Reliability-based multiobjective optimization for automotive ...

Struct Multidisc Optim (2007) 33: 255–268 DOI 10.1007/s00158-006-0050-x

I N D U S T R I A L A P P L I C AT I O N S

Kaushik Sinha

Reliability-based multiobjective optimization for automotive crashworthiness and occupant safety

Received: 18 November 2005 / Revised manuscript received: 22 April 2006 / Published online: 21 October 2006 © Springer-Verlag 2006

Abstract This paper presents a methodology for reliabilitybased multiobjective optimization of large-scale engineering systems. This methodology is applied to the vehicle crashworthiness design optimization for side impact, considering both structural crashworthiness and occupant safety, with structural weight and front door velocity under side impact as objectives. Uncertainty quantification is performed using two first order reliability method-based techniques: approximate moment approach and reliability index approach. Genetic algorithm-based multiobjective optimization software GDOT, developed in-house, is used to come up with an optimal pareto front in all cases. The technique employed in this study treats multiple objective functions separately without combining them in any form. It shows that the vehicle weight can be reduced significantly from the baseline design and at the same time reduce the door velocity. The obtained pareto front brings out useful inferences about optimal design regions. A decision-making criterion is subsequently invoked to select the “best” subset of solutions from the obtained nondominated pareto optimal solutions. The reliability, thus computed, is also checked with Monte Carlo simulations. The optimal solution indicated by knee point on the optimal pareto front is verified with LS-DYNA simulation results. Keywords Reliability-based multiobjective optimization · Uncertainty quantification · FORM · Nondominated points · GDOT · Pareto optimal solution · Knee point · Automotive crashworthiness · Occupant safety · Side impact · Monte Carlo simulation

1 Introduction Computer analysis of crashworthiness has become a powerful and efficient tool. It can substantially reduce the cost and time required for the development and certification of new K. Sinha (B) DaimlerChrysler Research and Technology, Bangalore, India e-mail: [email protected]

vehicle designs. Rapid increase in computing power, evolution, and development of theoretically sound, robust, and efficient technologies for the simulation of nonlinear structural dynamics have advanced vehicle crash simulation to the point where the results are trusted with a high degree of confidence. Presently, major automotive companies use finite element simulation to analyze and design vehicles to meet federal and corporate safety guidelines for frontal impact, side impact, roof crush, rear impact, etc. Crash simulation is fundamentally computation intensive and requires fast and powerful computational facilities to have reasonable turnaround time for the analyses. As the simulations are computationally intensive and highly nonlinear, special optimization techniques and processes are required. The design problems for vehicle crash safety are particularly complex in terms of physics of the problem and defining design requirements and design alternatives. Full vehicle design demands multidisciplinary design optimization coupled with computational crashworthiness analysis. A balance must be struck between designing the vehicle structure to absorb/manage the crash energy through structural deformation and maintaining the structural integrity of the passenger compartment. Techniques that can achieve both enhanced crashworthiness performance and weight reduction simultaneously are therefore key body technologies for future. In general, enhanced crashworthiness performance is associated with increased energy absorption efficiency. A major candidate that can have a big impact on achieving this is an application-specific, “synthesized” material with “optimal” mechanical properties. There has been some design studies applying singleobjective optimization to crashworthiness under a set of crash safety constraints (Yang et al. 1994). Simulation-based optimization generates deterministic optimum designs, which are often pushed to the limits of design constraint boundaries, living no room for tolerances/uncertainty in modeling, simulation and/or manufacturing uncertainties/imperfections. Consequently, deterministic optimum designs may result in infeasible design, advocating a definite need for consideration of uncertainty quantification as an ‘in-core’ design optimization process. However, reliability-based multiobjective optimization studies for structural crashworthiness and occupant safety, treating it as a vector optimization problem,

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have not been reported to the best of author’s knowledge. This paper presents an application of a nonlinear response surface-based, reliability-based multiobjective optimization, to reduce structural weight and front door velocity together with enhanced side impact performance of a vehicle. To reduce the computational effort, an archive of already computed solutions is maintained and referred if same point is encountered again in the process and no additional computation is performed in such case. All the responses for building the response surface are computed using LS-DYNA code. The statistical measures used to judge the model capability was R2 2 and Radj (Yang et al. 2000). The response surfaces developed using optimal LHS scheme were chosen for the subsequent study. All design variables are assumed to be normally distributed in this study. Numerical results are given and discussed for the benchmark vehicle side impact model.

curve = 1.0). The corresponding reliability for ith constraint is given as: Ri = 1 − P[G i (X ) ≤ 0] Now the reliability-based constraint can be reformulated in terms of the probability of failure such that the achieved probability of failure (Pfs ) is always smaller or equal to the target or acceptable probability of failure (Pft ): Pfs ≤ Pft ⇒ P(G i (X ) ≤ 0) ≤ 8(−βti ) ⇒ 8(−βsi ) ≤ 8(−βti ) ⇒ βsi ≥ βti

2 Methodology 2.1 Reliability-based design optimization In reliability-based optimization formulations, the constraints are made probabilistic to take care of uncertainties in design variables and still make sure that all desirable performance criteria are met. Let us consider a deterministic constraint of the form: gi (X ) ≤ gˆ where gˆ is the limiting value for this constraint. This constraint is transformed into probabilistic constraint as: G i (X ) ≡ gˆ − gi (X ) A typical probability constraint is represented as: P G i (X ) > 0 ≥ Ri where Ri are the desired reliability level and the probabilistic constraint are described by the performance function Gi (X) with Gi (X) ≤ 0 indicating failure. The probability of failure is statistically defined by a cumulative distribution function as: Z Z P G i (X ) ≤ 0 = FG i (0) = .... f X (X )d X G i (X )≤0

≡ 8(−β); where fx (X) is a joint probability density function, which needs to be integrated. Here, 8(.) represents the cumulative distribution function of standard normal distribution with β being the reliability index (Liu and Kiureghian 1991). This definition can be transformed in terms of probability of success and using the complementary property of cumulative distribution function (total area under the probability

