A Multi-Objective Discrete Design Optimization Algorithm for Portable

Working Paper NCAC 2007-W-005

June 2008

A Multi-Objective Discrete Design Optimization Algorithm for Portable Concrete Barriers by Coupling Grey Relational Analysis with Successive Taguchi Method Murat Buyuk, Dhafer Marzougui Cing-Dao (Steve) Kan The National Crash Analysis Center The George Washington University 20101 Academic Way, Ashburn, VA 20147 USA Email: [email protected] Email: [email protected]

Hasan Kurtaran Gebze Institute of Technology Department of Design and Manufacturing Engineering PK. 141, 41400 Gebze/Kocaeli, Turkey

This working paper summarizes recent efforts and findings derived from NCAC research. It is intended to solicit feedback on the approach, scenarios analyzed, findings, interpretations, and implications for practice reported by the research team. The statements contained herein do not necessarily reflect the views or policy of the FHWA. Please forward comments or questions to the authors noted above. These efforts will ultimately be documented and made available to advance research efforts related to this topic and guidance for practice.

ABSTRACT In this paper an effective successive multi-objective discrete optimization algorithm is developed to find the optimum design of Portable Concrete Barrier (PCB) systems. Multi-objective discrete optimization algorithm is attained by coupling Taguchi’s Orthogonal Arrays (OAs) with Grey Relational Analysis (GRA) method while nonlinear explicit dynamic Finite Element Analyses (FEA) are employed to generate response points and evaluate the performance of the PCB systems in accordance with National Cooperative Highway Research Program (NCHRP) Report No. 350 guidelines. GRA is used to convert multiple performance characteristics into a single performance criterion. Taguchi’s method is used to generate Design of Experiment (DOE) tables and prediction of the optimum design is accomplished by using Analysis of Means (ANOM) concept. Longitudinal and lateral ride down accelerations, the vehicle roll angle and dynamic barrier displacement are used as the design objectives to minimize while barrier safety shape, length, width, opening gap and hook distance in the connection are considered as discrete design variables.

A Multi-Objective Discrete Design Optimization Algorithm for Portable Concrete Barriers by Coupling Grey Relational Analysis with Successive Taguchi Method INTRODUCTION PCBs are temporary roadside appurtenances that are often used to separate vehicle traffic from workzone areas. These portable and versatile longitudinal barriers are advantageous and handy while keeping errant vehicles away from workers behind them. As a longitudinal barrier, each PCB configuration is required to comply with the National Cooperative Highway Research Program (NCHRP) Recommended Procedures for the Safety Performance Evaluation of Highway Features Report-350 guidelines, which require an evaluation after full scale crash testing [1]. According to Test Level 3 of NCHRP Report No. 350 and as it is represented in Tables 1 and 2, the longitudinal barriers must be subjected to two full scale vehicle crash tests and the intention is to evaluate the strength of the section for containing and redirecting the vehicle. Table 1. NCHRP Report No. 350 Evaluation Criteria for NCHRP Test 3-11 [1]. Test Article Longitudinal Barrier

Test Designation

Test Vehicle

3-10 3-11

820 kg Small Car 2000 kg Pickup Truck

Impact Conditions Speed (km/h) (mph) 100 62.1 100 62.1

Angle (degrees) 20 25

Evaluation Criteria (Table 2) A,D,F,H,I,K,M A,D,F,K,L,M

In this study, 820-kg small car tests and simulations, which are designated as Test 3-10, are ignored based on previous research that demonstrated the performance of small cars easily meet safety performance standards [2-5]. Consequently, to be on the conservative side, Test 3-11 is considered to be the most critical compliance test for evaluating the structural responses upon multi-objective criteria, which are listed in Table 2. Table 2. NCHRP Report No. 350 Evaluation Criteria for NCHRP Test 3-11 [1]. Structural Adequacy Occupant Risk

Vehicle Trajectory

A

Test article should contain and redirect the vehicle; the vehicle should not penetrate, under-ride, or override the installation although controlled lateral deflection of the test article is acceptable. D Detached elements, fragments or other debris from the test article should not penetrate or show potential for penetrating the occupant compartment, or present an undue hazard to other traffic, pedestrians, or personnel in a work zone. Deformations of, or intrusions into, the occupant compartment that could cause serious injuries should not be permitted. F The vehicle should remain upright during and after collision although moderate roll, pitching, and yawing are acceptable. G It is preferable, although not essential, that the vehicle remain upright during and after collision. H Longitudinal and lateral occupant impact velocities should fall below the preferred value of 9 m/s (29.53 ft/s), or at least below the maximum allowable value of 12 m/s (39.37 ft/s). I Longitudinal and lateral occupant ride-down accelerations should fall below the preferred value of 15 G’s, or at least below the maximum allowable value of 20 G’s. K After collision it is preferable that the vehicle's trajectory not intrude into adjacent traffic lanes. L The occupant impact velocity in the longitudinal direction should not exceed 12 m/s and the occupant ride-down acceleration in the longitudinal direction should not exceed 20 G’s. M The exit angle from the test article should be less than 60 percent of the test impact angle, measured at the time of vehicle loss of contact with the test device.

