Crashworthiness Optimization of Vehicle Structures Made of a ...

52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference
19th 4 - 7 April 2011, Denver, Colorado

AIAA 2011-1896

Crashworthiness Optimization of Vehicle Structures Made of a Lightweight Magnesium Alloy Andrew Parrish1, Masoud Rais-Rohani2, and Ali Najafi1 Mississippi State University, Mississippi State, MS 39762

This study uses material substitution of magnesium for steel and optimization to reduce the mass of a full-scale Dodge Neon finite element model while maintaining the crashworthiness of the vehicle. Crash simulations are performed in LS-DYNA to obtain vehicle responses used as measures for crashworthiness in the absence of a crash dummy model. These include intrusion distance, acceleration, and internal energy. An AZ31 magnesium alloy is substituted for the baseline steel in twenty-two parts of the vehicle model. Polynomial Response Surface, Gaussian Process, Radial Basis Function, Kriging, Support Vector Regression, and Optimized Ensemble metamodels are used to develop global surrogate models of crash-induced responses. The results show that lightweight magnesium alloy maintains or improves vehicle crashworthiness with an approximate 50% reduction in selected part mass.

I. Introduction

S

TRICTER regulations on fuel economy and growing concerns for automobile emissions have caused design engineers to develop new ways of reducing fuel consumption. One way of improving fuel economy is by reducing vehicle weight through optimization of existing designs, substitution of lighter weight materials, or a combination of the two. Fuel savings, however, should not come at a cost of occupant safety in a crash situation. During the past several years, the U.S. Department of Energy has sponsored many studies in the area of Automotive Lightweighting Materials1 under the Vehicle Technologies Program to investigate the application of lightweight materials such as magnesium in combination with improvements in structural design and manufacturing for lightweighting of vehicle structures. Ref. 2 proposes strategies for researching and incorporating more magnesium parts into automobile design due to magnesium‟s high strength-to-weight ratio, fewer parts due to larger castings, and the ability to tune magnesium parts to frequencies related to Noise, Vibration, and Harshness (NVH). This study explores the weight and crashworthiness possibilities of using a magnesium material. Multi-objective studies have been carried out to improve crash performance while maintaining weight in a full frontal impact3 and another considering multiple impact scenarios4. Another study solved a two-step optimization problem minimizing weight, offset impact intrusion distance, and an integration deceleration before optimizing the occupant restraint system based on head injury criteria5. Nonlinear crash responses such as acceleration and intrusion distance appear as design constraints or objectives in crashworthiness optimization. When considering that a single high-fidelity vehicle crash simulation for a brief time period of approximately 100 ms could take about 10 CPU hours on a multiprocessor-based parallel computing platform 6 and the fact that some crash responses such as acceleration can be very noisy, it is necessary to develop and use analytical surrogate models of crash responses in solving crashworthiness optimization problems. Metamodels are approximate mathematical models that are developed using sets of data obtained using finite element (FE) simulations or physical tests at multiple design or training points. These models are not as accurate at predicting responses as FE simulations, but are much less computationally intensive. The process of building a metamodel and thus performing design optimization first requires selection of a metamodel type and adjustment of metamodel parameters if applicable.

1

Graduate Research Assistant, Department of Aerospace Engineering, Member AIAA Professor, Department of Aerospace Engineering, Center for Advanced Vehicular Systems, Associate Fellow AIAA 2

1 American Institute of Aeronautics and Astronautics

Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Different metamodeling techniques have been developed. In this study a number of metamodeling techniques are studied to determine the most accurate ones for the desired responses. These models include Polynomial Response Surface (PRS), Radial Basis Function (RBF), Kriging (KR), Support Vector Regression (SVR), Gaussian Process (GP), and an optimized ensemble (ENS) containing multiple metamodels 7-9. Models such as RBF, KR, and SVR have parameters in their formulations that can be tuned altering the accuracy of the model. Tuning parameters, particularly for SVR, are crucial for an accurate metamodel. In this study, stand alone metamodels for each response are developed and tuned for maximum accuracy before being used to construct an optimized ensemble of metamodels. Vehicle crashworthiness optimization is performed by coupling nonlinear constrained optimization and metamodeling techniques with use of a lightweight magnesium alloy for a select group of steel parts. The paper is organized as follows: Section II briefly introduces the FE model and crash scenarios. Section III discusses design variables, responses, and material substitution. Section IV presents the metamodel construction and Section V shows the optimization problems and results with concluding remarks appearing in Section VI.

II. FE Model and Validation A full-scale 1996 Dodge Neon model (version 7) developed at the National Crash Analysis Center (NCAC) is used in this study. This model is made up of approximately 270,000 elements in 330 parts and the total vehicle mass is approximately 1300 kg. This model was validated by NCAC for a Full Frontal Impact (FFI) scenario 10. We used this vehicle model for Side impact (SIDE) and Offset Frontal Impact (OFI) simulations in LS-DYNA FE code with each crash scenario following the Federal Motor Vehicle Safety Standards (FMVSS) 11 for impact velocity and impact angle. Figure1 shows the Neon model in each crash simulation and Fig. 2 shows acceleration for the FE model compared to the actual test data.

