crashworthiness optimisation of vehicle structures with magnesium alloy parts

This article was downloaded by: [Gazi University]On: 07 December 2012, At: 20:15Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

International Journal of CrashworthinessPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/tcrs20

Crashworthiness optimisation of vehicle structureswith magnesium alloy partsAndrew Parrish a , Masoud Rais-Rohani b & Ali Najafi ba Department of Aerospace Engineering, Mississippi State University, MS, 39762, USAb Department of Aerospace Engineering and Center for Advanced Vehicular Systems,Mississippi State University, MS, 39762, USAVersion of record first published: 13 Jan 2012.

To cite this article: Andrew Parrish, Masoud Rais-Rohani & Ali Najafi (2012): Crashworthiness optimisation of vehiclestructures with magnesium alloy parts, International Journal of Crashworthiness, 17:3, 259-281

To link to this article: http://dx.doi.org/10.1080/13588265.2011.648518

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form toanyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses shouldbe independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims,proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly inconnection with or arising out of the use of this material.

http://www.tandfonline.com/loi/tcrs20

http://dx.doi.org/10.1080/13588265.2011.648518

http://www.tandfonline.com/page/terms-and-conditions

International Journal of CrashworthinessVol. 17, No. 3, June 2012, 259–281

Crashworthiness optimisation of vehicle structures with magnesium alloy parts

Andrew Parrisha, Masoud Rais-Rohanib∗ and Ali Najafib

aDepartment of Aerospace Engineering, Mississippi State University, MS 39762, USA; bDepartment of Aerospace Engineeringand Center for Advanced Vehicular Systems, Mississippi State University, MS 39762, USA

(Received 13 August 2011; final version received 7 December 2011)

This paper explores the effects of replacing the baseline steel with lightweight magnesium alloy parts on crashworthinesscharacteristics and optimum design of a full-vehicle model. Full frontal, offset frontal and side crash simulations are performedon a validated 1996 Dodge Neon model using explicit nonlinear transient dynamic finite element analyses in LS-DYNA toobtain vehicle responses such as crash pulse, intrusion distance, peak acceleration and internal energy. Twenty-two parts ofthe vehicle body structure are converted into AZ31 magnesium alloy with adjustable wall thickness while the remaining partsare kept intact. The magnesium alloy material model follows a piecewise linear plasticity law considering separate tensionand compression properties and maximum plastic strain failure criterion. Six different metamodelling techniques includingoptimised ensemble are developed and tuned for predictions of crash-induced responses within the design optimisationprocess. The crashworthiness optimisation problem is solved using the sequential quadratic programming method with mostaccurate surrogate models of structural responses considering both constrained single- and multi-objective formulations. Theresults show that under the combined crash scenarios with the selected material models and design constraints, the vehiclemodel with magnesium alloy parts can be optimised to maintain or improve its crashworthiness characteristics with up to50% weight savings in the redesigned parts.

Keywords: crashworthiness; design optimisation; magnesium alloy; material model; material substitution; finite elementanalysis; metamodels

1. Introduction

Stricter regulations on fuel economy and growing concernsover automobile emissions have led to an increased fo-cus on vehicle weight reduction through the application oflightweight materials, especially in auto body structures.The fuel economy improvements gained through weight re-duction must also be balanced with the requirements forcrash safety. Therefore, a simple one-to-one substitution ofthe original parts in absence of a thorough considerationof the overall crash characteristics may not result in a safedesign. Figure 1 describes the overall approach consideredin this study, where material substitution and design opti-misation techniques are combined to reduce vehicle weightwhile maintaining the vehicle’s crash performance or crash-worthiness.

The U.S. Department of Energy has sponsored manystudies in the area of Automotive Lightweighting Materi-als [7] under the Vehicle Technologies Program to promotethe application of lightweight materials such as magnesiumalloys and improvements in structural design and manufac-turing for lightweighting of vehicle structures. The strategicvision for magnesium [32] advocates opportunities for in-corporating more magnesium parts into automobile design.It suggests that magnesium is a good candidate material

∗Corresponding author. Email: [email protected]

to replace the heavier steel parts due to magnesium’s highstrength-to-weight ratio, potential reduction in part countdue to larger castings and the ability to tune magnesiumparts to frequencies related to noise, vibration and harsh-ness (NVH). That study also indicates the need for moreresearch to address corrosion and manufacturing issues aswell as brittle fracture at high strain rates before magnesiumis ready for widespread application in auto body structures.

Crashworthiness is the ability of the structure to protectoccupants during crashes. Performing real crash tests toobtain necessary data for crashworthiness optimisation isnot feasible. Finite element (FE) crash simulations are oftenused in place of physical experiments to obtain responsesneeded for optimisation or exploratory studies.

Critical responses for crashworthiness optimisation areoften selected to measure or estimate the likelihood ofoccupant injury. The U.S. Department of Transportationplaces requirements of crash responses felt by test dum-mies through the Federal Motor Vehicle Safety Standards(FMVSS) [24]. FMVSS contain specifics for multiple crashscenarios as well as for different classes of vehicles. Crashscenarios in these requirements include full frontal im-pact (FFI), offset frontal impact (OFI), side impact (SIDE),rollover or roof impact and rear impact.

ISSN: 1358-8265 print / ISSN: 1754-2111 onlineC© 2012 Taylor & Francis

http://dx.doi.org/10.1080/13588265.2011.648518http://www.tandfonline.com

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12

http://dx.doi.org/10.1080/13588265.2011.648518

http://www.tandfonline.com

260 A. Parrish et al.

Figure 1. Considerations for a lighter weight design.

Obtaining these values in FE simulations requires anaccurate crash occupant model set correctly in the vehiclemodel. Alternative responses must be chosen if an FEmodel with an occupant is not available. Internal energy ab-sorption, intrusion distance and peak acceleration measuredat selected locations are often used as substitutes. Theseresponses focus on the vehicle’s crash performance ratherthan occupant injury metrics. Fang et al. [8] used internalenergy absorption of selected components at two timesteps along with peak engine top acceleration. Liao et al.[15] used a combination of occupant- and vehicle-basedresponses including an integration of the deceleration curveand intrusion distance at the toeboard. Intrusion distance ofthe front panel for frontal impacts and intrusion distance ofthe door for side impacts were considered by Fang et al. [9].

Several studies have considered different approachesfor crashworthiness optimisation. For example, Akkermanet al. [3] used shape and sizing optimisation to improvecrashworthiness of an automotive instrument panel. Fanget al. [9] considered sizing optimisation of vehicle struc-tures whereas Rais-Rohani et al. [27] used shape and sizingoptimisation to improve crashworthiness of a vehicleby altering the geometry of the side rails. Alternatively,Mozumder et al. [20] examined the application of cellularautomata for topology-based crashworthiness optimisation.

A number of studies have used multi-objective formula-tion to solve crashworthiness optimisation problems. Fanget al. [8] improved crash performance while maintainingweight in a FFI whereas Fang et al. [9] improved crash per-formance considering multiple impact scenarios. In anotherstudy, Liao et al. [15] solved a two-step optimisation prob-lem minimising weight and vehicle crash responses beforeoptimising the occupant restraint system based on occupantinjury criteria.

Simulation runtime is a major obstacle in optimisationstudies. In many cases, hundreds or thousands of functionevaluations may be required for solving a design optimi-sation problem depending on the method selected as wellas the numbers of design variables and constraints used.The computational burden is considerably reduced whenthe objective and constraint responses are described by an-

alytical functions but the problem becomes significantlymore challenging when high-fidelity FE crash simulationsare required to obtain the response values. We have foundthat a single crash simulation using a full-vehicle modeltakes anywhere between one and a half and three hours, de-pending on the crash scenario, using 4 six-core Intel X5660processors with 48 GB of total RAM. Another importantconsideration is the non-smooth (noisy) behaviour of someof the crash responses that could pose problems in appli-cation of gradient-based optimisation methods. Hence, ap-proximate mathematical models or metamodels are neededto overcome the difficulties associated with computationalcost and noisy responses.

Metamodels or surrogate models that approximate func-tion values are widely used in automotive crashworthinessstudies [9, 8, 15, 30] to provide approximate responses at adesign point. Responses predicted using a metamodel willhave some error associated with them when compared tosimulation or test results at the same design point, but theseerrors can be reduced by selecting appropriate values forthe corresponding tuning parameters.

This study explores the application of AZ31 magne-sium alloy for vehicle body structures. Effects of materialsubstitution on crash responses at different vehicle sites arestudied considering three major crash scenarios (i.e. FFI,OFI and SIDE). Tension and compression stress–strain re-sponses are included to enhance the accuracy of the materialmodel. A maximum plastic strain criterion is used to iden-tify failed elements in FE simulations using LS-DYNA.

No previous study has been found comparing steel andmagnesium designs in full-scale automotive crash scenar-ios. This study provides an example of using equivalenttoughness as a way of determining substituted materialthickness with a magnesium material model based on sepa-rate stress–strain curves for tension and compression. Strainrate dependency in crash simulations is also explored. Se-lected results from designs with the steel and magnesiumstructures are compared. The study also introduces optimi-sation techniques to determine the weight savings of a mag-nesium design with crashworthiness similar to the heaviersteel design.

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12

International Journal of Crashworthiness 261

Figure 2. Dodge neon FE model (a) exterior view showing elements and (b) with exterior panels removed.

The remaining portion of the paper is organised as fol-lows: Section 2 introduces the FE model, crash scenariosand design responses. Section 3 discusses the magnesiumfor steel material substitution. Section 4 presents the meta-model construction. Section 5 shows the optimisation for-mulations and results, with concluding remarks and futurework appearing in Section 6.

