An ion Stratigy for Industrial Metal Forming

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Struct Multidisc Optim (2008) 35:571586DOI 10.1007/s00158-007-0206-3

INDUSTRIAL APPLICATION

An optimisation strategy for industrial metalforming processesModelling, screening and solving of optimisation problems in metal forming

M. H. A. Bonte A. H. van den Boogaard J. Hutink

Received: 29 September 2006 / Accepted: 7 September 2007 / Published online: 18 March 2008 The Author(s) 2008

Abstract Product improvement and cost reductionhave always been important goals in the metal formingindustry. The rise of nite element (FEM) simulationsfor processes has contributed to these goals in a ma- jor way. More recently, coupling FEM simulations tomathematical optimisation techniques has shown thepotential to make a further giant contribution to prod-uct improvement and cost reduction. Much research onthe optimisation of metal forming processes has beenpublished during the last couple of years. Althoughthe results are impressive, the optimisation techniquesare generally only applicable to specic optimisationproblems for specic products and specic metal form-ing processes. As a consequence, applying optimisationtechniques to other metal forming problems requires alot of optimisation expertise, which forms a barrier formore general industrial application of these techniques.In this paper, we overcome this barrier by proposing agenerally applicable optimisation strategy that makesuse of FEM simulations of metal forming processes.It consists of a structured methodology for modellingoptimisation problems related to metal forming. Sub-sequently, screening is applied to reduce the size of the optimisation problem by selecting only the mostimportant design variables. Finally, the reduced opti-misation problem is solved by an efcient optimisationalgorithm. The strategy is generally applicable in asense that it is not constrained to a certain type of metalforming problem, product or process. Also, any FEMcode may be included in the strategy. Furthermore,

M. H. A. Bonte (B ) A. H. van den Boogaard J. HutinkUniversity of Twente, P.O. Box 217,NL-7500 AE Enschede, The Netherlandse-mail: [email protected]

the structured approach for modelling and solvingoptimisation problems should enable non-optimisationspecialists to apply optimisation techniques to improvetheir products and processes. The optimisation strat-egy has been successfully applied to a hydroformingprocess, which demonstrates the potential of the op-timisation of metal forming processes in general andmore specic the proposed optimisation strategy.

Keywords Metal forming Finite element method Optimisation

1 Introduction

During the past few decades, nite element (FEM)simulations of metal forming processes have be-come important tools for designing feasible productionprocesses. In more recent years, coupling FEM simula-tions to mathematical optimisation techniques evolvedto address two industrial needs: (1) designingoptimal metal forming processes instead of onlyfeasible ones(better products, lower costs) and (2) solving problemsin manufacturing.

The basic concept of mathematical optimisation ispresented in Fig. 1. Basically, it consists of two majorphases: the modelling and the solving of the optimisa-tion problem. The modelling phase consists of:

1. Selecting a number of design variables the user isallowed to adapt

2. Choosing an objective function, i.e. the optimisa-tion aim

3. Taking into account possible constraints


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574 M.H.A. Bonte et al.

convergence to a local optimum generally requiresrelatively few iterations and is, hence, fairly efcient.Due especially to the former advantage, iterativealgorithms are quite often applied to optimise metalforming processes, see, e.g. Byon and Hwang (2001),Cao et al. (2001), Debray et al. (2004), Duan andSheppard (2002), Endelt and Nielsen (2001); Endelt(2003), Fann and Hsiao (2003), Jirathearanat andAltan (2004), Kim et al. (2001a, b), Kleinermann(2001), Kleinermann and Ponthot (2003), Lin et al.(2002, 2003), Naceur et al. (2001, 2004c), Ponthotand Kleinermann (2005), Shi et al. (2004), Sillekensand Werkhoven (2001), Yang et al. (2001), Zhaoet al. (2004).

