Sheet Metal Forming Optimization by Using Surrogate ...

CHINESE JOURNAL OF MECHANICAL ENGINEERING Vol. 30,aNo. 1,a2017

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DOI: 10.3901/CJME.2016.1020.123, available online at www.springerlink.com; www.cjmenet.com

Sheet Metal Forming Optimization by Using Surrogate Modeling Techniques

WANG Hu1, 2, *, YE Fan1, CHEN Lei1, and LI Enying3

1 State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, China

2 Joint Center for Intelligent New Energy Vehicle, China 3 School of Logistics, Central South University of Forestry and Technology, Changsha 41004, China

Received April 17, 2016; revised September 20, 2016; accepted October 20, 2016

Abstract: Surrogate assisted optimization has been widely applied in sheet metal forming design due to its efficiency. Therefore, to

improve the efficiency of design and reduce the product development cycle, it is important for scholars and engineers to have some

insight into the performance of each surrogate assisted optimization method and make them more flexible practically. For this purpose,

the state-of-the-art surrogate assisted optimizations are investigated. Furthermore, in view of the bottleneck and development of the

surrogate assisted optimization and sheet metal forming design, some important issues on the surrogate assisted optimization in support

of the sheet metal forming design are analyzed and discussed, involving the description of the sheet metal forming design, off-line and

online sampling strategies, space mapping algorithm, high dimensional problems, robust design, some challenges and potential feasible

methods. Generally, this paper provides insightful observations into the performance and potential development of these methods in

sheet metal forming design.

Keywords: surrogate, optimization, sheet metal forming

1 Introduction

Numerical simulations of sheet metal forming processes, based on finite element method(FEM) or others(such as meshless and other numerical algorithms), are powerful tools for predicting forming processes and are widely used for most of companies and institutes[1–3]. With the increasing of complexity of sheet metal forming design, in order to achieve enough accurate simulation procedure, more elements and less size of time step should be utilized. Despite continual advances in computing power, the complexity of analysis tools cannot keep pace with the request of the modern manufacturing companies[4]. Therefore, sheet metal forming design is still a computation-intensive problem and difficult to be handled. In the past three decades, surrogate modeling and corresponding optimization methods have attracted intensive attention[5]. This type of approach approximates computation-intensive functions with simple analytical models. Based on surrogates, some novel global optimization methods have been developed, which is therefore referred as surrogate assisted optimization (SAO)[6–8]. Due to efficiency and feasibility, the SAO

* Corresponding author. E-mail: [email protected] Supported by National Natural Science Foundation of China(Grant Nos.

11572120, 11172097, 11302266) © Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2017

provides potential plans for sheet metal forming design quickly, it is becoming a most popular optimization tool in sheet forming design field[9–11].

Surrogate model(also called metamodel) are known as response surfaces(RF), surrogates, etc. By definition, a surrogate is an approximation of relationship between design variables(inputs)/response functions(outputs) which is implied by an underlying simulation model. The method was originally introduced by Box and Wilson termed as response surface method(RSM)[12]. With the development of RSMs, the RSM is termed as surrogate model(metamodel). It means a model of models. In the past 20 years, the SAO method has been improved significantly, most of popular commercial software involve surrogate assisted optimization module as shown in Table 1. These software products commonly provide interface between optimization toolkits and simulation codes. Sheet metal forming design and other practically industrial problems can be easily implemented in this way. Hyperstudy, LS-OPT and iSight might be the most popular software for sheet metal forming design. Specially, Hyperstudy and LS-OPT can be incorporated into HyperForm and Dynaform in the same simulation platform, Hyperworks and LS-Dyna.

We used “Google Scholar” with keywords, surrogate, metamodel, RSM and metal forming since 2005 as shown in Fig. 1. It can be found, from 2004–2012, the number of literatures related to metal forming optimization by

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surrogate/metamodel continued to grow. In this period, the SAO is a hotspot in sheet metal forming design. Scholars and engineers realized the potential of SAOs. However, in 2013–2016, a lot of new problems emerge in engineering applications. Most of these problems are generic bottlenecks for other disciplines, such as reliability analysis, high dimensional problems[13–14]. We always ask ourselves how to use SAOs in sheet metal forming applications and what we need to do in the future. For this reason, we prepare a review paper for investigating some novel SAOs and corresponding applications and give our outlook for the further.

Table 1. Popular software and their surrogate modules

Software Developer Surrogate

HyperStudy Altair Engineering

Least squares regression for polynomials[15], Moving least square(MLS)[16], Kriging[17],

Radial basis function(RBF)[18]

Isight Dassault Systems,

Formerly-Engineous Software

Least squares regression for polynomials, Kriging, RBF,

Taylor series approximation[19], Neural

network(NN)[20]

Dakota Sandia National

Laboratories

Taylor series approximation, Least squares regression for

polynomials, MLS, NN, Kriging, RBF, Multipoint

approximations[21], Multivariate adaptive

regression splines(MARS)[22]

LS-OPT Livermore Software

Technology Corporation

Least squares regression for polynomials, Successive

response surface method(SRSM)[23]

BOSS/

Quattro LMS International RBF, NN, Kriging

Visual Doc Vanderplaats R&D,

Inc. Least squares regression for

polynomials

Model Center

Phoenix IntegrationLeast squares regression for

polynomials

Fig. 1. Approximate number of literatures

The rest of paper is organized as follows. Firstly, design

parameters, objective and constraint functions for sheet

metal forming design are introduced. In sections 3 and 4, off-line and online SAOs are presented and some potential feasible SAOs are introduced. Space mapping algorithm based on sheet metal forming design is introduced in section 5. In section 6, curse of dimensionality is discussed because it is a major bottleneck in the industry. Reliability based on sheet metal forming design is introduced in section 7, and some challenges in the final section are presented.