1

Determining the right combination in the PCB design, which requires advanced perception of the critical design parameters that play a significant role on the performance, is crucial to prevent crash related injuries and fatalities. Various PCB design configurations are already operational and have been tested for their compatibility with NCHRP Report-350 guidelines [6-20]. However, the number of possible design options makes it infeasible to test every alternative PCB formation. Therefore, non-linear explicit dynamic FE numerical solvers are used to simulate and validate full scale crash tests [21-24]. Subsequently, different PCB designs are remodeled and a better design is sought with engineering intuition. However, considering the amount and discrete characteristics of the design variables, engineering insight and intuition is often not enough and can even be computationally very costly. Consequently, advanced optimization algorithms are needed that are coupled with FEA to find the optimum designs with a reasonable amount of costly numerical simulations. These optimization schemes propose effective and efficient automated design methodologies for complex structural design applications resembling PCBs. Nevertheless, even these state-of-the-art algorithms need to be advanced since they only account for single objectives. As a matter of fact, NCHRP Report 350 necessitates multiobjective criteria for full compliance and certification. In this paper, an effective multi-objective discrete optimization methodology based on successive Taguchi method and Grey Relational Analysis is developed and applied to automatically design modular PCB systems. The optimum modular PCB design combination is found by improving several performance criteria simultaneously while satisfying the imposed constraints on responses. Multiobjective optimization method is coupled with LS-DYNA [25] to solve subsequent modular FE models of the PCB systems for Test 3-11 conditions with a non-linear explicit dynamic numerical scheme. MODULAR FE MODEL GENERATION The impact performance of the PCB systems are influenced by five discrete independent design variables with different levels that are listed in Table 3 with their corresponding abbreviated symbols. These variables are considered as the key decision points of the design for a pin-and-loop type of connection. PCB design configurations differ significantly in their segment safety shape, segment length, segment width, barrier gap between the segments and the distance between the connecting hooks. For a full factorial design space 160 different combinations can be achieved by using these design options. Table 3. Combination of Independent Discrete Design Variables. Sequence and Number of Independent Discrete Design Variables 1 2 3 4 Barrier Segment Barrier Segment Barrier Segment Gap Between Safety Shapes Lengths Widths Barrier Segments Variable Levels

Abbr.

Variable

Abbr.

1

F

F-Shape

6

2

N

New Jersey

10

3

V

Vertical

12

4

S

5

I

Single Slope Inverted

20

Variable 1829 (6’) 3048 (10’) 3658 (12’) 6096 (20’)

mm mm

Abbr. N W

mm mm

2

Variable Narrow Version Wide Version

Abbr. N W

Variable Narrow Gap Wide Gap

5 Distance Between Connecting Hooks Abbr. Variable N W

Narrow Distance Wide Distance

Segment Safety Shapes Some barrier shapes have shown that they have a tendency to cause vehicle rollover [5]. For that reason, several research studies have been conducted to evaluate the crash and rollover performance of various temporary PCB designs, such as: the New Jersey safety shape, F-shape, and Single Slope barrier [1215]. In this study, five different widely used PCB safety shapes that can be seen in Fig. 1 are investigated for their Test 3-11 performance. The models are generated with fairly fine and mapped meshing by using solid hexagonal elements. Due to their superior contact behavior, solid elements of the PCB are covered with shell elements to reflect vehicle-to-barrier and barrier-to-barrier impact load transfer accurately. This kind of modeling technology is numerically found to behave better since curvatures on the PCB can be modeled in more detail. To be able to predict the tendency of vehicle rollover or the post impact trajectory, barrier facial profiles are also need to be accurately modeled. There is also a need to estimate a reasonable friction coefficient. In this study, the experimentally determined and numerically tuned friction values between PCBs and pavement are adopted, which are suggested by Marzougui et al. [2123].

Figure 1. Five segment safety shapes.

There is no failure modeled for the concrete material. The rebars and other structural details inside the PCB are also not considered. The loop connections are coupled with the faces of each segment accordingly. Barrier Segment Lengths Considering that the reduction in the length of the barrier segments is an effective means of decreasing the weight and enhancing portability, it also generally results in increased barrier deflections. That means, for a given connection design, the dynamic barrier deflection increases as the barrier segment length decreases. This is primarily due to the additional number of joints along the length of the system. The additional joints can introduce more slack into the system that, during the impact, translates into additional lateral barrier deflection. In this study, four different lengths are modeled. Considering the advantageous handling capability of the shorter segments during the installation and to be on the conservative side in terms of the amount of 3

dynamic deflection, the design matrix for the segment lengths is started from 1829 mm (6 ft) long segments up to 6096 mm (20 ft) barriers as it can be seen from Fig. 2.

Figure 2. Four segment lengths.

Barrier Segment Widths Barrier width also has a major influence on the overall PCB system response. Reducing the width of the barrier can dramatically decrease its structural competence. The width parameter changes the mass of the single segment without playing with the facial profile of the concrete barrier. In this study, keeping the safety shapes of the existing barrier technology that is illustrated in Fig.1, wider versions of all five different safety cross-sections are considered and demonstrated in Fig. 3.

Figure 3. Five wide segment safety shapes.

4

Gap Between Barrier Segments PCB segments are connected end-to-end by using pin-and-loop type of connection. These segments are coupled by a contact algorithm. The end face design also plays an important role during the impact loading. The details of two possible designs are shown in Fig. 4. The amount of slack and rotation is reduced and moment carrying capacity of the connection is increased by closing the gap between the two barriers by introducing a small design detail at the barrier ending. Open gap style is usually preferred for its convenience during the installation. An additional part is designed to avoid the amount of slack and is also coupled by a contact algorithm without affecting the performance of the pin-andloop connection.

