the design and manufacturing processes optimization of...

UNIVERSITY OF MICHIGAN

THE DESIGN AND MANUFACTURING

PROCESSES OPTIMIZATION OF A CAR’S B-PILLARS By

Zheng Shen

Liang Xi

Ying Luo

Yue Jian

ME 555-12-03

Winter 2012 Final Report

The design and manufacturing processes optimization of a car‘s B-pillars was investigated in

detail. This project aims at optimizing the B-pillars to reduce the cost to build lightweight B-

pillar structure and decrease the structure weight subjected to feasible manufacturing process

and satisfactory crash worthiness in side impact. The B-pillars system has been divided into

four significant subsystems, this is, lightweight structure design, hot stamping optimization,

optimization of laser welding process and cost optimization. Each subsystem is optimized to

minimize its own objective function subjected to its own constraints. In the subsystems, at

least one of analysis, Finite Element simulation and metamodel (stepwise and linear

regression and neural networks) is applied to construct the corresponding model. Finally

subsystems are integrated and optimized as a single optimization problem, namely All-in-One

(AIO) approach. The project terminates by finding an optimal design with the tradeoff among

low cost, lightweight and applicable manufacturing

ABSTRACT

Contents

3

Contents

1 Design Problem Statement................................................................................................................. 8

2 METAMODEL-BASED LIGHTWEIGHT STRUCTURE OPTIMIZATION OF B-

PILLAR UNDER SIDE IMPACT LOADING --- Zheng Shen ............................................................... 9

2.1 Simplified Structure Model ...................................................................................................... 9

2.2 Simplified Structure Model .................................................................................................... 10

2.3 Nomenclature ......................................................................................................................... 12

2.4 Mathematical Models ............................................................................................................. 13

2.5 Model Analysis ...................................................................................................................... 14

2.6 Optimization Methodology .................................................................................................... 14

2.6.1 Metamodeling Method ....................................................................................................... 14

2.6.2 Design of Experiment ........................................................................................................ 15

2.6.3 Monotonicity Analysis ....................................................................................................... 17

2.6.4 Optimization Method ......................................................................................................... 18

2.7 Numerical Result ................................................................................................................... 18

2.8 System-level Tradeoffs .......................................................................................................... 20

2.9 Discussions ............................................................................................................................ 20

3 METAMODEL-BASED HOT STAMPING OPTIMIZATION OF B-PILLAR --- Ying Luo ....... 22

3.1 Problem Statement ................................................................................................................. 22

3.2 Nomenclature ......................................................................................................................... 25

Contents

4

3.3 MATHEMATICAL MODEL ................................................................................................ 27

3.3.1 Objective function .............................................................................................................. 27

3.3.2 Design Variables ................................................................................................................ 27

3.3.3 Parameters .......................................................................................................................... 27

3.3.4 Constraints ......................................................................................................................... 29

3.3.5 Summary of Model ............................................................................................................ 31

3.4 Optimization Model Analysis ................................................................................................ 32

3.4.1 Metamodeling .................................................................................................................... 32

3.4.2 Design of Experiment and Simulation ............................................................................... 32

3.4.3 Dependent Analysis ........................................................................................................... 34


3.4.5 Activeness Analysis ........................................................................................................... 35

3.4.6 Numerical Results .............................................................................................................. 36

3.4.7 Parameter Analysis ............................................................................................................ 38

3.5 System Trade-Off ................................................................................................................... 39

3.6 Problem Discussion ............................................................................................................... 39

4 Optimization of Laser Welding Process .......................................................................................... 40

4.1 Background ............................................................................................................................ 40

4.2 Subproblem Description ........................................................................................................ 41

Contents

5

4.3 Nomenclature ......................................................................................................................... 42

4.4 Mathematical Model .............................................................................................................. 43

4.4.1 Objective Function ............................................................................................................. 43

4.4.2 Constraints ......................................................................................................................... 44

4.4.3 Metamodel ......................................................................................................................... 46

4.4.3.1 .................................................................................................................. 46