2 Design parameters, Objective

and Constraint functions For each optimization problem, design variables,

objective and constraint functions should be given firstly. For most of sheet metal optimization problems, design parameters should be assigned according to the requirement of engineering design, if these design parameters can be modeled, such as geometric, process, material parameters, etc. For objective/constraint functions in this discipline, they are commonly criteria for prediction of defects of a blank. JAKUMEIT, et al[24], summarized four major objective functions for formability of blank as (1) The final sheet without cracks, (2) The final sheet without wrinkles, (3) The sheet with a stable form, and (4) Springback effect.

Although the summary might not be comprehensive, engineers often use other design variables required by the requirement of customs and market, such as weight of blank and cost of whole process[25]. However, these criteria are commonly used in practical sheet metal forming design. In this section, several important criteria are demonstrated and discussed as follows.

2.1 Thickness variation

The minimization of thickness variation allows improving on workpiece feasibility, because a rupture is always preceded by a high thinning and wrinkling is always preceded by a high thickening in practice. Thickness variation between the initial and final states and is given by BARLET, et al[26], as

1

0

1 0

( ) ( ) with ( )

pp eNe e

h h he

h hf x f x f x

h

/

=

æ öæ ö - ÷÷ çç ÷÷ ç= =ç ÷÷ çç ÷÷ç ÷çè ø è øå , (1)

where he, 0h are the initial and final thicknesses,

respectively. They are the principal stretch along the thickness. N is the number of elements in a blank FE model.

The coefficient p=2, 4, 6 is introduced to emphasize the

extremes of the objective function.

2.2 FLD(forming limit diagram) criterion A FLD, also known as a forming limit curves(FLC), is

used in sheet metal forming for predicting forming behavior of sheet metal[27–28]. The FLD provides a graphical description of material failure tests and widely used in

Y WANG Hu, et al: Sheet Metal Forming Optimization by Using Surrogate Modeling Techniques

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sheet metal forming simulation. Compared with the thickness criterion, the FLD criterion is more reasonable for objective and constraint functions. Therefore, various scholars used the FLD as objective or constraint function. Practically, most of FLD criteria are distance related functions.

JANSSON, et al[29], set the constraint that failure is not allowed by FLD criterion and objective to minimize the risk of wrinkles in a blank, and hence to obtain as far away from as possible from the wrinkling tendency line of FLD as shown in Fig. 2.

Fig. 2. Illustration of FLD and wrinkle criteria

BREITKOPF, et al[30], defined two FLCs in principal

plan of logarithmic strains by 1 2 1 2( ), ( )s w = = .

s and w are used to control rupture and wrinkle,

respectively. A safety FLC as shown in Fig. 3 is defined by

2 2

2 2

( ) ( ) ,

( ) ( ) ,s s

w w

s

s

ì = -ïïíï = -ïî (2)

and the final formulation can be written as

with

( )( )

1

1

1 2 1 2

2 1 1 2

( ) ( )

( ) for ( ),

( ) for ( ),

0, otherwise

pNe

he

pe e e e es s

pe e e e ew w

e

f x f x

f

f

f

/

=

æ ö÷ç= ÷ç ÷÷çè øìïï = - >ïïïïï = - <íïïïï =ïïïî

å

(3)

Fig. 3. FLD based criterion defined by BREITKOPF, et al[29]

2.3 Springback criterion As mentioned by JAKUMEIT, et al[24], springback effect

is another important criterion in sheet metal forming design. Springback is mainly an elastic deformation which occurs after sheet metal forming processes, when the stamped part is removed from the forming tools. The springback changes the geometry of a blank so it causes difficulties during a subsequent assembly process, or cause the twisting in an assembled part. The springback is always an important issue in sheet forming design, and this problem is still a critical problem in practice, especially for advanced high strength steel(AHSS). TENG, et al[31], proposed a new method of springback prediction using SVR and optimized variable stretch force trajectory to reduce springback.

Springback is commonly measured by the distance between nodes on a drawn part before and after springback. For a 2D case, it is easy to be defined. For example, maximum distance after removing tools should be the objective as shown in Fig. 4(a), springback angle as shown in Fig. 4(b). For a 3D case, due to the complexity of geometry, it is difficult for designers to define an objective function which can represent the physical characteristic of springback. WEI, et al[32], tried to evaluate springback with the sum of squares of the nodal distances on a drawn part before and after springback. In our opinion, this criterion can be used for the regular shape. However, for some complex geometric shapes, how to build a reasonable criterion still needs further study.

