tube

2 1- Introduction Since the 1920s, the metal sheet forming methods have been quickly developed in production of industrial parts due to their speed and cost-effectiveness in mass-production of parts in automotive and aerospace industries. The formation processes can be implemented on most of the metals. The metal forming has the advantages like the very high dimensional accuracy, same parts, high surface roughness similar to the first sheet (typically the use of cold-rolled sheet), and high-speed and cost-effective production in large numbers. Due to the increased number of parts in these methods, the price of part may be increased as much as the increase in the number with the same ratio. In metal sheet forming is, the type of applied stress on the sheet is usually unlike the large tensile forming. Therefore, the rupture is the major disadvantage of these processes. In contrast, wherever the stresses are towards the compressive stress in the sheet, the part began to fold out (Bardelcik et al, 2008). 2- Tube hydroforming process Generally, the primary raw part becomes in the shape of formation mold or punch due to the imposing the fluid pressure in hydroforming method. The primary raw part applied in this process is usually the sheet or tube, and this process is called the sheet hydroforming in the cased on using the sheet. In the case of using the tube as the primary raw part, this process is called the tube hydroforming. With regard to the part geometry, the hydroforming process is generally divided into three categories (Choi et al., 2007): 1) Tube hydroforming; 2) Shell hydroforming; 3) Sheet hydroforming. Tube hydroforming is a process in which the target shape is created under the high internal pressure. Nonetheless, if only the internal pressure is the force during the forming process, the geometry and quality of part will be limited because the tube will become very thin in some parts of tube. The material can be easily fed into the mold cavity by applying an axial force on both ends of tube. The following figure schematically shows different stages of this process. The straight or curved tubes give the target forms under the pressure of fluid. The expansion of tube under high pressure will reduce its wall thickness. This ultimately leads to the rupture of tube (Rimkus et al, 2000). In order to avoid the local reduction in tube wall thickness at the same time to the internal pressure, the tube will become under the axial compressive load from one side or both sides in order to axially import more materials into the deformation feeding zone. However, the inadequate axial feeding leads to the other defects such as wrinkling and buckling of tube or vacant corners of mold. Generally, it is essential to select the pressure path in terms of axial feed for producing a perfect part with target dimensions and tolerances (Fann et al, 2003).

3 Figure 1: Parts from the sheet and tube hydroforming process In this approach, the target part is formed through the internal pressure and axial force in a mold; and a counter punch has been recently applied for controlling the tube surface behavior in free zone of deformation. In the hydroforming process, the way of internal pressure change, displacement of axial, and counter punches over time are interpreted to loading path. By inappropriate application of loading path during the process, the possibility of defects such as thinning, tearing and wrinkling will be indispensable. Numerous researchers have studied the optimization of loading paths in this process. Rimkus et al (2007) have studied the design of load-curves for hydroforming applications. They have divided the pressure curve into three parts of yield, expansion, and ultimate pressure, and then determined the optimal values based on the pressure in terms of axial punch displacement. Their applied relations are very simple and based on the early conjecture. Fann et al (2003) have investigated the optimization of loading paths by combination of gradient method with finite element method. In this study, the authors have suggested the optimal path by considering the simple and limited pressure path with a few numbers of variables. Imaninejad et al (2005) have optimized the loading path by available codes in commercial software. They have chosen the best results in terms of thickness and height of branch by providing several different paths. Furthermore, Heo et al (2006) calculated the optimal paths through the approach similar to the previous paper. Aydemir et al (2005) and Li et al (2008) determined the optimal paths for loading parameters by combining the finite element method and fuzzy method using the gradual optimization method. There are not the analytical relations for formability criteria in terms of loading variables in most of the conducted studies, thus providing an optimization method for determining the optimal loading paths is faced with serious limitations. Therefore, most of the studies suggest the optional paths for internal pressure and axial counter displacement, and then determine the optimal path by comparing the results of path. There are not the analytical relations for formability variables based on loading variables in most of the studies, thus providing an optimization method for determining the optimal loading paths is faced with serious limitations. Therefore, most of the studies are conducted