Evolutionary and genetic algorithms A second groupof algorithms for which there is a direct coupling be-tween the algorithm and theFEM software (Fig.3a) aregenetic and evolutionary optimisation algorithms. Ge-netic and evolutionary algorithms look promising be-cause of their tendency to nd the global optimum andthe possibility for parallel computing. Furthermore, itis not difcult to calculate sensitivities with them. How-ever, the rather large number of function evaluationsthat is expected to be necessary using these algorithmsis regarded as a serious drawback (Emmerich et al.2002). Several authors have applied genetic and evolu-tionary algorithms to optimise metal forming processes,see Abedrabbo et al. (2004), Antnio et al. (2004),Castro et al. (2004), Do et al. (2004), Fourment et al.(2005a, b), Poursina et al. (2003), Schenk and Hillmann(2004), Sousa et al. (2003), and Weyler et al. (2004).

Approximate algorithms A third way of optimisationin combination with time-consuming function evalua-tions is using approximate optimisation algorithms, of which response surface methodology (RSM) is a well-known representative. RSM is based on tting a low-order polynomial metamodel through response points,which are obtained by running FEM calculations forcarefully chosen design variable settings and nallyoptimising this metamodel (Myers and Montgomery2002). Hence, for approximate optimisation, the di-rect coupling between the optimisation algorithm andthe FEM calculations is removed and a metamodelis placed in between as a buffer. This is schemati-cally presented in Fig.3b. Metamodels are sometimesalso referred to as response surface models or surro-gate models. Next to RSM, other metamodelling tech-niques are Kriging [or design and analysis of computer

experiments (DACE)] and neural networks. Allow-ing for parallel computing and lacking the necessityfor sensitivities, approximate optimisation is appeal-ing to many authors in the eld of metal forming,see Ben Ayed et al. (2004a, b, 2005), Breitkopf et al.(2004, 2005), Jansson (2002), Jansson et al. (2005), Kocet al. (2000), Lenoir and Boudeau (2003), Liew et al.(2004), Naceur et al. (2003, 2004a, b), Ohata et al.(2003), Repalle and Grandhi (2004), Revuelta andLarkiola (2004), Sahai et al. (2004), and Thiyagarajanand Grandhi (2004). Disadvantages include an approx-imate optimum as a result rather than the real globaloptimum and the curse of dimensionality: these algo-rithms tend to become very time consuming if manydesign variables are present.

Adaptive algorithms A fourth group is formed by so-calledadaptive algorithms. Adaptive algorithmsarenotcoupled to FEM in the same way as the other threegroups of algorithms. Adaptive algorithms are incor-porated within the FEM code and generally optimisethe time-dependent load paths of the metal formingprocess during each increment of the FEM calculation.An advantage is that the optimum is obtained in onlyone FEM simulation. However, access to the sourcecode of the FEM software is necessary and only time-dependent design variables can be taken into account.These disadvantages seriously limit the general applica-bility of these kinds of algorithms. Literature describesseveral applications of these algorithms to metal form-ing (Aydemir et al. 2005; Carrino et al. 2003a, b; DiLorenzo et al. 2004; Johnson et al. 2004; Labergreet al. 2004; Labergre and Gelin 2004a, b; Ray andMac Donald 2004; Sheng et al. 2004; Sillekens andWerkhoven 2001; Strano and Carrino 2004) especiallyto optimise the internal pressure and axial feeding loadpaths in hydroforming.

All groups of optimisation algorithms introduced inthe previous section have been applied to optimisationproblems in metal forming. In general, one can con-clude from literature that specic problems for specicmetal forming processes are sometimes quite arbi-trarily modelled and subsequently solved using analgorithm suitable for that specic application. In ouropinion, a generally applicable optimisation strategyfor modelling and solving optimisation problems inmetal forming problems is lacking. As a consequence,we developed such an optimisation strategy which canbe applied to model and solve all kinds of optimisationproblems for all kinds of metal forming processes usingany simulation code.


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An optimisation strategy for industrial metal forming processes 575

3 Optimisation strategy for industrial metalforming processes

In this section, we will propose an optimisationstrategy for industrial metal forming processes. InSection3.1, the requirements for the optimisation strat-egy are shortly introduced. Subsequently, the optimisa-tion strategy is presented in Sections3.2to 3.4.