In system parameter design, using the above relationship in terms of the reliability index, the reliability-based multiobjective design optimization (RBMODO) problem can be stated as: Minimize f (X ) = ( f 1 (X ), f 2 (X ), ..., f M (X )) Subject to: βsi ≥ βti where, i = 1, ..., p Here, β ti is the prescribed reliability target corresponding to ith constraint. This formulation is used for all reliability constraint formulations in this work. One striking difference between the deterministic- and reliability-based optimization is that, in the reliability-based optimization, the optimized parameters are the “mean” optimal values rather than “the” optimal values. This is because, reliability-based optimization formulation assumes variation about “mean” values and makes sure that the optimal design thus arrived do not fail in any performance criteria (e.g., constraints) due to these variations. In reliability-based optimization approaches, the additional computation step is computation of safety reliability index β si . Two approaches have been tried for computation of β si : Approximate moment approach Approximate moment approach (AMA) does not require information about the probability distribution of design variables, but require derivative information of performance constraints with respect to design variables about their mean values. The performance function G(X) is expanded at the mean value point using Taylor series:

G(X ) ∼ = G(µ) +

n X ∂G(µ) i=1

∂ Xi

(X i − µi )

(1)


The variance is approximated as: σG2

Z∞ =

either an MPP search algorithm that has been specifically developed for the first-order reliability analysis or a general optimization algorithm can be used. Due to its simplicity and efficiency, the HL-RF method is a popular choice for conducting a reliability analysis in RIA, and the same has been used in this work (Yang et al. 2000; Liu and Kiureghian 1991). The iterative algorithm of the HL-RF method is,

[G(X ) − G(µ)]2 d X

−∞ ∞

∼ =

Z n X ∂G(µ) 2 ∂ Xi

i=1

=

n X i=1

257

(X i − µi )2 d X

−∞

∂G(µ) σXi ∂ Xi

2 (2)

(k) ∇G u (k) n (k) = u .n n +

∇G u (k) (k)

(k)

(6)

where,

The safety reliability index is computed as: G(µ) βsi = σG

u

(k+1)

n (3)

This is the simplest possible approach and no detailed probabilistic distribution models are required. But this approximation may yield inaccurate results if X is not close to the mean value µ, which occurs if the standard deviation ó of the random variable X is large (Youn and Choi 2004). If the distribution of X is nonnormal with high nonlinearity (e.g., Gumbel distribution), the corresponding reliability using AMA is usually more conservative. First-order reliability index approach This is a detailed probabilistic analysis-based evaluation approach. In this case, the analysis is performed in two different random spaces: the original random design variable space (X-space) for design optimization and the independent standard normal space (U-space) for reliability analysis (Liu and Kiureghian 1991; Youn and Choi 2004). During the reliability-based multiobjective optimization process, a transformation between X and U space at design points must be carried out for estimating the probabilistic constraints. The transformation between two different random spaces at the design point dk is defined as:

(k)

∇G u (k) = −

∇G u (k)

is the steepest descent direction of the performance function G(u) at u(k) . The first term on the right side of the iterative relation above finds a direction with the shortest distance to the failure surface, and the second term is a correction term to reach G (U). Post optimization, the probability of failure (Pf ) is computed using the following formulae: .√ i 1h Pf = 8(−β) = 1 + er f −β 2 for β ≤ 8.0 2 =

β2 1 √ e− 2 for β > 8.0 β 2π

2.2 Multiobjective optimization technique

(5)

A multiobjective optimization problem (MOOP) involves a number of objective functions to be optimized simultaneously. One of the striking differences between single and multiobjective optimization is that in MOOP, the objective functions lie in a multidimensional space, in addition to the usual multidimensional design variable space. For each solution, X in the design variable space, there exists a point in the objective space Z = [f1 (X), f2 (X), ...., fm (x)]. Mathematically, this becomes a partial order and we require some higher level information to transform this into a total order to enable comparison operations (Kreyszig 1989; Miettinen 1999). The output of a MOOP is not a single point, but a set of nondominated points designated as pareto optimal set, which conforms to a partial order relation. A pareto optimal solution that is better with respect to one objective requires a compromise in at least one other objective (Kreyszig 1989). There are two principal goals of multiobjective optimization:

The minimum point on the failure surface is called the most probable point (MPP) u ∗G(u)=0 and the safety reliability

index is defined by βs = u ∗G(u)=0 . To find the solution to

1. Find a set of solutions close to the pareto optimal solutions, and 2. Find a set of solutions, which are diverse enough to represent the entire spread of the pareto optimal front.

U = T (X ) where d k = µk (X )

(4)

It is assumed that, no correlation exists among the design variables. This requires probability distribution information for each input random variable. Most of the transformations from X to U-space are nonlinear, except the normal distribution. In reliability index approach (RIA), the first-order safety reliability index is obtained by formulating an optimization problem with an equality constraint in U-space, which is the failure surface, as minimize kU k subject to G(U ) = 0

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Initially, a random population in P0 is created. This parent population is sorted into multiple nondominated levels with each solution assigned a unique nondomination level with 1 being the best. Then crowded tournament selection operator (to maintain diversity), recombination using SBX operator, and polynomial mutation operator are used to create an offspring population Q0 . The basic algorithm works as follows (refer to Figs. 1 and 2):

Fig. 1 A schematic of the algorithm for elitist selection

After a set of nondominated (constrain nondominated in constrained cases) optimal solutions in objective space is found, user can then use higher level qualitative information (e.g., specify an additional importance specification/utility vector for a particular variant of a vehicle to generate an optimal design) to make specific choices. For an ideal multiobjective optimization procedure, there are two steps (Deb 2001): 1. Find a diverse pareto optimal set of nondominated solutions. 2. Choose one of the candidate optimal solutions based on higher level information. The algorithm (Deb et al. 2002) used for finding pareto optimal set has three basic features: 1. It uses elitist principle for population generation. 2. It uses an explicit diversity preserving mechanism, and 3. It emphasizes nondominated solutions in a population.