Fig. 4. Illustrations of springback


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3 Off-line Sampling

Early SAO method involves three phases: (1) Selecting a Design of Experiment(DoE or sampling

strategy) for generating sampling points. (2) Constructing an approximate model based the

generated sample points. (3) Optimizing objective functions based on approximate

model. For this scenario, the sampling procedure is isolated

from modeling procedure. Once surrogate model is established, it cannot be changed unless to reconstruct surrogate. If the surrogate is a not enough accurate, new sample points should be added. The key issues are how to generate new sample points and how many sample points should be added. Therefore, the only way is to reuse DoE with more sample points for a more accurate surrogate. Although some generated sample points can be reused, the performance of the surrogate might be degraded in term of efficiency. Considering the isolation of modeling and sampling, such strategy can be termed as off-line sampling SAO.

The off-line strategy generates sample points by considering mathematical characteristics, such as orthogonality, density and uniformity. Widely used off-line experimental designs include factorial or fractional factorial(FF)[36], Central composite design(CCD)[37], Box- Behnken design(BBD), alphabetical optimal, and Plackett- Burman design(PBD)[36]. These classical methods tend to spread sample points around boundaries of design space and leave a few samples at the center of design space. SACKS, et al[38], stated that in the presence of systematic rather than random errors. A good experimental design tends to fill design space rather than to concentrate on boundary. Sequentially, space filling design methods, such as orthogonal arrays[39], Latin hypercube designs(LHS)[40], Hammersley sequences[41], and uniform design[42], were developed. Generally, more information of a function can be gathered from more sample points. However, computational time should be determined by the size of sample points.

For sheet metal forming design, most problems are physically nonlinear. It is difficult for the off-line sampling scenario to obtain a reasonable solution in one cycle. Due to this defect, off-line strategies are seldom applied to engineering applications. However, for some complex geometrical problems, it is very difficult for optimizers to integrate CAD and CAE model seamlessly[43]. Therefore, several scholars still used off-line strategy for sheet metal forming optimization owing to this bottleneck. For example, OHATA, et al[44], utilized polynomial RSM to find the annealing conditions suitable for a sheet metal forming condition. TANG, et al[45], used polynomial RSM and one step FEM for numerical simulation and drawbead optimization. KAREN, et al[46], used RSM assisted

improved differential evolution(DE) algorithm to handle intelligent die design based shape and topology optimization. However, it is easy to observe that the number of design variables is low and most of physical models are weak nonlinear in these problems. Generally, the major drawback of existing off-line strategies is limited applications.

4 Online(adaptive) Sampling

Due to the difficulty of knowing the “appropriate” sampling ahead of design, intelligent or sequential sampling has gained popularity recently[47]. Compared with the off-line strategy, these methods generate new sample points adaptively by different criteria. Because the location of a sample point generated by such strategies is determined by evaluations(simulations) of surrogates, such a strategy is thereby called online sampling. Recently, various online approaches, such as sequential exploratory experiment design(SEED) method[48], simulated annealing(SA)[49], Bayesian method[50] and boundary-based method[51–52] have been developed. Compared with off-line strategies, the location of a new sample point is determined by some specific criteria, such as accuracy, reliability. Because of its dynamical characteristic, it prevails in terms of the efficiency and accuracy than the off-line one. Therefore, online sampling strategies have been widely used in sheet metal forming design. In this section, several novel online sampling strategies are introduced and discussed.

4.1 Successive response surface method

The successive response surface method(SRSM) is an important achievement for sheet metal forming optimization which was originally proposed by STANDER, et al[53]. The famous optimization tool LS-OPT was also developed from the SRSM. The SRSM uses a region of interest, a subspace of the design space, to obtain an approximate optimum. A range is chosen for each variable to determine its initial size. A new region of interest centers on each successive optimum. Progress is made by moving the center of a region of interest as well as reducing its size. The size of the subregion for each design variable jx will be scaled by a variable parameter 1i

j + which is calculated as

*( ) ( ) ( )U L

1( ) ( )U L

1( )

20.5

i i ij j j

ij i i

j j

x x x

x x +

- += +

-, (4)

where Ljx and Ujx are the lower and upper bounds of

jx , respectively. *jx denotes the present approximated

optimum. The lower and upper bounds can be updated by

( )( )

( 1) *( ) ( ) ( )L U L

( 1) *( ) ( ) ( )U U L

0.5 ,

0.5 ,

i i i ij j j j j

i i i ij j j j j

x x x x

x x x x

+

+

ìï = - -ïïíï = + -ïïî

(5)


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where ai(i=3, 4, 5, 6) is the force arm of the shoulder ring, elbow, elbow ring and wrist joints, respectively.

The basic pattern of the SRSM can be easily understood by Fig. 5.

Fig. 5. Adaptation of subregion in SRSM[50]

The SRMS was firstly applied to design the die shape

used in the manufacture of an automotive wheel center pressing from a sheet metal blank by KOK, et al[25]. According to their results, the performance of the SRSM seems good. A weight savings of up to 9.4% could be realized for four shape variables. Later, the SRMS has been developed to a popular optimizer, LS-OPT. Then, STANDER, et al[53], used LS-OPT for springback compensation in sheet metal forming. The standardized NUMISHEET’96 S-Rail was used as a benchmark in their study. A converged optimal design was obtained in four or

five cycles and springback trends were found to be consistent with changes in the die shape.

