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Struct Multidisc OptimDOI 10.1007/s001580100590y
INDUSTRIAL APPLICATION
Identification of validated multibody vehiclemodels for crash analysis using a hybridoptimization procedure
Marta Carvalho · Jorge Ambrósio · Peter Eberhard
Received: 29 May 2010 / Revised: 29 August 2010 / Accepted: 22 October 2010c© SpringerVerlag 2010
Abstract The design of components, or particular safetydevices, for road vehicles often requires that many analyses are performed to appraise different solutions. Multibodyapproaches for structural crashworthiness provide alternative methodologies to the use of detailed finite elementmodels reducing calculation time by several orders of magnitude while providing results with the necessary quality.The drawbacks on the use of multibody approaches forcrashworthiness are the cumbersome model developmentprocess and difficulty of model validation. This work proposes an optimization procedure to assist multibody vehiclemodel development and validation. First the topologicalstructure of the multibody system is devised representingthe structural vehicle components and describing the mostrelevant mechanisms of deformation. The uncertainty in themodel development resides on the constitutive behavior ofthe plastic hinges used to represent the structural deformations of the vehicle, identified by using the classic multibody approach. The constitutive relations are identified byusing an optimization procedure based on the minimization of the deviation between the observed response of thevehicle model and a reference response, obtained by experimental testing or using more detailed models of the vehicle.
M. Carvalho · J. Ambrósio (B)Institute of Mechanical Engineering, Instituto Superior Técnico,Technical University of Lisbon, Lisbon, Portugalemail: [email protected]
P. EberhardInstitute of Engineering and Computational Mechanics,University of Stuttgart, Stuttgart, Germany
The identification is supported by a genetic optimizationalgorithm that enables the characterization of several initialsets of parameters that characterize the plastic hinge constitutive relations in the vicinity of an EdgeworthPareto front.Afterwards, a deterministic optimization algorithm is usedto identify the accurate minimum. The procedure is demonstrated by the identification of the multibody model of alarge family car suitable for front impact.
Keywords Multibody models · Multicriteria optimization ·Model validation · Crashworthiness
1 Introduction
In vehicle modeling, details concerning the design are ofmajor importance. However, such aspects often should bekept confidential to external developers, either because theyrepresent technological advances or due to legal reasonsassociated to liability or, therefore, they cannot be disclosed even to partners. A solution to this problem is thedevelopment of virtual vehicle models that have dynamicresponses similar to the original ones for selected crashscenarios (Sousa et al. 2008). Such responses can be measured in terms of accelerations of given points in the vehiclestructure, energy absorption characteristics of subsystems,intrusion measures or by other measurable characteristics.Alternative approaches to the use of detailed finite elementmodels, which reduce calculation time while providingmeaningful results, are of upmost importance. For a typical frontal crash simulation, with the duration of 150 ms,a finite element model with the appropriate level of detailrequires about 3 days of computational time in a current desktop computer while a multibody model able todescribe the same mechanisms of deformation, kinematics

M. Carvalho et al.
of the vehicle, contact forces and energy deformation levels requires about 3 min. Similar ratios in computation timebetween finite element and multibody models are observedfor other types of crash scenarios. These alternative modelsare developed to be used at the conceptual design development stage for quick analysis (Kim et al. 2001). A typicalusage of this virtual model would allow a developer to tacklethe task of devising a protective system for a selected part ofthe vehicle being assured that the overall behavior of such avehicle is reasonable, even without being a detailed matchof the original (Sousa et al. 2008).