Fig. 2 A set of solutions and the corresponding best nondominated front

Step 1: Combine parent and offspring population, Rt = Pt ∪ Q t Step 2: Perform nondominated sorting of Rt into multiple fronts F = (F1 , F2 ,...), Step 3: Set Pt+1 = null and i = 1 Step 4: Until |Pt+1 | + |Fi | ≤ N compute crowding distance for each Fi Pt+1 = Pt+1 ∪ Fi check the next nondominated front for inclusion by setting: i = i + 1 Step 5: Perform a sorting operation on crowding distance (in descending order of the crowding distance). Include the most widely spread (N − |Pt+1 |) solutions of Fi in Pt . Step 6: Create an offspring solution Qt + 1 from Pt+1 by applying the crowded tournament selection, crossover (SBX operator) and mutation (polynomial mutation) operators: Qt+1 = make-new-population (Pt+1 ) Step 7: Increment the generation counter: t = t+1 The optimization code is capable of handling different kinds of variables occurring in the same problem. For example, real variables, enumerated variables, integer variables, and binary variables can be used in the same optimization problem. The way recombination operation is performed


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remains the same. But crossover and mutation operators used for nonbinary variables are different. Simulated binary crossover (SBX) and polynomial mutation operators are used in this study for all nonbinary variables (Deb 2001). Here, a brief introduction is provided about the three genetic operators used in this framework and start with the crowded tournament selection operator. This selection operator is a modified variant of tournament selection operator (Deb 2001), with an additional emphasis on diversity preservation in addition to fitness value based ranking (resembles to sharing function approach (Deb 2001) used for handling multimodal problems in single objective). The crowded tournament selection operator compares two solutions at a time and returns the winner based on two attributes: 1. A nondominated rank ri in the population. 2. A local crowding distance di in the population, indicating diversity. It is a measure of search space around ith solution not occupied by any other solution in the population (Deb 2001). A solution i wins a tournament against solution j if it has a better rank (ri < rj ) or has a higher crowding distance (di > di ), provided they share the same rank (ri = rj ). Now a brief overview of the crossover operator called the simulated binary crossover (SBX) for modeling nonbinary variables is given. The SBX operator applied between two parent values p1 and p2 is as follows: Step 1: Choose a random number u ∈ [0,1). Step 2: Calculate β q using  1 ηc +1    (2u) , if u ≤ 0.5; η 1+1 βq = c 1   , otherwise  2(1 − u) Step 3: Compute two offspring solutions as follows: C1 = 0.5 (1 + βq ) p1 + (1 − βq ) p2 , C2 = 0.5 1 − βq p1 + 1 + βq p2 . Please note that the random number u cannot take the value of 1. This will avoid any numerical problem (divide by zero) while computing β q in step 2. Here, η c is the crossover distribution index. It is any positive real number and is usually kept in the range (Yang et al. 1994; Deb et al. 2002) and controls the spread of offspring solutions. It can be observed that a larger value of η c gives a higher probability for creating solutions “close” to the parents and a smaller value of η c allows distant solutions to be selected as offspring. It has been observed that in case of highly nonlinear responses, a smaller values of η c yield better results (avoids suboptimal distribution of nondominated points). Note that two offspring are symmetric about the parent solutions and thus avoids any “biasing” towards a particular parent solution. For a constant

Fig. 3 The point (K) on the pareto front having largest distance from line AB, connecting individual minima, is termed knee point

η c the offspring have a spread proportional to that of the parent solutions:

C1 − C2 = βq ( p1 − p2) This relationship allows solutions to converge under the action of genetic operators because distant solutions are not allowed focusing the search to a narrow region. Now the mutation operator used for modeling the nonbinary variables, termed polynomial mutation operator, is described in a nutshell [refer (Deb 2001) for details]. The polynomial mutation operator perturbs the solution c (with a random number r ∈ [0,1]) to create a new solution cnew , as follows: cnew = c + x (u) − x (l) δ, where x(u) and x(l) are upper and lower bounds of the variable x and

δ=

 1  (2r ) ηm +1 − 1, if r ≤ 0.5; 1



1 − [2(1 − r )] ηm +1 , if r ≥ 0.5

Here, η m is the mutation distribution index controlling the spread of mutated solution cnew from c and is kept in the range [10, 100] in general. For highly nonlinear responses, smaller values of η m should be used. One has to select one pareto optimal point instead of many and this requires a higher level, user-specified information. Mathematically, this operation converts a partial order relation (pareto optimal set) to a total order relation to enable comparison. Most of the time, the chosen point belongs to a region in the pareto optimal front that “bulges” out the most and lie somewhat “in the middle” of the front. This point is termed as the “knee point”. It is characterized by a point that lies farthest from the surface connecting each individual optimal point for each objective (refer to Fig. 3). Such a design point generally demarcates distinct regions in objective space. A point given by the optimal values of each individual objective is termed as the “utopia point”. This point cannot be attained in practice in the presence of conflicting objectives.

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2.3 Reliability-based multiobjective optimization formulation In system parameter design, the reliability-based multiobjective design optimization (RBMODO) problem can be stated as: Minimize f (X ) = ( f 1 (X ), f 2 (X ), ..., f M (X )) Subject to: P(G i (X ) ≥ 0) ≤ 8(−βti ) where, i = 1, ... p 8(.) represent the cumulative distribution function of standard normal distribution with β ti being the prescribed reliability target corresponding to ith constraint. The constraints can also be cast in another format relating safety reliability index β si to prescribed reliability target such that β si ≥ β ti . This formulation is used for all reliability constraint formulations in this work. One striking difference between the deterministic- and reliability-based optimization is that, in the reliability-based optimization, the optimized parameters are the “mean” optimal values rather than “the” optimal values. This is because, reliability-based optimization formulation assumes variation about “mean” values and makes sure that the optimal design thus arrived do not fail in any performance criteria (e.g., constraints) due to these variations. In reliability-based optimization approaches, the additional computation step is computation of safety reliability index β si . In this study, safety reliability index β is computed using two different approaches described before: AMA (approximate moment approach) and RIA (reliability index approach). Other reliability computation methods like the performance measure approach (PMA) could also be used. But because only normally distributed design variables are used, only the AMA and RIA were deemed adequate (Youn and Choi 2004).