2. Vehicle model and crash simulations

A full-scale 1996 Dodge Neon model, FE model (v07LS-DYNA) developed at the National Crash AnalysisCenter (NCAC) is used in this study. This model, shownin Figure 2, contains no interior panels or seats and ispublicly available from the NCAC website [23]. Moststructural components are made of steel defined usingMat 24 (piecewise linear plasticity) in LS-DYNA [17] andmodelled using Belytchko-Tsai reduced integration shellelements. The vehicle is made up of 270,768 elements in336 parts and the total vehicle mass is 1333 kg. This modelwas validated by NCAC for an FFI scenario [22].

This vehicle model is used for FFI, SIDE and OFI sce-narios by following the FMVSS for impact velocity andimpact location/angle. The FFI scenario models an impactinto a rigid wall at a speed of 56 km/h. Velocity for vali-dation and testing for SIDE was 52 km/h to coincide with

test data. The impact occurs at a 27◦ angle from an impactvehicle with honeycomb material simulating the front ofanother automobile. The OFI simulations were validated at60 km/h based on available test data and used for this studyat 56 km/h to coincide with FFI simulations. The Neonmodel impacts a honeycomb material barrier in front of arigid wall at 40% offset. Figure 3 shows the Neon model ineach crash simulation.

Simulation-based acceleration curves at the left rear sillfor OFI and at the middle of the B-pillar for SIDE are shownin Figure 4 and compared to physical test results [14, 19].These curves show that the general shapes of the curves arethe same but peak values may differ as a result of filteringand the method used to capture the data. A Butterworthfilter with a frequency of 60 Hz was used for the simulation-based results. The location of these observation points wasdetermined by the position of accelerometers in the actualtesting.

We identified 22 parts as shown in Figure 5 with sig-nificant contributions to energy absorption and structuralstiffness in our preliminary investigation. These parts ac-count for approximately 40% of the energy absorption inall three crash scenarios and have a mass of 105 kg or 8%of the vehicle mass at 1333 kg.

Responses considered include the intrusion distancesat the toeboard and dashboard for FFI and OFI and at the

Figure 3. Crash scenarios (a) FFI, (b) SIDE, and (c) OFI scenarios.

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12


Figure 4. Acceleration for model validation (a) OFI a-accel. at left rear sill and (b) SIDE y-accel. at middle b-pillar.

door for SIDE (Int Toe, Int Dash and Int Door), resul-tant acceleration at a location on the B-pillar in all threescenarios (Accel) and internal energy absorption of the se-lected parts in all three scenarios (Int Eng). These responseswere chosen because of their relevance to occupant safetyand the acceleration response location was chosen to benear the approximate head location of an occupant duringa crash. Mass is determined by dividing the initial mass ofthe baseline part by the initial thickness. This coefficient ismultiplied by a new thickness for that part to determine thepart’s new mass. Selected response locations can be seen inFigure 6.

Intrusion distance was calculated by measuring the dis-tance between 20 nodes at each response location and a ref-erence node on the opposite side of the car before and afterthe crash and then finding the difference. An average overthe 20 nodes was used as an intrusion distance response. AButterworth filter at 60 Hz was applied to the accelerationcurve at 20 nodes in each direction, the resultant found, anaverage over the 20 nodes taken and the maximum valuefound to represent an acceleration response. The internalenergy response is the sum of the internal energy at the endof the simulation for the selected parts.

Figure 5. Selected vehicle components and associated designvariables.

The simulated response values determined using themethods described will be referred to as the true or actualresponse in the remainder of this paper.

3. Material substitution

The goal of material substitution here is to replace some ofthe baseline steel parts of the Neon body with those madeof a lightweight magnesium alloy. Magnesium alloys arebeing seriously considered for auto body applications bythe automotive industry as a way of reducing the vehicleweight and improving the fuel economy. The manufacturingprocesses being considered for magnesium parts includesheet forming, extrusion and die-casting [32]. In this paper,we are assuming that the parts are made of sheet formedwith AZ31 magnesium alloy.

Experimental studies show that magnesium alloysdemonstrate different behaviours under tension, compres-sion and shear, and that the failure points for tension andcompression are different due to anisotropic plastic defor-mation. Magnesium alloys also show considerable sensitiv-ity to strain rate.

In a vehicle crash, plastic deformation is consideredas the major mechanism for energy absorption in ductilemetallic parts. Studies at the component level show that thelocalised regions in the deformed component experiencedifferent load paths [26]. In this study, Material Type 124(MAT 124) in LS-DYNA [17] is used to model the materialbehaviour of magnesium parts based on the work by Wagneret al. [34]. This material model is capable of making adistinction between tension and compression behaviour byconsidering the piecewise linear isotropic hardening model.

The strain rate sensitivity is considered throughCowper-Symonds model that scales the instantaneous yieldstress as:

σ

σ0= 1 +

(ε

C

)1/p

(1)

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12


Figure 6. Selected response locations.

where σ and σ 0 are the von-Misses stress at equivalentstrain rate ε and quasi-static yield stress, respectively. Thestrain rate sensitivity parameters p and C for tension andcompression are assumed to be identical with C = 24,124and p = 3.09 for AZ31 [21].

The stress–strain curves for AZ31 under tensile loadingwere obtained from experimental data. Since similar datawas unavailable for AZ31 under compressive loading,the ratio of AM30 magnesium alloy extrusion undercompression and tension along with AZ31 tension datawere used to approximate the stress–strain curve for AZ31compression as:

AZ31(RD)comp = AZ31 (RD)tenAM30 (ED)ten

∗ AM30(ED)comp

(2)

where RD and ED represent the rolling and extruded di-rections, respectively, under tension (ten) and compression(comp). AZ31 properties in the transverse direction (TD)for tension and AZ31 (RD) for compression were usedto define the MAT 124 material card. Figure 7 shows thestress–strain curves for AM30 and AZ31.

Material failure in the model is captured through fail-ure strain limit, which is assumed to be identical for bothtension and compression. However, in the material used in

this study, tensile failure strain is higher than compressivevalue. In order to model this difference, a softening por-tion is added to the compression curve. By having threeintegration points through the shell thickness, the energyabsorption as a result of bending can be captured well byusing different curves for the region of the shell that is intension or compression. In using MAT 124, we are exclud-ing the presence and effect of anisotropic texture.

Material substitution was performed by changing the22 parts from steel to magnesium and leaving the otherparts as is in the original model. Whereas seven differentsteel materials are used to define the 22 selected parts inthe original Neon model, the same magnesium material isused for all of the parts. To establish a baseline model ofNeon with magnesium parts, the thicknesses of selectedparts were adjusted so as to maintain the same total internalenergy absorption as the original steel parts. This was doneby using the relationship between toughness and thicknessgiven as:

tMg = tStTSt

TMg(3)

where tMg and tSt are the thicknesses of magnesium andsteel parts, respectively, and TMg and TSt are the toughnessvalues of steel and magnesium derived from integrating the

Figure 7. Magnesium quasi-static stress-strain curves at 22◦C.

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12


Figure 8. Tensile loading on the single element model.

stress–strain curves for each material from 0 to 0.3 strain.An average of the thicknesses found using the toughnessunder compression and the toughness under tension wasused. Thicknesses of the selected parts are found inTable 1 for the steel and magnesium designs.

To examine the importance of strain rate dependencyfor AZ31 with MAT 124 representation, the deformationresponse of a single shell element under tension and com-pression was examined. To simulate a constant strain rateloading, a prescribed displacement was applied on one edgeof the element while the other edge was held fixed as shownin Figure 8. The displacement was applied by consideringthe actual time step size of the model using the relationship:

x = xl0(exp(ε(t +�t) − 1) (4)

where �t is some fraction of the solution time step and xand xl0 are the displacement and initial element length, re-spectively. Therefore, the amount of applied displacement isvaried within each time step increment. Figure 9 shows thestress–strain response of a single element at different strainrates for tensile and compressive loading conditions. Thismaterial model also allows the definition of plastic strainto failure, which is important in modelling the response ofparts made of magnesium alloy.

To examine the importance of including strain rate sen-sitivity parameters for magnesium alloy parts of the Neonmodel, the effective value of the strain rate tensor at eachintegration point for all of 22 parts were evaluated with the

Table 1. Part thicknesses (in mm) for baseline designs.

Part Design variable St base Mg base

A-pillar x1 1.611 2.597Front bump x2 1.956 5.975Firewall x3 0.735 1.072Front floor panel x4 0.705 1.136Rear cabin floor x5 0.706 1.138Outer cabin x6 0.829 1.366Cabin seat reinf. x7 0.682 1.099Cabin mid rail x8 1.050 1.692Shotgun x9 1.524 3.620Inner side rail x10 1.895 3.966Outer side rail x11 1.522 3.186Side rail exten. x12 1.895 3.966Rear plate x13 0.710 1.144Roof x14 0.702 1.157Susp. frame x15 2.606 5.342

results shown in Table 2. For parts that have left and rightcomponents such as the side rail, only that with the largestmeasured strain rate is shown in Table 2. The strain ratechanges during the crash simulation; therefore, the maxi-mum values are checked throughout the crash duration at0.005 s time intervals for total of 32 points for each elementintegration point. It is seen that the maximum value of effec-tive strain rate varies from 12/s to 447/s within the differentparts considered in these simulations. However, the averageof the maximum strain rates in the elements located in thelocalised regions are below 10/s. This shows that the strainrate dependency of the material can be neglected in the carcrash scenarios considered here. Table 2 also lists the partsthat experienced failure based on the failure strain limit of38% as defined in the material card. It was observed thatthe baseline model experiences a severe failure in the outercabin components. Additionally, the failure is affected bythe crash scenario. For instance, suspension frame has somefailed elements in OFI whereas it does not fail in the SIDEor FFI cases.