Figure 4. Two design options for gap between the barrier segments.

Hook Distances The design of the joint connection has a direct control on the impact response of PCBs, particularly on the amount of dynamic barrier displacement. An inadequate joint design can induce vehicle instability, can lead to an unacceptable deflection and/or can cause failure of the connection and penetration of the vehicle through the protected area [5]. The overall performance of the PCB system is generally limited by the strength of the joint details, since the capacity of the connection is often less than that of the reinforced concrete barrier section [6]. In this study, a pin and loop type of connection, which is the most commonly used version, is selected for joining the barrier segments since prior research showed that the pin and rebar connection provided adequate structural capacity and it was mentioned that this connection type is approximately 50% less expensive than other available connections that could provide the necessary required strength [6, 18]. Fig. 5 represents two alternative locations for the rebars that will be used as alternative discrete designs in this study.

5

Figure 5. Two different barrier pin-and-loop connection details.

FE MODELING OF NCHRP REPORT 350, TEST 3-11 CONDITIONS The full scale test conditions, which are required by NCHRP Report 350 Test 3-11, dictate that a 2000 kg pickup truck will be impacting the longitudinal barrier at an impact speed of 100 km/h and with an oblique angle of 25 degrees [1]. So, to create such an impact scenario, a very widely used and accepted FE model of a Chevrolet-C2500 pickup model from FHWA/NHTSA- National Crash Analysis Center (NCAC) is employed for the analysis [26]. In this study, a non-linear explicit dynamics FE code, LS-DYNA [25], is utilized for the numerical simulations. This FE code has successfully been used several times before for some similar research [2124]. The overall simulation time is accepted as 0.5 seconds. The details of the vehicle model can be seen in Fig. 6 and a complete test setup is illustrated in Fig. 7, where a random selection of PCB design parameters from Table 3 is installed in front of the vehicle model with a 25 degree impact angle. No. of Parts No. of Nodes No. of Elements

Figure 6. Chevrolet C2500 FE model [20].

6

: 248 : 66050 : 57850

Figure 7. Numerical modeling of NCHRP Report 350, Test 3-11 Condition.

MULTI-OBJECTIVE OPTIMIZATION PROBLEM FORMULATION The Multi-objective Optimization (MOO) design problem can be defined as determining the optimal values of the design parameters in order to obtain maximum or minimum of different objective functions simultaneously. The general MOO problem can be formulated in the following standard mathematical format: Find: Minimize/Maximize: Subject to:

(1) (2) (3) (4) (5)

x F (x) = [f1 (x), f 2 (x), …, fi (x)] g j (x) ≤ 0

hk ( x ) = 0 xiL ≤ xi ≤ xiU

j = 1,...m

k = 1,...q i = 1,...n

where x is the design variable vector, F(x) is the vector of multi-objective functions, g j ( x ) and hk ( x ) are the inequality and equality constraint functions respectively with m and q are being the number of constraints, xiL and xiU are the lower and upper bounds of the design parameters. Objective and constraint functions may correspond to accelerations, rotations, displacements, stresses, mass or weights in structural optimization problems. Multi-objective functions are often converted into an equivalent single objective function before the solution using weighted sum approach as [27]: F (x) = w1

f1 f1*

+ w2

f2 f 2*

+ .... + wi

fi f i*

(6)

where wi indicates i-th weight coefficient and fi* is the scaling value for i-th objective. Selection of appropriate scaling values for normalizing objectives are extremely important to find an optimum that simultaneously improves all objectives. These values are often chosen by experience or from the solution of single optimization problems that are time consuming.

7

However, this methodology has some drawbacks. Although this approach looks relatively simpler, it relies on the appropriate selection of the weighting coefficients, where initially described relative weight balance of objective functions may not remain still due to the varying objective values during the optimization iterations especially when highly nonlinear objective functions are considered. Therefore, a dynamic or adaptive approach should be adopted to keep the weight balance of the objectives constant. During the solution of Equations (1)-(5), the idea of Pareto optimality is used since a single point that minimizes or maximizes all objectives simultaneously usually does not exist. A solution point is Pareto optimal if it is not possible to move from that point and improve at least one objective function without detriment to any other objective function. Although Pareto optimality is helpful in selecting a compromising optimum for all objectives, it requires repetitive solutions of the optimization problem, which will be prohibitively computationally costly in structural optimization especially when coupled with Finite Element simulations. In this study, a Pareto optimal point is effectively and efficiently sought by transforming discrete MOO problem that is described in Equations (1)-(5), into an equivalent unconstrained single objective problem through penalty functions and GRA based Taguchi method. GRA provides an effective way to keep relative importance of the objective functions as initially assigned by normalizing each objective function separately so that maximum value equals unity. GREY RELATIONAL ANALYSIS Grey Relational Analysis (GRA) is adopted as an effective multi-criteria decision making method. The method was developed by Deng in the mid 80’s and it has been used in the solution of several multiobjective industrial optimization problems since then [28]. In the GRA, data series are first normalized. Each data series corresponds to discrete values of an objective function and represented in the form of multiple objective functions. A linear normalization between null and unity is usually performed. The normalization step is often referred to generating Grey Relations. To explain the normalization procedure for an objective function with GRA, let fij to indicate the value of function f i for j-th experiment. If objective function fi is to be minimized, then normalized value for j-th experiment ( fij' ) is expressed as: fij' =

{ } max { fij } − min { fij } max fij − fij

(7)