Table 1 Details of CoV for design variables assuming normal distribution

X1

Variables

X2

X3

X4

X5

X6

X7

X8

X9

(CoV) = σ µ 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.02 0.02

Subject to: Abdomen load (kN) ≤ 1.0 Rib deflection upper (mm) ≤ 28.0 Rib deflection middle (mm) ≤ 30.0 Rib deflection lower (mm) ≤ 38.0 Public symphysis force (kN) ≤ 4.4 B-pillar velocity(m/s) ≤ 10.0 V*C upper(m/s) ≤ 0.28 V*C middle(m/s) ≤ 0.32 V*C lower(m/s) ≤ 0.35 The corresponding reliability based multiobjective design optimization (RBMODO) statement becomes: Minimize

Structural weight = f 1 (X )

Minimize

Door velocity = f 2 (X )

Subject to: βt1 ≥ 6.0 - - - - - Constraint on abdomen load βt2 ≥ 6.0 - - - - - Constraint on rib deflection upper βt3 ≥ 6.0 - - - - - Constraint on rib deflection middle βt4 ≥ 6.0 - - - - - Constraint on rib deflection lower βt5 ≥ 6.0 - - - - - Constraint on pubic symphysis force βt6 ≥ 6.0 - - - - - Constraint on B-pillar velocity βt7 ≥ 6.0 - - - - - Constraint on V*C upper βt8 ≥ 6.0 - - - - - Constraint on V*C middle βt9 ≥ 6.0 - - - - - Constraint on V*C lower

3 Application: side impact problem statement In a side impact scenario, reducing the door intrusion velocity (Vdoor ) (hence, its rate of change of linear momentum) via structural upgrading of the body side help reduce the severity of the subsequent momentum exchange between the door and the occupant. Therefore, to optimize the side impact performance along with minimum structural weight, under the crash safety constraints, both structural weight and door intrusion velocity are minimized. This requirement can be formulated as a multiobjective optimization problem. The multiobjective optimization problem for side impact is posed as follows: Minimize

Structural weight = f 1 (X )

Minimize

Door velocity = f 2 (X )

In this case, all the design variables are assumed to be normally distributed with coefficient of variation given in Tables 1, 2, and 3. The multiobjective optimization is performed using inhouse GDOT (Generic Design Optimization Toolkit) code, Table 2 Details of GDOT specific parameters used

GA parameter name Population size Number of generations Probability of crossover (ρc ) Probability of mutation (ρm ) Distribution index for crossover (η c ) Distribution index for mutation (η m ) Generation number for initialization of archival database Required precision of design variables for archival

Value 100 100 0.9 0.1 2.0 20.0 20 4


Table 3 Side impact requirements as per EEVC

Performance measure HIC Abdomen load (kN) Pubic symphysis load (kN) Rib deflection upper (mm) Rib deflection middle (mm) Rib deflection lower (mm) V*C upper (m/s) V*C middle (m/s) V*C lower (m/s)

EEVC regulation Constraint values used 650 1.0 4.4

1,000 2.5 6.0 42

28

42

30

42

38

1.0 1.0 1.0

0.28 0.32 0.36

developed based on NSGA-II algorithm (Deb et al. 2002). The real parameter GA (Genetic Algorithm) parameters and GDOT specific variables chosen are as follows for all cases considered in this study:

4 Meta-model development for automotive side impact A typical vehicle side impact model is shown in Fig. 4. With a finite element dummy model, the total number of elements in this model is about 92,000 while total number of nodes is around 96,800. The initial lateral velocity of the moving deformable barrier (MDB) is 50 kph and has mass of 950 kg. The struck vehicle and MDB are at 90 degrees with the vehicle moving at a velocity of 56 kph. For side impact protection, the vehicle design must meet internal and regulatory safety requirements specific to vehicle market. In this study, European Enhanced Vehicle-Safety Committee (EEVC) side impact test configuration is used. The dummy safety performance is used as the main metric for side impact that includes head injury criterion1 (HIC), abdomen load, pubic symphysis force1, rib deflections and viscous criterion1 (V*Cs). These dummy responses must meet at least EEVC requirements. Other concerns from the structural viewpoint are the velocity of B-pillar at the middle point and front door velocity at the B-pillar. There are nine design variables used for this optimization study, making the design space nine dimensional. Of these, there are seven-thickness variable, which are continuous and remaining design variables relate to material yield stress (σ y ). The material design variables can be treated as discrete or continuous. The design space bounds adopted in this study are shown in (Table 4). This study looks at “optimal” material properties of alloy steels, including the advanced high strength steel (AHSS), for this application. This can subsequently be taken as an input for new material synthesis, if required. A response surface model is built based on the design domain defined above. An indirect benefit of the response models is that the main effects

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of the responses are determined explicitly and can be used as preliminary design rules by the designers. The response surface model or the metamodel is employed to develop a “model-free” nonlinear model based on a set of multivariate data. Optimal Latin hyper cube sampling (LHS) was used for generating the sample design points. The optimal LHS design tries to minimize bias part of the mean square error by distributing the sample points uniformly over the entire design domain, using minimization of a nonnegative entropy criterion given by H(X) = E[−ln fx (x)]. This usually results in a better global representation. The number of CAE simulations in the optimal LHS is determined by total number of design variables involved. The minimum number of sample points for creating the response models is 3N, where N is the number of design variables. In this study 4.5N sample points are used for building the response surfaces. There are many computational methods available for building the response model using regression analysis and the quadratic backwardstepwise regression technique (Gu et al. 2001) is used for building the response models in this study. The quadratic backward-stepwise regression begins with a model containing all quadratic regressors. Deleting trivial regressors one at a time arrives at a final regression model. This model contains sets of regressors that have large effects on the response. In the above stepwise regression formulations, the structural weight is assumed to be a linear function of design variables while all other responses are assumed to be quadratic function with first-order interaction terms. Note that the structural weight considered here is calculated only from those parts in which thickness are defined as design variables. The response surfaces used for the side impact study are as follows (Gu et al. 2001): Structural weight = 1.98 + 4.90*X 1 + 6.67*X 2 + 6.98*X 3 + 4.01*X 4 + 1.78*X 5 + 2.73*X 7