Table 3 compares the total internal energy absorptionof the selected parts for each crash scenario using 19%

Figure 9. Material behaviour (a) under tensile load and (b) under compressive load for single element model.

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12


Table 2. Effective strain rate and the element failure observed in the base line model with magnesium material.

Maximum effective strainrate (1/s)

Average of maximum effectivestrain rate (1/s) Element failure

Part FFI OFI SIDE FFI OFI SIDE FFI OFI SIDE

A-pillar 73 103 38 3 12 4 No No NoFront bump 142 103 0.5 6 4 N/A No No NoFirewall 269 137 105 9 15 0.2 Yes Yes NoFront floor panel 311 231 158 15 11 2 Yes Yes YesRear cabin floor 53 44 119 2 4 5 No No YesOuter cabin 337 223 347 6 15 35 Yes Yes YesCabin seat reinf 12 11 41 2 1 3 No No NoCabin mid rail 142 86 17 3 1 3 No No NoShotgun 145 110 46 8 15 10 No No NoInner side rail 166 216 49 6 10 0.1 Yes Yes NoOuter side rail 132 98 22 5 6 0.2 Yes Yes NoSide rail exten. 113 24 57 3 7 0.001 Yes No NoRear plate 447 9 118 7 0.5 4 No No NoRoof 39 45 48 0.5 4 2 No No NoSusp. frame 117 39 18 3 0.1 0.5 Yes No No

and 38% plastic strain to failure for magnesium parts andthe baseline steel parts. This table shows that magnesiumwith 38% plastic strain to failure has total internal energyabsorption closer to the steel baseline than the 19%.

Simulations of each crash scenario using the mag-nesium material model defined above were performed.Figure 10 shows that the acceleration curves at the B-pillarresponse location for each scenario generally match inshape and peak values. Table 4 shows an overview of theresponses for baseline simulations with both materials; seeFigure 5 for a labelled picture of the parts. The magnesiumparts have significantly less mass than the steel partsand generally maintain or improve the internal energyabsorption of the replaced parts but the intrusion distancesare significantly larger than the steel parts. These responsesand designs represent the baseline models for comparisonwith the optimum designs found later.

4. Response approximation

Response approximation is necessary in crashworthinessoptimisation as FE crash simulations tend to be verycomputationally expensive and many responses havenon-smooth (noisy) behaviour. Surrogate models in theform of analytical functions can be developed using a

Table 3. Total internal energy of selected parts for steel andmagnesium at 19% and 38% plastic strain to failure.

St Mg 19% Mg 38%

FFI 6.23E + 07 6.03E + 07 6.36E + 07OFI 3.94E + 07 3.69E + 07 3.94E + 07SIDE 2.24E + 07 1.94E + 07 2.09E + 07

number of metamodelling techniques. A metamodel isbuilt using data sets obtained through FE simulations orphysical tests at the training points selected using a designof experiments (DOE) model. Two DOE types frequentlyused for crashworthiness optimisation are Taguchi orthog-onal arrays [9, 8] and Latin hypercube sampling (LHS)[15, 25]. In this study we chose LHS because it provides amore uniform sampling of the design space based on thespecified bounds of the design variables.

The number of training points depends on severalfactors. These include the problem type, number of vari-ables, metamodel technique and the level of effort (cost)involved in obtaining the desired responses at each trainingpoint. Both Yang et al. [36], using an impact example, andFang and Wang [10], using analytic benchmark problems,

Table 4. Comparison of baseline steel and baseline magnesiumdesigns.

St Mg Mg relative to St

FFIInt toe (mm) 157 295 87.9%Int dash (mm) 122 186 52.3%Accel (g’s) 63.5 49.2 −22.5%Int eng (kJ) 62.3 62.3 0.1%

SIDEInt door (mm) 314 420 33.7%Accel (g’s) 47.9 40.8 −14.9%Int eng (kJ) 22.4 21.4 −4.3%

OFIInt toe (mm) 273 349 27.7%Int dash (mm) 247 386 56.2%Accel (g’s) 35.0 36.2 3.3%Int eng (kJ) 39.4 39.2 −0.4%Mass (kg) 105.2 42.7 −59.4%

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12


Figure 10. Acceleration at b-pillar response location (a) x-dir. FFI, (b), y-dir. SIDE, and (c) x-dir. OFI.

show that metamodels are generally more accurate as thenumber of training points increases but this trend is nottrue for all functions and metamodel types. In their study,Wang et al. [35] examined response functions of low tohigh nonlinearity consisting of 1–101 input variables using7–527 training points, and found that the required numberof response samples for metamodel accuracy depends inlarge part on the response considered and not so much onthe number of input variables. For example, they could notfind a good approximation to a two-dimensional responseeven after increasing the number of training points to1000, whereas in some other examples, they were ableto generate very good surrogates by using three times asmany training points as the input variables.

Different metamodelling techniques have been devel-oped with different levels of complexity and accuracy.Turner [31] shows metamodels divided into three maingroups. These groups are: Geometric, containing responsesurface models and spline-based models; Stochastic,containing Kriging models and radial basis functions; andHeuristic, containing kernel model, frequency domainmethods and neural networks. Jin et al. [13] and Wanget al. [35] compared several metamodelling techniquesusing metrics such as accuracy, robustness, efficiency,transparency, flexibility and conceptual simplicity for

benchmark as well as more industry relevant exampleproblems. Metamodel techniques explored in this study in-cluded Polynomial Response Surface (PRS), Radial BasisFunction (RBF), Kriging (KR), Support Vector Regression(SVR) and Gaussian Process (GP). An optimised ensemble(EN) containing multiple metamodels developed by Acarand Rais-Rohani [1] was created using the five modelslisted above. This technique was used previously in Acarand Solanki [2] for a crashworthiness problem and shownto be more accurate than stand-alone metamodels.

With the thicknesses of selected parts as input variables,45 training points were found in the range of ±50% of thebaseline thicknesses. The baseline design point was addedfor a total of 46 design points. Due to similarity of the partin the right and left sides of the vehicle model, only 15thickness values are needed for the 22 parts. Three separate(FFI, OFI, SIDE) LS-DYNA simulations were performedat each design point to provide the responses necessary formetamodel construction.

Stand-alone metamodels for each response weredeveloped and some (i.e. RBF, KR and SVR) were tunedfor maximum accuracy before being used to construct anoptimised ensemble of metamodels. The mathematicaldescription of each metamodel is provided in Appendix1 for completeness. The models were tuned by selecting

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12


Table 5. Metamodel tuning parameters for steel design.

FFI SIDE OFI

ScenarioResponse Int toe Int dash Accel Int eng Int door Accel Int eng Int toe Int dash Accel Int eng

RBFc 1 1 0.2 1 1 0.05 0.05 1 0.95 0.25 0.75� M M TP M M M G IM M TP M

KRUB, θ 0.1 0.011 1 1 0.1 0.011 0.1 0.011 0.1 1 0.1LB, θ 0.01 0.001 0.001 0.001 0.01 0.001 0.01 0.001 0.001 0.01 0.01C S Sp E L G L G Sp G S SRPD 0 1 0 1 0 1 0 0 1 0 0

SVRc 1 1 100 10 0.01 10 10 10 1 10 1P 10 2 0.1 2 10 2 5 5 5 0.1 5

Note: M = multiquadric; TP = thin plate; G = Guassian; IM = inverse multiquadric; UB = upper bound; LB = lower bound; C = correlation function;S = spline; E = exponential; L = linear; Sp = spherical; RPD = regression polynomial degree; P = penalty parameter.

the parameter or combination of parameters that producedthe least error for each response for each material.Cross-validation generalised mean square error (GMSE)was used as the error metric. A metamodel is created usingall except one design point and the predicted responseis compared to the actual response at that design pointto measure error. This process is repeated for all designpoints and the average is used as the overall error of themetamodel. GMSE was calculated using:

GMSE ≡ 1

N

N∑i=1

(fi − fi)2 (5)

where N is the number of design points, fi is the actualresponse and fi is the response predicted by the model.

Tables 5 and 6 show the tuned metamodel parameters forsteel and magnesium, respectively, and Table 7 shows theoptimised weight factors for the ensemble.

Normalised GMSE for steel and magnesium can befound in Tables 8 and 9, respectively. The optimised ensem-ble metamodel was then used to approximate the responsesat the baseline design point and compared to the simulationresponse value at the initial design for both materials. Thisserved as a baseline error when evaluating the accuracy ofeach optimum design.

The maximum per cent error for the steel baseline oc-curred for the intrusion at the toeboard for OFI with 15%error. This was followed by 9% error for intrusion at thedashboard for OFI and 5% error for intrusion at the toe-board for FFI. The remaining responses for steel had lessthan 2% error at the baseline design when predicted with

Table 6. Metamodel tuning parameters for magnesium design.

FFI SIDE OFI

ScenarioResponse Int toe Int dash Accel Int eng Int door Accel Int eng Int toe Int dash Accel Int eng

RBFc 0.05 0.05 0.2 0.05 1 1 1 1 0.15 0.05 1� G G M G M IM M M G M M

KRUB, θ 0.011 0.1 0.011 0.011 0.011 0.1 0.011 1 0.1 1 1LB, θ 0.01 0.001 0.001 0.001 0.001 0.01 0.001 0.001 0.001 0.01 0.001C Cu G S E L S Cu Sp S S CuRPD 0 0 0 0 1 0 1 1 0 0 1

SVRc 1 1 10 0.01 10 0.01 10 10 0.01 1 10P 5 10 0.1 2 2 0.1 0.1 10 10 0.1 2

Note: M = multiquadric; G = Guassian; IM = inverse multiquadric; UB = upper bound; LB = lower bound; C = correlation function; S = spline; E =exponential; L = linear; Cu = cubic; Sp = spherical; RPD = regression polynomial degree; P = penalty parameter.