The grey relational coefficient γ i ( j ) is then calculated to express the relationship between the optimum and the actual normalized data as: γ i ( j) =

Δ min + ξΔ max Δ i ( j ) + ξΔ max

(8)

where Δ i ( j ) is the j-th value in difference data series Δi and ξ ∈ [1, 0] . The coefficient ξ is used to compensate the effect of Δ max when it gets too big in the data series, and thus enlarges the difference 8

significance of the relational coefficient. In general, the value for ξ is set to 0.5 [28]. Difference data series Δi can be expanded as:

(

Δi = f01 − fi1' , f02 − fi'2 ,..., f 0 j − fij' ,

)

(9)

where f01 , f02 ,..., f0 j represent reference values for the normalized objective function values for comparison. Δ max and Δ min represent respectively the global maximum value and global minimum value in the difference data series: Δ max = max ( max Δi ) Δ min = min ( min Δi )

(10) (11)

The final step of GRA is the calculation of Grey relational grade by averaging the grey relational coefficients corresponding to each performance characteristics as: Γi =

1 m

m

∑ γ ( n) i

(12)

n =1

where Γ i is the grey relational grade for the j-th experiment and m is the number of performance characteristics (i.e. multiple objective functions). Grey relational grade at an experiment corresponds to a single equivalent normalized value of multiple objective functions. A higher grey relational grade represents that the corresponding experimental result is closer to the ideally normalized value. In the literature, optimum values are often chosen as the experiment with the highest grey relational grade. In this study, Taguchi’s analyses of means concept as explained in the following section is applied to grey relational grades to select the optimum design parameters. Upon application of GRA, function values for experiments will be ranked between zero and one. One will correspond to the best objective value; zero will correspond to the poorest objective value within design set. Also Upon normalization all objectives are now of the same order. Objective goals are converted into maximization of all normalized penalized objective functions. MULTI-OBJECTIVE DISCRETE OPTIMIZATION ALGORITHM WITH TAGUCHI’S METHOD Taguchi’s optimization methodology is a powerful method in finding robust optimum designs. It was originally developed to solve single-objective unconstrained optimization problems. Later it was extended to the solution of constrained optimization problems through the penalty method [29]. In this study, Taguchi’s optimization method is enhanced to effectively solve an optimization problem in its most general form (multi-objective constrained optimization problems) by incorporating GRA. GRAbased Taguchi optimization methodology is designed and explained in the following five successive steps: Selection of Orthogonal Array Taguchi’s OAs provide well planned sets of experiments, in which all parameters of interest are varied over a specified range. OAs are balanced as if they have equal number of levels for each factor in every column of the OA. 9

In Taguchi’s method, an appropriate OA is selected according to the minimum number of candidate values for discrete design parameters. If all parameters have the same number of levels, standard OAs can be directly used. Otherwise, mixed-level OAs are chosen as the DOE method. Regarding the level number selection in OAs, two-level OA investigates the linear effects of the characteristics with respect to the factors, whereas three-level OA investigates a quadratic effect. After selection of the suitable OA, candidate values of discrete parameters are assigned to the columns of the OA table. During the assignment, Level 2 is allocated by the initial design. Level 1 is allocated by the discrete value, which is one step higher than the initial design. Similarly, Level 3 is allocated by the discrete value, which is one step lower than the initial design [29]. Calculation and Evaluation of Results for the Orthogonal Array Multi-objective and constraint values, which are obtained through conducting structural analyses, are collected as the experiment numbers from the OA table. In structural optimization, objective and constraint functions often come from Finite Element analysis. Using penalty functions, multiple objectives of: f1 ( x ) , f 2 ( x ) ,..., f n ( x )

(13)

f1 ( x ) + P1 ( x ) , f 2 ( x ) + P2 ( x ) ,..., f n ( x ) + Pn ( x )

(14)

are penalized as:

where Pi ( x ) are penalty functions defined as: P1 ( x ) = r1

m

∑ j =1

(

)

max 0, g j ( x ) , P2 ( x ) = r2

where ri are the penalty scale factors and

m

∑

(

)

max 0, g j ( x ) ,…, Pn ( x ) = rn

j =1

m

∑ max ( 0, g

j

j =1

m

∑ max ( 0, g j =1

j

( x ))

(15)

( x ) ) is the sum of the constraint violations.

The objective functions in Equation 14 are often referred to as penalized or transformed objective functions. Using penalized multi-objective functions, an equivalent single objective function is created as stated in Equation 6. In this study, to develop an effective penalty approach, multiple-penalty scale factor approach is adopted since optimization is to be conducted with multiple objective functions. Each penalty scale factor is determined independently considering the specific objective function for which penalty is applied if the constraint is violated. Each penalty scale factor is determined to be one order higher than the objective function under consideration. Implementation of Grey Relational Analysis Grey relational coefficients for each penalized objective function are calculated. Using Grey relational coefficients, Grey relational grades for the unified objective function (single equivalent objective function) are produced as described in Section 5. 10