Figure 1. FFI, SIDE, and OFI scenarios OFI Left Rear sill X-acceleration

SIDE Middle B-Pillar Y-acceleration 100

20

-20

0.00

0.05

0.10

-40

0.15

Accel. (G's)

Accel. (G's)

50 0

0 -500.00

0.05

0.10

0.15

-100 -150

-60 seconds OFI Sim. Actual

-200

seconds SIDE Sim. Actual

Figure 2. Acceleration Curves Simulation and Actual Simulation-based acceleration curves for a location in the SIDE and OFI models are shown in Figure and compared to physical test results9, 10. These curves show that the general shape of the curves is the same but peak values may differ as a result of filtering and the method used to capture data. A Butterworth filter with the frequency 2 American Institute of Aeronautics and Astronautics

of 60 Hz is used. The location of these observation points was determined by the position of accelerometers in the actual testing.

III. Problem Discussion and Definition A. Design Variables and Responses We identified twenty-two parts as shown in Fig. 3 with significant contributions to energy absorption and structural stiffness in a preliminary investigation. These parts in the model are made of Belytchko-Tsai reduced integration shell elements allowing for simple manipulation of the thicknesses. Thicknesses of twenty-two selected vehicle parts are used as design variables in this study with thicknesses of the parts in the NCAC model serving as the baseline values, see Table 8 at the end of section V. These parts were selected based on their contributions to energy absorption, mass, and vehicle stiffness. Figure 3. Selected vehicle components and associated part The twenty-two selected parts account for numbers. approximately 40% of the energy absorption in each crash scenario and have a mass of approximately 105 kg compared to the vehicle mass of approximately 1300 kg. A total of fifteen design variables result from the twenty-two parts because of symmetry in the vehicle. Responses from the three crash scenarios are used in the design optimization problem. These are intrusion distances at the toeboard and dashboard for FFI and OFI and at the door for SIDE (Int Toe, Int Dash, and Int Door), resultant acceleration at a location on the B-pillar in all three scenarios (Accel), and internal energy absorption of the selected parts in all three scenarios (Int Eng). These responses were chosen because of their relevance to occupant safety and the acceleration response location was chosen to be near the approximate head location of an occupant during a crash. Approximate response locations can be seen in Fig 4.

Figure 4. Approximate Response Locations. These eleven responses and the mass are used as objectives or design constraints in the optimization problems discussed later in Section V. Mass is determined by dividing the initial mass of the baseline part by the initial thickness. This coefficient is multiplied by a new thickness for that part to determine the part‟s new mass. A metamodel is not used for calculating mass while a metamodel is used to calculate each nonlinear response. Intrusion distance was calculated by measuring the distance between twenty nodes at each response location and a reference node on the opposite side of the car before and after the crash and then finding the difference. An average over the twenty nodes was used as an intrusion distance response. A Butterworth filter (60 Hz) was applied to the acceleration curve at twenty nodes in each direction, the resultant found, an average over the twenty nodes taken, and the maximum value found to represent an acceleration response. The simulated response values determined using the methods described will be referred to as the true or actual response and the metamodels are the approximations or predications for the remainder of this paper. 3 American Institute of Aeronautics and Astronautics

B. Material Substitution AZ 31 magnesium alloy was selected to replace the baseline steel sheet formed parts in the Neon model and Mat 124 in LS-DYNA was used to model it. This material model allows the definition of separate stress-strain curves for compression and tension14. These stress-strain curves were taken from experimental data. This data was unavailable for AZ 31 under compressive loading so a relation of the ratio of AM 30 under compression and tension along with AZ 31 was used to approximate a curve for AZ 31 compression. This material model also allows the definition of plastic strain to failure. Magnesium‟s behavior under compressive loads requires this parameter be defined to provide an accurate material response. We have used a plastic strain to failure of 38% in this study. The material defined in this manner is not an exact magnesium material model but a surrogate. Material substitution was performed by changing the material of the twenty-two design parts, from steel to magnesium and leaving the other parts steel as in the baseline model. The thicknesses of these magnesium parts were selected to maintain the same total internal energy absorption as the steel baseline parts. An average of the thicknesses found using the toughness under compression and the toughness under tension was used. Magnesium thickness was found from (1) where and are the thicknesses of magnesium and steel respectively, and and are the toughness values of steel and magnesium which were derived from the stress strain curves for each material. Several baseline simulations of the vehicle model with magnesium parts having thicknesses defined using Eq. (1) were performed to verify the internal energy absorption when compared with steel. Figure 5 shows the total internal energy of the selected parts after each crash simulation along with acceleration curves at the B-pillar response location. Table 1 shows an overview of the responses for baseline simulations with both materials. The magnesium parts have significantly less mass than the steel parts and generally maintain or improve the internal energy absorption of the replaced parts but the intrusion distances are significantly larger with the magnesium parts. FFI x-Acceleration at MidB-pillar

OFI x-Acceleration at MidB-pillar 10

0 0.00

0.05

0.10

0.15

-20 MG ST

-40 -60

Acceleration (G's)

Acceleration (G's)

20

0 -100.00

Acceleration (G's)

MG ST

-30 Seconds Internal Energy Comparison

SIDE

MG ST

OFI

0 0.05

0.10

-40

0.15 MG ST

Seconds

0.15

-40

Seconds

20

-60

0.10

-20

SIDE y-Acceleration at MidB-pillar 40

-200.00

0.05

FFI 0

20 40 60 Internal Energy (kJ's)

Figure 5. Magnesium to steel comparison graphs for baseline simulations

4 American Institute of Aeronautics and Astronautics

80

Table 1. Response Comparison.