(7)

Door velocity = 16.45 − 0.489*X 3 *X 7 −0.843*X 5 *X 6

(8)

Abdomen load = 1.163 − 0.3717*X 2 *X 4 −0.484*X 3 *X 9

(9)

Rib deflection upper = 28.98 + 3.818*X 3 − 4.2*X 1 *X 2 + 6.63*X 6 *X 9 − 7.70*X 7 *X 8

(10)

Rib deflection middle = 33.86 + 2.95*X 3 −5.057*X 1 *X 2 −11.0*X 2 *X 8 − 9.98*X 7 *X 8 +22.0*X 8 *X 9

(11)

Rib deflection lower = 46.36 − 9.90*X 2 1

Refer to Appendix for definition of these response quantities.1.0.

−12.9*X 1 *X 8

(12)

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K. Sinha

Fig. 4 Vehicle side impact model

V *C middle = 0.214 + 0.00817*X 5 − 0.131*X 1 *X 8

Pubic symphysis force = 4.72 − 0.5*X 4 − 0.19*X 2 *X 3

(13)

−0.0704*X 1 *X 9 + 0.031*X 2 *X 6 −0.018*X 2 *X 7 + 0.021*X 3 *X 8 +0.121*X 3 *X 9 − 0.00364*X 5 *X 6 (16)

B-pillar velocity = 10.58 − 0.674*X 1 *X 2 − 1.95*X 2 *X 8

(14) V *C lower = 0.74 − 0.61*X 2 − 0.163*X 3 *X 8 −0.18*X 7 *X 9 + 0.227*X 22

(17)

V *C upper = 0.261 − 0.0159*X 1 *X 2 − 0.188*X 1 *X 8 − 0.019*X 2 *X 7 + 0.0144*X 3 *X 5 + 0.08045*X 6 *X 9

(15)

Table 4 Design space definition

Design variable Thickness of B-pillar inner (X1 ) Thickness of B-pillar reinforcement (X2 ) Thickness of floor side inner (X3 ) Thickness of cross members (X4 ) Thickness of door beam (X5 ) Thickness of door belt line reinforcement (X6 ) Thickness of roof rail (X7 ) Material yield stress for B-pillar inner (Gpa) (X8 ) Material yield stress for floor side inner (Gpa) (X9 )

Lower bound Upper bound Baseline design 0.5 1.5 1.0 0.5

1.5

1.0

0.5

1.5

1.0

0.5

1.5

1.0

0.5

1.5

1.0

0.5

1.5

1.0

0.5

1.5

1.0

0.192

0.750

0.192

0.192

0.750

0.192

These explicit relations are used to compute responses during deterministic and robust multiobjective optimization. These models are validated at six more random validation points using LS-DYNA simulation and variations were found to be less than 5% including all responses with structural weight being the most accurate of all responses modeled in this work. The optimized configuration (corresponding to reliability target of 6.0 using RIA) considering the knee solution chosen is used to update the LS-DYNA deck and confirmation runs are performed. HIC is not imposed as a constraint explicitly in the formulation, but computed for optimal solution found above. In subsequent discussion, the utopia or ideal point signifies a point in objective space where both objectives attain their individual minimum. This point is not achievable in practice for problems involving conflicting objectives. All the variables are considered continuous in this work. The design variables used in this study can be categorized into common and local design variables. The common design variables exist as parameters in both objective functions while local variables appear in any one of the objectives. A functional dependence table is given below illustrating the design variables involved in each computed response in this work. Looking at Table 5 (see Appendix) and referring back to (7) and (8), one can notice that structural weight is a linear function of all thickness variables except X6 (door belt


line reinforcement) and all are additive. Hence, all variables should tend to attain lower values to minimize weight. But in case of door velocity, higher values of design variables would minimize it.

5 Results and discussions Looking at the knee solution point data in Table 7 (see Appendix), design variables (X1 through X4 ) attain lower values, while (X5 through X7 ) attains higher values. This behavior can be explained by looking at the response (7)–(17) derived in the “Conclusion” section as the global response surface for this problem. Structural weight is a linear function of all thickness variables except for X6 (door belt line reinforcement) and all are additive. Hence, all variables should tend to attain lower values to minimize weight. But in case of door velocity, higher values of design variables would minimize it. The deterministic multiobjective optimization shows that both weight and front door velocity can be reduced simultaneously (by 34.08 and 5.55%, respectively), as compared to the baseline design with improved safety performance (see Table 6). In case of reliability-based multiobjective optimization with reliability target of 6.0 using RIA, the results show a remarkable reduction of 22.43% in structural weight and 6.73% in door velocity as compared to baseline design at the knee design point (see Table 11). Here, the pareto optimal solutions’ corresponding to the knee point is taken for both RIA-based and AMA-based solution as before and reliability is computed for each constraint using Monte Carlo simulations (MCS). In all cases 1,000,000 samples were used. The minimum computed reliability for any constraint observed was RMCS = 0.999997 for solution obtained using RIA. In case of AMA, a minimum reliability of RMCS = 0.999992 was observed for the constraint set (RMCS = 1.0—probability of failure). In case of reliability-based optimization, the constraints are not the performance measures themselves but their respective reliability index values or sigma levels. Figure 5 gives the pareto optimal fronts for various cases. The ranges and shape of the pareto optimal front changes with varying target reliability indices as compared to the deterministic case. All the pareto optimal fronts are obtained with the same population size (100) and number of generations (100). It is expected that reliability-based multiobjective problems could take more number of generations to “converge” relative to its deterministic counterpart with all parameters remaining the same. To understand this phenomena, two cases with 200 generations (all other parameters detailed in Table 4 remain the same) with target reliability of 6.0 and 3.0 using RIA. The resulting pareto optimal fronts are shown in Figs. 9 and 10. The knee design points change substantially for case with target reliability of 6.0, but not by much in the case of target reliability index of 3.0 (refer to Table 6). This quite