4.4.3.2 ................................................................................................................. 49

4.4.4 Model summary ................................................................................................................. 51

4.4.5 FDT Analysis ..................................................................................................................... 54


4.5 Optimization Study ................................................................................................................ 57

4.5.1 Process Description ............................................................................................................ 57

4.5.2 Algorithm Compare ........................................................................................................... 58

4.5.3 Parametric Studies.............................................................................................................. 61

4.5.4 Discussion of results .......................................................................................................... 62

5 COST OPTIMIZATION OF B-PILLAR PRODUCTION .............................................................. 63

5.1 Problem Statement ................................................................................................................. 63

5.2 Nomenclature ......................................................................................................................... 64

5.2.1 Parameters: ......................................................................................................................... 64

Contents

6

5.2.2 Variables ............................................................................................................................ 66

5.3 Mathematical Model .............................................................................................................. 67

5.3.1 Constraints ......................................................................................................................... 67

5.3.2 Modeling ............................................................................................................................ 69

5.3.2.1 The cost in the stamping process, .............................................................................. 70

5.3.2.2 The cost in the welding process ................................................................................. 71

5.3.2.3 Material cost ............................................................................................................... 73

5.3.2.4 Model Sumary ............................................................................................................ 73

5.4 Optimization Model Analysis ................................................................................................ 75


5.5 Numerical Optimization ......................................................................................................... 77

5.6 Discussion of Results ............................................................................................................. 80

6 System Integration Study ................................................................................................................. 81

6.1 System Interactions ................................................................................................................ 81

6.2 Problem Formulation ............................................................................................................. 82

6.3 Optimization Approaches ...................................................................................................... 84

6.4 Dicussion................................................................................................................................ 85

7 REFERENCES ................................................................................................................................ 86

8 Apprendix ........................................................................................................................................ 88

Contents

7

8.1 MATLAB Code for the structure optimization ...................................................................... 88

8.2 MATLAB Code for the stamping optimization ................................................................... 101

8.3 MATLAB Code for the welding optimization ..................................................................... 110

8.4 MATLAB Code for the cost optimization ........................................................................... 116

8

1 Design Problem Statement

Pillars are the vertical or near vertical supports of an automobile's window area or greenhouse —

designated respectively as the A, B, C or D-pillar moving in profile view from the front to rear. In

American and British English, the pillars are sometimes referred to as posts (A-post, B-post etc.).This

project focus on the design and manufacturing processes optimization of a car‘s B-pillars.

The division of the system and the team member responsible for each subsystem is as follows:

Table 1.1 Subsystem

Subsystem Team member

Metamodel-based Lightweight Structure Optimization of

B-Pillar under Side Impact Loading

Zheng Shen

Metamodel-based Hot Stamping Optimization of B-Pillar Ying Luo

Optimization of Laser Welding Process Liang Xi

Cost Optimization of B-Pillar Production Yue Jian

http://en.wikipedia.org/wiki/Greenhouse_(automotive)

http://en.wikipedia.org/wiki/British_English

9

2 METAMODEL-BASED LIGHTWEIGHT STRUCTURE OPTIMIZATION OF B-

PILLAR UNDER SIDE IMPACT LOADING --- Zheng Shen

2.1 Simplified Structure Model

Structure Structure design is the first step in the whole developing and manufacturing process of a B-

pillar. A B-pillar is mainly involved in side impact crash and roll over crash conditions (Figure 1). In

order to limit the injury of passengers during the impact, and ensure the side door could open for

evacuation after the crash, a B-pillar should be of high strength to limit the intrusion and deformation

(Figure 2). Meanwhile, it also needs to cut down the total mass for lightweight design purpose.