3.1 Requirements

Our aim has been to develop an optimisation strategyfor metal forming processes using FEM models of theseprocesses, but which processes? For which (type of)product? And which FEM codes should be included?We have tried to develop a strategy which is as gen-erally applicable as possible. In this context, generalapplicability has four dimensions, as schematically pre-sented in Fig.4:

Processes: the strategy should be able to modeland solve all kinds of optimisation problems for allkinds of different metal forming processes.

Products: the same holds for different products.The products mentioned in Fig. 4 are all deep-drawn products, but quite different demands are setfor them.

Users: although both can be related to metal form-ing, a product designer is confronted with totallydifferent challenges from those of a manufacturingengineer.

FEM codes: many FEM codes are available. Thechoice for specic simulation software dependsheavily on the process, the product and the pref-erence of the user.

Whereas the general applicability is the rst require-ment, a second requirement is the application of math-ematical optimisation techniques: the modelling needsto result in a specic, mathematically formulated opti-

Fig. 4 General applicability of the optimisation strategy

misation problem to subsequently solve it by a mathe-matical optimisation algorithm. Both requirements arequite contradictory: the optimisation strategy needs tobe generally applicable, yet solving it mathematicallyrequires a detailed and specic optimisation model.

We propose a three-stage optimisation strategy thatlives up to both of the above requirements:

1. Modelling the optimisation problem2. Screening to determine the most important design

variables3. Solving the optimisation problem

The different stages are presented in Sections3.2to 3.4.

3.2 Modelling

The rst stage is to model the optimisation problem.It is quite a challenge to design a structured method-ology that is, on the one hand, generally applicable toany kind of metal forming problem but, on the otherhand, yields a specic mathematical formulation of theoptimisation problem.

We adopted the following approach for tacklingthis problem:

1. Brainstorming for industrially relevant objectives,constraints and design variables.

2. Structuring these quantities by means of the gener-ally applicable product development cycle.

3. Applying this product development cycle to metal

products and their forming processes.4. Dening a seven-step methodology for modellingoptimisation problems. This methodology is gen-erally applicable to any metal forming problemfor any process, product, FEM code and user.However, after having followed the seven steps, itresults in a specic mathematical optimisationmodel.

Brainstorming Several brainstorm sessions have beenorganised at several large metal forming compa-nies. Different users joined the sessions and different

products and processes have been considered. Thebrainstorming sessions resulted in a large number of industrially relevant objectives, constraints and designvariables.

Product development cycle At the basis of the struc-tured methodology for modelling optimisation prob-lems for industrial metal forming processes is the product development cycle , which is a part of theprod-uct life cycle . A schematic of the product life cycle


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Fig. 5 The product life cycle

is presented in Fig.5 (Yang and El-Haik 2003). Theproduct development cycle is the stages 0 through 5,i.e. the product life cycle of Fig.5 minus the stages 6(product consumption) and 7 (product disposal).

Three groups of quantities are indicated in Fig.5:

Functional requirements (FRs): these are productproperties that are critical to customer satisfactionand product functionality.

Design parameters (DPs): these dene the productdesign.

Process variables (PVs): these are process settingsnecessary to manufacture the product.

The product life cycle and product development cy-cle are both very generally applicable. The next stepis to make this generally applicable concept somewhatmore specic by applying it to metal forming. This hasbeen done by confronting the product developmentcycle to the metal forming quantities resulting from thebrainstorm sessions.

Product development cycle applied to metal formingThe industrially relevant metal forming quantities which are possible objective functions, constraints anddesign variables for optimisation have been cate-gorised in FRs, DPs and PVs. Two additional categoriesare costs and defects. They have been added to theproduct development cycle as presented in Fig.6.

Fig. 6 The product life cycle applied to metal forming

Subsequently, top-down structures have been de-ned for each of the ve categories. The top-downstructures for the DPs and PVs are presented in Figs.7

and 8, respectively. Typical examples of FRs for metalforming are crashworthiness properties, fatigue prop-erties, stiffness, strength, etc. Examples of the categorycosts are material and process costs. Metal formingdefects such as necking and wrinkling in sheet metalforming are comprised in the category defects.