4 in a way that the optimal paths are suggested for internal pressure and axial punch displacement, and then the optimal path is determined by comparing the results. In this study, the results of applied loading paths on the tube deformation are calculated by the finite element simulation. The optimal mathematical models for estimating the deformation are determined through the regression analysis of response surface in statistical software, and then the simulated annealing algorithm is applied for designing the optimal loading paths. Therefore, the main question of this study investigates about the nature of optimal loading paths for optimal mathematical models in order to estimate the deformation through the regression analysis of response surface. 3- Loading parameters 3-1- Internal pressure The amount of internal pressure of fluid has a special impact on the deformation of tube and at the same time is a key parameter for determining the capacity of system. The effect of this parameter has been studied by numerous researchers and each of these researchers has proposed different paths for applying the pressure. Generally, the three-stage curve is the most common pressure curve which is based on the yield, expansion and the final pressure. Here, the calibration pressure is the necessary pressure to achieve the ultimate rate of deformation in expansion and final pressure. Numerous researchers have shown the improved tube formability by applying this curve compared to the other curves. Cherouat et al (2002) and Hwang et al (2007) have studied the hydroforming process by applying this general curve on pressure Axial punch displacement Two axial punches are applied for conducting the metal flow into the mold in order to prevent the tube thinning in open space of mold. Given the symmetry of deformation, the displacement curve of two axial punches is similar, and thus a curve is considered for this parameter. A curve with two-stage displacement is usually chosen for this parameter. This curve has been proposed by numerous researchers. Lang et al (2007) and Ray and Mac Donald (2004) have utilized this curve as a reference for axial punch Counter punch displacement curve The use of counter punch has become recently common for controlling the deformation behavior in free zone of mold. This parameter has been applied by Boudeau et al (2002) and Cheng et al (2009) for hydroforming the connections. The path of change for this parameter has a total of four variables including the initial and final positions of punch, and counter punch start and stop time. 4- Research method According to the above-mentioned cases, 11 variables should be determined for loading paths and for accurate design of these curves in order to design the loading paths based on them. In this study, the experimental design method is used to design these loading paths. The experimental design means to determine the number of tests and studied variables in each test through investigating several variables of process. In other words, the implementation of full

5 tests to control the whole process requires the extreme implementation of test, and if only a few levels of each variable are studied for reducing the number of tests, the number of tests is increased exponentially and this number of test is not also justifiable. The fitted linear models to data obtained from implementing the plan of Taguchi tests are introduced separately for two output variables of branch maximum height and the minimum tube thickness. Minitab software is applied for calculating the regression models. In addition to the base model (without exclusion), the models are modified with stage methods and calculated as follows. The efficiency of response surface method has a direct relationship with the selection of a right design for experiments. The design of experiments includes the use of geometric principles for statistical sampling, and the most important aim of this design is to minimize the number of required tests as well as maintaining the accuracy of model. If the response surface is properly modeled with a linear equation of variables, the approximation function will be a first order model, and if there is curve in the system, there will be a need for polynomial with high levels of unit. After determining the loading variables, the formability criteria, and the relationship between these parameters, we should investigate the optimal conditions of loading data for designing the loading paths. Therefore, we should first determine the appropriate method for optimizing this issue. In this regard, we use the simulated annealing algorithm for optimizing the loading paths Experiment design The effectiveness of response surface method has a direct relationship with selection of appropriate design for tests. The design of tests includes the use of geometric principles for statistical sampling; and the main aim of this design is to minimize the number of necessary tests and at the same time preserving the accuracy of model. If response surface is properly modeled with a linear equation of variables, the approximation function will be a first-order model, and if there is a curve in the system, there will be a need for polynomial system with higher degrees. In this study, it is assumed that the approximation function is a first-order linear equation. In this case, the relationship between the system and design variables will be linear as follows: In this study, the levels of levels are shown by (-1) low, (0) intermediate and (1) high levels which facilitates the fit of regression curve. To reduce the number of variables, and thus reducing the number of experiments, the numerical test design is done based on the pressure-displacement curve. The design variables are here the control points located on the pressure-displacement curve. The smaller ranges of change in these variables, the more accurate prediction model resulting in decreased error. The initial range of variables is selected by doing several simulation tests by applying different pressure-displacement curves and evaluating the changes in their results. Table 1 represents different levels of design variables. Based on the data of this table, the simulation

6 test is done by applying different force to build the model. The results of tests are presented in Table (2). Table 1- Design variables at the first stage of optimization Levels Variable a Variable a Variable a Table 2- Results of simulation at the first stage of optimization Test a1 a2 a3 Height Thickness extrusion dispersion As shown in the table, the more the variables are moved to the upper bound, the more the height of extrusion and the non-uniform wall thickness.