a) Roll over crash conditions b) Side impact crash conditions

Fig. 1 Two main crash conditions that a B-pillar will get involved in

a) Deformation mode sketch b) Deformation in FE simulation

Fig. 2 the deformation mode of a B-pillar in side impact

Side impact

force

Upper of the B-

pillar

Maximum

Intrusion

Maximum

Intrusion

10

Due to the geometric complexity, a B-pillar is formed in several components, and then gets jointed by

welding process. In order to decrease the total mass, different materials are applied in different

components according to their requirements of mechanics behavior. In nowadays, even one

component can be break down into different pieces with different materials to maximally reduce the

redundant weight and make fully use of material properties (Figure 3).

a) Assembly components of B-pillar b) Outer components divided into

partitions with different materials

Fig. 3 Assembly components design of a B-pillar

2.2 Simplified Structure Model

a) Before crash simulation b) deformation during crash simulation

Fig. 4 Simplified FE Structure Model with side barrier

In this project, a simplified FE structure model is generated. The main frame structure of passenger

cabin is constructed with distributed nodal mass loading (0.6 ton on the front wheel axis, 0.4 ton on

the rear wheel axis) to simulate the real situation of a compact passenger sedan. The test sample of B-

11

pillar is represented by a simplified square-box welded up by 2 U-section bars. And the outer bar is

actually formed with tailor-welded blank method, with which the lightweight design is accessible by

applying different materials to different pieces of panels. The B-pillar is mounted on the main frame

structure on the inner sheet at both ends by rigid-body connection.

a) Intersection design of B-pillar b) Tailor welded blanks design of outside of B-pillar

Fig. 5 Geometric design variables of B-pillar

The side impact scenario is built up according to IIHS side-impact protocol with certain simplification.

The initial impact velocity of main frame is 5566.6 mm/s. The barrier with the same size of that in real

IIHS test is instead made with rigid material.Main design variables are focus on the geometry size and

material selection of B-pillar.

The material for components of B-pillar is selected as high strength steel with properties shown in

following tables.

Table 1 Material property

ID Young's

Module

Poisson

Ratio Density

Yield

Stress

Plastic

curve

Mpa Ton/mm^2 Mpa Strain Stress

HS 1400 210000 0.3 7.89E-09 411 0 411

7.27350-3 404.20001

0.030141 465.73001

0.15315 548.25

0.2 615.38

0.22 800

0.2461 1444.4

𝑙𝑙

𝑙𝑚2

𝑙ℎ

𝑎

ℎ𝑙

𝑏

𝑑 𝑐

𝑡

12

2.3 Nomenclature

In this sub-system problem

Objective Value

The mass of the B-pillar

Design Variables

Thickness of lower component

Thickness of higher component

ℎ Stamping depth of lower component

ℎ Stamping depth of higher component

intermediate variable

The maximum intrusion of the inner board of B-pillar.

The peak acceleration during side impact

Constraint parameters

The maximum intrusion of the inner board of B-pillar of baseline model

The peak acceleration during side impact of baseline model

Geometric Design Parameters (see Fig 5)

Length of lower component

Length of middle component

ℎ Length of higher component

Total width of component

Stamping width of component

Stamping Angle

Material Parameters

13

Sy Material Yield Strength

E Material Elongation

Strain of the Material

Stress of the Material

Density of material

Number of spot-welds

The failure stress of spot-weld

2.4 Mathematical Models

The ultimate goal in this structure design problem is to minimize the total mass of the B-pillar, and

ensure the crashworthiness performance is better than the baseline model. The crashworthiness

performance, including maximum side intrusion and maximum side impact acceleration, can be

obtained during FE-simulation. Due to the complexity of plastic deformation, the mathematical model

of intrusion can only be obtained in surrogate model created on the result of preliminary DoE study.