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12


Table 7. Optimised ensemble metamodel weight factors.

FFI SIDE OFI

Int toe Int dash Accel Int eng Int door Accel Int eng Int toe Int dash Accel Int eng

PRS 0 0 0 0 0 0 0 0 0.56 0 0GP 0.57 0.17 0 0.33 0.48 0 0 0.51 0 0 0RBF 0.30 0 0 0 0.01 0 0 0 0 0.16 0KR 0.13 0.59 0.41 0.54 0.11 0.57 0.59 0.14 0.44 0.54 0.58SVR 0 0.24 0.59 0.13 0.40 0.43 0.41 0.35 0 0.30 0.42PRS 0 0 0 0 0 0 0 0.01 0 0 0GP 0 0 0 0 0.34 0 0.14 0 0.07 0.27 0.15RBF 0.33 0 0 0.09 0.08 0 0 0 0 0 0KR 0.67 1.00 0.30 0.36 0.55 1.00 0.86 0.57 0.56 0.10 0.31SVR 0 0 0.70 0.55 0.03 0 0 0.42 0.37 0.63 0.55

Table 8. Normalised GMSE for steel design.

FFI SIDE OFI


PRS 1.60 1.41 1.64 1.12 1.56 1.53 1.49 1.31 1.04 2.10 1.79GP 1.12 2.09 1.57 1.29 1.09 2.26 2.31 1.09 1.49 1.35 1.95RBF 1.39 1.75 1.32 2.14 1.39 1.60 1.46 1.51 1.26 1.13 1.44KR 1.13 1.15 1.00 1.12 1.18 1.06 1.12 1.22 1.04 1.05 1.01SVR 1.34 1.20 1.19 1.13 1.34 1.15 1.19 1.24 1.07 1.31 1.27ENS 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

the ensemble. The maximum per cent error for the magne-sium baseline was 4% for the acceleration at mid B-pillarfor FFI. The remaining magnesium responses had less than2% error.

5. Crashworthiness optimisation problems

5.1. Single-objective

With mass as the objective function, both the steel andmagnesium models are optimised such that the constrainedresponses do not exceed the corresponding (material spe-cific) baseline values. The single-objective (SO) nonlinearconstrained minimisation problem is formulated as:

min F (x)

s.t.Rj (x) ≤ Rjbase ; j = 1..8Rj (x) ≥ Rjbase ; j = 9..11

0.5xbase ≤ x ≤ 1.5xbase

(6)

where F(x) is the objective function, x is the vector of 15 de-sign variables (part thicknesses), Rj (x) is the jth response,and Rjbase is the jth response of the baseline model (materialspecific) as described by the corresponding optimised en-semble metamodel. In Equation (6), Rj, j = 1..8 representsthe intrusion distances at the toeboard and dashboard (IntToe, Int Dash) for FFI and OFI and at the door (Int Door)for SIDE and resultant acceleration (Accel) at a locationon the B-pillar in all three scenarios, whereas Rj, j = 9..11represents the internal energy (Int Eng) responses of theselected parts in all three scenarios.

The optimisation problem in Equation (6) was solvedusing the Sequential Quadratic Programming (SQP)method implemented in the VisualDOC [33] software.Among the optimisation methods available in VisualDOC,we chose SQP because of its efficiency and accuracy as alocal optimiser. However, to expand the search of feasibledesign space, multiple (i.e. eight) initial design points were

Table 9. Normalised GMSE for magnesium design.

FFI SIDE OFI


PRS 1.73 1.84 1.50 1.70 1.52 1.07 2.07 1.48 1.91 3.00 1.07GP 2.71 1.58 1.77 1.84 2.15 1.84 1.38 1.50 2.59 2.44 1.64RBF 1.30 1.54 1.31 1.59 1.34 1.32 1.65 1.25 1.40 3.33 1.53KR 1.00 1.01 1.25 1.04 1.00 1.06 1.08 1.13 1.11 1.05 1.06SVR 1.55 1.61 1.05 1.37 1.26 1.03 1.49 1.16 1.12 2.85 1.07ENS 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12


Table 10. Metamodel prediction errors relative to LS-DYNAresults at the SO optimum.

St Mg

FFIInt toe 4.1% 4.9%Int dash 13.8% 1.2%Accel 14.9% 3.0%Int eng 3.1% 1.6%

SIDEInt door 4.4% 2.2%Accel 15.8% 1.9%Int eng 2.9% 0.1%

OFIInt toe 5.1% 10.1%Int dash 24.1% 6.7%Accel 0.4% 23.3%Int eng 0.9% 1.8%

used and results are shown only for the best case. We chose alimited number of initial design points to explore the impactof non-convex design space on the optimum design as thesearch for global optimum would require a wider array ofinitial design points. The initial design points were selectedat random from different regions of the design space.

In the steel design, mass of designed parts droppedfrom 105.2 kg to 88.0 kg while in the magnesium design,the same mass dropped from 42.7 kg to 37.2 kg. Thatis a 16% and 13% reduction for steel and magnesium,respectively, relative to the corresponding baseline models.Metamodel-based responses at the optimum design pointare compared to the LS-DYNA simulation results inTable 10, where the per cent error is that of the metamodelpredictions relative to the simulation results. For the steeldesign, the relative errors are in the range of 0.4% to 24.1%for an average of 8.1% while those for the magnesiumdesign vary from 0.1% to 23.3% for an average of 5.2%.

Table 11. Comparison of surrogate-based crash responses at theSO optimum for St and Mg.

St SOopt

Relative toSt base

MgSO opt

Relative toMg base

FFIInt toe (mm) 163 −1.1% 261 −11.7%Int dash (mm) 123 −0.1% 165 −11.1%Accel (g’s) 59 −1.4% 51 0.4%Int eng (kJ) 64 1.9% 61 −1.2%

SIDEInt door (mm) 319 0.1% 423 0.8%Accel (g’s) 45 −7.6% 40 −1.0%Int eng (kJ) 24 7.8% 21 0.3%

OFIInt toe (mm) 221 −5.1% 352 0.1%Int dash (mm) 205 −11.5% 268 −28.0%Accel (g’s) 36 1.29% 38 −0.7%Int eng (kJ) 40 −0.4% 39 0.0%Mass (kg) 88 −16.4% 37.2 −13.0%

It took total of 2483 function calls and nine optimisationcycles to find the optimum point for the steel designwhereas for the magnesium design, 2095 function callswere used for the same number of optimisation cycles.An optimisation cycle refers to a complete solution of thequadratic (direction-finding) subproblem and associatedstep size calculation in SQP.

Figure 11 shows the normalised design variables at theSO optimums. All except two parts (outer cabin and outerside rail) saw their thickness decrease for the steel opti-mum design while for the magnesium optimum, nine partsdecreased in thickness while the rest became thicker. It isinteresting to note that the variations in part thickness arequite different in the steel and magnesium optimum designs.Surrogate-based response values at the SO optimum pointsfor both steel and magnesium can be found in Table 11

Figure 11. Normalised design variables for SO optimums (a) steel and (b) magnesium.

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12


along with a per cent difference relative to their respectiveconstraint baselines. A per cent decrease for intrusion dis-tance, acceleration and mass along with a per cent increasefor internal energy is favourable. A constraint violationtolerance of 1.5% was defined when solving the optimisa-tion problem. The intrusion distances using the magnesiumparts are significantly larger than their steel counterparts.The accelerations are lower or about the same for magne-sium compared to steel while internal energy absorption isslightly less. This difference is understandable as the mag-nesium design was optimised based on its correspondingbaseline model and not that of the steel design.

5.2. Multi-objective

Results in the previous section suggest that the lighter mag-nesium designs, baseline and SO optimum, maintain orimprove upon the steel baseline responses for accelerationand internal energy but exceed the steel baseline intrusiondistances. For this reason, another optimisation problem isformulated and solved to determine the mass of a mag-nesium design that meets the crashworthiness properties ofthe steel baseline. The goal here is to find the weight savingsoffered by the lightweight magnesium to find a design withsimilar crashworthiness characteristics as the steel base-line design without focusing on alternative combinationsof responses to optimise or constrain.

This optimisation problem uses the baseline steel in-trusion distances as target values while constraining theacceleration and internal energy to the steel baseline val-ues. This multi-objective (MO) constrained optimisationproblem is formulated as:

s.t.

min F (x) = f (Rj (x) , j = 1..5)Rj (x) ≤ RjSt ; j = 6..8Rj (x) ≥ RjSt ; j = 9..110.5xMg ≤ x ≤ 1.5xMg

(7)

Table 12. Metamodel prediction errors relative to LS-DYNAresults at the MO optimum.

Metamodel LS-DYNA Error

FFIInt toe (mm) 166 186 −10.6%Int dash (mm) 136 131 4.4%Accel (g’s) 52.0 59.3 −12.3%Int eng (kJ) 65.3 62.0 5.2%

SIDEInt door (mm) 328 361 −9.1%Accel (g’s) 44.2 48.0 −8.0%Int eng (kJ) 22.1 21.9 0.8%

OFIInt toe (mm) 256 212 20.7%Int dash (mm) 247 236 4.7%Accel (g’s) 35.0 37.0 −5.4%Int eng (kJ) 39.0 37.9 2.7%

Figure 12. Normalised design variables for MO optimum.

where F(x) is the vector of objective functions representingthe differences between the intrusion distances of the mag-nesium design and the corresponding values in the baselinesteel model, RjSt for j = 6..8 are the baseline steel accel-erations, RjSt for j = 9..11 are the baseline steel internalenergies and xMg are the baseline magnesium part thick-ness values.