IV. Metamodel Construction

Metamodel construction begins with building a Mg St % of St Design of Experiments (DOE). Forty-five design Base Base Base points were constructed using Latin Hypercube Sampling with a range of plus or minus 50% of the 295 157 87.86% FFI Int Toe (mm) baseline thickness. The baseline design point was 186 122 52.25% Int Dash (mm) added for a total of forty-six design points. An LS49.2 63.5 -22.48% Accel (g's) DYNA simulation was performed at each of the design points to provide the responses necessary for 62.3 62.3 0.06% Int Eng (kJ) metamodel construction. The same DOE table was 420 314 33.65% SIDE Int Door (mm) used for magnesium and steel. RBF, KR, and SVR‟s tuning parameters were 40.8 47.9 -14.90% Accel (g's) adjusted to find the most accurate combination for 21.4 22.4 -4.25% Int Eng (kJ) each response. The degree of the polynomial in the 349 273 27.72% OFI Int Toe (mm) PRS model was also tuned. Metamodel accuracy was measured using cross validation generalized 386 247 56.22% Int Dash (mm) mean square error (GMSE), see Eq. (2). These 36.2 35.0 3.29% Accel (g's) tuning parameters can drastically affect the accuracy of the metamodels and finding appropriate 39.2 39.4 -0.43% Int Eng (kJ) parameters is important. See the Appendix for a 42.74 105.23 -59.39% MASS discussion of these metamodeling techniques including tuning parameters. The tuned metamodels were used to construct an optimized ensemble of metamodels. This was created by minimizing GMSE to determine global weight factors for each response 8. This ensemble is more accurate than the individual metamodels and is used to approximate the responses during optimization. GMSE was calculated using (2) where N is the number of data sets, fi is the actual response, and is the model predicted response. Normalized GMSE for steel and magnesium can be found in Tables 2 and 3 respectively. The optimized ensemble metamodel was then used to approximate the responses at the baseline design point and compared to the simulation response value at the initial design for both materials. This served as a baseline error when evaluating the accuracy of each optimum design once determined. The maximum percent error for the steel baseline occurred for the intrusion at the toeboard for OFI with 15% error. This was followed by 9% error for intrusion at the dashboard for OFI and 5% error for intrusion at the toeboard for FFI. The remaining responses for steel had less than 2% error at the baseline design when predicted with the ensemble. The maximum percent error for the magnesium baseline was 4% for the acceleration at midBpillar for FFI. The remaining magnesium responses had less than 2% error. Table 2. Normalized GMSE for steel. FFI Int Toe

Int Dash

Accel

Int Eng

SIDE Int Door

Accel

Int Eng

OFI Int Toe

Int Dash

Accel

Int Eng

PRS

1.60

GP

1.12

1.41

1.64

1.12

2.09

1.57

1.29

1.56

1.53

1.49

1.31

1.04

2.10

1.79

1.09

2.26

2.31

1.09

1.49

1.35

1.95

RBF

1.39

1.75

1.32

2.14

1.39

1.60

1.46

1.51

1.26

1.13

1.44

KR

1.13

1.15

1.00

1.12

1.18

1.06

1.12

1.22

1.04

1.05

1.01

SVR

1.34

1.20

1.19

1.13

1.34

1.15

1.19

1.24

1.07

1.31

1.27

ENS

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

5 American Institute of Aeronautics and Astronautics

Table 3. Normalized GMSE for magnesium. FFI Int Toe

Int Dash

PRS

1.73

SIDE Accel

Int Eng

Int Toe

OFI Int Dash Accel

Int Eng

1.52

1.07

2.07

1.48

1.91

3.00

1.07

Accel

Int Eng

Int Door

1.84

1.50

1.70

GP

2.71

1.58

1.77

1.84

2.15

1.84

1.38

1.50

2.59

2.44

1.64

RBF

1.30

1.54

1.31

1.59

1.34

1.32

1.65

1.25

1.40

3.33

1.53

KR

1.00

1.01

1.25

1.04

1.00

1.06

1.08

1.13

1.11

1.05

1.06

SVR

1.55

1.61

1.05

1.37

1.26

1.03

1.49

1.16

1.12

2.85

1.07

ENS

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

V. Optimization Problems This section defines and presents results for two optimization problems using the design variables, responses, and metamodels defined previously. One problem finds the design for each material that minimizes mass while the second focuses on magnesium crashworthiness using a multi-objective optimization approach. A. Weight minimization problem (Steel and Magnesium) The first optimization problem explored is formulated as

Table 4. Metamodel percent error at the single objective optimum point. The percent error shown is relative to the LS-DYNA simulation at the optimum point for steel and magnesium.

(3) % of Mg DYNA -4.93%

% of St DYNA -4.11%

FFI Int Toe where F(x) is the objective function, x is the input vector of 15 -1.22% 13.79% Int Dash design variables, is the jth response, with representing the response of the baseline model. In Eq. (3), -3.03% -14.91% Accel represents the intrusion distances and accelerations 1.62% -3.08% Int Eng and represents the internal energy responses. This 2.16% -4.37% SIDE Int Door formulation searches for the design that has the lowest weight but maintains or improves the baseline responses. -1.88% 15.78% Accel This optimization problem was solved using Sequential -0.06% 2.88% Int Eng Quadratic Programming (SQP) implemented in the 10.14% -5.07% OFI Int Toe VisualDOC software package. The steel and the magnesium cases were solved using their respective baseline response 6.67% -24.07% Int Dash values and metamodels. Metamodel errors at the optimum 23.26% -0.35% Accel point are shown in Table 4. The percent errors vary from one 1.77% -0.91% Int Eng response to another with noticeable dependence on the crash scenario as well as the material used. A mass decrease of approximately 16% and 13% for steel and magnesium respectively was observed resulting in a total mass of the design parts of 88.0 kg and 37.2 kg down from the initial design with 105.2 kg and 42.7 kg for steel and magnesium respectively. Comparisons of individual responses can be seen in Table 5 and the optimum design point can be found at the end of this section in Table 8. It should be noted that the responses in Table 5 are the results from the LS-DYNA simulation at the optimum point found using the metamodels and not the metamodel predictions. Results in Table 5 show that overall responses were improved or maintained with a negative percent indicating an improvement for intrusion distance, acceleration, and mass with a positive percent indicating an improvement for internal energy.