The original SRMS is based on the polynomial RSM, although the polynomial regression can filter some noise points and outliers, it cannot approximate some high nonlinear cases. To improve the performance of the SRMS, the SRMS has been integrated with other surrogates and used for some real industrial cases. BREITKOPF, et al[30], used the MLS approximation for the SRSM. They extended pattern search algorithms with a fixed pattern panned and zoomed in a continuous manner across the design space as shown in Fig. 6. Moreover, the region of interest moves across a predefined discrete grid of authorized experimental designs. The suggested MLS-SRSM was combined with one-step forming and finally checked by the explicit dynamical algorithm, so the computational time was significantly saved. Compared with the original SRSM, a complicated model, the deep drawing of an oil pan was optimized. The objective function is to determine the drawbead restraining forces minimizing the risk of thickening at the end of the drawing process and hence to reduce the global thickness variations. Both thickness and FLD were objective functions. After optimization, the maximum variation of the thickness was reduced and the FLD of final formed blank was much better than the initial design. NACEUR, et al[30, 54], also successfully applied the MLS-SRSM to design of tool’s radii for springback and hood outer panel deep forming. They also utilized the one-step inverse approach(IA) as a surrogate model during the optimization. The optimization of the thickness distribution and the material parameters in order to minimize the springback effects of U bending sheet.

However, it is difficult for the SRSM to handle high nonlinear and high dimensional problems. The major drawback of the SRSM is easy to fall into local convergence, and it is quite possible when no feasible solution can be found. Moreover, the trade-off between exploration(global) and exploitation(local) is not considered.

4.2 Adaptive response surface method(ARSM)

ARSM was developed by WANG G G[55], which disregards regions with large function values as predicted by the surrogate, and build a new DoE using central composite design(CCD) or Latin hypercube sampling(LHS) in the reduced region. The basic pattern of ARSM is demonstrated in Fig. 7. Similar to SRSMs, the ARSM is also a space reduction method. WANG H, et al[56], firstly improved this method and applied to sheet metal forming optimization. Compared with the original ARSM, they proposed particle swarm optimization intelligent sampling (PSOIS) strategy for enhancing the efficiency. ARSM and PSOIS were integrated and employed for optimizing the initial blank shape and blank hold force(BHF) in sheet metal forming process. The results showed that the improved ARSM method might be potential for highly


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nonlinear problems with more design variables. WEI, et al[35], used the ARSM for optimization of the feasible processing parameters and prediction of the performance

Fig. 6. Search patterns of the MLS-SRSM[29]

tolerance which is induced by the fluctuation of noise factors. The results showed that the ARSM is an effective method to reduce the computational cost while to keep a satisfactory accuracy for the optimization. Although higher dimensional problem can be handled by the ARSM, it is difficult for the ARSM to apply to multimodal problems due to its assumption.

Fig. 7. Illustration of the original ARSM[51]

4.3 Boundary and best neighbor sampling

Boundary and best neighbor sampling proposed by WANG, et al[51], is also a kind of space reduction sampling strategy. The BBNS strategy generates new samples derived from information of boundaries and the best sample points of the initial sparse distributed sample points. It is easy to obtain better samples and avoid local convergence due to consideration of the boundary information. The pattern of the BBNS is presented in Fig. 8. The advantage of the BBNS is to consider infill criterion, boundary and infilling information simultaneously. Therefore, more useful information can be gathered to generate sample points for the next cycle.

Fig. 8. Search pattern of the BBNS[52]

Sequentially, WANG, et al[52], integrated PSO, BBNS


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and one-step forming to optimize blank formability. To improve the efficiency of the BBNS-based surrogate model, it has been parallelized by using MPI[56]. The major bottleneck of the BBNS is also to consider boundary information due to lack of priori information. For the BBNS, the boundary of design space is commonly assigned by designers. However, for most underlying problems, prior information is difficult to be obtained due to lack of prior knowledge. With the increasing of dimensionality, the BBNS needs to be improved for more complicated real engineering problems.

SRSM, ARSM and BBNS can be categorized as the space reduction method. For most of engineering optimization applications, it is difficult to achieve a real global optimum, tradeoff between global(exploration) and local(exploitation) should be investigated. Therefore, the space reduction method is suitable for sheet forming design due to its efficiency. Although such strategy can’t find the real global optimum, such as high dimensional problems, it has great potential for nonlinear problems in the future.

4.4 Nature-inspired surrogate assisted method

Most nature-inspired optimization methods, such as genetic algorithms(GAs), PSO and DE are population based algorithms. Artificial neural network(ANN) is another kind of nature inspired method and is also a kind of SAO.

GA, PSO and other similar strategies can be considered as the population-based method. The advantage of them is to find global optimization. Obviously, the major bottleneck is low efficiency and is difficult for expensive simulation-based problems. WANG, et al[52], proposed a PSO intelligent sampling for reducing the number of sampling points and has been discussed previously. SENER, et al[57], used GA combined with Taguchi method to present the optimization of process parameters in reducing the risk of failure due to rapture and fracture in deep drawing of a rectangular cup. LIU, et al[58–59], integrated the second order polynomial RSM and multi-objective GA(MOGA) to optimize fracture and wrinkle, control springback of stamping parts. LIAO, et al[60], used MOGA to investigate the stamping direction of an automotive part for the sheet metal forming. Although the surrogate can be instead of simulation in nature-inspired optimization method, the number of sample points used in these methods is still high compared with SAOs. Fortunately, for most of nature-inspired methods, they are easy to be parallelized due to their natural essential. Therefore, the feasibility of the nature-inspired SAO can be enhanced by the parallel strategy and will be discussed in the next Section.