Determination of Optimum Levels with Taguchi’s Analyses of Means Concept Optimum levels for discrete parameters are calculated by applying Taguchi’s ANOM concept to the Grey relational grades found in the previous step. The optimum level for a variable is chosen as the level that yields maximum mean of Grey relational grades. Confirmation analysis is carried out with the optimum levels to ensure that the design feasibility or the statistical validity of additivity in Taguchi’s concept. The result of optimum levels from ANOM is compared with that of best combination of variables in the OA and best levels are selected as the new levels (i.e. initial design) for the next optimization iteration. The analysis and interpretation of data stage covers determination of optimum levels by applying Taguchi’s ANOM concept to the results column attached to the OA. Optimum value for each factor (factor means variable in Taguchi method) is calculated using OA values. Taguchi optimization methodology can be defined with an example, where a L9 OA is shown in Table 4 and is used to find optimum values of four variables. Optimum value for Factor A: Mean values for all levels of factor A are calculated. The level with minimum or, depending on the objective, maximum mean is adopted as the optimum level. (16) (17) (18)

meanA1 = ( f1 + f 2 + f 3 ) / 3 meanA2 = ( f 4 + f 5 + f 6 ) / 3 meanA3 = ( f 7 + f8 + f 9 ) / 3

Optimum level for A = max ( meanA1 , meanA2 , meanA3 ) (for maximization) or Optimum level for A = min ( meanA1 , meanA2 , meanA3 ) (for minimization) This procedure is carried out likewise for the remaining factors of B, C and D, respectively. Table 4. L9 (34) Orthogonal Array.

Experiment no. 1

Factors A Column no. 1 1

2

1

2

2

2

f2

3

1

3

3

3

f3

4

2

1

2

3

f4

5

2

2

3

1

f5

6

2

3

1

2

f6

7

3

1

3

2

f7

8

3

2

1

3

f8

9

3

3

2

1

f9

B

C

D

2 1

3 1

4 1

Results f1

Check for Convergence Criteria Since Taguchi method is applied successively, appropriate criteria are needed to stop the optimization process, where the process is terminated if the difference between the last two iterations is less than the specified tolerance. In this study, 1 % is used as the convergence tolerance. 11

Unlike single objective optimization, MOO history may show oscillation (zig-zag) behavior, since it may not be possible to improve an objective without deteriorating others. In the case of conflicting objective functions, an additional convergence criterion that takes care of zig-zag behavior is also used in this study. Optimization is terminated if optimum has showed three successive oscillations. Fig.8 illustrates the steps of the multi-objective discrete optimization algorithm that is developed for this study.

Start Select Candidate Values for Design Variables

Select Orthogonal Array

Select Initial Design

Calculate Objective and Constraint Values

Perform Structural Analyses at Orthogonal Array Design Points

Assign Initial Design and Neighboring Design Values in Orthogonal Array

Calculate Penalty Scale Factors and Penalized Objective Values

Calculate Grey Relational Grades for Penalized MultiObjective Values (GRA)

Select Optimum Design using Analysis of Means of Grey Relational Grades (ANOM) Compare the Optimum Design with the Best Design in the OA

NO Check if Convergence Criterion is Satisfied ? YES Stop

Figure 8. Steps of successive multi-objective discrete optimization algorithm.

DISCRETE DESIGN OPTIMIZATION OF PCB SYSTEMS Accelerations, rotations of the vehicle and the displacement of the barrier upon vehicle impact are good indicators of a good PCB design and are considered as important performance criteria. An effective PCB design usually should minimize these response values and/or keeps under the specified limits as recommended in Report 350 [1]. In this study, PCB design combinations are changed by varying the barrier type, barrier length, barrier width, barrier gap and hook distance to achieve an optimum PCB design while satisfying multi-objective performance measures that are used simultaneously for both objectives and constraints. The PCB design problem corresponds to the following multi-objective discrete constrained optimization problem: 12

Minimize: Ride-Down Acceleration in x-direction: Ride-Down Acceleration in y-direction:

(19) (20)

ax ay

Ride-Down Velocity in x-direction: Barrier Displacement:

(21) (22)

vx Disp

or as an equivalent single composite objective function: (23)

Minimize: w1 ax + w2 a y + w3 vx + w4 Disp

Subjected to: Ride-Down Acceleration in x-direction: Ride-Down Acceleration in y-direction:

ax ≤ 15 G ay

≤ 15 G

Ride-Down Velocity in x-direction: Ride-Down Velocity in y-direction:

vx ≤ 9 m/s

Rotation Around x-axis (Rollover):

Rx ≤ 15

vy

≤ 9 m/s o

Using candidate values of discrete variables: Barrier Segment Safety Shape: Barrier Segment Length: Barrier Segment Width: Gap Between Barrier Segments: Distance Between Connecting Hooks:

{ F, N, V, S, I } { 6, 10, 12, 20 } { N, W } { N, W } { N, W }

(24) (25) (26) (27) (28)

(29) (30) (31) (32) (33)

where w1 , w2 , w3 and w4 are weighting factors (importance) for objectives. Objectives and constraints are imposed as absolute values of certain components for accelerations, velocities and rotations considering the vehicle coordinate system and impact direction. Design variables present a discrete set from which an optimum value is chosen. Orthogonal Array Design for Discrete PCB Variables Considering the discrete values of PCB design variables, an appropriate OA is selected for iterative Taguchi optimization process. Since there are five design variables, three of which have only two levels, a modified L36 OA, which is referred to as mixed-level L36 OA, is utilized in this study. Mixed-level OAs are usually obtained by merging mutually interactive columns in standard OAs. L36 mixed-level OA is used successively in PCB optimization. If the number of candidate designs for all design variables were more than three, standard OAs with three levels would be normally sufficient. Candidate values for design variables from Equations 29-33 are adapted to the OA levels as shown in Table 2, where interaction columns are ignored. L36 mixed-level OA can handle up to 11 variables with two levels and 12 variables with three levels, as well as, 3 variables with two levels and 13 variables with three levels (i.e. 32 x 23 ). Column numbers for main factors in Table 5 correspond to column numbers of 12, 13, 1, 2 and 3 in L36 mixed-level OA. 13