Advances in computer tools support the design of vehiclestructures for crashworthiness. However, in industrial applications trial and error procedures are still more popular thanthe systematic use of the optimization procedures, especially for highly nonlinear problems, such as those presentin passive safety applications. Structural optimization forcrashworthiness considerations in the automotive industriesare mostly based on finite element models. Bennett et al.(1997) present a methodology for the identification of thefrontend structural stiffness for barrier impact standards.The constitutive relations of the structural components areused in the integration of the equations of motion, beingcoupled to algorithms of optimization in order to improvethe designs obtained by the trial and error procedure. Adrawback of this work was that the component dimensions could not be directly manipulated during optimization.Later, Song (1986) used boxbeam elements in the numerical model and included a new optimization capability, whichdetermines the dimensions of the structural components tominimize the structural mass while meeting given safety criteria. Lust (1992) developed an optimal structural designapproach considering two criteria, simultaneously, associated to the elastic loads and crashworthiness constraints.In order to reduce the computational effort, the crashworthiness constraints are nonlinear approximations. Dias andPereira (1994) presented some of the first applications ofoptimal design for crashworthiness using multibody modeling approaches. They presented the optimization problemby modeling the vehicle and restraint system by means of asimplified mechanical system. Another approach used byBennett and Park (1995) starts from a detailed and complex finite element model for crash and uses an optimizationstrategy, based on the sequential linear programming withsomewhat large finitedifference perturbations, to calculatethe sensitivities. Etman et al. (1996) developed a designoptimization tool for road vehicle crashworthiness that isfirst demonstrated for an analytical test problem with animpact absorber, and applied afterwards to an industrialcrashworthiness design problem consisting in the development of a restraint system for frontal impact. The crashworthiness optimization problem is solved by Etman et al.(1996) using sequential approximate optimization in order
to deal with the noisy functional behavior of the objective function and with the high computational costs of theevaluation of the objective functions and constraints. Thesequences of linear programming problems used in theapproach are generated by means of multipoint approximations. Following their earlier work, a design methodologyfor crash structures is presented by Ambrósio et al. (1996)and Dias and Pereira (2004). Complex vehicle structures,such as train sets, are modeled using multibody dynamicsformulations and simulated in crash events. The design andsimulation of these models are linked to deterministic andevolutionary optimization algorithms, which are used in theconceptual design phase to obtain the best characteristics ofthe crash structures and energy absorption devices of trainstructures. The design of railway crash tests with multibodymodels of railway vehicles and deterministic optimizationalgorithms has been proposed by Milho et al. (2004). In allcases of application of optimization methodologies to support the vehicle design for passive safety, the major issueis the identification of proper objective functions and constraints, which are problem dependent. Also, as far as theoptimization method is concerned, there is no clear favoritewhich one to use.
This work attempts at not only proposing a solution forthe identification of reliable vehicle multibody models forcrashworthiness, but also at identifying, from our experience, a numerically robust optimization procedure to handlethis type of problems with the objective of devising properly validated vehicle models. The input data required forthe model validation is limited to the accelerations, velocities and/or intrusions of the reference vehicle for the crashtest configurations in which the vehicle model is to be used(Seiffert and Wech 2003). It is assumed that the initial modelof the vehicle is obtained using standard good modeling procedures (Nikravesh et al. 1983; Ambrósio 2001; Sousa et al.2008) and that the validated vehicle model is to be used onlyin the crash scenario configurations for which it is validated.
2 Multibody equations
The multibody models are a collection of rigid and/orflexible bodies connected by kinematic joints that restraintheir relative motion. Several types of loads are applied,including spring and damper forces, contact forces resulting from the impact with the barriers and from the relative motion between the bodies, inertia loads and gravity(Nikravesh 1988). A multibody system can be representedschematically as shown in Fig. 1.
The most commonly used sets of coordinates to describethe motion of each body of a multibody system are Cartesian, natural and joint coordinates, which are applieddepending on the type of application, complexity of the

Identification of validated multibody vehicle models for crash analysis using a hybrid optimization procedure
Fig. 1 Generic multibody system
multibody system, preference of the research group inwhich the applications are being developed, or, as in thecase of this work, the commercial code which is used. Theefficiency of the numeric methods selected to solve theequations depends on the complexity of the system and onthe set of coordinates chosen (see e.g., Nikravesh 1988;Haug 1989; Schiehlen 1990; Jálon and Bayo 1994).