Several approaches may be used to solve a constrainedMO problem such as the one in Equation (7). Due to the con-flict that often exists among the different objectives, thereare multiple optimal solutions that fall on the Pareto fron-tier. Depending on the level of preference (weight) givento an individual objective, a different point on the Paretofrontier may be selected as the desired Pareto optimum de-sign. In this problem, the Pareto frontier would be definedin a five-dimensional space. Given the nonlinearity of theconstraint functions involved, we chose compromise pro-gramming formulation to combine the multiple objectivesinto a single composite objective function expressed as:

F (x) =

√√√√√ 5∑j=1

Wj

[fj (x) − f Tj

f Wj − f Tj

]2

, j = 1..5 (8)

Figure 13. Intrusion distance at the baselines and MO optimum(F is FFI, S is SIDE and O is OFI).

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12


Table 13. Comparison of surrogate-based responses at MO optimum to St constraints.

FFI SIDE OFI

Accel Int eng Accel Int eng Accel Int eng Mass

Mg MO 52 g’s 65 kJ 44 g’s 22 kJ 35 g’s 39 kJ 50.7kg

Relative to St base −12.6% 4.5% −8.6% −0.4% −2.8% −0.1% −51.8%

Table 14. Design variable summary (thicknesses in mm).

Part Part no. Design variable St base Mg base St SO Mg SO Mg MO

A-pillar 310, 311 x1 1.611 2.597 0.915 2.561 1.984Front bump 330 x2 1.956 5.975 1.204 2.987 6.649Firewall 352 x3 0.735 1.072 0.545 0.867 1.515Front floor panel 353 x4 0.705 1.136 0.612 1.211 1.592Rear cabin floor 354 x5 0.706 1.138 0.477 0.569 1.696Outer cabin 355, 356 x6 0.829 1.366 0.988 1.482 2.049Cabin seat reinf 357 x7 0.682 1.099 0.622 1.649 1.649Cabin mid rail 358, 359 x8 1.050 1.692 0.633 1.792 1.636Shotgun 373, 374 x9 1.524 3.620 0.762 1.810 1.810Inner side rail 389, 391 x10 1.895 3.966 1.887 3.436 4.141Outer side rail 390, 392 x11 1.522 3.186 1.834 3.145 2.754Side rail exten. 398, 399 x12 1.895 3.966 1.780 4.805 5.950Rear plate 415 x13 0.710 1.144 0.417 1.559 1.717Roof 416 x14 0.702 1.157 0.351 0.739 0.791Susp. frame 439 x15 2.606 5.342 1.303 4.367 4.931

where Wj is the weight factor for the jth objective, fj (x) isthe jth objective, f Tj is the target value of the jth objectivetaken to be equal to the corresponding response value

in the steel baseline design (RjSt ) and fWj is the worstknown value of the jth objective taken to be equal to thecorresponding response value in the magnesium baseline

Figure 14. Optimisation response results summary: (a) accelerations and internal energies, and (b) intrusion distances. (F is FFI, O isOFI, and S is SIDE)

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12


Figure 15. Simulation images of the designs: FFI at 150 ms for (a) St Base, (b) Mg Base, (c) St SO, (d) Mg SO, and (e) Mg MO.

design (RjMg ). Given the importance of all intrusiondistances to vehicle safety, we chose equal weight factorsfor all the objectives in Equation (8). As in the case ofSO, all responses are represented by the correspondingoptimum ensemble metamodels.

The optimisation problem in Equation (8) is solved us-ing SQP in VisualDOC software with eight different ini-tial design points. The crash responses predicted by themetamodels and the corresponding LS-DYNA values atthe MO optimum design point are shown in Table 12. Themetamodel errors are generally less than 10% with onlytwo responses above 11%. The MO optimum design was

found in nine optimisation cycles with total of 2909 functioncalls.

Figure 12 shows the normalised optimum design vari-able values with five reaching the upper bound, one at thelower bound and the rest scattered between the two bounds.It appears that the parts defined by design variables x5, x6,x7, x12 and x13 play a more critical role in this MO optimi-sation problem than the rest. The chart in Figure 13 showsthat the intrusion distance responses at the MO optimumare considerably less than those of the magnesium baselinemodel and most are near to or less than those of the steelbaseline design with IntToe of FFI and IntDoor of SIDE

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12


Figure 16. Simulation images of the designs: SIDE at 150 ms for (a) St Base, (b) Mg Base, (c) St SO, (d) Mg SO, and (e) Mg MO.

being about 16–18% higher. The values shown in Figure 13are the LS-DYNA simulation results at the MO optimumrather than the metamodel predictions. This MO optimumdesign with magnesium parts has a mass that is about half ofthat of the baseline steel design, 50.7 kg compared to 105.2kg, and is 8 kg heavier than the baseline magnesium design.

Surrogate-based results of the other responses of inter-est are found in Table 13 along with a percentage com-parison to the steel baseline constraint values. A per centdecrease for intrusion distance, acceleration and mass alongwith a per cent increase for internal energy is favourable.A constraint violation tolerance of 1.5% was defined whensolving the optimisation problem. Table 13 shows that theMO optimum design produces better results for the accel-

eration constraints and FFI internal energy response witha minimal reduction in the OFI and SIDE internal energyresponses.

5.3. Optimisation results summary

Results show that the magnesium designs have lower massthan the steel designs and similar or better crashworthi-ness when considering acceleration and internal energy asdesign constraints. The all steel model was 17.2 kg lighterwhen optimised for mass than the steel baseline model. Theoptimised model with selected parts replaced with magne-sium was 54.5 kg lighter than the steel baseline and 37.3 kglighter than the optimised steel model with similar crash

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12


Figure 17. Simulation images of the designs: OFI at 150 ms for (a) St Base, (b) Mg Base, (c) St SO, (d) Mg SO, and (e) Mg MO.

responses. The LS-DYNA based response values at thepoint of optimum for SO and MO problems are summarisedin Figure 14.

Design variable values representing wall thickness ofindividual parts in mm at the baseline and optimised de-signs can be found in Table 14. The shotgun wall thicknesswas reduced to the lower bound in all three optimisationproblems solved. This suggests that this part is at its en-ergy absorption limit and altering the thickness within thesebounds does not alter the absorption significantly. Resultsshow that magnesium for steel substitution based on onlymaintaining energy absorption does not produce a designthat satisfies the other crashworthiness characteristics. De-sign optimisation is needed to find a magnesium design

meeting steel crashworthiness. Figures 15, 16 and 17 showsimulation images of the five designs for FFI, SIDE andOFI, respectively. Differences can be observed in the doorfor the FFI scenario, the door and roof for SIDE, and theroof for OFI. The overall crash response of the magnesiumat the point of MO optimum appears to be fairly close tothat of the steel baseline model.

6. Summary and conclusions

This study used metamodelling and optimisation techniquesto explore the application of a lightweight magnesium alloyfor a group of energy absorbing parts in a vehicle bodystructure.

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12


A full-scale Dodge Neon model developed and vali-dated for FFI at the NCAC was incorporated into SIDE andOFI scenarios. Vehicle-based responses from these threescenarios along with selected part mass were used as con-straints and objectives during design optimisation with partthicknesses as the design variables. All crash simulationswere performed using LS-DYNA finite element solver.

AZ31 magnesium alloy was used to replace the base-line steel material in the FE model. Due to the complex-ity of the failure characteristics of magnesium, a limit onmaximum plastic strain was used as a failure criterion todisable failed elements. This material substitution coupledwith design optimisation reduced the mass of the selectedparts by approximately 50% of the baseline steel model andmaintained or improved the crashworthiness for most of thefactors considered.

A direct substitution of magnesium for steel using thick-nesses defined by maintaining the internal energy absorp-tion does not give the same crash characteristics or re-sponses as the steel components. Material substitution canchange the deformation mode and folding mechanism ofenergy absorbing parts under crash loading as indicated bysimilar internal energies but significantly larger intrusiondistances for the models using steel and magnesium parts.

Shotgun part thickness was reduced to the lower boundin all of the optimum designs. This suggests that this part isat its energy-absorbing limit for these bounds and increas-ing part thickness does not necessarily increase energyabsorption. Parts such as the roof were chosen as designparts primarily because of their contributions to vehiclestiffness. Roof thickness was significantly reduced in allof the designs because of its small contribution to energyabsorption. Inclusion of vehicle stiffness responses duringoptimisation would likely show less weight reduction inthe optimised designs.

Future work will consider inclusion of vehicle stiffness(as a measure of overall rigidity as well as NVH charac-teristic of the vehicle) in the design optimisation process,including a dummy model to provide dummy-based re-sponses, adding a microstructure-based material model toprovide a more representative model of magnesium, andusing alternative optimisation strategies.

AcknowledgementsThis material is based on the work supported by the U.S. Depart-ment of Energy under Award Number DE-EE0002323.

DisclaimerThis report was prepared as an account of work sponsored byan agency of the United States Government. Neither the UnitedStates Government nor any agency thereof, nor any of their em-ployees, makes any warranty, express or implied, or assumes anylegal liability or responsibility for the accuracy, completeness,or usefulness of any information, apparatus, product, or processdisclosed, or represents that its use would not infringe privatelyowned rights. Reference herein to any specific commercial prod-

uct, process, or service by trade name, trademark, manufacturer,or otherwise does not necessarily constitute or imply its endorse-ment, recommendation, or favouring by the United States Gov-ernment or any agency thereof. The views and opinions of authorsexpressed herein do not necessarily state or reflect those of theUnited States Government or any agency thereof.

References[1] E. Acar and M. Rais-Rohani, Ensemble of metamodels with

optimised weight factors, Struct. Multidiscip. Optim. 37(2008), pp. 279–294.

[2] E. Acar and K. Solanki, Improving the accuracy of vehiclecrashworthiness response predictions using an ensemble ofmetamodels, Int. J. Crashworthiness 14 (2009), pp. 49–61.