7 step methodology The product development cycleapplied to metal forming is the basis for a seven-stepmethodology for the modelling of optimisation prob-lems in metal forming:

1. Determine the appropriate optimisation situation.

2. Select only the necessary responses.3. Select one response as objective function, the oth-ers as implicit constraints.

4. Quantify the objective function and implicit con-straints.

5. Select possible design variables.6. Dene the ranges on the design variables.7. Identify explicit constraints.

Fig. 7 Design parameters of metal formed parts


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Fig. 8 Process variables for metal forming processes

The rst step is to determine the appropriate op-timisation situation. Four situations in which the op-timisation of metal products and their manufacturingprocesses can play a role are distinguished:

Part design, where it is aimed to optimise the metalformed products FRs by determining the DPs.

Process design type 1, where it is aimed to ex-actly obtain the DPs set by the part designer bydetermining the PVs. Alternatively, one can aimto manufacture a defect-free product or minimiseproduction costs by determining the PVs.

Process design type 2, where it is aimed to optimisethe metal formed products FRs by determining thePVs. DPs, defects and costs can still play a role inparallel to the FRs.

Production, where optimisation techniques can beused to solve manufacturing problems.

These situations and their relations to the productdevelopment cycle are presented in the left part of Fig.6. To demonstrate how optimisation can be appliedto one of these four situations, compare the inputresponse model for optimisation in Fig.2 (repeated inFig. 9) to that for a process design type 1 problem inFig. 10. Note the resemblance: one can immediatelyobserve that, for a process design type 1 problem, theobjective function and implicit constraints are the DPs,defects and costs, whereas the design variables arerelated to the PVs.

In step 2 of the seven-step methodology, the top-down structures such as the ones in Figs.7 and 8 can beused to select the necessary responses for the specicproblem. These responses are output quantities of theFE simulation.

Fig. 9 Inputresponse model for FEM in relation to optimisation

Fig. 10 Inputresponse model for FEM during the processdesign type 1 situation

According to step 3, one of the dened responsequantities is selected as objective function, the others asimplicit constraints. We prefer selecting one objectiveand dening other responses as constraints rather thanmulti-objective approaches such as a weighted sum of the responses. The latter approach compares, in ouropinion, difcult to compare quantities and is some-what arbitrary in the selection of weight factors.

Now, it is clear which FEM responses are formulatedas objectives and which as constraints. However, theexact mathematical formulation of the responses is stillnot clear. This is done in step 4 of the seven-stepmethodology. For response quantication, Table1 isproposed. It assists in selecting the nal mathemati-cal formulation of the objective function and implicitconstraints. Objective functions are further subdividedin objectives that aim to reach a target and those thatdo not. Upper- and lower-limit implicit constraints aredistinguished. Because the response quantityX is aFEM output, it can be a nodal or element value or not.Examples of nodal/element values are strains, stresses,thickness, etc. Quantities such as forming energy arenot nodal/element related: one number results fromone FEM calculation. Furthermore, Table1 subdividesthe element/nodal value related responses further intocritical and non-critical values. Critical values are thevalues for which none of the nodal/element values areallowed to exceed a specied level. If it is acceptablethat some of the element/nodal values exceed this spec-ied level, but important that the average responsevalue performs well, the response is non-critical. Con-straints are, by denition, critical values as one can seein Table 1.

Steps 5, 6 and 7 of the modelling methodologyconcern the FEM inputs, the design variables in caseof optimisation. Step 5 comprises the design variableselection. The optimisation situation selected in step 1determines the groups of design variables to be takeninto account. For example, Fig.10 shows that the PVsare the group of design variables for a process designtype 1 situation. The top-down structure in Fig.8 cansubsequently be used to select the design variables forthe specic optimisation problem.

Step 6 comprises the selection of the ranges (up-per and lower bounds) on all design variables. This is


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Table 1 Response quantication

Type of response No nodal/element value Nodal/element value, critical Nodal/element value, non-critical

Objective function, no target min X minmax N X min N ( X )N

Objective function, target= X 0 min | X X 0 | minmax N | X X 0 | min X X 0 2

N Implicit constraint, USL X U SL 0 max N ( X U SL ) 0Implicit constraint, LSL LSL X 0 max N ( LSL X ) 0

something that needs to be done based on the experi-ence and insight of the user.