When the position and orientation of each rigid body isdescribed by Cartesian coordinates, the kinematics relationsmust be described by algebraic constraints. The equationsof motion for the system of nb unconstrained bodies arerepresented by
M q̈ = g (1)
where M is the mass matrix, which includes the masses andinertia tensors of the individual bodies, q̈ is the accelerationvector and g the vector with applied forces and gyroscopicterms. The relative motions between the bodies are constrained by kinematic joints, which can be mathematicallydescribed by a set of nc algebraic equations, written as
�(q, t) = 0 (2)
The first and second time derivatives of (2) constitute thevelocity and acceleration constraint equations,
�̇(q, t) ≡ �q q̇ − ν = 0 (3)
�̈(q, q̇, t) ≡ �q q̈ − γ = 0 (4)
where λ is a vector with nc unknown Lagrange multipliers, i.e.
M q̈ = g − �Tq λ (5)
Equation (5) has 6nb+ nc unknowns that must be solved together with the second time derivative of the constraint (4).The resulting system of differentialalgebraic equations is
[M �Tq�q 0
] [q̈λ
]=
[gγ
](6)
The stabilization or corrections of the kinematic constraint drift observed during the integration of (6) can beobtained, e.g., by using the Baumgarte stabilization, theAugmented Lagrangean formulation, or the Coordinate Partioning method (Baumgarte 1972; Wehage and Haug 1982;Bayo and Avello 1994; Shabana and Sany 2001; Neto andAmbrósio 2003).
3 Plastic hinge approach to multibody vehicle modeling
The multibody vehicle model for passive safety analysis isbased on the use of the plastic hinge approach (Nikraveshet al. 1983; Ambrósio 2001). This modeling strategyis based on the observation that the structural components experience plastic deformations in localized areaswhen overloaded. These deformations, presented in Fig. 2,develop at points where maximum bending moments occur,where loads are applied, in structural joints or in locallyweak areas. Some of the first applications of the plastichinge approach are proposed by Nikravesh et al. (1983).Later, Pereira et al. (1987) presented the application to
Fig. 2 Plastic hinge bendingmoment applied in positive (a)and negative (b) axis and itsconstitutive relationship
(a) (b)10000
8000
6000
4000
2000
0
2000
4000
6000
8000
10000
1.5 1 0.5 0 0.5 1 1.5Rotation (rad)
Mo
men
t (N
m)
FE MODELMB MODEL
Joint ID 3 Top View
X2
3 1
2
3 1
2
3 1
2
3 1

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Fig. 3 Plastic hinge models fordifferent load joints
vehicle rollover. The application of the plastic hingeapproach to railway vehicle design has been proposed byMilho et al. (2004). Plastic hinges are now available inmultibody codes such as MADYMO (2004). A furtherextension on this modeling strategy in crashworthinessdesign has been investigated by Pedersen (2003) who proposed a finite element with the ability to develop plastichinges on its nodes. The methodology briefly describedherein is also known in the automotive, naval and aerospaceindustries as conceptual modeling (Sousa et al. 2008). Theplastic hinge is modeled by a kinematic joint, describing the kinematics of the deformation, and a generalized
springdamper element, which is used to represent the constitutive characteristics of the elasticplastic deformation ofthe member.
The characteristics of the springdamper that describesthe properties of the plastic hinge are obtained by experimental component testing, localized finite element nonlinear analysis or simplified analytical methods. For a flexuralplastic hinge the spring stiffness is expressed as a functionof the change of the relative angle between two adjacentbodies connected by the plastic hinge as shown in Fig. 2.