[3] A. Akkerman, M. Burger, B. Kuhn, H. Rajic, N. Stander,and R. Thyagarajan, Shape optimisation of instrument panelcomponents for crashworthiness using distributed comput-ing, Proceedings of the 6th International LS-DYNA Confer-ence, Detroit, MI, 2000.

[4] C.M. Bishop, Neural Networks for Pattern Recognition, Ox-ford University Press, New York, 1995.

[5] V. Cherkassky and Y. Ma, Practical selection of SVM pa-rameters and noise estimation for SVM regression, NeuralNetw. 17 (2004), pp. 113–126.

[6] S. Clarke, J. Griebsch, and T. Simpson, Analysis of sup-port vector regression for approximation of complex en-gineering analyses, J. Mech. Des. 127 (2005), pp. 1077–1087.

[7] EERE, Annual progress reports, Vehicle Technologies Pro-gram, U.S. Department of Energy, Washington, DC, (2011).Available at http://www1.eere.energy.gov/vehiclesandfuels/resources/fcvt_reports.html.

[8] H. Fang, M. Rais-Rohani, Z. Liu, and M.F. Horstemeyer,A comparative study of metamodeling methods for multi-objective crashworthiness optimisation, Comput. Struct. 83(2005), pp. 2121–2136.

[9] H. Fang, K. Solanki, M.F. Horstemeyer, and M. Rais-Rohani,Multi-impact crashworthiness optimisation with full-scalefinite element simulations, Proceedings of the 6th WorldCongress of Computational Mechanics, Beijing, China,2004.

[10] H. Fang and Q. Wang, On the effectiveness of assessingmodel accuracy at design points for radial basis functions,Commun. Numer. Meth. Eng. 24 (2008), pp. 219–235.

[11] S.R. Gunn, Support vector machines for classification andregression, Technical Report, University of Southampton,Southampton, 1997.

[12] C.W. Hsu, C.C. Chang, and C.J. Lin, A Practical Guide toSupport Vector Classification, National Taiwan University,Taipei, 2010.

[13] R. Jin, W. Chen, and T.W. Simpson, Comparative studies ofmetamodeling techniques under multiple modeling criteria,Struct. Multidiscip. Optim. 23 (2001), pp. 1–13.

[14] Karco Engineering, National Highway Traffic Safety Ad-ministration Frontal Barrier Forty Percent Offset ImpactTest: 1996 Dodge Neon, U.S. Department of Transportation,Washington, DC, 1997.

[15] X. Liao, Q. Li, X. Yang, W. Li, and W. Zhang, A two-stagemulti-objective optimisation of vehicle crashworthiness un-der frontal impact, Int. J. Crashworthiness 13 (2008), pp.279–288.

[16] S.N. Lophaven, H.B. Nielsen, and J. Søndergaard, DACE: AMATLAB Kriging Toolbox, Informatics and MathematicalModeling, Technical University of Denmark, Lyngby, 2002.

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12

http://www1.eere.energy.gov/vehiclesandfuels/resources/fcvt_reports.html

http://www1.eere.energy.gov/vehiclesandfuels/resources/fcvt_reports.html


[17] LSTC, LS-DYNA Keyword User’s Manual: Volume II, Ma-terial Models, Version 971, Livermore Software Technol-ogy Corporation, Livermore, CA, 2007, pp. 110–114 and473–475.

[18] Mathworks, Inc., MATLAB: The Language of TechnicalComputing, Software Package, Ver. 7.8.0347 (R2009a), TheMathworks, Inc., Natick, MA, 2009.

[19] MGA, Saftey Compliance Testing for FMVSS 214 ‘SideImpact Protection – Passenger Cars’: 1997 Dodge Neon,prepared by MGA proving Grounds, U.S. Department ofTransportation, Washington, DC, 1997.

[20] C. Mozumder, P. Bandi, N. Patel, and J. Renaud, Thick-ness based topology optimisation for crashworthiness de-sign using hybrid cellular automata, Proceedings of the 12thAIAA/ISSMO Multidisciplinary Analysis and OptimisationConference, Victoria, British Columbia, Canada, 2008.

[21] A. Najafi and M. Rais-Rohani, Mechanics of axial plasticcollapse in multi-cell, multi-corner crush tubes, Thin-Wall.Struct. 49 (2011), pp. 1–12.

[22] NCAC, Finite Element Model of Dodge Neon: Model Year1996 Version 7. FHWA/NHTSA National Crash AnalysisCenter, Ashburn, VA, 2006.

[23] NCAC, Finite Element Model Archive, National Crash Anal-ysis Center, Ashburn, VA. Available at http://www.ncac.gwu.edu/vml/models.html.

[24] NHTSA, Federal Motor Vehicle Safety Standards and Regu-lations, U.S. Department of Transportation, National High-way Traffic Safety Administration, Washington, DC, 1998.

[25] F. Pan, P. Zhu, and Y. Zhang, Metamodel-based lightweightdesign of B-pillar with TWB structure via support vec-tor regression, Comput. Struct. 88 (2010), pp. 36–44.

[26] N. Peixinho and C. Doellinger, Characterisation of dy-namic material properties of light alloys for crashwor-thiness applications, Mater. Res. 13 (2010), pp. 471–474.

[27] M. Rais-Rohani, K. Solanki, E. Acar, and C. Eamon, Shapeand sizing optimisation of automotive structures with deter-ministic and probabilistic design constraints, Int. J. VehicleDes. 54 (2010), pp. 309–338.

[28] C. Rasmussen and C. Williams, Gaussian Processes for Ma-chine Learning, MIT, Cambridge, MA, 2006.

[29] T. Simpson, T. Mauer, J. Korte, and F. Mistree, Krigingmodels for global approximations in simulation-based mul-tidisciplinary design optimisation, AIAA J. 39 (2001), pp.2231–2241.

[30] J. Sobieszczanski-Sobieski, S. Kodiyalam, and R.Y. Yang,Optimisation of car body under constraints of noise, vibra-tion, and harshness (NVH), and crash, Struct. Multidiscip.Optim. 22 (2001), pp. 298–306.

[31] C. Turner, Design space analysis with hyperdimensionalmetamodels, Proceedings of the NSF Engineering Researchand Innovation Conference, Honolulu, HI, 2009.

[32] USAMP, Magnesium Vision 2020: A North American Au-tomotive Strategic Vision for Magnesium, U.S. AutomotiveMaterials Partnership, Southfield, MI, 2006.

[33] Vanderplaats R&D, VisualDOC 6.2: Advanced ExamplesManual, Vanderplaats Research and Development, Inc.,Novi, MI, 2009, p. 93.

[34] D.A. Wagner, S.D. Logan, K. Wang, and T. Skszek, FEApredictions and test results from magnesium beams in bend-ing and axial compression, Proceedings of the SAE 2010World Congress & Exhibition, Detroit, MI, 2010.

[35] L. Wang, D. Beeson, G. Wiggs, and M. Rayasam,A comparison of meta-modeling methods using prac-

tical industry requirements, Proceedings of the 47thAIAA/ASME/ASCE/AHS/ASC Structures, Structural Dy-namics, and Materials Conference, Newport, RI, 2006.

[36] R.J. Yang, N. Wang, C.H. Tho, J.P. Bobineau, and B.P.Wang, Metamodeling development for vehicle frontal im-pact simulation, J. Mech. Des. 127 (2005), pp. 1014–1020.

[37] L. Zerpa, N.V. Queipo, S. Pintos, and J. Salager, An optimi-sation methodology of alkaline-surfactant-polymer floodingprocesses using field scale numerical simulation and mul-tiple surrogates, J. Petrol. Sci. Eng. 47 (2005), pp. 197–208.

Appendix 1

Polynomial response surface (PRS)PRS is one of the most widely used and simplest metamodellingtechniques, but it may not provide an accurate prediction for cer-tain responses. The most typical form of PRS is a second-degreepolynomial function of the form:

f (x) = bo +L∑i=1

bixi +L∑i=1

biix2i +

L−1∑i=1

L∑j=i+1

bij xixj (A.1)

where f (x) is the metamodel prediction at point x, L is the num-ber of design variables in the design vector x and b0, bi, bii,bij are the unknown coefficients found using the least squarestechnique.

As a regression model, PRS may not pass through the train-ing points. The degree of the polynomial can be changed orsome of the terms appearing in Equation (A.1) can be omitteddepending on the nonlinearity of the response function beingmodelled.

Gaussian process (GP)Descriptions of Gaussian Process (GP) in this discussion fol-low that in Wang et al. [35] and Acar and Rais-Rohani[1]. The GP metamodel is a group of output variables fN ={fn(x1

n, x2n, . . . , x

1n)Nn=1} with a Gaussian joint probability distri-

bution:

P (fN |CN,XN )

= 1√(2π )N |CN | exp

[−1

2(fN − µ)T C−1

N (fN − µ)

](A.2)

whereXN ≡ {xn}Nn=1 are N pairs of L-dimensional input variablesxn = x1

n, x2n, . . . , x

Ln , CN is the covariance matrix with elements of

Cij = C(xi, xj ) and µ is the mean output vector.Elements of the covariance matrix CN are calculated from:

Cij = θ1 exp

⎡⎢⎣−1

2

L∑l=1

(x

(l)i − x

(l)j

)2

r2l

⎤⎥⎦+ θ2 (A.3)

Cij = θ1 exp

⎡⎢⎣−1

2

L∑l=1

(x

(l)i − x

(l)j

)2

r2l

⎤⎥⎦+ θ2 + δij θ3 (A.4)

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12

http://www.ncac.gwu.edu/vml/models.html

http://www.ncac.gwu.edu/vml/models.html


where θ1, θ2, θ3 and rl are referred to as ‘hyperparameters’ withrl being the length scale. θ3 is an independent noise parameterand δij is Kronecker’s delta (equal to one when i = j and zerootherwise). These hyperparameters are selected to maximise log-arithmic likelihood of the predictions matching the training data.This is given by:

LL = −1

2log |CN | − 1

2f TN C

−1N fN − N

2log(2π ) + ln(P (θ ))

(A.5)

where P (θ ) is the prior distribution of the hyperparameters. Thisis usually uniform because no prior knowledge is available andcan be equated to zero for optimisation.