Min ∑

(Equation 5.1)

Subject to ℎ (Equation 5.2)

ℎ ℎ ℎ (Equation 5.3)

ℎ ℎ (Equation 5.4)

(Equation 5.5)

(Equation 5.6)

ℎ (Equation 5.7)

(Equation 5.8)

(Equation 5.9)

(Equation 5.10)

(Equation 5.11)

ℎ (Equation 5.12)

14

2.5 Model Analysis

It is helpful to do the preliminary analysis of mathematic model before solving it using optimization

method. The monotonicity analysis will tell the trend of model and guide the local computation. The

problem discussed here is a typical lightweight structure optimal design problem, and the structure

mass is the monotonically increasing function of component thickness and stamping depth. In the

preliminary study, the mass function is obtained as

ℎ ℎ ℎ ℎ ℎ ℎ (Equation 5.13)

Where ,

2.6 Optimization Methodology

2.6.1 Metamodeling Method

Metamodel-based optimization is an effective approach for engineering design problems, especially

when the problem has highly complex relationships between variables, constraints and objects.

The common approach to access Metamodel-based optimization includes several following steps (

Fig. 6):

1. Define the optimization problem: objectives, constraints and design variables.

2. Generate samples of design variables using Design of Experiment method, including Latin

hypercube method, Uni-variant method, Orthogonal arrays method, Central composite methods

etc.

3. Conduct numerical simulations using sampled points as input, and extract the interesting values of

responses.

4. Use the extracted result values to generate the metamodel. Techniques for metamodeling includes

regression, neural network, etc.

5. Assess the predictive capabilities of the metamodel by testing samples in unsampled design space.

If the model is ill constructed, refine the model by these testing samples.

6. Search the optimization over the design space using the constructed metamodel. If the

convergence is achieved, stop and output the result, otherwise, refine the model by increasing

samples.required.

15

Fig. 6 Metamodel-based optimization procedure

2.6.2 Design of Experiment

To generate the metamodel, several real results are required to do the training. Samples are constructed

in Catia, meshed and prepared in Hypermesh. FE simulations are conducted using LS-Dyna and run

on CAEN advanced computing center. Each sample uses 8 clusters and 30 minutes to compute. A

batch script is generated to extract the interesting data out of result automatically. Matlab is called to

do the post-processing, including integration and smooting.

In the first round of DoE study, quasi-full-factorial sampling method is applied to gather global

information of design space. The variables and their levels are shown in Table 2

Thus, there are 75 samples generated and run in side impact scenario via Finite Element Simulation.

And the validation result shows good consistency between metamodel and real situation whenh_l is

close to 3 sampling levels, that means it goes well along the surface of t_l and t_2 once h_l is fixed

close enough to its sampling levels. However, in the dimention of h_l due to the sparse levels in h_l

dimension, the metamodeling results turn to vary dramatically, showing obvious over-fitting problem.

Yes

No No

Yes

Define the

optimization problem

DoE study over the

design space

FE simulation at

sampling points

Refine the

model by more

Construct the metamodel

usually not analytical

Validate the model Perform the optimization

Check the convergence

Output

16

Table 2 Quasi-full-factorial sampling

Design Variables Level 1 Level 2 Level 3 Level 4 Level 5

ℎ

2

In order to improve the quality of metamodel, a second round of 25 samples is generated using Latin

Hypercube Sampling method and added into the training data. In the meantime, the spread of radial

basis functions of neural network is controlled at 1.5, and the total amount of neurons is limited at 50.

Getting trained in such way, the new metamodel shows higher training error but the validation error is

remarkably improved, which shows in Table 3.

Table 3 Validation result for improved Metamodel

Design

Variables Sample_1 Sample_2 Sample_3 Sample_4 Sample_5 Sample_6

ℎ 39.4649 41.9507 40.2964 35.7155 44.0470 37.8566

2.2368 1.0475 1.3636 1.5958 2.0580 1.8682

2 1.7757 1.0225 2.1880 1.9923 1.6392 2.3058

Error of max

intrusion 0.1094 0.1347 0.0816 0.0023 0.0134 0.0838

Error of peak

acceleration 0.1582 0.1693 0.0272 0.1412 0.1509 0.0054

17

2.6.3 Monotonicity Analysis

Fig. 7 Intrusion response surface model

Fig. 8 Acceleration response surface model

In the monotonicity analysis based on metamodel, the highly nonlinear response of maximum

intrusion, peak acceleration are shown in fig.7 and fig 8. And they together with the structure mass in

fig. 9 have different trends in performance according to 3 design variables. Thus there shall be several

local optimization results of structure mass inside the design space under the nonlinear constraints of

intrusion and acceleration.