Step 7 concludes the input modelling by identifyingexplicit constraints. Explicit constraints describe impos-sible combinations of the design variables, e.g. it is notpossible to run a FEM simulation for those settings, orthe combination is beforehand an infeasible solution of the problem.

Without going into further detail, we conclude by re-

emphasising that this seven-step methodology is gener-ally applicable to any metal forming problem and yieldsa specic mathematical optimisation model, which cansubsequently be solved using a suitable optimisation al-gorithm. The seven-step methodology is demonstratedin the next section when it is applied to a simple hydro-forming example.

3.3 Screening

After modelling, many design variables may be present,which makes the problem time-consuming to solve. It isworthwhile to invest some time in reducing the numberof design variables before applying the optimisationalgorithm. This is done in the screening stage.

We propose to screen the importance of the designvariables by applying factorial DOE strategies (Myersand Montgomery2002). Figure11a shows a well-knownfull factorial design for two levels and three factors(design variables), a so-called2 3 full factorial design.It allows for estimation of the linear and interactioneffects of the design variables and requires2 3 = 8 FEMcalculations.

If one likes to estimate quadratic effects, a DOE of at least three levels is required. Figure11b shows a3 2full factorial design for two design variables on threelevels each. 3 2 = 9 FEM calculations are required toestimate the linear, interaction and quadratic effectsof the design variables. For more design variables, thenumber of FEM calculations explodes exponentially,which makes the application of full factorial designsprohibitively time-consuming.

However, if one is only interested in linear effects,the number of necessary FEM calculations can be sig-

nicantly reduced by applying fractional factorial de-signs. Figure11c shows a2 3 1III fractional factorial DOEfor three design variables. Resolution III denotes that itis possible to independently estimate the linear effects.Interaction effects cannot be independently estimated.The loss of information with respect to applying the2 3 full factorial design returns a decrease in the time-consuming FEM simulations that need to be run: the2 3 1III fractional factorial design requires only four FEM

calculations in contradiction to the eight calculationsfor the full factorial design. For more design variables,the difference becomes signicantly larger in favour of the resolution III fractional factorial designs.

For screening purposes, we are willing to considerlinear effects only. Of course, neglecting interactionand other non-linear effects is a crude assumption,but the increase in efciency is at least during thescreening stage more important than the accuracy.Moreover, the amount of the linear effects should givean indication of the importance of the different designvariables and can thus be used to omit the less impor-

tant design variables.After having applied the resolution III fractionalfactorial design and having run the corresponding FEMsimulations, the linear effects can be estimated by ap-plying statistical techniques such as analysis of variance(ANOVA), see e.g. Myers and Montgomery (2002).The amount and direction of the effect of each variableon each response can be nicely displayed in Pareto andeffect plots (Yang and El-Haik2003).

Using these techniques, the variables with the largesteffects may be kept in the optimisation model, whereasthe variables having less effect may be omitted. In

Fig. 11 a 2 3 full factorial design;b 3 2 full factorial design;c 2 3 1III fractional factorial design


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such a way, the amount of design variables may besignicantly decreased while maintaining control overobjective function and constraints during optimisation.Screening and the use of Pareto and effect plots isfurther demonstrated in Section4.

3.4 Solving

The nal stage of the optimisation strategy is to solvethe optimisation problem by a suitable algorithm.

Based on the literature study presented in Section2.2, we propose to use an approximate optimisationalgorithm: it is efcient because several calculations canbe run at the same time on parallel processors and itconverges to the global optimum, which mostly resultsin better results than local algorithms. The disadvan-tage that this type of algorithm is not efcient in casemany design variables are present is overcome by thescreening stage of the optimisation strategy. Addition-ally, the algorithm is applicable to all kinds of metalforming processes, products and problems because theFEM simulations are included as a black-box.

An overview of the algorithm is presented in Fig.12.Here, we will shortly summarise the different steps of the algorithm. For more details, refer to publicationson the sequential approximate optimisation (SAO)algorithm (Bonte et al.2005a, b, 2007; Bonte 2007).