Equations (A.3) and (A.4) define the interpolation and re-gression modes of the Gaussian process model, respectively. Theformer passes through all training points while the latter providesa smoother surface to help with noisy data. The prediction surfacewith noise filtered out is less complex and might not pass throughall training points but this has better predictions at non-trainingpoints.

The response value at a prediction point xp =(x1p, x

2p, . . . , x

Lp ) is estimated as:

f (xp) = kT C−1N fN (A.6)

where k = [C(x1, xp), . . . , C(xN, xp)]. Standard deviation at theprediction point is available without requiring additional simula-tions or tests and can be calculated from:

σf (xp) = κ − kT C−1N k (A.7)

where κ = C(xp, xp).No tuning parameters are explored within GP. The MATLAB

[18] toolbox from Rasmussen and Williams [28] is used to developthe GP metamodels.

Radial basis function (RBF)This formulation of RBF requires normalised training and testpoints in the range of 0 to 1. This is done by dividing each variableby the maximum value of that variable in the DOE table. The basicform of RBF is given as:

f (x) =N∑i=1

λi�(‖x − xi‖) (A.8)

where f (x) is the metamodel prediction at point x, N is the numberof training points, x is the input vector of normalised variables, xi

is vector of normalised design variables at the ith training pointand ‖x − xi‖ = √(x − xi)T (x − xi) is the Euclidean norm or dis-tance r from point x to the training point xi. The λi parameters arethe unknown interpolation coefficients that must be calculated.�is the radially symmetric basis function that can take on a numberof forms. Equation (A.8) represents a linear combination of a fi-nite number of basis functions. Typical radial basis functions arelisted below.

• Thin Plate Spline: �(r) = r2 ln�(cr)• Gaussian: �(r) = exp(−cr2)• Multiquadric: �(r) = √

r2 + c2

• Inverse Multiquadric: �(r) = 1/√r2 + c2

The value c is a constant that the user determines. Values ofr are between 0 and 1 because the training and test points arebetween 0 and 1 resulting in 0 ≤ c ≤ 1. In general, Multiquadricwith c = 1 gives good results for many function types.

The interpolation coefficients, λi, can be found by minimisingthe residual (the sum of the squares of the deviations) as:

R =N∑j=1

[f (xj ) −

N∑i=1

λi�(‖xj − xi‖)

]2

(A.9)

In matrix form, this is expressed as:

[A]{λ} = {f } (A.10)

where [A] = �(‖xj − xi‖) with j = 1, N , and i = 1, N . SolvingEquation (A.11) for λ and inputting into Equation (A.8) generatespredictions. Error analysis typically relies on data at test pointsoutside of the training set since RBF is an interpolation model thatpasses through all the training points.

RBF tuning parameters are c and �. All four radial ba-sis functions discussed here were used with values of c ={0.05, 0.1, 0.15, . . . , 1} to determine the combination with lowestcross-validation GMSE.

Kriging (KR)Descriptions of Kriging follow those in Acar and Rais-Rohani [1],Lophaven et al. [16] and Simpson et al. [29]. For this discussion,the design (training) point matrix is s, the corresponding trainingpoint function evaluations are Y , the test point or prediction pointis x and general variables w and v are introduced. This descrip-tion of Kriging requires the training set to have zero mean and acovariance of 1. This is done by:

s = (soriginal − µs)

σs(A.11)

Y = (Yoriginal − µY )

σY(A.12)

where original indicates the un-normalised values, µs and µY arethe means of the training points and their responses, respectively,and σs and σY are the standard deviations of the training pointsand their responses, respectively.

Kriging models assume that the function takes the form of:

f (x) = P (x) + Z(x) (A.13)

where f (x) is the approximate function, P (x) is a polynomialfunction that globally approximates the actual function and Z(x) isthe stochastic component that accounts for deviations or Z(x) =r(x)T γ .

First, the polynomial portion of the model is explored, i.e.P (x). This is a linear combination of np polynomial functions,p(x), and regression parameters β:

P (x) = p(x)T β (A.14)

The degree of the polynomial is chosen to be 0, 1 or 2 re-sulting in np polynomial equations, with a 1st degree polynomial

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12


generally being used. The 2nd degree (quadratic) formulation issimilar to the 2nd degree PRS model discussed previously. Thatis, with L = number of variables using general variable w:

degree 0(constant)np = 1p1(w) = 1

(A.15)

degree 1(linear)np = L+ 1

p1(w) = 1, p2(w) = w1, . . . , pL+1(w) = wL

(A.16)

degree 2(quadratic)

np = 1

2(L+ 1)(L+ 2)

p2(w) = w1, . . . , pL+1(w) = wLpL+2(w) = w2

1, . . . , p2L+1(w) = w1wLp2L+2(w) = w2

2, . . . , p3L(w) = w2wL. . . . . . pnp(w) = w2

np

(A.17)

Polynomial function evaluations at each training point mustbe performed to fit the training data. Define P(s) or just P as:

P (s) = P = [p(s1), . . . , p(sN )]T (A.18)

where N is the number of training points. This is a vector of onesif the 0 degree polynomial is used.

The stochastic component Z allows the Kriging model tointerpolate the response value and has a zero mean and covarianceof:

COV[Z(xi), Z(xj )] = σ 2R(θ, xi, xj

)i, j = 1, ..., N

(A.19)

where σ 2 is the variance, R(θ, xi, xj ) is the correlation functionbetween sample points xi and xj and θ represents correlation pa-rameter that must be calculated. The user defines the correlationfunction, R(θ, xi, xj ), with a Gaussian correlation function gen-erally chosen. Correlation functions use general variables w and vand are in the form:

R(θ,w, v) =L∏k=1

Rk(θk, dk) (A.20)

where L is the number of design variables and dk = wk − vk isthe distance between wi and vj at the kth component of the points.Some Kriging correlation functions are listed below:

• Gaussian: Rk(θk, dk) = exp(−θkd2k )

• Exponential: Rk(θk, dk) = exp(−θk|dk|)• Exp. General: Rk(θk, dk) = exp(−θk|dk|θL+1 ),

0 < θL+1 ≤ 2• Linear: Rk(θk, dk) = max{0, 1 − θL+1|dk|}• Spherical: Rk(θk, dk) = 1 − 1.5ξ 2

k + 0.5ξ 3k ,

ξk = min{1, θk|dk|}• Cubic: Rk(θk, dk) = 1 − 3ξ 2

k + 2ξ 3k ,

ξk = min{1, θk|dk|}• Spline: Rk(θk, dk) = ς (ξk), ξk = θk|dk|

• ς (ξk) =⎧⎨⎩

1 − 15ξ 2k + 30ξ 3

k for 0 ≤ ξk ≤ 0.21.25(1 − ξk)3 for 0.2 < ξk < 1

0 for ξk ≥ 1

⎫⎬⎭

The correlation matrix R is defined as

R(θ, s) = R = R(θ, si , sj ), i, j = 1, ..., N (A.21)

Maximum likelihood estimation is used to estimate the cor-relation parameter, θ . θ is found by solving the following optimi-sation problem if a Gaussian correlation model is selected:

min ψ(θ ) ≡ |R|1/N σ 2 ormaxψ(θ ) = − [N ln(σ 2) + ln |R|] /2

s.t.θ > 0

(A.22)

where |R| is the determinant of R and σ 2 is the process variancedefined as:

σ 2 = [(Y − P β)T R−1(Y − P β)]/N (A.23)

The initial value of θ as well as its upper and lower boundsinfluence the calculation of θ from Equation (A.22) and the accu-racy of the resulting Kriging model. θ usually falls between 0 and2 with the initial value being taken as the midpoint. Almost anyvalue of θ will produce a Kriging model and results will be pre-dicted, but these predictions are not necessarily accurate. Solvingthe optimisation problem in Equation (A.22) will result in a moreaccurate Kriging model.

The steps to derive Equations (A.24)–(A.27) are not presentedhere; see Lophaven et al. [18] for a description of the derivation.Once regression and correlation functions are chosen, the responseis predicted as:

f (x) = p(x)T β + r(x)T γ (A.24)

where p(x) is the polynomial term found from Equation (A.15)to Equation (A.17), r(x) is the correlation vector (see Equation(A.25)), β comes from a generalised least squares solution andseen in Equation (A.26) and γ is computed from the residual andseen in Equation (A.27).

r(x) = [R(θ, s1, x) . . . , R(θ, sm, x)]T (A.25)

β = (P T R−1P )−1P T R−1Y (A.26)

γ = R−1(Y − P β) (A.27)

Below are some key points about the discussion of Kriging.

1. β and γ are functions of training points (not test points). Thismeans β and γ are constant for a given training set, so fromEquation (A.24) only the correlation and regression functionsneed to be evaluated at the test point(s) to get the Krigingprediction(s) once β and γ are found.

2. β and γ are functions of R(θ, s) and therefore functions ofcorrelation parameter theta.

3. The quantity R−1 appears in multiple places. This can be-come very computationally expensive as the number of trainingpoints (N) increases (R is an N by N matrix), which is commonin a number of metamodelling applications. For this reason, theinverse should be calculated using linear algebra techniques.