This springdamper is associated with the common typeof joints, applied in multibody modeling as depicted in
Fig. 4 Side structure of avehicle modeled using theplastic hinge approach

Identification of validated multibody vehicle models for crash analysis using a hybrid optimization procedure
Fig. 5 Accelerometers positionin GCM3 model: (a) FE model;(b) MB model
Front_SsillAP
BP_MiddleBP_Bottom
FW_Center
Front_Ssill Front_SsillAP
BP_MiddleBP_Bottom
FW_Center
AP
BP_Middle
BP_Bottom
FW_Center
AP
BP_Bottom
FW_Center
(a) (b)
Fig. 3, for one axis bending, two axis bending, torsion andaxial, forming what is defined as plastic hinge (Ambrósio2001). Then, a structural component can be modeled as acollection of rigid bodies connected by plastic hinges, asshown in Fig. 4 for the side structure of a road vehicle.By implementing a proper discretization of the structureit is possible to model its most relevant mechanisms ofdeformation, i.e., not only each structural component of thevehicle must be modeled by a correct number of rigid bodies and plastic hinges according to good modeling practices(Nikravesh et al. 1983; Ambrósio 2001; Sousa et al. 2008)but also the model has to be able to develop the mechanismsof deformation characteristics of the crash configurations inwhich it is used.
Note that although the approach is deemed as ‘plastic’hinge, the deformations observed may also be elastic only.Although the formulation implied here only considers rigid
bodies connected by kinematic joints of the plastic hinges,the methodology can be extended to include flexible bodies(Dias and Pereira 1994).
4 Multibody models for vehicles
The generic car model developed in this work is to be usedin crash test scenarios described by the regulations approvedin Europe. Of particular relevance are the ECE regulationsfor frontal and side impact that must be fulfilled by any newvehicle that seeks approval for release in the EU countries.In the application described here the GCM3 model is testedaccording to the ECE R33 European Regulation (1995) forfull frontal rigid barrier impact. Due to the impossibility toaccess the experimental data a surrogate FE model testedaccording to the ECE R33 regulation, validated for frontal
Fig. 6 Deformations of GCM3MB model developed withnonoptimal plastic hingeconstitutive relations andthe FE model for the ECE R33crash test 0 ms
20 ms
40 ms
60 ms

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Fig. 7 Dynamic responsemeasured at the APillar for theMB model obtained aftermodeling structural deformationwith plastic hinge approach
impact as reported in (Puppini et al. 2005), is used to supplythe data required to the construction and validation of theMB model. Note that real vehicle data can be used directlyas reference data with the proposed procedure when available. The frontal crash test is conducted with the GCM3 FEmodel for the regulatory impact speed of 48.3 km/h.
The crash response of the vehicle can be measured interms of accelerations, velocities and intrusions in selectedpoints of the structure such as those shown in Fig. 5. It iscommon practice to filter the acceleration signals with the
CFC 60 filter and the velocities and displacements with CFC180 filter, for both FE and MB models.
The deformations of the finite element reference modeland of the multibody vehicle model for ECE R33 crash scenario are shown in Fig. 6 for a sequence of instants in time.It is observed that although the mechanisms of deformationare similar, the multibody model shows here a much stifferappearance than that of the finite element model.
In order to quantify the outcome of the dynamic analysis of the vehicle model obtained with the recommended
Fig. 8 Dynamic responsemeasured at the middle BPillarfor the MB model obtained aftermodeling structural deformationwith plastic hinge approach

Identification of validated multibody vehicle models for crash analysis using a hybrid optimization procedure
multibody modelling approaches, the accelerations, velocities and displacements in the longitudinal direction forthe pickup points located in APillar and middle BPillar,are presented in Figs. 7 and 8, respectively. These crashresponses show that the dynamic responses of the multibodyvehicle model are not yet well correlated with the referenceresponse of the reference vehicle. Such lack of correlationis also observed in all other measured crash responses notshown here. In an industrial environment the analyst manually tunes the model parameters associated with the plastichinges until its response matches, or is sufficiently close,that of the reference. This procedure, besides being far fromoptimal, relies completely in the experience of the analyst.Therefore, a systematic approach to the calibration of themodel by using optimization is introduced in this work.