DOE After having modelled and perhaps screenedthe optimisation problem, the rst step of the algo-rithm is to apply another DOE strategy. This DOEstrategy indicates for which design variable settings

Fig. 12 Sequentialapproximate optimisation(Bonte et al. 2005a, b,2007; Bonte 2007)

to run the rst FEM simulations for optimisation. Aspacelling Latin hypercubes design (LHD) is a goodand popular DOE strategy for constructing metamod-els from deterministic computer experiments such asFEM calculations (McKay et al. 1979; Santner et al.2003). The developed DOE takes into account explicitconstraints and uses a so-called maximin criterion forspacellingness (Bonte2007).

Running the FEM simulations For the design variablesettings determined by the LHD, FEM simulations areperformed on parallel processors. Although FEM sim-ulations for metal forming processes can be very time-consuming, running the FEM simulations in parallel isquite time-efcient. Any FEM code for any productand any process may be applied, which guarantees thegeneral applicability of the SAO algorithm.

Fitting the metamodels The FEM simulations resultin a number of measurement points for each one of the modelled responses (objective function and implicitconstraints). For each response, several metamodelsare constructed using response surface methodology(RSM) (Myers and Montgomery 2002) and Kriging(Sacks et al. 1989a, b, Santner et al. 2003) metamod-elling techniques. Both RSM and Kriging are statisticaltechniques: the metamodels of the responses can beinterpreted as means. Additionally, it is also possibleto determine a standard deviation at any design vari-

able setting. To explain this, Fig.13 presents a Krigingmetamodel tted through three response pointsy(i) thatresulted from running three FEM calculations for thedesign variable settingsx(i) . At an untried design vari-able setting x, the predicted objective function valueis y, which can be interpreted as a mean value atthat location. Because we have not run a simulationat that location, there is a probability that the value isdifferent. For both Kriging and RSM, a standard devi-ation s can be calculated that reects this uncertainty,see Santner et al. (2003) for Kriging and Myers andMontgomery (2002) for RSM. Figure13 visualises this

standard deviation at the untried design variable settingx. For Kriging, being an interpolative technique, thestandard deviation equals 0 at the DOE points.

Validating the metamodels For each response, differ-ent metamodels have been tted now. Metamodel val-idation techniques are employed to assess the accuracyof the different metamodels. Metamodel validation forRSM is based on analysis of variance (ANOVA) andresidual plots, see e.g. Myers and Montgomery (2002).


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Fig. 13 A Kriging metamodel and its standard deviation

Metamodel validation for Kriging is based on cross vali-

dation (Bonte 2007; Bonte et al.2007). Using these vali-dation techniques, the best metamodels (either RSM orKriging) for each response are selected and included asan approximation in the optimisation problem.

Optimisation Theoptimisation problemis subsequent-ly optimised using a standard sequential quadraticprogramming (SQP) algorithm. In case constraints orKriging metamodels are present in the nal optimi-sation problem, there is a risk of ending up in a lo-cal optimum. This problem is overcome by initialisingthe SQP algorithm at multiple locations. This impliesperforming many function evaluations, but this ishardly a problem because both RSM and Kriging meta-models being explicit mathematical functions canbe evaluated thousands of times within a second. TheDOE points are used as initial locations for the SQPalgorithm.

The obtained approximate optimum is nallychecked by running one last FEM calculation with theapproximated optimal settings of the design variables.Next to metamodel validation, the difference betweenthe approximate objective function value and the realvalue of the objective function calculated by the last

FEM run is an additional measure for the accuracy of the obtained optimum. If the user is not satised withthe accuracy, the SAO algorithm allows for sequentialimprovement, as can be seen in Fig.12.