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12


4. The best regression and correlation functions are dependenton the problem, but a 1st degree polynomial and a Gaussiancorrelation function serve as a good starting point.

The MATLAB toolbox developed in Lophaven et al. [16] isused in this study. The tuning parameters of KR explored are thepolynomial degree, regression function, as well as the upper andlower bounds on θ . The upper and lower bound of θ are the finaltwo tuning parameters of KR.

Support vector regressionDescriptions of SVR in this discussion follow that in References[1, 5, 6, 11, 12]. The literature suggests that the design variablesshould be normalised to a range of (−1,1) or (0,1). Simply, SVRconstructs a hyperplane that passes near each design point suchthat they fall within a specified distance of the hyperplane. Intwo dimensions, this hyperplane is simply a line. The hyperplaneis then used to predict other responses. SVR estimates the realfunction as:

y = r(x) + δ (A.28)

where δ is an independent random noise, x is the multivariableinput, y is the scalar output and r is the mean of the conditionalprobability (regression function); see Cherkassky and Ma [5] andGunn [11] for more information. SVR technique selects the ‘best’approximate model from a group of selection models that min-imise the prediction risk. Linear or nonlinear regression can beperformed. When a linear regression is used, the pool of approxi-mation models is given by:

f (x) = 〈ω · x〉 + b (A.29)

where b is the bias term and 〈ω · x〉 is the dot product of ω andx. Minimising empirical risk using the ε-insensitive loss func-tion allows regression estimates. It is desirable to have a ‘flat’approximation function and this is achieved by minimising |ω|2.Non-negative slack variables (ξi and ξ ∗

i ) are introduced to accountfor training points that fall outside of the ε-insensitive zone. Thatis:

minimize1

2|ω|2 + C

N∑i=1

(ξi + ξ ∗i )

s.t.

⎧⎪⎨⎪⎩yi − 〈ω · xi〉 − b ≤ ε + ξ ∗

i

〈ω · xi〉 + b − yi ≤ ε + ξi

ξ ∗i , ξi ≥ 0

(A.30)

where C is a positive constant and ε is the insensitive zone, bothchosen by the user. C is also referred to as the regression parameteror penalty parameter. Cherkassky and Ma [5] propose that C bechosen as:

C = max(|µY + 3σY |, |µY − 3σY |) (A.31)

where µY and σY are the mean and standard deviation of thetraining point responses. Hsu et al. [12] suggest a cross-validationapproach to find C.

The parameter ε determines the width of the ε-insensitivezone and affects the complexity/flatness of the model. The values

of ε should be tuned to the input data, but a reasonable startingvalue is found using:

ε = c

100|ymax − ymin| (A.32)

with c = 1 where |ymax − ymin| is the range of the responses atthe training points. The value of c can be tuned for the function.Cherkassky and Ma [5] propose that ε be chosen as:

ε = 3σ

√ln N

N(A.33)

where σ is the standard deviation of the noise associated withthe training point response values and N is the number of trainingpoints. This assumes that the noise is known or can be determined.Cherkassky and Ma [5] suggest the following to estimate theunknown variance of noise using a k-nearest neighbour technique:

σ 2 = N 1/5k

N 1/5 − 1· 1

N

N∑i=1

(yi − yi)2 (A.34)

where k is in the range [9, 32] and∑N

i=1 (yi − yi)2 is the squaredsum of the residuals.

The optimisation problem in Equation (A.31) written as aLagrangian function is

L = 1

2|ω|2 + C

N∑i=1

(ξi + ξ ∗

i

)−N∑i=1

αi (ε + ξi − yi

+ 〈ω · xi〉 + b)

−N∑i=1

α∗i

(ε + ξ ∗

i + yi − 〈ω · xi〉 − b)

−N∑i=1

(ηiξi + η∗

i ξ∗i

)(A.35)

where ηi and η∗i are additional slack variables. From Lagrangian

theory, necessary conditions for α to be a solution are listed below:

∂bL =N∑i=1

(α∗i − αi) = 0 (A.36)

∂ωL = ω −N∑i=1

(α∗i − αi

)xi = 0 (A.37)

∂ξiL = C − αi − ηi = 0 (A.38)

∂ξ∗iL = C − α∗

i − η∗i = 0 (A.39)

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12


Substituting Equations (A.36)–(A.39) into Equation (A.30)gives the dual form optimisation problem:

Maximize

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

− 12

N∑i,j=1

(αi − α∗

i

) (αj − α∗

j

) ⟨xi · xj

⟩

−εN∑i=1

(αi + α∗

i

)+N∑i=1yi(αi − α∗

i

)

s.t.

⎧⎪⎨⎪⎩

N∑i=1

(αi − α∗

i

) = 0(αi − α∗

i

)ε[0, C]

(A.40)

Equation (A.37) is rewritten as:

ω =N∑i=1

(α∗i − αi

)xi (A.41)

The linear regression first expressed in Equation (A.29) iswritten as:

f (x) =N∑i=1

(αi − α∗

i

) 〈xi · xi〉 + b (A.42)

Clarke et al. [6] (page 1079) summarise the process of trans-forming the problem into dual form by stating:

Transforming the optimisation problem into dual formyields two advantages. First, the optimisation problem isnow a quadratic programming problem with linear con-straints and a positive definite Hessian matrix, ensuring aunique global optimum. For such problems, highly efficientand thoroughly tested quadratic solvers exist. Second, ascan be seen in Equation [(A.40)], the input vectors onlyappear inside the dot product. The dot product of each pairof input vectors is a scalar and can be preprocessed andstored in the quadratic matrixMij = (〈xixj 〉)ij . In this way,the dimensionality of the input space is hidden from theremaining computations, providing means for addressingthe curse of dimensionality.

A nonlinear regression model can be developed by replacingthe dot product 〈xi · xi〉 with a kernel function, k, rewriting theoptimisation problem in Equation (A.40) as

Maximize

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

− 12

N∑i,j=1

(αi − α∗i )(αj − α∗

j )k(xi, xj )

−N∑i=1

(αi + α∗i ) +

N∑i=1yi(αi − α∗

i )

s.t.

⎧⎪⎨⎪⎩

N∑i=1yi(αi − α∗

i ) = 0

(αi − α∗i )ε[0, C]

(A.43)

Replacing the dot product with a kernel function in the ap-proximation function Equation (A.43) gives the nonlinear SVR

approximation as:

f (x) =N∑i=1

(αi − α∗

i

)k(xi, xj ) + b (A.44)

Common kernel functions include the following:

• Linear: k(xi, xj ) = xTi , xj• Polynomial: k(xi, xj ) = (γ 〈xi · xj 〉 + r)d > 0• Gaussian: k(xi, xj ) =

exp(−‖xi−xj ‖2

2p2 )• Radial Basis Function: k(xi, xj ) =

exp(−γ ‖xi − x2j ‖) γ > 0

• Sigmoid: k(xi, xj ) = tan h(γ 〈xi · xj 〉 + r)

where γ , d, p and r are kernel parameters and should be adjustedby the user for each data set. Listed below are some insights aboutsetting kernel parameters:

• The polynomial degree d is typically chosen to be 2. Gunn[11] uses r = 1 to ‘avoid problems with the Hessian be-coming zero’.

• Hsu et al. [12] use a cross-validation approach to determineγ for RBF. This procedure could be applied to γ for otherkernels as well as other kernel parameters.

• Cherkassky and Ma [5] suggest p for the Gaussian kernel(they call it RBF, the formulation is the same as ‘Gaus-sian’ here) as P ∼ (0.1 − 0.5)∗range(x) for single variableproblems and PL ∼ (0.1 − 0.5) for multivariable problemswhere L is the number of variables and all input variablesare normalised to [0,1].

It should be noted that the ‘Gaussian’ kernel here is some-times referred to as ‘Radial Basis Function’ and ‘Gaussian RadialBasis Function’ in the literature. Experience shows that SVR meta-models are highly sensitive to tuning parameters. The suggestionsexpressed in this discussion may not result in an acceptable ac-curacy level. The reader is encouraged to perform SVR tuning toensure an accurate model for the selected response.

The SVR MATLAB toolbox developed by Gunn [11] was usedin this study and the Linear kernel is used. The tuning parametersexplored within SVR are penalty parameter, C, and ε-insensitivezone parameter, c.

Optimised ensembleAn ensemble of metamodels combines predictions from severalstand-alone metamodels such as the ones presented above. Ensem-ble metamodels are more accurate than the individual membersbut also more computationally expensive. The general form of anensemble is a weighted sum of the predictions of separate meta-models. In mathematical form, this is expressed as [4, 37]:

f (x) =M∑i=1

wi(x)fi(x) (A.45)

where f (x) is the ensemble prediction, x is the vector of inputvariables, M is the number of metamodels used to build the en-semble, wi is the weight factor for the ith metamodel and fi is theprediction of the ith metamodel. The weight factors must sum to

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12


one:

M∑i=1

wi(x) = 1 (A.46)

Selection of the weight factors is the most important stepwhen constructing an accurate ensemble. Acar and Rais-Rohani[1] developed an ensemble minimising the error by finding theoptimal weight factors. In mathematical form, this is expressedas:

min ε = Err{f (wi, fi(xk)), f (xk), k = 1 ∈ N}

s.t.

M∑i=1

wi = 1 (A.47)

where Err{} is the error metric that finds error of the ensemblepredictions, f , f (xk) is the actual response at the training point xk,and N is the number of training points. In this study the predictionerror was based on the GMSE metric defined as:

GMSE ≡ 1

N

N∑i=1

(f i − fi)2 (A.48)

where N is the number of design points, fi is the actual responseand fi is the predicted response from each metamodel.

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 2

0:15

07

Dec

embe

r 20

12

crashworthiness optimisation of vehicle structures with magnesium alloy parts

Documents