The model already contains the mechanisms of deformation of the vehicle, which is a fundamental featurefor the success of what follows. The real deformation ofeach member of the structural setup includes the generalized elastic deformation of the member plus the plasticdeformation of the plastic hinge region. The plastic hingeapproach disregards the generalized elastic deformation ofthe structural members and considers the plastic deformation region as lumped. Two corrective measures can beforeseen for the MB models; one is to increase the number of plastic hinges in each structural member in orderto better distribute the deformation and, the second is tomodify some plastic hinges constitutive functions so thatthe generalized elastic deformation that develops in theirneighbouring regions can be accounted for. The use of thefirst corrective measure implies changing the topology ofthe multibody model by modifying the number of rigid bodies and kinematic joints used in the model. This change ofthe model structure must always be done by manually withgreat care. As no automatic procedure exists in the currentmultibody methodology to modify the level of discretizationof the structural elements a suitable computational procedure should be devised first. This is no trivial task, sincein multibody systems no mathematically welldefined errormeasures are known. Therefore, in this work the secondcorrective measure, in which the plastic hinge constitutiverelations are modified by using optimization approaches, isselected.
5 Identification of the multibody vehicle modelusing optimization methods
The model adaptation is the next step in the model development and constitutes in fact an optimization process toidentify the MB model that best represents the vehicle. Theobjective of the identification problem is to achieve a MBmodel that has displacements, velocities and accelerations
that correlate well with those measured in the same structural locations in a reference vehicle. In what follows theidentification of the vehicle model is pursued only for thefront crash impact scenario. Although a MB vehicle modelsuitable for several crash impact scenarios can be devisedusing the proposed methodology, front and side impact forinstance (Carvalho and Ambrosio 2010), the idea of a modelsuitable for omnidirectional impacts, if achievable, wouldbe a complex task.
5.1 Design variables
At this stage, the parameter values in the constitutive equations and the locations of the plastic hinges are adjustedwithin a given range of variation, to increase the correlation between the behavior of the reference and multibodymodels. This has been the procedure used (Mooi et al.1999; Gielen et al. 2000; van der Zweep and Lemmen2002; van der Zweep et al. 2005). The acceptable rangeof variation of the model data is related to the approximations made when defining the discretization of the modeland the constitutive equations of the plastic hinges. Theseranges are represented as corridors inside which differentforcedisplacement behaviors are accepted, see e.g. Fig. 9.
For the MB model, the plastic hinges are identifiedtogether with their variation ranges. The locations of theplastic hinges that supply the design variables for the optimization are represented for the left side of the car inFig. 10. To demonstrate the procedure a simplified situationwith just five design parameters is used in what follows. Dueto the ability to change the model crash behavior by scaling the plastic hinge constitutive relation (Sousa et al. 2008)and to the complexity of the vehicle multibody models
Angle
Moment
Fig. 9 Constitutive relation for a generic plastic hinge and variationcorridor

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Fig. 10 Joint localization and respective design variables for theGCM3MB model
(Ambrósio and Dias 2007), the design variables x used inthis work are the scaling factors of selected constitutivefunctions:
1. Forces on the plastic hinge of the body structure,2. Forces on the plastic hinges of the frame,3. Forces on the plastic hinges of the bumper frame,4. Forces on the plastic hinges of the subframe,5. Forces on the plastic hinges of the bumper subframe.
It must be noticed that each of the parameters selectedas design variable affects several plastic hinges that sharethe same constitutive relation. Therefore, each parameterinfluences the structural behavior of parts of substructuresof the vehicle model.