ANN is another kind of SAO and it often combined with other natural-inspired strategies, such as GA, PSO. The ANN has been well studied for over several decades. They were widely engaged in sheet metal design. INAMDAR, et al[61], used a ANN based on back propagation(BP) of error, surrogate for training involving 5 input, 10 hidden and 2

output(punch displacement and springback angle). According to the test result, it was found that accuracy of predictions depended more on the number of used training patterns than on ANN architecture. LIU, et al[62], used the optimum BPNN found by Levenberg-Marquardt algorithm to predict the bending radius in sheet metal forming. Cheng and Lin concluded that the RBF neural network(RBF-NN) model is superior to other NN models in predicting bending angle[63]. WANG J, et al[64], used a feed forward(FF) - BP-NN with a set of geometrical variables as its inputs and the possibility of the occurrence of wrinkling as its output to investigate the geometrical influence on wrinkling in sheet metal forming. KAZAN, et al[34], used the ANN approach to predict springback in wipe-bending process. SIVASANKARAN, et al[65], developed a FF-BP-NN model to map the mechanical properties and instantaneous geometry features of the deep drawing process(as shown in Fig. 9).

Fig. 9. FF-BP-NN model for the deep drawing process

of aluminum sheet[65]

Some scholars integrated the ANN and other natural-inspired method for sheet forming design. LIEW, et al[66], combined the ANN and evolutionary algorithm(EA) to reduce springback. Compared with the original EA, the ANN-EA is capable of identifying optimal process parameters for the sheet metal forming operation with significant reduction in the number of springback computational cost. LIU, et al[33], also used the GA-ANN for springback prediction. Compared with LIEW et al s’ work[66], an improved GA was used to optimize the weights of NN. The results showed that more accurate prediction of springback can be well acquired with the GA-ANN model. FU, et al[67], also used the GA for predicting the weights of a neural network for minimizing the error between the predictive punch radius and the experimental one. KITAYAMA, et al[68], integrated sequential approximation and the RBF-NN to design variable blank holder force trajectory.

Generally, the ANN approach is seldom used to sheet forming optimization directly. ANN based EA methods are commonly used to avoid the local convergence and enhance the diversity of sample points. On the other hand,


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the surrogate assisted ANN methods were also commonly used to improve the efficiency of modeling procedure. Considering the differences of performance criteria, it is difficult to determine which combination is the best one. More importantly, another major issue is to filter noises and outliers and construct more robust model.

4.5 Parallel sampling strategies

Parallel strategy is another important method to improve the efficiency of surrogate assisted optimization. Recently, some effective parallel strategies were suggested for RBF[69–70], Kriging[71] and EA-based optimization methods[72], and some commercial software have been released their parallel versions, such as DAKOTA[73]. However, these software products seldom can be used for sheet metal forming design specifically. In our opinion, the main obstacles are how to build the bridge between CAD and CAE models and how to define objective and constraint functions objectively. Therefore, scholars and engineers need to program specific codes for different problems. Recently, JAKUMEIT, et al[74], proposed an iterative parallel Kriging algorithm for sheet metal forming design. WANG H, et al[56, 75], combined parallel sampling strategy with least square support vector regression(LS-SVR) and Kriging surrogates, respectively. IVANOV, et al[76], applied a parallel optimization algorithm based on FANOVA decomposition to the sheet metal forming.

Another important issue is parallel nature-inspired strategies, such as parallel GA, PSO. Parallel nature- inspired optimization methods are particularly easy to be implemented and promise substantial gains in performance. The basic idea behind most parallel programs is to divide a task into chunks and to solve chunks simultaneously by multiple processors. This divide-and-conquer approach can be applied to nature-inspired methods in various ways, and the literatures contain many examples of successful parallel implementations. If the nature-inspired methods are integrated with other surrogates, the efficiency should be improved significantly. Surrogate assisted memetic algorithm(SAMA) is such a typical surrogated assisted nature-inspired optimization method. Unlike traditional Evolutionary Algorithm(EA) methods, MAs are intrinsically concerned with exploiting all available knowledge about the problem under study. ONG, et al[72], proposed a RBF-based MA; the RBF is used for predicting fitness instead of computationally expensive fitness function and general constraints. ZHOU, et al[77], suggested parallel strategy based on the MA for computationally expensive cost problems and it could be applied to sheet metal forming design.

4.6 Potential sampling strategies

Theoretically, all sampling strategies can be used in sheet metal forming design, but some of them might not be feasible in practice due to limits of expensive evaluations.

In this subsection, some potential surrogate assisted sampling/optimization methods are introduced.

Efficient global optimization(EGO)[78] is an important method for global optimization. The EGO uses a Kriging surrogate to fit the data and direct the compromise between exploration and exploitation. Kriging does not only provide the estimate of an objective function everywhere, but also a normal distribution around that value which characterizes the uncertainty. The commonly used version of EGO employs the uncertainty by selecting as the next simulation the point that maximizes the expected improvement(EI) over the present best solution. The exploration part of the algorithm is enhanced by the fact that EI is actually a conditioned expected improvement, conditioned on improvement taking place. This condition favors sample points with large uncertainty to balance the advantage of points with low values of the surrogate(exploitation points). Therefore, these advantages can enforce the reliability and efficiency for nonlinear problems. The major bottleneck of this problem is for high dimensional problems. If the number of design parameters is limited, the EGO and its related methods can be used for sheet metal design.