6 American Institute of Aeronautics and Astronautics

Table 5. Multi-crash responses at the point of optimum.

FFI

SIDE

OFI

Mg Base

Mg Opt

% of Mg Base

St Base

St Opt

% of St Base

Int Toe (mm)

295

274

-7.12%

157

170

8.00%

Int Dash (mm)

186

167

-9.94%

122

108

-11.16%

Accel (g's)

49.2

53.1

7.79%

63.5

69.0

8.58%

Int Eng (kJ)

62.3

60.1

-3.58%

62.3

65.7

5.44%

Int Door (mm)

420

414

-1.24%

314

333

6.22%

Accel (g's)

40.8

41.2

0.98%

47.9

38.6

-19.48%

Int Eng (kJ)

21.4

21.5

0.27%

22.4

23.2

3.78%

Int Toe (mm)

349

320

-8.40%

273

233

-14.72%

Int Dash (mm)

386

252

-34.76%

247

269

9.08%

Accel (g's)

36.2

30.7

-15.04%

35.0

36.6

4.42%

Int Eng (kJ)

39.2

38.3

-2.35%

39.4

40.2

1.89%

MASS

42.74

37.19

-12.99%

105.23

88.01

-16.37%

B. Multi-objective targeting steel baseline responses (Magnesium) Results in the previous section suggest that the magnesium designs, both baseline magnesium and single objective optimum, maintain or improve upon the steel baseline responses for acceleration and internal energy but exceed the steel baseline intrusion distances. Another optimization problem is formulated and solved in this section to determine the mass savings using magnesium and maintaining the original crash responses. Overall baseline refers to the initial Dodge Neon model made of steel parts. This optimization problem uses the magnesium model and targets the baseline steel intrusion distance with constraints to maintain or improve the other responses without consideration of the mass. This problem is formulated as Table 6. Metamodel error at the targeted optimum. MM is the metamodel prediction and DYNA is the LSDYNA simulation result at the optimum point. Percentages are calculated relative to the simulation

FFI

SIDE

OFI

Int Toe (mm)

Mg MM 166

Mg DYNA 186

% of DYNA -10.63%

Int Dash (mm)

136

131

4.40%

Accel (g's)

52.0

59.3

-12.32%

Int Eng (kJ)

65.3

62.0

5.23%

Int Door (mm)

328

361

-9.07%

Accel (g's)

44.2

48.0

-7.97%

Int Eng (kJ)

22.1

21.9

0.76%

Int Toe (mm)

256

212

20.68%

Int Dash (mm)

247

236

4.67%

Accel (g's)

35.0

37.0

-5.44%

Int Eng (kJ)

39.0

37.9

2.73%

(4) where F(x) represents a composite objective function representing the intrusion distances of the baseline steel model, for are the baseline steel accelerations, for are the baseline steel internal energies, and are the baseline magnesium design point values. VisualDOC15 software is once again used to solve this problem. Compromise programming was used inside VisualDOC to convert this multi-objective into a single-objective optimization problem solved using SQP. Using the compromise programming formulation, the composite objective function in Eq. (4) is given by (5)

7 American Institute of Aeronautics and Astronautics

where is the number of targeted objective functions (five intrusion distances in this case), is j-th objective, is the target value of the j-th objective, and is the worst known value of the j-th objective (the responses at the initial magnesium design point in this case). The response values predicted by the metamodels and the LS-DYNA verified values at the optimum design point using this formulation are found in Table 6. The metamodel error shows that six responses had error at or lower than 5%, three near 10%, and two above 10%. Table 7 shows the responses at the targeted Table 7. Mg targeted optimum compared to steel optimum using the magnesium design compared to baseline. Percentages are relative to the steel baseline. the steel baseline responses. The values displayed are the LS-DYNA simulation results rather than the Mg St % of St metamodel predictions. The targeted optimization Target Base Base found a design that improves the magnesium 186 157 18.23% FFI Int Toe (mm) models intrusion distance responses but does not meet the steel responses in general. The intrusion 131 122 7.08% Int Dash (mm) distances at the toeboard (FFI), door (SIDE), and to 59.3 63.5 -6.58% Accel (g's) a lesser extent at the dashboard (FFI) were larger 62.0 62.3 -0.41% Int EngF(kJ) than the steel baseline response. The intrusion distances for the OFI simulation were less than the 361 314 14.95% SIDE Int Door (mm) steel baseline. This magnesium design has a mass 48.0 47.9 0.24% Accel (g's) that is about half of that of the baseline steel design 21.9 22.4 -2.08% Int Eng (kJ) parts and 8 kg heavier than the baseline magnesium. 212 273 -22.55% OFI Int Toe (mm) Int Dash (mm)

236

247

-4.44%

Accel (g's)

37.0

35.0

5.61%

C. Optimization results summary Response values for the single-objective and multi-objective, targeted optimizations discussed in 37.9 39.4 -3.73% Int Eng (kJ) the previous sections are summarized in Fig. 6 and 50.73 105.23 -51.79% MASS Fig. 7. Results show that the magnesium designs have much less mass than the steel designs and the targeted optimum design is near or better than the baseline steel design for most responses. Design variable values representing wall thickness of individual parts in mm at the baseline and optimized designs can be found in Table 8. Figures 8 through 10 show simulation images of the five designs for FFI, SIDE, and OFI respectively. Table 8. Design variable summary. Part

Part No.