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clearly brings out the “convergence” issues with higher target reliability indices. Reliability-based multiobjective optimization problems with higher target reliability would require higher number of generations with other parameters being held constant. The optimal pareto fronts with a target reliability target of 3.0 with 100 and 200 generations (both with 100 population size) almost coincide (see Fig. 6 below). But the same is not true while the reliability target is higher (see Fig. 7). The range of optimal pareto front with reliability target of 6.0 is very different for 100 and 200 generations. Similar behavior was observed even while using AMA method for reliability index computation. The structural weight is much lesser as target reliability index is lowered from 6 to 3. This is understood because lowering of target reliability leads to less safer design. But front door velocity does not indicate any such definitive pattern. Lowering of target reliability does not seem to impact the door velocity in the same way as structural weight at the knee design point. This also indicates that there is a transformation of optimal pareto front for different target reliability index rather than a simple translation as usually perceived. But this could be a result of the behavior of some constraints dominating over the second objective function, ultimately leading to such behavior. Generically, it can be concluded that different target reliability indices would result in transformation of optimal pareto fronts. To investigate further into the observed behavior described above, a deterministic multiobjective run is performed with GDOT. The results are tabulated in Tables 6 and 9 (see Appendix). In this case, the structural weight is reduced in comparison to those obtained using in-built reliability, corresponding to the tabulated significant points on the optimal pareto front. But the door velocity matches with those obtained for target reliability index β ti = 3. As can be observed from Table 8 (see Appendix), second, fifth, and ninth constraints are “active” in terms of their target reliability index. Looking at the functional dependence table (Table 5), X2 , X3 , and X7 are the most important crossfunctional thickness variables and X8 and X9 become important material property variables for meeting occupant safety specific performance. A look at the constraint values (refer to Table 9) corresponding to the important points (knee and individual optima points) on the pareto optimal front indicate the active constraints. This information used in conjunction with (9)–(17), indicates that the second objective (door velocity) is overshadowed by active constraints (Papalambros and Wilde 1993). That is, the constraint boundary is playing a “limiting” role on the minimum door velocity. A closer look at Tables 7 and 10 would indicate that the thickness values of door beam (X5 ) and belt reinforcement (X6 ) are always 1.5 mm for knee solution points in all cases, that of rail roof (X7 ) reaches its upper limit for target reliability of six while that of floor side inner (X3 ) reaches its lower limit for all cases except for the case with target reliability of 6.0 using RIA. Therefore, look-

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Bi-objective pareto solution in objective space

16

Pareto front with RIA 3.0 Pareto front with RIA 6.0 Pareto front with AMA 6.0 Deterministic pareto front

b 15.5

Door velocity (m/s)

15

c 14.5

d

a 14

13.5

13 16

18

20

22 24 Structural weight

26

28

30

Fig. 5 Pareto optimal front for multiple cases considered: a deterministic pareto front; b pareto front for reliability target 3.0 (RIA); c pareto front for reliability target 6.0 (AMA); and d pareto front for reliability target 6.0 (RIA)

ing at (8) would indicate that the front door velocity is lower for the case with target reliability of 6.0 using RIA at the knee point. Tables 11, 12, and 13 (see Appendix) details the case where the reliability-based multiobjective problem with

reliability target 6.0 using RIA is run for 200 generations with all other algorithm specific parameters remaining the same. A close look reveals that the thickness of floor side inner (X3 ) design value reduces substantially, reducing the

Fig. 6 Pareto optimal front for reliability target 3.0 (RIA) with 100 and 200 generations


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Fig. 7 Pareto optimal front for reliability target 6.0 (RIA) with 100 and 200 generations

structural weight, but increasing the front door velocity corresponding to the knee design point on the optimal pareto front. One salient point that is observed in all cases, is that the optimized designs require higher yield advanced high strength steel (AHSS) for B-pillar inner, while relatively lower yield strength AHSS is adequate for floor side inner. This requires for heterogeneous grades of steel and one needs to look at joining technology to be used and associated cost for the same. In addition, forming simulation studies have to be performed to ascertain the formability aspects of this design.

6 Conclusion This study illustrates use of an integrated probabilistic design optimization framework capable of supporting multidisciplinary, multiobjective optimization process. This theme is central to any simulation based design synthesis approach. It has been shown that both weight and front door velocity can be reduced simultaneously (by 34.08 and 5.55%, respectively), as compared to the baseline design with improved safety performance for deterministic case (without accounting for uncertainty) and by 22.43 and 6.73%, respectively, with reliability-based optimization strategy. Different uncertainty quantification techniques have been studied in the context of multiobjective optimization. When

uncertainty is taken into consideration, the “optimal” pareto front shifts towards a “safer” region where parameter uncertainties no longer impact the feasibility of the optimal design solutions. It is natural to anticipate that a higher β ti -based “optimal” front is more “conservative”. But this does not seem to be true for both objectives in this study. This is because of the constraints in this formulation and the functional dependence table involving all responses. In addition, in reliability-based approaches, it is reported (Youn and Choi 2004) that AMA is conservative in terms of computing the safety reliability index as compared to RIA approach in general in case of singleobjective optimization. But this observation does not seem to hold for multiple objective case. In this study, it was observed that RIA based result seem more conservative with respect to structural weight, while AMA based result is more conservative with respect to front door velocity. It is demonstrated that reliability-based multiobjective optimization requires higher number of generations to “converge”, as expected. But this still initial days and more numerical investigation involving varying categories of problems are required before a “strong” claim can be made. What could perhaps be claimed with a degree of certainty is that the resulting “reliable” pareto optimal front with different target reliabilities does not result in only translational shit, but is combination of “translational” and “rotational” shifts. But this could only be valid for this problem using surrogate models. One needs to test different types of probability distributions for modelling design parameters to put forth a claim in this regard. Comparison with Monte Carlo simulation (MCS) was done for all cases

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and excellent closeness of reliability values was obtained. Results obtained suggest higher yield strength material perform better from energy absorption viewpoint (for the same structural weight) for materials having similar plastic strain to failure. This study indicates the “optimal” material for this application, in terms of the material yield stress value of 0.75 GPa, which maps to AHSS for B-pillar inner panel, while a relative lesser yield strength is adequate for the floor inner material. In the future, performance measure approach (PMA) needs to be implemented for formulating the probabilistic constraints as another reliability-based approach. In addition, single loop reliability estimation approaches for efficiency enhancement also needs to be investigated in this context. The accuracy of adopted global response surface for probabilistic constraint modelling also should be investigated in detail. Looking at a broader picture, this methodology can potentially fill the gap between numerically optimized system development and simulation-driven digital product development. This, in turn, helps realize numerical simulation-driven product development process by aiming to achieve designs that are “first time right”.