Mode pursuing sampling(MPS)[79] is a global optimization method which systematically generates more sample points in the neighborhood of the function mode while statistically covering the entire search space. The MPS can find a feasible solution without enough efficient surrogate models. The final solution derived by the MPS might not be mathematical global optimum but it could be potentially used for real engineering problems.

Another branch is parallel nature-inspired SAOs, such as SAMA discussed in the previous section, compared with EGO and MPS, these types of algorithms are easy to be parallelized. If the number of sample points can be controlled, the parallel nature inspired methods might be used for high dimensional nonlinear problems.

Recently, surrogate model selection for different distributed sampling points is also a matter of urgent concern. The common way is to construct multiple surrogates based on a training data set, evaluate their accuracy, and then to use only a single model perceived as the best while discarding the others[80–81]. This strategy has some shortcomings as it does not take full advantage of the resources devoted to constructing different surrogates, and it is based on the assumption that changes in the training data set will not jeopardize the accuracy of the selected model. Therefore, GOEL, et al[82], proposed an ensemble surrogate to overcome these drawbacks. They suggested that the prediction accuracy of a surrogate model can be improved if the separate stand-alone surrogates are combined to form an ensemble. Compared with single surrogate strategies, they stated that ensemble of surrogates can be used to identify expected large uncertain regions and selection of the best surrogate model based on error statistics for more robust approximation than individual surrogates. For these reasons, ensemble of surrogate should


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be valuable for sheet metal forming design.

5 Space Mapping Algorithm

Space mapping(SM) originally developed by BANDLER[83] is an important method for sheet metal forming application. The purpose of the SM algorithm is to achieve a satisfactory solution with a minimal number of computationally expensive "fine" model evaluations (simulations). The SM method procedures iteratively update and optimize surrogates based on a fast physically based "coarse" model as demonstrated in Fig. 10. More details of the SM method and corresponding improvements can be found in BANDLER’s literature[84].

Fig. 10. Illustration of SM[84]

For SM algorithms, the major advantage is that the coarse model can be instead of the fine model during the optimization procedure. Therefore, the computational cost can be significantly reduced. Although performances of SM algorithm and corresponding modified algorithms have been improved significantly in the past 20 years, the range of their applications was limited in electromagnetics and circuit optimization. Due to nonlinearity and computational expensive evaluations of sheet metal forming, it is difficult to determine whether this method can be used for such complicated problems. JANSSON, et al[85], firstly introduced the SM algorithm for sheet forming design. They studied the accuracy and efficiency of optimization by using the SM algorithm in nonlinear transient dynamical problems, such as vehicle impact and nonlinear quasi-static problems. Traditional polynomial regression was integrated with SM algorithm. In their work, two sheet forming cases have been investigated. The first is a simple case to obtain the restraining force of an equivalent drawbead(one design variable case). The second one is much more complicated model(Fig. 11) and the number of design variable is 10. The fine model was evaluated using LSDYNA with a running time of two hours. The model consisted of 64,874 elements. The coarse model was constructed by linear RS based on the initial evaluations of a fine model. Compared with the real evaluation based surrogate model, the efficiency of optimization was improved and the results seemed quite good. Sequentially, JANSSON, et al[86], successfully applied SM algorithm to real engineering problems. They claimed that the SM was a very effective and accurate method when calibrating the draw-in of a sheet metal forming process.

Fig. 11. FE model of sheet forming and corresponding FLD[85]

Is the SM really feasible for engineering problems?

WANG, et al[87], deemed that the SM algorithm was difficult to converge for highly nonlinear cases and they also pointed out three bottlenecks of SM algorithm, accuracy, convergence and uncertainties of Parameter Extraction(PE) procedure. According to their suggestions, they developed an adaptive SM algorithm based on response function(objective function) as shown in Fig. 12. Compared with the original SM algorithm, two important strategies were proposed. The first one is to reproduce the coarse model in each cycle if certain conditions are not satisfied. Compared with the original SM algorithm, the coarse model is updated according to the error estimation. Therefore, the accuracy of SM expression can be gradually improved. The second strategy of the proposed algorithm is to establish a mapping between response(objective) function of the coarse and fine models directly instead of projection between design space, and the corresponding expression is updated in each cycle. The second strategy is to avoid the PE procedure and is easy to converge accordingly.

CAD model of corner backstop of forint floor of a truck was optimized as shown in Fig. 13(a) and geometrical parameters of an addendum surface and BHF were design variables as shown in Fig. 13(b). Compared with the


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commercial optimization software Hyperstudy, the method achieved better results by using less computational time.

Fig. 12. The response based SM algorithm[87]

Fig. 13. Corner backstop of forint floor of a truck[87]

6 Curse of Dimensionality

Curse of dimensionality is one of the most challengeable issues in optimization especially for most expensive evaluation engineering problems. For most of low dimensional problems, some popular surrogates can well approximate[88]. The major problem associated with these models(e.g. RBF, polynomial, Kriging) is that the modeling effort or resource demand, in order to obtain acceptable accuracy, grows exponentially with the dimensionality of

the underlying problems. A general set of quantitative model assessment and

analysis tool, termed high dimensional model representation(HDMR), is a potential useful method for high dimensional modeling[89–90]. WANG G G, et al[88], used RBF as basis for HDMR to model more than 16-dimensions problems. WANG H, et al[91], suggested MLS-based HDMR to build more reliable surrogates for high dimensional problems. For HDMR methods, the adaptive sampling strategy cannot be integrated directly because the sample points should be located in the axial of hyper space. Therefore, a project strategy was proposed to be integrated with the HDMR. The number of evaluations can be reduced significantly.