Design Variable

St Base

Mg Base

St Min

Mg Min

Mg Target

A-Pillar

310,311

x1

1.611

2.597

0.915

2.561

1.984

Front Bump

330

x2

1.956

5.975

1.204

2.987

6.649

Firewall

352

x3

0.735

1.072

0.545

0.867

1.515

Front Floor Panel

353

x4

0.705

1.136

0.612

1.211

1.592

Rear Cabin Floor

354

x5

0.706

1.138

0.477

0.569

1.696

Outer Cabin

355,356

x6

0.829

1.366

0.988

1.482

2.049

Cabin Seat Reinf

357

x7

0.682

1.099

0.622

1.649

1.649

Cabin Mid Rail

358,359

x8

1.050

1.692

0.633

1.792

1.636

Shotgun

373,374

x9

1.524

3.620

0.762

1.810

1.810

Inner Side Rail

389,391

x10

1.895

3.966

1.887

3.436

4.141

Outer Side Rail

390,392

x11

1.522

3.186

1.834

3.145

2.754

Side Rail Exten.

398,399

x12

1.895

3.966

1.780

4.805

5.950

Rear plate

415

x13

0.710

1.144

0.417

1.559

1.717

Roof

416

x14

0.702

1.157

0.351

0.739

0.791

Susp. Frame

439

x15

2.606

5.342

1.303

4.367

4.931

8 American Institute of Aeronautics and Astronautics

Optimization problems in this study focused on reducing mass and maintaining crash performance. Improved crashworthiness beyond the steel baseline was accepted but not an objective. Additional optimization problems could be formulated and solved to minimize mass and improve crashworthiness simultaneously or target a specific safety rating for the vehicle structure while reducing mass. 120

Acceleration and Internal Energy Comparison

100 80 60 40 20 0 AccelF Int EngF AccelS Int EngS AccelO Int EngO Mass (kg) (g's) (kJ) (g's) (kJ) (g's) (kJ) St Base Mg Base St Min Mg Min Mg Target Figure 6. Optimization Results Comparison: Acceleration and Internal Energy. F represents FFI, S represents SIDE, and O represents OFI 500

Intrusion Distance Comparison

400 300 200 100 0 Int ToeF Int DashF Int DoorS Int ToeO Int DashO Mass (kg) (mm) (mm) (mm) (mm) (mm) St Base Mg Base St Min Mg Min Mg Target Figure 7. Optimization Results Comparison: Intrusion Distance. F represents FFI, S represents SIDE, and O represents OFI.

9 American Institute of Aeronautics and Astronautics

Figure 8. Simulation images of the five designs, FFI

(a) FFI St Base

(b) FFI Mg Base

(c) FFI St Min

(d) FFI Mg Min

(e) FFI Mg Target

10 American Institute of Aeronautics and Astronautics

Figure 9. Simulation images of the five designs, SIDE

(a) SIDE St Base

(c) SIDE St Min

(b) SIDE Mg Base

(d) SIDE Mg Min

(e) SIDE Mg Target

11 American Institute of Aeronautics and Astronautics

Figure 10. Simulation images of the five designs, OFI

(a) OFI St Base

(b) OFI Mg Base

(c) OFI St Min

(d) OFI Mg Min

(e) OFI Mg Target VI. Summary and Conclusions This study used metamodeling and optimization techniques to explore the application of a lightweight magnesium alloy for a group of energy absorbing parts in a full vehicle model. A full-scale Dodge Neon model developed and validated for Full Frontal Impact (FFI) at the National Crash Analysis Center (NCAC) was incorporated into Side impact (SIDE) and Offset Frontal Impact (OFI) scenarios. Responses from these three scenarios along with selected part mass were used as constraints and objectives during design optimization with part thicknesses as the design variables. All crash simulations were performed using LS-DYNA finite element solver. Due to the complexity of the failure characteristics of magnesium a limit on maximum plastic strain was used as a failure criterion to disable failed elements. AZ 31 magnesium was used to replace the baseline steel material in the finite element model. This material substitution reduced the mass of the vehicle by approximately 50% of the baseline steel model and maintained or improved the crashworthiness for most of the factors considered. We found that material substitution can change the deformation mode and folding mechanism of energy absorbing parts under crash loading. Therefore, modification of the cross-sectional geometry as well as other geometric attributes should be considered along with the thickness 12 American Institute of Aeronautics and Astronautics

modifications in vehicle design optimization. Future considerations include accounting for vehicle stiffness in the design optimization process and including a crash test dummy for more accurate occupant based responses.