Table 5. L36 Mixed-Level OA Design. Experiment Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Discrete Design Variables Barrier Segment Barrier Segment Safety Shape Length F 6 N 10 V 12 F 6 N 10 V 12 F 6 N 10 V 12 F 6 N 10 V 12 F 10 N 12 V 6 F 10 N 12 V 6 F 10 N 12 V 6 F 10 N 12 V 6 F 12 N 6 V 10 F 12 N 6 V 10 F 12 N 6 V 10 F 12 N 6 V 10

Barrier Width N N N N N N N N N N N N N N N N N N W W W W W W W W W W W W W W W W W W

Segment

Gap Between Barrier Segments N N N N N N N N N W W W W W W W W W N N N N N N N N N W W W W W W W W W

Distance Between Connecting Hooks N N N N N N W W W N N N W W W W W W W W W W W W N N N W W W N N N N N N

OPTIMIZATION RESULTS AND DISCUSSION Results of multi-objective optimization are often presented in terms of Pareto optimal designs. In solving the MOO problem, weighting coefficients (i.e. importance) are varied for their respective objectives. Pareto optimal designs produced with GRA-based successive Taguchi optimization method are shown in Tables 6 and 7 for two randomly chosen starting design points. In the first case, the random initial PCB combination corresponds to a feasible design. MOO algorithm is used to produce a better design with improved objective values. Optimization history for the multiobjectives is demonstrated with respect to the number of iterations in Fig. 9.

14

Table 6. MOO Results for Case-I.

Safety Shape Barrier Length Barrier Width Barrier Gap Hook Distance ax (G’s) ay (G’s)

F 6 N N N 5.47

Optimum Design Points for Corresponding Weights w1 = 1 w1 = 0 w1 = 0 w1 = 0 w1 = 1 w2 = 0 w2 = 1 w2 = 0 w2 = 0 w2 = 1 w3 = 0 w3 = 0 w3 = 1 w3 = 0 w3 = 1 w4 = 0 w4 = 0 w4 = 0 w4 = 1 w4 = 1 F F V F F 10 6 12 12 20 N N W W W N N N N N N N W N N 4.10 5.47 12.63 4.94 4.17

9.93

11.04

9.93

12.27

10.24

10.15

vx

(m/s)

4.25

4.41

4.25

6.51

4.45

4.71

vy

(m/s)

5.64

5.79

5.64

6.30

6.40

6.33

Rz

(Degrees)

5.95

5.48

5.95

12.46

3.10

6.56

ax

(G’s)

5.47

4.10

5.47

12.63

4.94

4.17

ay

(G’s)

9.93

11.04

9.93

12.27

10.24

10.15

Rx Disp

(Degrees)

1.51

4.18

1.51

0.72

6.11

8.76

(mm)

1100.95

1131.68

1100.95

1042.75

621.26

624.66

20

Ride Down Acceleration ay (G's)

Ride Down Acceleration ax (G's)

Objectives

Constraints

Discrete Design Parameters

Random Initial Design Case-I

15 10 5 0 0

1

2

20 15 10 5 0 0

1

Initial Design [F-6-N-N-N]

1. Iteration [F-12-W-N-N]

2. Iteration [F-20-W-N-N]

Constraint Upper Limit

Initial Design [F-6-N-N-N] 2. Iteration [F-20-W-N-N]

10 7.5 5 2.5 0 0

1

2

1. Iteration [F-12-W-N-N] Constraint Upper Limit

1500 1250 1000 750 500 250 0 0

1

2

Number of Iterations

Number of Iterations Initial Design [F-6-N-N-N]

2

Number of Iterations

Dynamic Displacement Disp (mm)

Roll-Over Rx (Degrees)

Number of Iterations

Initial Design [F-6-N-N-N]

1. Iteration [F-12-W-N-N]

1. Iteration [F-12-W-N-N]

2. Iteration [F-20-W-N-N]

2. Iteration [F-20-W-N-N]

Figure 9. Change of objective values with optimization iterations for Case-I considering multi-objectives.

15

Table 7. MOO Results for Case-II.

Safety Shape Barrier Length Barrier Width Barrier Gap Hook Distance ax (G’s) ay (G’s)

V 12 N N W 18.89

Optimum Design Points for Corresponding Weights w1 = 1 w1 = 0 w1 = 0 w1 = 0 w1 = 1 w2 = 0 w2 = 1 w2 = 0 w2 = 0 w2 = 1 w3 = 0 w3 = 0 w3 = 1 w3 = 0 w3 = 1 w4 = 0 w4 = 0 w4 = 0 w4 = 1 w4 = 1 V S V S S 20 12 12 12 6 N W W W W W N N N N W N W N N 8.13 14.30 12.63 14.30 11.68

9.89

13.79

8.84

12.27

8.84

9.42

vx

(m/s)

5.41

6.53

4.59

12.27

4.59

5.54

vy

(m/s)

4.85

5.11

6.76

6.30

6.76

6.85

Rz

(Degrees)

2.81

13.61

5.76

12.46

5.76

3.82

ax

(G’s)

18.89

8.13

14.30

12.63

14.30

11.68

ay

(G’s)