7 Reliability Design with Surrogate Model

In engineering applications, reliability plays a more

important role compared with accuracy and efficiency criteria[92]. Due to nonlinearities and uncertainties in sheet metal forming design, the reliability deserves more attention in surrogate modeling research field. As mentioned in Section 2, designers can use any design parameters according to the requirement of applications, such as geometric, BHF, drawbead and others, these parameters can be easy controlled and deterministic values can be given during the forming process. Conversely, there are some parameters which are not possible to be controlled due to their uncertain variations. This kind of parameters can be regarded as noise parameters and might represent the main reason of instability and failures of sheet forming design, such as friction, some material properties, thickness and temperature, etc. In order to enforce the reliability of sheet metal forming, several strategies are presented in this Section.

7.1 Surrogate with uncertainties

For approximation methods, even the simplest surrogate, polynomial RSM, can be used for filtering noises and outliers in expensive evaluations, but these kinds of method can’t construct highly accurate model due to its’ weakness. Other popular surrogates, interpolated-based surrogates, such as Kriging, RBF, commonly thought to be high performance method, can’t filter the noises and outliers. Therefore, how to build a reliable surrogate which concerns both of reliability and accuracy is a challenge. The SVM is such an alternative method for robust regression[93]. TANG, et al[94], proposed a robust design models for sheet metal forming process integrated adaptive importance sampling. To minimize impact of variations and achieve reliable process parameters, the SVM with nonlinear capability in both pattern recognition and regression was adopted to map the relation between input process parameters and part quality. The major shortcoming of the SVM is about the efficiency that it should be much more sample points for construction a high quality model according to our tests.


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Therefore, the parallel strategy might be applied to the SVM for sheet metal forming optimization.

Another way is to construct a hybrid surrogate, both uncertain and deterministic design parameters are involved in a surrogate. WANG, et al[52], proposed a hybrid fuzzy regression-based metamodeling technique is proposed to optimize the sheet metal forming design. Compared with other surrogate techniques, the distinctive characteristic of this approach is to involve both uncertain and deterministic design parameters in a surrogate. This can be regarded as a real uncertain surrogate model. Because it considers the uncertainties while modeling, much more sample points should be used. To overcome it, they used one-step forming for accelerating the speed of construction. Obviously, although the reliability of the model is enforced, the accuracy might not be guaranteed in some complicated cases. Furthermore, due to its distinctiveness, such surrogate is complicated and the robustness is difficult be guaranteed.

7.2 Stochastic design

It is difficult to find a proper way for constructing an uncertain surrogate due to lack of a mathematical breakthrough. It is also difficult to use mathematical criteria to estimate the reliability of surrogates. Therefore, stochastic approach of the uncertainties of sheet forming design is a very important tool to avoid the risk of failure.

At the beginning of a stochastic design, designers should investigate objective functions without any uncertain parameters at the early stage and the robustness evaluation should be carried out by probabilistic methods later. This kind of method can be called hybrid methods which involve deterministic and stochastic analysis. For a hybrid method, the Monte Carlo(MC) simulation technique or just MC analysis is the most important tools.

The idea of MC analysis is that a number of experiments are constructed from the defined distributions of probabilistic variables. These experiments are evaluated by computer simulations, from which the results then are used to determine the probabilistic response. Because one evaluation of the sheet metal forming process usually takes several hours to complete, it is impossible for engineers to use simulation evaluations directly. As pointed out in KLEIBER, et al[95], there is a need to use an approximation, a surrogate, of the real model for the reliability assessment. JANSSON, et al[96], evaluated the use of linear and quadratic approximating RS as surrogates in a reliable assessment of a sheet metal forming process using MC simulation technique. MC simulation was used to determine the probability for springback and thickness variation in a sheet metal part. ZHANG, et al[97], used this hybrid method to quantify the uncertainties and to incorporate them into the surrogate model. A surrogate model of the process mechanics is generated using FEM-based high-fidelity models and DoEs, and a simple linear weighted approach is used to formulate objective

function or quality index(QI). They successfully used this method to find the optimal combination of BHF and friction coefficient under variation of material properties. The results showed that with the suggested approach, the QI improved by 42% over the traditional deterministic design. It also shows that by further reducing the variation of friction coefficient to 2%, the QI will improve further to 98.97%. Shivpuri and Zhang used the same method to investigate optimal design of spatially varying frictional constraints in reducing the risk of failure due to wrinkling and thinning[97]. The deterministic and probabilistic designs were compared to the conventional spatially uniform friction case. The results showed that in the drawing of a Hishida oil pan part, an overall improvement of 33% in strain distribution can be obtained and a further 12% improvement can be obtained with probabilistic design. A covariance based approach is used to explore a reliable design that is robust to process uncertainties. NAJAFI, et al[98], used surrogate to develop appoximate models for manufacturing and performance response. Moreover, a multi-objective robust design optimization problem is formulated and solved to illustrate the methodology and the influence of uncertainties on manufacturability and energy absorption of a metallic double-hat tube.