Appendix: Description of Metamodels Polynomial Response Surface (PRS) PRS is one of the most widely used and simplest metamodeling techniques. It is also less computationally expensive than other metamodels. The most typical form of PRS is a second-degree polynomial function of the form (A1) where is the metamodel prediction at point x, L is the number of design variables in the design vector x, and b0, bi, bii, bij are the unknown coefficients found using the least squares technique. As a regression model, PRS does not pass through the training points and as such error analysis can be performed without the use of additional data at test points. The degree of the polynomial can be changed or some of the terms appearing in Eq. (A1) can be omitted depending on the nonlinearity of the response function being modeled. In this study a first-degree polynomial was used for all responses for both materials because it had a lower GMSE than second or third-degree. Radial Basis Function (RBF) This discussion follows that in Ref. (8). The basic form of RBF is given as (A2) where is the metamodel prediction at point x, N is the number of training points, x is the input vector of normalized design variables, is vector of normalized design variables at the ith training point, and is the Euclidean norm or distance r from design point x to the training point . The parameters are the unknown interpolation coefficients that must be calculated. Φ is the radially symmetric basis function that can take on a number of forms. Eq. (A2) represents a linear combination of a finite number of basis functions. Typical radial basis functions, Φ, are listed below.    

Thin Plate Spline: Gaussian: Multiquadric: Inverse Multiquadric:

The value c is a constant that the user determines. Values of r and therefore the training points should be normalized to values between 0 and 1 resulting in . In general, Multiquadric with c = 1 gives good results for many function types. The interpolation coefficients, , can be found by minimizing the residual (the sum of the squares of the deviations see Eq. (A3). (A3) In matrix form this is expressed as (A4) where with j=1,N, and i=1,N. Solving Eq. (A4) for λ and Eq. (A2) generates predictions. Since RBF represents an interpolation model that passes through all the training points, error analysis typically relies on data at test points outside of the training set.

13 American Institute of Aeronautics and Astronautics

Table A1shows the RBF parameters used in this study. Table A1. RBF parameters used in this study. Multi = Multiquadric, Gauss =Gaussian , and Inv Multi = Inverse multiquadric.

St

const., c Basis Funct.,ϕ const., c Basis Funct.,ϕ

Mg

FFI Int Toe 1

Accel 0.05

Int Eng 0.05

Multi

Mult

Multi

Gauss

0.05

0.2 Thin Plate 0.2

Int Eng 1

SIDE Int Door 1

0.05

1

1

Gauss

Multi

Gauss

Multi

1 Inv Multi

OFI Int Toe 1 Inv Multi 1

Int Dash 1

Multi

Multi

0.05 Gauss

Multi

Multi

Accel

Int Dash 0.95

Accel

Int Eng 0.75 Multi

0.15

0.25 Thin Plate 0.05

Gauss

Multi

Multi

Multi

1

Kriging (KR) Descriptions of kriging for this discussion follow those reported in Refs. (8, 16, and 17). Kriging models assume the function takes the form of (A5) where f(x) is the approximate function, p(x) is a polynomial function that globally approximates the actual function, and Z(x) is the stochastic component that accounts for deviations and allows the Kriging model to interpolate the response value. The stochastic component has a zero mean and covariance of (A6) where is the variance, R is an N by N symmetric correlation matrix where N is the number of sample points, R is the correlation function between sample points and . The user defines the correlation function, R , with a Gaussian correlation function generally chosen. Some options for correlation function are listed below with all of them having the form (A7) where L is the number of design variables and being the distance between component of the points, and are unknown correlation parameters that must be determined. Kriging Correlation Functions are as follows:       

Gaussian: Exponential: Exp. General: Linear: Spherical: Cubic: Spline:

and

at the kth

) ), , ,

o

The response is predicted by the following equation once a correlation function has been chosen (A8) 14 American Institute of Aeronautics and Astronautics

where is a column vector of responses for the N sampling points, p is a column vector of length N which comes from the section of the global polynomial p(x) (this is a vector of ones if p(x) is taken as a constant as is commonly done), is the correlation vector of length N of an untried x and the sampled data points see Eq. (A9), and is estimated using Eq. (A10). (A9) (A10) Variance,

, between the global model

and y is estimated as (A11)

Maximum likelihood estimation is used to estimate the correlation parameter done by solving the following optimization problem:

necessary to fit a KR model. This is

(A12) Any value of will produce a kriging model and results will be predicted, but these predictions are not necessarily the most accurate model. Solving the optimization problem in Eq. (A12) will result in a more accurate kriging model. The initial value and bounds on θ also play a role in determining the most accurate model for a given data set. Another tuning parameter the user can control is the degree of the polynomial, generally 0, 1, or 2. We used the MATLAB18 kriging toolbox developed in Ref. 16 and the KR parameters used can be found in Table A2. Table A2. KR parameters used in this study. Spl = Spline, Spher = Spherical, Exp = Exponential, Lin = Linear, Gau = Guass, UBnd = Upper Bound, LBnd, Lower Bound, Corr = Correlation, and R.P. Deg = Regression Polynomial Degree Mat

Param

St

UBnd, θ

FFI Int Toe 0.1

St

LBnd, θ

0.01

0.001

0.001

0.001

0.01

0.001

0.01

0.001

St

Corr.

Spl

Spher

Exp

Lin

Gau

Lin

Gau

Spher

Int Dash 0.011

SIDE Int Door 0.1

Accel

1

Int Eng 1

Accel

0.011

Int Eng 0.1

OFI Int Toe 0.011

Int Dash 0.1

1

Int Eng 0.1

0.001

0.01

0.01

Gau

Spl

Spl

Accel

St

R.P. Deg

0

1

0

1

0

1

0

0

1

0

0

Mg

UBnd, θ

0.011

0.1

0.011

0.011

0.011

0.1

0.011

1

0.1

1

1

Mg

LBnd, θ

0.01

0.001

0.001

0.001

0.001

0.01

0.001

0.001

0.001

0.01

0.001

Mg

Corr.