9.89

13.79

8.84

12.27

8.84

9.42

Rx

(Degrees)

0.40

1.08

7.43

0.72

7.43

1.25

Disp

(mm)

1661.17

1661.17

700.55

1042.75

700.55

707.96

Ride Down Acceleration ay (G's)

Ride Down Acceleration ax (G's)

Objectives

Constraints

Discrete Design Parameters

Random Initial Design Case-I

20 15 10 5 0 0

1

2

3

20 15 10 5 0

4

0

1

Initial Design [V-12-N-N-W] 2. Iteration [S-6-N-N-N]

1. Iteration [V-10-W-N-N] 3. Iteration [S-6-W-N-N]

4. Iteration [S-6-W-N-N]

Constraint Upper Limit

Initial Design [V-12-N-N-W] 2. Iteration [S-6-N-N-N] 4. Iteration [S-6-W-N-N]

3

2

1

0 0

1

2

3

1. Iteration [V-10-W-N-N]

2. Iteration [S-6-N-N-N]

3. Iteration [S-6-W-N-N]

3

4

1. Iteration [V-10-W-N-N] 3. Iteration [S-6-W-N-N] Constraint Upper Limit

1500 1250 1000 750 500 250 0 0

4

1

2

3

4

Number of Iterations

Number of Iterations Initial Design [V-12-N-N-W]

2

Number of Iterations

Dynamic Displacement Disp (mm)

Roll-Over Rx (Degrees)

Number of Iterations

Initial Design [V-12-N-N-W] 2. Iteration [S-6-N-N-N]

1. Iteration [V-10-W-N-N] 3. Iteration [S-6-W-N-N]

4. Iteration [S-6-W-N-N]

4. Iteration [S-6-W-N-N]

Figure 10. Change of objective values with optimization iterations for Case-II considering multi-objectives.

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In the second case, the random initial PCB combination corresponds to an infeasible design. MOO algorithm is then first used to eliminate infeasibility as well as to produce a better design with improved multi-objective values. Optimization results by using different weight factors for objectives are listed in Table 7. Optimization history for the multi-objectives is demonstrated with respect to the number of iterations in Fig. 10. From Figures 9 and 10 it is seen that a significant amount of improvement is achieved on certain objective responses through the optimization method with only limited number of FE analyses. It is also discovered that F-12-W-N-N is the global optimum and there is a local optimum just around that point, which is F-20-W-N-N. For all cases the optimum design is found as F-Shape concrete barrier. It is very well known that NewJersey shape barriers can lead to vehicle rollover and over-riding the barriers. The other safety shapes are also causing violations of imposed design constraints. So, with the given constraints it is shown that the algorithm successfully chooses the F-Shape barrier design as the most effective safety shape. In accordance with an objective function defined to minimize the maximum deflection of the concrete barrier system, the algorithm has chosen 12’ as the optimum design length. It is noticeable to show that the amount of deflection for 12’ barrier is less than the amount found for 20’ barrier. The only reason for that is pointing out the amount of forces that the barrier connections are confronting. With no exception, it is also found that the optimum barrier performance, which leads to the minimum amount of barrier deflection, is provided with wider versions of the F-Shape barriers that are connected to each other with a closed hook type of pin-and-loop connection, where the gap between the two segments are closed. CONCLUSIONS A modular FE model pool is developed to assess the impact performance of PCB roadside hardware systems from candidate values of discrete variables to obtain different combinations of PCB designs with five different concrete barrier safety shapes, four different barrier segment lengths, two different barrier segment widths, two different barrier gaps and two different pin-and-loop designs. These numerical models are then coupled with a successive discrete design optimization algorithm to improve the performance of the randomly chosen baseline designs by trying to find the best combination of discrete variables that can lead to the multiple objectivity criteria of the PCB system while keeping the imposed constraints dictated by Report - 350. The MOO method effectively coupled with Taguchi’s OAs, GRA and FE analyses. GRA has provided a way of normalizing responses towards minimum or maximum values as well as converting multiple objective performance measures into a single performance value. The mixed-level OAs that are developed for the algorithm serves as an efficient design of experiment method to reduce the number of structural analyses in optimization. The successive nature of the method provides an efficient way of solving nonlinear optimization problems especially when large number of candidate values in the optimum search is considered. Penalty functions are used to employ the required constraints with multiple objectives, where a new concept of multiple penalty scale factors is used for the formulations. Multi-objective performance measures are obtained through the explicit dynamic FE method. 17