7 4-2- Modeling At this stage, the response surface models are created by fitting a surface to the obtained responses in the previous step. The fit models are created based on the least sum of difference squares, approximated functions of wall thickness, and the height of extrusion in terms of pressure-displacement linear curve points. The response surface obtained from the numerical tests is shown in the following equations. The presented equations indicate the estimated model for thickness variance and the height of extrusion (H) Verification of obtained equations Normality test of residual terms The normal histograms of errors in model is applied and shown in figures 1 and 2 in order to assess the normality of residual terms in two models (thickness variance and extrusion height models). According to the results of both figures, the significance levels of Jarque-Bera statistic are equal to 0.42 and 0.39, respectively, thus the null hypothesis about the non-normality of residuals is rejected in both models. Therefore, the obtained results of Jarque-Bera statistic are confirmed for normality of errors in models. Figure 1: Normal histogram of errors in thickness variance model Series: Standardized Residuals Sample Observations 780 Mean 2.06e-17 Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability

8 Figure 2: Normal histogram of errors in extrusion height model Series: Standardized Residuals Sample Observations 780 Mean -7.33e-17 Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Autocorrelation test between error terms (residuals of model) The autocorrelation between the error terms is investigated according to Durbin-Watson Statistics and then drawing the diagram for the residuals of model. As shown in description of method, if Durbin-Watson Statistics is close to 2, it shows the lack of autocorrelation in model. In this research, the results of Durbin-Watson Statistics in estimating the theoretical research model for both models are presented in the following table, so that the results of both models indicate the lack of autocorrelation between the errors of model. Table 3- Results of Durbin-Watson Statistic Results of estimating the equations Equation 1: Thickness variation Durbin-Watson statistic 2.07 Equation 2: The height of extrusion (H) 1.91 Source: Eviews software output Heteroscedasticity test White test is utilized to investigate the heteroscedasticity of models. According to the values of F-statistic (7.89) and Obs*R-squared statistics (21.14) for tables and the probability of accepting the H0 for both statistics (0.07 and 0.001) and comparing the values of F-statistic and Obs*R-squared with chi-square statistic of table, it can be concluded that the H0 based on the lack of heteroscedasticity is accepted for both models, thus the models do not have the problem of heteroscedasticity. Table 4- White test for thickness variance model White Heteroskedasticity Test: F-statistic Probability Obs * R-squared Probability

9 Table 5- White test for extrusion height model White Heteroskedasticity Test: F-statistic Probability Obs* R-squared Probability Table 6- Results of tests for two models Name Model Normality of residual terms Correlation Homoscedasticity Thickness variance Normal autocorrelation homoscedasticity Extrusion height Normal autocorrelation homoscedasticity Therefore, it can be claimed that the equations have enough validity, and thus the estimated equations can be calculated: 4-4- Solution of estimated equations At this stage, the obtained equations in previous step are put in solver to achieve the optimal solution. Here, the minimum wall thickness variance is considered with equation (1) as the aim of optimization and design factors, limited at the range of and, and the extrusion height with equation (2) as a qualification. The extrusion height should be higher than the minimum acceptable value of mm. Excel solver should be applied for solving the equations. Despite the fact that the obtained value is the optimal response of Equations 1 and 2, it is not necessarily the optimal response of whole system. In other words, this graph is the local optimal state at the initial range for variables. The response surface is a successive method, thus the next step is to search an optimal point of system in optimization zone. Table 7- Obtained values from solving the response surface equations Design variables a1 a2 a3 Normal values obtained from solver Actual values of variables Multi-stage optimization The optimization continues by selecting the new borders at this stage. It is expected that the optimal neighborhood is obtained from the pressure-displacement curve at the previous

10 stage. At this stage, a more narrow range of changes is considered around the pressuredisplacement curve at the previous stage in order to access the more precise power curve. The design variables are similar to the previous model, and the range of new changes in these variables is shown in Table 8. Here, the factorial 3k design is utilized to create the statistical model and finding the optimal curve. Table 8- Design variables at the second stage of optimization Levels Variable a Variable a Variable a The obtained results of simulation tests for wall thickness dispersion and extrusion height functions are shown in the following table. The alternative model is built as same as the previous stage. The fitted equations to experimental data are obtained at the second stage of optimization as follows: Similar to the previous case, we first investigate the accuracy of model: Normality test of residual terms Figure 3- Normal histogram of errors in thickness variance model Series: Standardized Residuals Sample Observations 780 Mean -6.98e-17 Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability

11 Figure 4- Normal histogram of errors in extrusion height model Series: Standardized Residuals Sample Observations 780 Mean -3.92e-08 Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Autocorrelation test between error terms (residuals of model) The results of Durbin-Watson Statistics in estimating the theoretical model of research for both models are presented in the following table. The obtained results of both models indicate the lack of autocorrelation between errors of model. Table 9- Results of Durbin-Watson Statistic Results of estimating the equations Durbin-Watson statistic Equation 1: Thickness variation, 2.12 Equation 2: Height of extrusion (H) 2.02 Source: Eviews software output Heteroskedasticity test: According to the comparison of F-statistics and Obs*R-squared values with chi-square statistic of table, it can be concluded that the H 0 based on the lack of heteroskedasticity is accepted for both models, thus the models of second stage do not have the heteroskedasticity problem. Table 10- White test for thickness variance model of second stage White Heteroskedasticity Test: F-statistic Probability Obs*R-squared Probability Table 11- White test for extrusion height model of second stage White Heteroskedasticity Test: F-statistic Probability Obs*R-squared Probability

12 Table 12- Results of tests for two models of second stage Name Model Normality of residual terms Correlation Homoscedasticity Thickness variance Normal autocorrelation homoscedasticity Extrusion height Normal autocorrelation homoscedasticity Therefore, it can be claimed that the equations have enough validity, and thus the estimated equations can be calculated: Like the previous stage, the equation 3 is solved for the minimum variance of wall thickness using the solver and imposing the height constraints in equation 4 and constraints of variable ranges. Table 14 represents the optimal values obtained from the alternative model. According to Table 14, it is found that the obtained pressure-displacement curve at optimization stage is totally at the defined range for variables and does not have the boundary point. Table 13- Results of simulation at the second stage of optimization Test a1 a2 a3 Extrusion height Thickness dispersion

13 Table 14- Obtained values from solving the response surface equations Design variables a1 a2 a3 Normal values obtained from solver Actual values of variables Therefore, with respect to the criterion, which has already been mentioned, this curve can be considered as the optimized pressure-displacement curve and the optimization stages stopped because if this small range of variables was not in the neighborhood of absolute optimal response, the response graph would converge to absolute optimization and be placed on the borderline points. Similar to what happened at the first stage of optimization, as the accuracy of model is increased by reducing the scope of variables, a broader range is selected for finding the neighborhood of optimal point at this stage, but a smaller range of variables selected for optimal accurate selection at the next stage. 5- Summary and conclusion The main aim of this research is to optimize the loading curve in tube hydroforming process through the multilevel response surface, designing the minimization of numbers of required tests, and at the same time maintaining the precision of response surface model. If the surface the response surface is properly modeled with a linear equation of variables, the approximate function will be a first-order model; and if there is curve in the system, there will be a need for polynomial with higher degrees of unit. The response surface models are created by fitting a level on the obtained responses of previous stage. The fit models are expressed based on the least sum of squares of differences, approximation functions of wall thickness, and extrusion height according to the spots of multi-linear pressure-displacement curve. In order to access the more precise power curve, we consider a more narrow range of variation around the pressure-displacement curve at the previous stage. Therefore, with respect to the criterion, which has already been mentioned, this curve can be considered as the optimal pressure-displacement curve, and the optimization stages can be stopped. We can argue that if this small range of variables is not the neighborhood of optimal response, the response graph converges to the absolute optimization and located on the borderline points. Similar to what occurred in optimization of model at the first stage, since the model precision is increased by reducing the range of variables, a broader range is selected at this stage to find the optimal point neighborhood, and then the smaller range of variables for accurate optimal selection.

14 In this limited research, the results of optimization pressure-displacement curve are extracted for deformed tube, and a mathematical model calculated based on the input variables. The effect of loading parameters on the deformation is studied after determining the accuracy of mathematical models. By applying the mathematical models in response surface algorithm, we measure the optimal values for each loading variable and design the loading paths for each parameter based on these values. The capacity of machine is reduced and the favorable results obtained from the hydroformed tube in terms of formability criteria by comparing the results of this method with experimental method Suggestions The results of this research indicate that the response surface method is a useful tool for predicting the behavior of a system and can be applied in similar cases. By changing the ranges of variables, the optimization will be continued at multistages. The aim of this work is to eliminate the dependence on the borders of variables and improvement of accuracy in estimating the optimal graph. Different criteria have been introduced for stopping the algorithm in multi-stage optimization. In this study, the optimal status of applied method occurs when obtained response is completely within the range of variables. Since the application of this method practically requires the programmer presses, it is suggested using this curve. The simulation techniques based on the finite element method can be utilized in similar studies, and the results of research can be compared with the obtained results.

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