5.2 Optimization criteria
In this work the time interval of the analysis is discretizedinto time points and the objective functions are evaluated asthe sum of the square deviation of the vehicle responses insuch instants, as implied by Fig. 11. For each of the dynamicresponses selected, the error between the reference and the
AyModel Response
Reference Response
timet0 t1 t2 t3 t4 t5 t6 t7
Fig. 11 Time responses of the MB and reference models measured ina selected point of the vehicle
MB model response, at a finite number N of instants, is calculated as errors of acceleration, fi,acc(xxx), velocity, fi,vel(xxx)and intrusion, fi,disp(xxx). These errors are defined as
fi,acc(xxx) =N∑
j=1
√(acciMB(xxx, t j ) − accire f (t j )
)2 (7)
fi,vel(xxx) =N∑
j=1
√(veliMB(xxx, t j ) − velire f (t j ))2 (8)
fi,disp(xxx) =N∑
j=1
√(dispiMB(xxx, t j ) − dispire f (t j )
)2 (9)Due to the potentially conflicting responses or due to theimpossibility of minimization of all responses simultaneously an appropriate tradeoff between objectives has to befound. Usually these problems do not have a unique solution. The concept of EdgeworthPareto (EP) optimality canbe used to characterize the objectives (Censor 1977; Cunhaand Polak 1967; Deb 2001).
The purpose of a multicriteria optimization problem isdefined as finding the design variables xxx that
Optimize F(xxx) nvector of objective functions
subject to gi (xxx) ≤ 0 inequality constraints,i = 1, . . ., m
hj (xxx) = 0 equality constraints,j = 1, . . ., p
xL ≤ x ≤ xU bounds
Instead of a single minimum of a scalar optimizationproblem, the multiobjective optimization problem leads toa set of EPoptimal designs, also called EP front. In multicriteria problems there are several typical approaches: thefirst is to find a solution on the EP front, and the second is tofind a set of solutions as diverse as possible. Mathematicallyit is not possible to prefer one EP solution over another without further userspecified information. If this information isavailable, it can be used to make a choice. In the absence ofsuch information, EPoptimal solutions are not comparable(Deb 2001).
Genetic algorithms do not require gradient informationand that they also allow reaching an EP front (Deb 2001).Their disadvantage is the high number of function evaluations required. Therefore, several generations of the geneticalgorithm are used in this work to find sets of starting designvariables that are close to the EP front. Then, these designsare used as starting points for a gradient based optimization method to obtain efficiently minima of the problem.

Identification of validated multibody vehicle models for crash analysis using a hybrid optimization procedure
The best solution among all minima obtained is deemed asthe solution of the optimization problem. This is a hybridapproach that takes advantage of the ability of the geneticalgorithms to find suitable starting designs and the suitability of the gradient based algorithms to converge to minimafrom each one of those designs.
5.3 Formulation of the optimization problem
In this work the objective is to identify a vehicle multibody model that has prescribed crash responses. First, theproblem can be defined as a multiobjective optimizationfor the search of EP front (Ambrósio and Eberhard 2009).Afterwards the model identification is reduced to a scalaroptimization problem
minx
F(xxx) =n∑
i=1
N∑j=1
√(acciM B(xxx, t j ) − accire f (t j )
)210−4(10)
subject to
1
N
N∑j=1
√(veliM B(xxx, t j )−velire f (t j )
)2 ≤ f ∗i,vel
, i = 1, . . . , n
(11)
0.1 ≤ xi ≤ 2, i = 1 to 5
Table 1 Values for the scalar objective function and constraints foreach design of the MB model
Design F(xxx)N∑
j=1
√(veliMB(xxx, t j ) − velire f (t j ))2
Apillar Fire wall Front Bpillar Bpillar
center sill (middle) (bottom)
1 3.57 38 47 43 57 41
2 5.48 46 72 46 64 44
3 5.31 72 91 70 98 71
4 4.17 43 43 49 47 47
5 4.36 45 65 47 74 46
6 5.60 54 81 47 88 49
7 4.24 42 46 47 52 45
8 4.84 81 96 78 115 80
9 4.64 62 78 54 99 57
where the objective function is defined as the sum of themean square errors of the n accelerations measured indifferent points of the vehicle, represented in Fig. 5, andsampled on N instants in time for the duration of the crashevent, as represented in Fig. 11. In the case of this studyn = 5 and N = 60, which means a time sampling of 1 ms.
It is assumed here that all acceleration criteria are equallyimportant. The errors in the intrusion velocity profiles of thevehicle are used as constraints. The summation of the meansquare deviation between the model and the goal velocitiesfor each sensor is depicted by f ∗i,vel.