PCB system is optimized to minimize ride-down accelerations ( ax , a y ), rollover ( Rx ) and dynamic barrier displacement ( Disp ) upon impact while keeping certain response values under the specified safety limits. The MOO algorithm is started from two different randomly chosen initial designs, where the first design point corresponds to a feasible and the other corresponds to an infeasible design, to test and address the ability of the algorithm in removing initial infeasibility and improving the performance functions. It is shown that the MOO algorithm successfully improved the objectives after removing initial infeasibility. Findings of this study show that the proposed state-of-the-art iterative Taguchi method based Grey relational analyses coupled with a non-linear explicit dynamic FE solver can be effectively used in automated multi-objective robust design optimization of other complex engineering problems with discrete variables and can be utilized as an effective decision making tool. REFERENCES [1] Ross HEJ, Sicking DL, Zimmer RA, Michie JD. Recommended procedures for the safety performance evaluation of highway features, national cooperative highway research program report 350. Transportation Research Board, National Research Council, Washington, D.C., 1993. [2] Bronstad ME, Calcote LR, Kimball CEJ. Concrete median barrier research. Vol. 2, Report No. FHWA-RD-77-4, Federal Highway Administration, San Antonio, TX, 1976. [3] Buth CE, Campise WL, Griffin-III LI, Love ML, Sicking DL. Performance limits of longitudinal barrier systems. Vol. I: Report No. FHWA/RD-86/153, Texas A&M University, 1986. [4] Fortuniewicz JS, Bryden JE, Phillips RG. Crash tests of portable concrete median barrier for maintenance zones. Report No. FHWA/NY/RR-82/102, New York, 1982. [5] Mak KK, Sicking DL. Rollover caused by concrete safety shape barrier. Volume I-II: Report Nos. FHWA-RD-88-219/220, Federal Highway Administration, Texas A&M University, 1989. [6] Faller RK, Rosson BT, Rohde JR, Smith RP, Addink KH. Development of a TL-3 F-shape temporary concrete median barrier. MwRSF Research Report No. TRP-03-64-96, 1996. [7] Graham JL, Loumiet JR, Migletz J. Portable concrete barrier connectors. Report No. FHWATS-88-006, Independence, MO, 1987. [8] Institute of Transportation Engineers. Portable concrete barrier connectors. ITE Journal 1988; 58(4). [9] Ivey DL. Barriers in construction zones – the response of atypical vehicles during collisions with concrete median barriers. Vol. 4, Report No. FHWA/RD–86/095, Washington, D.C., 1985. [10] Bligh RP, Sheikh NM, Menges WL, Haug RR. Development of low-deflection pre-cast concrete barrier. FHWA Report No. FHWA/TX-05/0-4162-3, 2005. [11] Bligh RP, Sheikh NM, Menges WL, Haug RR. Portable concrete traffic barrier for maintenance operations. FHWA Report No. FHWA/TX-05/0-4692-1, 2005. [12] Bronstad ME, Calcote LR, Kimball CEJ. Concrete median barrier research. Vol. 2, Report No. FHWA-RD-77-4, Federal Highway Administration, San Antonio, TX, 1976. [13] Buth CE, Campise WL, Griffin-III LI, Love ML, Sicking DL. Performance limits of longitudinal barrier systems. Vol. I: Report No. FHWA/RD-86/153, Texas A&M University, 1986. 18

[14] Fortuniewicz JS, Bryden JE, Phillips RG. Crash tests of portable concrete median barrier for maintenance zones. Report No. FHWA/NY/RR-82/102, New York, 1982. [15] Beason WL, Ross HEJ, Perera HS, Marek M. Single-slope concrete median barrier. Transportation Research Record No. 1302, Transportation Research Board, National Research Council, Washington D.C., 1991. [16] Ivey DL, Ross HE, Hirsch TJ, Buth CE, Olson RM. Portable concrete median barriers: structural design and dynamic performance. Transportation Research Record No. 769, Transportation Research Board, National Research Council, Washington D.C., 1980. [17] Ivey DL, Buth CE, Robertson RG, Koppa RJ, Beason WL, Pendleton OJ, Ross HEJ. Barriers in construction zones. Vol.I: Summary Report, Texas A&M University, April 1985. [18] Beason WL, Ivey DL. Structural performance levels for portable concrete barriers. Transportation Research Record No. 1024, Transportation Research Board, National Research Council, Washington D.C., 1985. [19] Dresback TL. NCHRP Report 350 Test 3-11 of the Indiana Department of Transportation temporary pre-cast concrete barrier wall. Report No. 010213, Transportation Research Center Inc., Indiana Department of Transportation, 2001. [20] Dresback TL. NCHRP Report 350 Test 3-11 of the Ohio Department of Transportation NewJersey shape portable concrete barrier. Report No. 011012, Transportation Research Center Inc., Ohio Department of Transportation, 2001. [21] Marzougui D, Bahouth G, Eskandarian A, Meczkowski L, Taylor H. Evaluation of portable concrete barriers using finite element simulation. In: Proc. Pres. FHWA Conference, Washington, D.C., 1998. [22] Marzougui D, Kan CD, Eskandarian A, Meczkowski L. Finite element simulation and analysis of portable concrete barriers using LS-DYNA. In: Proc. 2nd European LS-DYNA Users Conference, Stockholm, Sweden, pp.119-125, 1999. [23] Marzougui D, Kan CD, Eskandarian A. Safety performance evaluation of roadside hardware using finite element simulation. In: Proc. ASCE, EM2000, 14th Engineering Mechanics Conference, Austin, Texas, 2000. [24] Atahan A. O., Finite-element crash test simulation of New York portable concrete barrier with I-shaped connector, Journal of Structural Engineering 2006; 132 (3): 430-440. [25] Hallquist JO. LS-DYNA theoretical manual. Livermore Software Technology Corporation, Livermore, CA, USA, 1997. [26] http://www.ncac.gwu.edu/vml/models.html [27] Fang H. , Rohani M. R., Liu Z., Horstemeyer M.F, “A comparative study of metamodeling methods for multiobjective crashworthiness optimization”, Computers and Structures, 83 (2005) 2121–2136. [28] Deng JL. Introduction to grey systems. J Grey Syst 1989; 1(1):1-24. [29] Lee KH, Yi JW, Park JS, Park GJ, An optimization algorithm using orthogonal arrays in discrete design space for structures, Finite Elements in Analysis and Design, 2003; 40 (1): 121135.

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