Appendix

Table 5 Details of design variable categorization—a functional dependence table

Description Design variables for structural weight (f1 ) Design variables for door velocity (f2 ) Abdomen load (G1 ) Upper rib displacement (G2 ) Middle rib displacement (G3 ) Lower rib displacement (G4 ) Pubic force (G5 ) B-pillar velocity (G6 ) VC upper (G7 ) VC middle (G8 ) VC lower (G9 )

X1 X2 X3 X4 X5 X6 Yes Yes Yes Yes Yes Yes

X7 X8 Yes

X9

Table 7 Results for reliability-based multi-objective optimization for 3σ level using RIA approach

Description

Baseline design

1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.192 0.192 29.05

Knee point Minima of structural weight only 0.501 0.5 0.6 0.624 0.507 0.5 0.55 0.546 1.5 0.501 1.5 1.344 1.1042 0.986 0.742 0.745 0.45 0.434 19.85 17.85

Minima of door velocity only 0.5 0.58 1.5 0.516 1.5 1.5 1.5 0.727 0.24 27.60

X1 (mm) X2 (mm) X3 (mm) X4 (mm) X5 (mm) X6 (mm) X7 (mm) X8 (GPa) X9 (GPa) Structural weight (kgs) Front door velocity (m/s) Abdomen load (kN) Upper rib intrusion (mm) Middle rib intrusion (mm) Lower rib intrusion (mm) Pubic force (kN) B-pillar velocity (m/s) VC upper VC middle VC lower G1 (sigma level) G2 (sigma level) G3 (sigma level) G4 (sigma level) G5 (sigma level) G6 (sigma level) G7 (sigma level) G8 (sigma level) G9 (sigma level)

15.118

14.28

13.453

0.6984

0.93

15.64 0.931

0.879

28.39

27.834

27.8

27.466

28.54

28.16

28.433

25.113

33.98

35.654

35.37

35.93

4.03

4.387

4.39

4.297

9.53

9.515

9.463

9.563

0.22 0.22 0.293

0.24 0.205 0.223 11.838 3.027 3.575 10.0 3.037 15.14 15.47 25.6 3.116

0.225 0.2 0.219 11.630 3.07 3.397 10.88 3.02 15.98 22.02 26.83 3.04

0.233 0.24 0.15 17.48 3.28 6.02 9.1 10.484 14.26 18.576 16.94 6.0

Yes Yes Yes

Yes Yes Yes Yes Yes Yes

Yes Yes Yes Yes Yes

Yes Yes Yes

Yes Yes Yes

Yes Yes

Yes

Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

Yes Yes Yes Yes Yes Yes Yes

Table 6 Results of deterministic multiobjective simulation

Description Baseline design Knee solution Minima of structural weight only Minima of door velocity only Utopia point

X1 1.0 0.5 0.5 0.5 NA

X2 1.0 0.5 0.64 0.5 NA

X3 1.0 0.5085 0.5 1.5 NA

X4 1.0 0.548 0.519 0.535 NA

X5 1.0 1.5 0.5 1.5 NA

X6 1.0 1.5 1.5 1.5 NA

X7 1.0 1.088 0.7385 1.5 NA

X8 0.192 0.748 0.747 0.749 NA

X9 0.192 0.4338 0.258 0.267 NA

Structural weight (kg) 29.05 19.15 17.18 27.15 17.18

Door velocity (m/s) 15.118 14.28 15.64 13.453 13.453


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Table 8 Results for reliability-based multiobjective optimization for 6σ level using AMA approach

Table 10 Results for reliability-based multi-objective optimization for 6σ level using RIA approach

Description

Baseline design

Description

Baseline design

X1 (mm) X2 (mm) X3 (mm) X4 (mm) X5 (mm) X6 (mm) X7 (mm) X8 (GPa) X9 (GPa) Structural weight (kg) Front door velocity (m/s) Abdomen load (kN) Upper rib intrusion (mm) Middle rib intrusion (mm) Lower rib intrusion (mm) Pubic force (kN) B-pillar velocity (m/s) VC upper VC middle VC lower G1 (sigma level) G2 (sigma level) G3 (sigma level) G4 (sigma level) G5 (sigma level) G6 (sigma level) G7 (sigma level) G8 (sigma level) G9 (sigma level)

X1 (mm) X2 (mm) X3 (mm) X4 (mm) X5 (mm) X6 (mm) X7 (mm) X8 (GPa) X9 (GPa) Structural weight (kgs) Front door velocity (m/s) Abdomen load (kN) Upper rib intrusion (mm) Middle rib intrusion (mm) Lower rib intrusion (mm) Pubic force (kN) B-pillar velocity (m/s) VC upper VC middle VC lower G1 (sigma level) G2 (sigma level) G3 (sigma level) G4 (sigma level) G5 (sigma level) G6 (sigma level) G7 (sigma level) G8 (sigma level) G9 (sigma level)