Dual RSM(DRSM) is another typical application for sheet metal forming design. It is well known that the focus of RSM is on the mean value of response where the variance is assumed to be small and constant. However, if this assumption does not hold, only constructing a surrogate model for the mean may not be adequate and optimization results can be misleading. Kim and Lin noted that the assumption of constant variance is often not practically valid and classical RSMs can be misleading in many cases[99]. An alternative to classical RSMs is the dual response surface approach. Initially proposed by Vining and Myers, this approach seeks to build two models: one for the mean and one for the standard deviation[100]. SUN, et al[101], introduced DRSM to build surrogate assisted multi-objective functions to obtain robust Pareto solutions. In their study, Equivalent drawbead restraining forces(DBRF) were obtained by developing a multi- objective robust PSO method, and the DBRF model was integrated into a single-objective PSO to optimize geometric parameters of drawbead. The optimal design showed a good agreement with the physical drawbead geometry and remarkably improve the formability and robust.

8 Challenges and Potential Feasible Ideas

For all engineering optimization methods, the purpose is

to obtain reliable and accurate with cheap computational cost. Can SAO method lead engineers to a feasible solution in practice? According to the previous discussions, the SAO method still plays an important role in sheet metal forming design. In our opinion, several critical issues are


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major challenges in this research field.

8.1 Time related surrogates Most of sheet metal forming design problems are related

with time, such as wrinkling, rupture and springback, but current popular objective/constraint functions are not time dependent. Therefore, the quality of a blank cannot be controlled during forming. Therefore, if a time related surrogate can be constructed for sheet metal forming design, the entire procedure of forming can be controlled and the objective/constraint functions should be more reasonable.

8.2 Aggressive ensemble surrogates

As we mentioned before, ensemble surrogate might obtain more reliable model than a single one. The present ensemble strategy commonly uses weight functions to combine different surrogates. There are various strategies to determine the weights; most of them are according to some statistical metrics which are derived from generated sampling data. These strategies are passive and are difficult to determine which kind of surrogates should be involved in combination. Theoretically, if surrogates can be selected automatically and aggressively, the ensemble surrogates should be more reliable and feasible with limited sample points.

8.3 Surrogates of constraint and objective functions

For most of SAO methods used in sheet metal forming optimization, Lagrange, penalty methods commonly used for constraint functions. Unfortunately, not all constraint functions’ response values can be obtained directly. It requires us to construct constraint surrogate while surrogate of objective functions are constructed. Obviously, much more sample points should be generated. How to generate the constraint and objective functions efficiently and how about the correlations between them are not well studied yet.

8.4 Parallel Surrogate Assisted Optimization on GPU

platform A novel parallel technique based on graphical processing

unit(GPU) has developed rapidly in recent years. This novel parallel approach of solving general purpose problem is known as general purpose computing on GPUs(GPGPU). GPUs have an inherent parallel throughput architecture that focuses on executing many concurrent threads simultaneously. A significant speedup of GPU-based computation over traditional CPU-based computation has been reported in different areas[102–103]. On the one hand, several FEA problems are accelerated by running on GPU in parallel, such as fluid mechanics[104], molecular dynamics[105] and wave propagation[106]. Meanwhile, CAI, et al[107], developed a GPU-based sheet forming simulation system, which is greatly increased the computation efficiency with the same computation precision as the CPU-based system. On the other hand, some

nature-inspired optimization methods have implemented on GPU in parallel to enlarge the optimization population and problem dimension. LI, et al and ZHOU, et al[108–109] accelerated the computation of the traditional PSO algorithm and fine-grained PSO algorithm on GPU in parallel, respectively. POSPICHAL, et al[110–111], designed an efficient parallel genetic algorithm model running entirely on the GPU and used it previously for acceleration 0/1 knapsack problem.

In our opinion, any SAO can be implemented on GPU parallel platform integrated with traditional parallel technique, such as symmetric multiprocessing(SMP) parallel computing structure with open Multi-Processing (OpenMP) interface. The parallel SAMO method can be summarized as follows. Firstly, the initial samples can be evaluated and generated on GPU in parallel. Secondly, CPU used in clustering the better-sample-set into K groups and scattering them to each GPU by OpenMP interface. Thirdly, GPU used in parallel solving the direct problem with the assigned sample by GPGPU interface, then, send back the information of these sample points to the CPU.

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Biographical notes WANG Hu, born in 1975, is currently a professor and a PhD candidate supervisor at State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, China. He received his PhD degree from Hunan Universtiy, China, in 2006. His research interests include mechanical design, engineering optimization, numerical computer method and parallel optimization. Tel: +86-731-88821417; E-mail: [email protected] YE Fan, born in 1991, is currently a PhD candidate at State Key Laboratory of Advance Design and Manufacturing for Vehicle Body, Hunan University, China. He received his bachelor degree from Yangzhou University, China, in 2013. His research interests include composite material and engineering optimization. E-mail: [email protected] CHEN Lei, born in 1991, is currently a master candidate at State Key Laboratory of Advance Design and Manufacturing for Vehicle Body, Hunan University, China. E-mail: [email protected] LI Enying, born in 1975, is currently a associate professor at Central South University of Forestry and Technology, China. Her main research interests include engineering optimization, mechanical design and nonlinear dynamic problem. E-mail: [email protected]

Sheet Metal Forming Optimization by Using Surrogate ...

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