Fig. 12 Deformation of thereference and multibody vehiclemodels 0 ms
20 ms
40 ms
60 ms

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6 Results of the application: validated multibodymodels
In this implementation, the genetic algorithm is used in firstplace to identify the initial designs that are later exploredby using a Sequential Quadratic Programming (SQP) algorithm. The genetic algorithm stops after a preset numberof generations, which are deemed enough to characterizea group of individuals that constitute the EP front. Deb’selitist algorithm NSGAII is the selected genetic algorithmused in this work (Deb et al. 2002). This algorithm is implemented in the function ‘gamultiobj’ available in MATLABGenetic Algorithm Toolbox (MATLAB 2008). This geneticalgorithm is part of the MOEA (Multiobjective Evolutionary Algorithm) family, which have the ability to handlecomplex problems involving features such as discontinuities, multimodality, disjoint feasible spaces and noisy functions evaluations, reinforcing their potential effectivenessin multiobjective search and optimization (Fonseca andFleming 1995; Eberhard et al. 2003).
The number of sets of design variables used here is 15times the number of design variables, i.e., 75 in the optimization problem proposed in this work. The initial designis randomly created in the feasible region. In each generation a selection function chooses parents for the nextgeneration, based on their scaled values from the fitnessfunctions in which a ‘tournament’ is used. Each parent isselected by choosing two individuals at random, and thenselecting the best individual among them. Reproductionis done by scattered crossover with a rate of 95%, being1% of mutation rate introduced to provide genetic diversity and to enable the genetic algorithm to search a broader
Table 2 Measure of the criterion for the initial design and optimizedMB model
Scalar obj. fi,acc(×104) for accelerometer ifunction Apillar Fire wall Front Bpillar BpillarF(xxx) center sill (middle) (bottom)
Initial 4.36 0.69 2.00 0.61 0.43 0.63
Optimal 3.26 0.58 0.96 0.60 0.54 0.58
space. The EdgeworthPareto front population fraction isanother parameter that keeps the fittest population downto the specified fraction, which is 35% in this application,to maintain a diverse population. In this implementation,the genetic algorithm is used to provide the initial designsfor SQP algorithm. The number of generations in this caseis set to 10. The number of stall generations is set to 1.The improvement of the objective function, which is theweighted average change in fitness value over stall generations, is measured. If this value is less than a functiontolerance, the algorithm stops.
The first nine candidates for the EP front are computedhere after the calculation of three generations in 7.5 h ofCPU time. Each crash analysis of the vehicle MB modeltakes about 3 min of CPU time. The values of the scalarobjective function, described by (10), and velocity errorconstraints are listed in Table 1. The locations of the sensors used to formulate the objective functions, referred to inTable 1, are listed in Fig. 5.
Instead of continuing to generate more designs by considering more generations, which is a time consuming procedure, the next step is to continue with an SQP algorithm.
Fig. 13 Dynamic responsemeasured at the APillar for theMB models obtained after GAand SQP versus the FE modelfor the front impact test

Identification of validated multibody vehicle models for crash analysis using a hybrid optimization procedure
Fig. 14 Dynamic responsemeasured at the middle BPillarfor the MB models obtainedafter GA and SQP versus the FEmodel for the front impact test
The initial designs used for each analysis with deterministicalgorithm are the already computed intermediate solutions.The SQP optimization algorithm is available in the MATLAB Optimization Toolbox (MATLAB 2008) as function‘fmincon’, based on a sequential programming approachwith activeset strategy.
The F(xxxopt ) = 3.26 design is achieved in the 5th iteration for the vector of design variables xxxopt = [0.10, 0.25,0.10, 0.10, 0.10]. The patterns of deformation of the reference and optimized vehicle multibody models are depictedin Fig. 12. It is observed that the major characteristics of thedeformation are now well correlated. This qualitative information gives an impression of the improved behavior of thevalidity of the model but it is not enough to deem it as beingvalidated.