Sequential improvement Three possible ways of se-quential improvement have been presented and com-pared to each other by (Bonte et al. 2006; Bonte 2007).The basic idea of sequential improvement is to increasethe accuracy of the optimum by adding new DOEpoints to the original spacelling LHD. In this paper,

we employ sequential improvement by minimising amerit function (SAO-MMF). The merit function is:

f merit = y w s (1)

where y and s are, for both RSM and Kriging, given bymetamodels from previous iterations of the algorithm,

see Fig.13. w is a weight factor. If one selectsw= 0

, thenew DOE points equal the optima of the metamodel y. If w , the new DOE points are simply addedin a spacelling way. We found thatw = 1 providesa good compromise between both extreme cases. Themerit function is minimised by the same multistart SQPalgorithm introduced in the previous section. Again,the DOE points are used to initiate the SQP algorithm.Because the merit function in(1) is also a metamodel,minimising the merit function is very time-efcient.

Implementation The optimisation algorithm presentedin Fig. 12 and the previous sections was imple-mented in MATLAB and can be used in combinationwith any FEM code for any metal forming process.It may also be applied to other applications forwhich performing many function evaluations is time-consuming or otherwise prohibitive. For the tting of the DACE/Kriging metamodels, use was made of theMATLAB Kriging toolbox implemented by Lophaven,Nielsen and Sndergaard (Nielsen 2002; Lophavenet al. 2002a, b). The efciency of the algorithm hasbeen assessed by comparing it to other optimisationalgorithms and applying all algorithms to two forging

processes, see (Bonte et al. 2006; Bonte 2007).

4 Application to hydroforming

We will demonstrate the proposed optimisation strat-egy and the potential of optimisation in general by applying it to a simple hydroforming example, seeFig.14.

4.1 Modelling

We follow the seven-step methodology for modellingthe optimisation problem.

Step 1: Determine the appropriate optimisation situationAim of optimisation is to design the manufacturingprocess to produce the part presented in Fig.14. Thisis a process design type 1 situation.

Step 2: Select only the necessary responses The pos-sible groups of responses for a process design type 1situation are indicated in Fig.10. These areDPs, defects


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x4 x2 x1 x5 x3 x6 x7 Error0

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0%

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x4 x5 x6 x7 x3 x2 x1 Error0

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1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1

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3.8

4

x7

f 1

a b c

Fig. 16 a Pareto plot for the objective function;b Pareto plot for the implicit constraint;c effect plot

The structured seven-step methodology yieldedthe following mathematically formulated optimisationmodel:

min f (t 1 , t 2 , , umax , R , t , ) =h h 0 2

N s.t. gimpl = maxN

(d tool product) 0

gexpl1 = t 1 t 2 0

gexpl2 = umax 9 (t 1 t 2 ) 0

0 s t 1 5 s2 .5 s t 2 10 s

9 .5 MPa/ s 12 MPa/ s0 mm umax 9 mm

40 mm R 43 .5 mm0 .8 mm t 1 .5 mm

0 .10 0 .15 (2)

where h is the thickness throughout the nal part,h 0 the nal thickness dened by the part designer(1mm), N the number of nodes throughout the partand d tool product the distance between nal product andthe die. R and t are the initial tubes radius and thick-ness, t 1 and t 2 are the time when axial feeding starts andstops, umax is the amount of axial feeding and is theincrease in internal pressure. denotes the coefcientof friction.

4.2 Screening

Seven design variables is not that many; hence, itis possible to apply the SAO algorithm immediately.However, for demonstration purposes, we will performscreening anyway.

A 2 (7 4 )III fractional factorial DOE is applied, whichimplies that eight FEM simulations have been run to

screen the importance of the seven design variables. A2D axisymmetric FEM model has been made of theproduct. The in-house FE code DiekA has been usedas FEM solver.

The resulting Pareto plot for the objective functionis presented in Fig.16a. Based on this plot, one mayestimate that keeping the three most important vari-ables umax , R and t and omitting the other four variableswill still result in about 80% control over the objectivefunction. A same Pareto plot has been generated forthe implicit constraint, see Fig.16b. umax , t 2 and t 1 werethe most important variables for the lling of the die.It is important to keep control over all responses: if the most important variables are taken into accountfor the objective function only, it is possible that theless important variables are set to a level for whichthe implicit constraints are not satised. If the controlover the constraints has been lost after screening, it is,during optimisation, not possible anymore to yield afeasible solution. Without losing too much control over

0 20 40 60 80 100 1200.5

1

1.5

2

2.5

3

Number of FEM calculations

F e a s

i b l e o b

j e c t

i v e f u n c

t i o n v a

l u e

Convergence plot

Fig. 17 Convergence of the optimisation algorithm


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Table 2 Optimised processsettings t 1 t 2 umax R t f g1

1.97 2.50 12 4.33 42.32 1.11 0.15 0.615 0.071

both responses, it is, in this case, possible to reduce thenumber of design variables to ve.