Cub

Gau

Spl

Exp

Lin

Spl

Cub

Spher

Spl

Spl

Cub

Mg

R.P. Deg

1

1

1

1

2

1

2

2

1

1

2

Support Vector Regression (SVR) Descriptions of SVR in this discussion follow that in Refs. (8, 19, 20, 21, and 22). Simply, SVR constructs a hyperplane that passes near each design point such that they fall within a specified distance of the hyperplane. In two dimensions, this hyperplane is simply a line. The hyperplane is then used to predict other responses. SVR estimates the real function in the equation (A13)

15 American Institute of Aeronautics and Astronautics

where δ is independent random noise, x is the multivariable input, y is the scalar output, and r is the mean of the conditional probability (regression function). For more information, see Ref. (19 and 20). SVR methods select the “best” approximate model from a group of selection models that minimize the prediction risk. Linear or nonlinear regression can be performed. When a linear regression is used, the pool of approximation models is given by +b

(A14)

where b is the bias term. Minimizing empirical risk using the ε-insensitive loss function allows regression estimates. It is desirable to have a “flat” approximation function and this is achieved by minimizing . Non-negative slack variables are introduced to account for training points that fall outside the ε-insensitive zone.

(A15)

where C is a positive constant, and ε is the insensitive zone, both are chosen by the user. C is also referred to as the regression parameter or penalty parameter. Ref. 19 propose C be chosen as (A16) where and are the mean and standard deviation of the training point responses. Ref. 21 suggests a crossvalidation approach to find C. The ε parameter determines the width of the ε-insensitive zone and affects the complexity/flatness of the model. The values of ε should be tuned to the input data, but a reasonable starting value is found using (A17) where

is the range of the responses of the training points. Ref. 19 proposes ε be chosen as (A18)

where ζ is the standard deviation of the noise associated with the training point response values and N is the number of training points. This assumes the noise is known or can be determined. Ref. 19 suggests the following to estimate the variance of noise if it is not known using a k-nearest neighbor technique (A19) where k is in the range 2-6 and is the squared sum of the residuals. The optimization problem in Eq. (A15) written as a Lagrangian function is

(A20) where and are additional slack variables. From Lagrangian theory, necessary conditions for α to be a solution are listed below (A21) (A22) 16 American Institute of Aeronautics and Astronautics

(A23) (A24) Substituting Eqs. (A21) - (A24) into Eq. (A11) gives the dual form optimization problem seen below

(A25)

Eq. (A22) is rewritten as (A26) The linear regression first expressed in Eq. (A14) is written as (A27) Ref. 22 summarizes the process of transforming the problem into dual form “Transforming the optimization problem into dual form yields two advantages. First, the optimization problem is now a quadratic programming problem with linear constraints and a positive definite Hessian matrix, ensuring a unique global optimum. For such problems, highly efficient and thoroughly tested quadratic solvers exist. Second, as can be seen in Eq. [(A25)], the input vectors only appear inside the dot product. The dot product of each pair of input vectors is a scalar and can be preprocessed and stored in the quadratic matrix . In this way, the dimensionality of the input space is hidden from the remaining computations, providing means for addressing the curse of dimensionality [20].” A nonlinear regression model can be developed by replacing the dot product discussed previously with a kernel function, k, rewriting the optimization problem in Eq. (A25) as

(A28)

Replacing the dot product with a kernel function in the approximation function Eq. (A27) gives the nonlinear SVR approximation as (A29) Common Kernel functions include  

Linear: Polynomial:



Gaussian:

 

Radial Basis Function: Sigmoid: 17 American Institute of Aeronautics and Astronautics

where γ,d, p, and r are kernel parameters and should be adjusted by the user for each data set. Listed below are some insights about setting kernel parameters:   

The polynomial degree d is typically chosen to be 2. Ref. 16 uses r = 1 to “avoid problems with the hessian becoming zero”. Ref. 17 suggests a cross-validation approach to determine γ for RBF. This procedure could be applied to γ for other kernels as well as other kernel parameters. Ref. 15 suggests p for the Gaussian kernel (they call it RBF, the formulation is the same as “Gaussian” here) as for single variable problems and for multi variable problems where L is the number of variables and all input variables are normalized to [0,1].

It should be noted that the “Gaussian” kernel here is sometimes referred to as “Radial Basis Function” and “Gaussian Radial Basis Function” in the literature. Experience shows that SVR metamodels are highly sensitive to tuning parameters. The suggestions expressed in this discussion may not result in an acceptable accuracy level. The reader is encouraged to perform SVR tuning themselves to ensure an accurate model. We used the SVR MATLAB toolbox from Ref. 20 and the SVR parameters used are found in Table A3. Only the Linear, Guassian, and Radial Basis Function kernels were considered with p and γ in the Gaussian and Radial Basis Functions respectively being represented by p1 in Table A3. Table A3. SVR parameters used in this study. c = coefficient in the insensitive zone determination equation ( ), p1=Kernel Parameter, Kern = Kernel, Lin = Linear, RBF = Radial Basis Function, and Pen = Penalty Parameter Mat