The errors on the dynamic response for the initial designand for the optimized MB model are summarized in Table 2.The results show an improvement of almost all dynamicresponses, except accelerations on the middle of Bpillar, asseen in Table 2 and illustrated by Figs. 13 and 14 that presentselected dynamic responses of the reference model and thebest MB model obtained using the gradient based algorithmand the initial model defined by the 5th design identifiedwith the genetic algorithm. The vehicle MB model obtainedwith the hybrid algorithm shows a good correlation withthe reference model and all acceleration responses are better correlated with the reference responses than those of theinitial model.
The hybrid algorithm approach applied to the problemof validating the multibody vehicle model for front impacttakes advantage of the diversity and potential location for
the minima with the genetic algorithm in few generations,and after that applies the SQP method to get faster convergence from there. The genetic algorithm takes 7.5 hof computational time to obtain designs suitable for thenext step of optimization undertaken by the SQP algorithm.Based on good initial designs obtained with the geneticalgorithm, the convergence by the SQP method is reached inonly five iterations, taking about 1 h of additional CPU time.Based on the methodology described in this work moredesign variables and criteria can be easily accommodated,leading to better vehicle models.
7 Conclusions
A methodology for the construction of validated multibodymodels of vehicles for impact analysis has been demonstrated in this work, being the optimal model identifiedby the hybrid approach proposed. First the genetic optimization algorithm generates a group of designs usuallyclose to the EP front. The use of gradient based algorithm,in this case the SQP, explores each one of these designsallowing to find the best of them, deemed here as the validated vehicle multibody model. It has been observed thatthe identification of the design variables as proportionalfactors of the plastic hinge constitutive relations is suitable not only because they represent what is uncertain inthe construction of multibody models for crash but alsobecause they allow obtaining a vehicle model with good predictability. Although the objective function can be expressedby intrusion, velocity and acceleration errors between the

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reference and multibody models dynamic responses, it hasbeen shown that in the case of frontal impact by usingthe acceleration errors as the objective functions and thevelocity errors as constraints leads to a robust optimizationprocess with results that show a good correlation betweenmodels while improving the convergence of the optimization problem. The drawback of this approach is that theconstraints of the problem become nonlinear. The methodology was demonstrated in the development of a multibodymodel for a large family car for which a validated finiteelement model exists and is used as reference. Becausethe approach only uses the dynamic response of the vehicle that can be obtained in a crash test, the methodologyproposed can be applied to the development of multibodymodels when only generic models exist and the exact models are unavailable for the user. The MB vehicle modelpresented in this work is suitable for a front crash scenarioonly. However, by using reference data for other scenarioconfigurations it is possible to obtain a vehicle model suitable for multidirectional impact or to obtain specializedmodels valid for multiple crash scenarios. The two important issues to be taken into account when using the validatedmodel are that in can only used in the crash scenario inwhich it has been validated and that no modifications to themodel of the structural components responsible for its crashworthiness behavior can be done without further validation.Under these conditions, its use for the development of safetysystems such as airbags, seats and any other protectivedevice is possible without further need for validation.
Acknowledgments The support of EU through project APROSYSSP7 with the contract number TIP3CT2004506503, having aspartners Mecalog (F), CRF (I), TNO (NL), Politecnico di Torino(I), CIDAUT (ES), Technical University of Graz (A) is gratefullyacknowledged. The financial support of the first author received fromFCT—Fundação para a Ciência e Tecnologia through grant SFRH/BD/23296/2005 is highly appreciated.
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Identification of validated multibody vehicle models for crash analysis using a hybrid optimization procedure1 Introduction2 Multibody equations3 Plastic hinge approach to multibody vehicle modeling4 Multibody models for vehicles5 Identification of the multibody vehicle model using optimization methods5.1 Design variables5.2 Optimization criteria5.3 Formulation of the optimization problem
6 Results of the application: validated multibody models7 Conclusions
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