A question remaining is what to do with the twounimportant design variables and . For this, thereare two options:

Set them to a nominal value based on experience of the user.

Use effect plots of the objective function.

In this case, we use the effect plots of the objectivefunction. Figure 16c presents the effect of on thewall thickness. Because we are aiming to minimise themodelled objective function, one can best set tothe maximal value. Analogously, one can set to themaximal value, too.

4.3 Solving

The SAO algorithm has been applied to the reducedoptimisation problem. Figure17shows the convergencebehaviour of the algorithm. The simulations have beenperformed on 16 parallel processors; hence, the totalcalculation time was much shorter than running the 110simulations sequentially. These 110 simulations includethe eight screening simulations.

The optimised design variable settings are displayedin Table 2. The optimal objective function value is0.615 and the negative value for the implicit constraintdenotes that the nal product properly lls out the die.Note theoptimal initial tube thickness of 1.11mm. Con-

0 10 20 30 40 50 60 700.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Nodal position

W a l

l t h i c k n e s s

Screen 1 f=4.09Screen 2 f=2.30Screen 3 f=2.49Screen 4 f=2.40Screen 5 f=2.40Screen 6 f=2.49Screen 7 f=2.83Screen 8 f=3.51Target f=0Optimum f=0.615

Fig. 18 Wall thickness distribution

sidering the perfect wall thickness of 1 mm and materialthinning due to ination of the tube, this slightly thickerinitial tube thickness is indeed the result one wouldintuitively expect to be optimal.

Figure 18 presents the wall thickness distributionthroughout the nal product for the perfect product(uniform wall thickness of 1 mm), some arbitrary set-tings of the design variables (the eight screening cal-culations) and the optimised process. The optimisedprocess yields a signicantly better product, whichdemonstrates the high potential of optimisation inmetal forming, and specically the optimisation strat-egy proposed in Section3.

5 Conclusions

A generally applicable optimisation strategy for metalforming processes has been proposed. It includes threestages: modelling, screening and solving.

The strategy includes a seven-step methodology formodelling optimisation problems in metal forming. Inliterature, this modelling is often done quite arbitrarilyfor a specic problem, product, or process. The pro-posed seven-step methodology is generally applicableto all kinds of optimisation problems, products andprocesses. Moreover, the structured methodology al-lows also non-optimisation specialists to generate aproper mathematical optimisation model of the metalforming problem they are facing.

The second stage is screening to reduce the optimi-sation problem size by selecting only the most impor-tant design variables. Resolution III fractional factorialdesign-of-experiment strategies provide a nice balancebetween effect estimation and efciency. Analysis of variance and Pareto and effect plots are employed toestimate the importance of the design variables. Onlythe couple of most important design variables are takeninto account in the nal, reduced optimisation problem.

The reduced problem is subsequently solved by agenerally applicable optimisation algorithm. This SAOalgorithm has been developed by the authors to ef-ciently solve optimisation problems comprising time-consuming function evaluations, e.g. FEM simulationsfor metal forming processes.

The optimisation strategy has been successfully ap-plied to a hydroforming process. Modelling, screeningand solving this metal forming problem demonstrated


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the potential of the optimisation of metal formingprocesses in general and, more specically, the pro-posed optimisation strategy.

Acknowledgements This research has been carried out in theframework of the project Optimisation of Forming ProcessesMC1.03162. This project is part of the research programmeof the Netherlands Institute for Metals Research (NIMR). Theindustrial partners co-operating in this project are gratefullyacknowledged for their useful contributions to this research.

Open Access This article is distributed under the terms of theCreative Commons Attribution Noncommercial License whichpermits any noncommercial use, distribution, and reproductionin any medium, provided the original author(s) and source arecredited.

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