1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.192 0.192 29.05



15.118

14.13

13.454

0.6984

0.92

15.54 0.927

0.912

28.39

25.716

26.44

25.72

28.54

25.6

24.97

22.8

33.98

36.12

34.6

31.68

4.03

4.34

4.34

4.296

9.53

9.602

9.323

9.38

0.22 0.22 0.293

0.232 0.204 0.305 11.75 6.637 8.931 7.896 6.01 11.77 14.53 46.62 6.156

0.211 0.202 0.31 9.86 6.11 11.5 12.538 6.06 15.58 22.66 49.1 6.05

0.161 0.184 0.232 12.745 6.074 13.313 17.275 9.912 15.09 22.63 34.63 13.0

1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.192 0.192 29.05



15.118

13.56

13.45

0.6984

0.914

14.23 0.866

0.86

28.39

26.08

26.16

26.0

28.54

22.79

23.27

21.0

33.98

35.028

32.47

32.3

4.03

4.3

4.35

4.231

9.53

9.4

8.935

9.02

0.22 0.22 0.293

0.22 0.228 0.311 13.867 6.155 8.204 11.9 10.24 16.9 23.54 19.7 6.11

0.224 0.212 0.3143 17.79 6.58 7.97 17.64 6.04 21.61 20.14 23.48 6.11

0.2 0.228 0.224 19.283 6.15 9.3 18.37 14.77 21.02 24.965 18.03 6.115

Table 9 Results of deterministic multiobjective simulation (constraint response values)

Description

Abdomen Load (kN) 0.6984

Upper rib intrusion (mm) 28.39

Middle rib intrusion (mm) 28.54

Lower rib intrusion (mm) 33.98

Baseline design Constraint values for knee solution Constraint values for minimum of structural weight Constraint values for minimum of front door velocity

0.96

27.92

29.02

0.977

27.86

0.87

27.67

Pubic Force (kN)

VC upper VC middle VC lower

4.03

B-pillar velocity (m/s) 9.53

0.22

0.22

0.293

36.58

4.397

9.69

0.24

0.21

0.3495

27.2

35.2

4.398

9.432

0.211

0.202

0.349

26.1

36.57

4.31

9.68

0.237

0.241

0.24

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Table 11 Comparison between baseline and knee point for reliability target 6.0 using RIA (with 200 generations)

Description

X1

X2

X3

X4

X5

X6

X7

X8

X9

Baseline design Knee solution for reliability of 6σ (RIA)

1.0 0.5

1.0 0.71

1.0 0.62

1.0 0.57

1.0 1.5

1.0 1.5

1.0 1.5

0.192 0.75 (AHSS)

0.192 0.486 (AHSS/HSS)

a All

Structural weight (kg) 29.05 22.534 (22.32)a

Door velocity (m/s) 15.118 14.1 (13.97)

values within parentheses indicate validated results using LS-DYNA simulation.

Table 12 Constraint values at knee design point (200 generations)

Description

Abdomen load (kN) 0.6984

Upper rib intrusion (mm) 28.39

Middle rib intrusion (mm) 28.54

Lower rib intrusion (mm) 33.98

Baseline design Constraint values for knee solution

0.866 (0.874)

26.0 (25.91)

24.79 (25.12)

34.47 (34.22)

Pubic force (kN)

VC upper

VC middle

VC lower

4.03

B-pillar velocity (m/s) 9.53

0.22

0.22

0.293

4.35 (4.19)

9.30 (9.34)

0.24 (0.242)

0.212 (0.204)

0.222 (0.24)

G8 (sigma level) 23.12


Table 13 Sigma level for constraint values at knee design point (200 generations)

Description G1 (sigma level) Sigma level 18.618 of constraint values for knee solution







Definitions of some responses and notions used in this study:

References

1. Head injury criteria (HIC): It is the resultant acceleration at the CG of dummy head and is given by the quantity:

Deb K (2001) Multi-objective optimization using evolutionary algorithms. Wiley, Chichester, UK Deb K, Agrawal S, Pratap A, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197 Gu L, Yang RJ, Cho CH, Makowski M, Faruque M, Li Y (2001) Optimization and robustness for crashworthiness. Int J Veh Des 26(4) Kreyszig E. (1989) Introductory functional analysis with applications. Wiley Classics Library Liu PL, Kiureghian AD (1991) Optimization algorithms for structural reliability. Struct Saf 9:161–177 Miettinen K (1999) Nonlinear multiobjective optimization. Kluwer Academic Publishers, Boston Papalambros PY, Wilde DJ (1993) Principles of optimal design: modeling and computation. Cambridge University Press Yang RJ, Tseng L, Nagy L, Cheng J (1994) Feasibility study of crash optimization. ASME 69(2):549–556 Yang RJ, Gu L, Liaw L, Gearhart C, Tho CH (2000) Approximations for safety optimization of large systems. 26th ASME Design Automation Conference, DETC2000/DAC-14245, Baltimore, MD Youn BD, Choi KK (2004) Selecting probabilistic approaches for reliability-based design optimization. AIAA 42(1)

HIC = max

2.

3. 4.

5.

t2 1 ∫adt (t2 − t1 ) t1

2.5

(t2 − t1 ).

Where, a is the resultant acceleration expressed as a multiple of g and t1 and t2 are any two points in time during the crash of the vehicle which are separated by not more than a 15 ms time interval (e.g., t2 −t1 ≤ 15 ms). Viscous criteria (V*C): It is an injury criterion for chest and is defined as the maximum value of the instantaneous product of chest wall velocity (V) of the SID and chest compression (C) expressed in percent of chest depth. Pubic symphysis force: This refers to force measured at the pubic symphysis (region of the pelvis) of the side impact dummy (SID). Nondominated points: Let J1 , J2 ∈ Rm be two objective vectors where Ji =[J1 , J2 ..., Jm ]T . The J1 is said to dominate J2 strongly (in a minimization sense) if Ji1 < Ji2 ∀ i. If Ji1 < Ji2 ∀ i and Ji1 < Ji2 for at least one i, then J1 is said to weakly dominate J2 in a minimization sense. The corresponding point X1 is termed as an efficient solution. Reliability index (β ): In standard normal space or u-space, the most probable point (MPP) for failure is defined as the minimum distance from the origin and the “limit-state” or constraint function. This minimum distance is called the Hasofer and Lind “reliability index” (β ) (Liu and Kiureghian 1991).