St

Param

c

FFI Int Toe 1

Int Dash 1

100

Int Eng 10

Accel

SIDE Int Door 0.01

10

Int Eng 10

Accel

OFI Int Toe 10

Int Dash 1

Accel 10

Int Eng 1

St

p1

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

St

Kern

RBF

Lin

Lin

Lin

RBF

RBF

Lin

RBF

RBF

Lin

RBF

St

Pen, C

10

2

0.1

2

10

2

5

5

5

0.1

5

Mg

c

1

1

10

0.01

10

0.01

10

10

0.01

1

10

Mg

p1

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

Mg

Kern

Lin

RBF

Lin

RBF

RBF

Linear

Lin

Linear

RBF

Lin

Lin

Mg

Pen, C

5

10

0.1

2

2

0.1

0.1

10

10

0.1

2

Gaussian Process Descriptions of Gaussian Process (GP) in this discussion follow that in Refs. (7 and 8). The GP metamodel is a group of output variables with a Gaussian joint probability distribution (A30) where with elements of

are N pairs of L-dimensional input variables , is the covariance matrix , and is the mean output vector. The output is estimated at a prediction point as (A31)

where . Standard deviation at the prediction point is available without requiring additional simulations or tests. The standard deviation can be calculated from 18 American Institute of Aeronautics and Astronautics

(A32) where Elements of the covariance matrix

are calculated from (A33)

(A34) where , and are referred to as “hyperparameters”. is an independent noise parameter and is Kronecker‟s delta (equal to one when i = j and zero otherwise). These hyperparameters are selected to maximize logarithmic likelihood of the predictions matching the training data. This is given by (A35) where is the prior distribution of the hyperparameters. This is usually uniform because no prior knowledge is available. The final term in Eq. (A35) can be equated to zero for optimization. Eq. (A33) and Eq. (A34) define the interpolation and regression modes of the Gaussian process model respectively. The former passes through all training points while the latter provides a smoother surface to help with noisy data. The prediction surface with noise filtered out is less complex and might not pass through all training points but this has better predictions at non-training points. We use the MATLAB code from Ref. 23. Optimized Ensemble The type of ensemble used here is that developed in Ref. 8 and the description here follows the discussion in that reference. An ensemble combines other metamodel predictions in a manner that gives better predictions than a single standalone model. The general form of an ensemble is a weighted sum of the predictions of separate metamodels (such as PRS, GP, RBF, KRIG, and SVR). In mathematical form, this is expressed as (A36) where is the ensemble prediction, is the vector of input variables, M is the number of metamodels used to build the ensemble, is the weight factor for the ith metamodel, and is the prediction of the ith metamodel. The weight factors must meet the constraint that they sum to one (A37) Selection of the weight factors is the most important step when constructing an accurate ensemble. Ref. 8 developed an ensemble minimizing the error of the ensemble by finding the optimal weight factors. In mathematical form, this is expressed as (A38)

where Err{ } is the error metric that finds error of the ensemble predictions, , is the actual response at the training point , and where N is the number of training points. A generalized mean square cross-validation error (GMSE) metric or a root mean square error (RMSE) metric was used in 1 to build an ensemble. GMSE is expressed in this context as

19 American Institute of Aeronautics and Astronautics

(A39) where is the actual response at and is the ensemble prediction using metamodels built with all training points except . To compute GMSE for ensemble, N metamodels of each type must be built and this metric is sensitive to the number and distribution of training points. This or other cross validation error metrics are used when computational costs for a true response are large and limited number of points exists. Ref. 8 used RMSE at validation points to measure accuracy. This formulation of RMSE is given by (A40) where is the number of validation points and is the input variable vector of the validation points. This study used this ensemble method with GMSE minimization to determine the weight factors for each metamodel.

Acknowledgments This material is based on the work supported by the US Department of Energy under Award Number DEEE0002323. Disclaimer: This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

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14 ”LS-DYNA Keyword User‟s Manual: Volume II Material Models,” Livermore Software Technology Corporation, Version 971, 2007, pp. 473-475. 15 ”VisualDOC 6.2 Advanced Examples Manual,” Vanderplaats Research and Development, Inc., 2009, pp. 93. 16 Lophaven SN, Nielsen HB, Søndergaard J “DACE – A MATLAB Kriging Toolbox,” Informatics and mathematical modeling, Technical University of Denmark, Lyngby, 2002 17 Simpson, T. Mauer, T. Korte, J. Mistree, F. “Kriging Models for Global Approximations in Simulation-Based Multidisciplinary Design Optimization ,”AIAA Journal, Vol. 39, No. 12, 2001, pp. 2231-2241. 18 MATLAB: The Language of Technical Computing, Software Package, Ver. 7.8.0347 (R2009a), The Mathworks, Inc., 2009. 19 Cherkassky, V.,Ma, Y. “Practical selection of SVM parameters and noise estimation for SVM regression,” Neural Networks, Vol. 17, 2004, pp. 113-126. 20 Gunn, SR. “Support vector machines for classification and regression,” Technical Report, University of Southampton, 1997. 21 Hsu, CW., Chang CC., Lin CJ. “A Parctical Guide to Support Vector Classification,” National Taiwan University, last updated 2010. 22 Clarke, S., Griebsch, J., Simpson, T. “Analysis of Support Vector Regression for Approximation of Complex Engineering Analyses,” Journal of Mechanical Design, Vol. 127, November, 2005, pp. 1077-1087. 23 Rasmussen, C., Willams, C. “Gaussian Processes for Machine Learning,” MIT, Cambridge, Massachusetts, 2006.

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