18

Fig. 9 Structure mass response surface model

2.6.4 Optimization Method

The matlab function fmincon is applied to solve the optimization problem. Since the numerical

metamodel has no analytical function, the Active-Set Optimization method is used to find the local

optimal. In order to search around the whole design space, the fmincon is iterated for 5 times, and

each time the 6 initial searching points are generated again by Latin Hypercube Sampling method.

2.7 Numerical Result

The 1st round optimization result is shown in Table 4, the average iteration time is 35.

Table 4 Comparison of 1st round optimization results

Design

Variables Sample_1 Sample_2 Sample_3 Sample_4 Sample_5 Baseline

ℎ 38.3712 36.4330 40.6616 37.0970 38.8566 40.0000

1.6236 2.3895 1.7008 2.0780 2.2735 1.80000

2 2.2770 1.4531 1.0000 1.9966 1.2300 2.40000

max

intrusion 82.9471 76.6247 83.0011 82.1529 74.5444 83.4110

peak acc. 5.4619 5.2423 5.5377 5.8168 5.8037 5.8169

Mass 0.0035 0.0027 0.0019 0.0033 0.0024 0.0038

19

From the result, it is easy to tell the notable local optimum phenomenon, which proves that the model

is highly nonlinear and non-monotony. The 3-D plot of solution space (Fig. 6) also reveals such

property in the metamodel of constraint functions. Another reason that leads the result to be non-

converged is that the unequal constraints from Baseline model is relatively slack so that there are more

local optimums that satisfy the constraints, and the optimization algorithm will stop at those local

optimums.

Fig. 10 The local optimum situation in problems with nonlinear constraint

The figure above (fig.10) sketches the local optimum situation caused by nonlinear constraint

function. From the figure we can see, a tighten constraint value will eliminate those local optimum

with higher object value and force the optimization converge to a lower solution, which is more likely

the global optimum. According to this understanding, in the second round optimization running, a

tighten upper bound of baseline model is set at max intrusion = 70.000, and max acc. = 5.5000. Then

the result is shown in Table 5, the average iteration time is 37.

In this result, a better convergence is achieved at , and another better

local minimal is at . These results are then checked in FE simulation. The

design point at has a simulation result at ,

which is worse than the design constraints. However, the minimal at has

a performance at , which is quite close to the result from metamodel. In

comparisons of these samples with of closer samples that are used in DoE study, their simulation

results are relatively closer to sample results, showing reasonable continuity of response in local

regions. The false result at from metamodel is probably caused by local

overfitting of metamodel.

20

Table 4 Comparison of 2nd

round optimization results

Design

Variables Sample_1 Sample_2 Sample_3 Sample_4 Sample_5 Baseline

ℎ 40.8301 44.6990 44.6989 41.6077 44.7009

2.3763 1.8800 1.8800 2.0050 1.8800

2 1.5988 1.5442 1.5442 1.0000 1.5441

max

intrusion 64.7752 70.0000 70.0000 70.0000 70.0000 70.000

peak acc. 5.3588 5.5000 5.5000 4.4169 5.5000 5.5000

Mass 0.0030 0.0028 0.0028 0.0021 0.0028

In the optimization running afterward, more results are founded closely around at

, which are encouraging to choose this local optimum as a reasonable

global solution under given constraints.

2.8 System-level Tradeoffs

The subsystem of structure optimization is closely related to other subsystem problems in this topic.

Currently the 3 design variables ℎ ，， 2 are only bounded by empirical design regions. While in

reality, ℎ ，， 2 all will strictly constrained by forming process of stamping (subsystem 3), the

， 2 will affect the quality of spot-welding (subsystem 2), and the total mass of the structure as well

as the manufacturability of such design in stamping process and spot-welding jointing will be

observably constrained by the manufacturing cost control (subsystem 4).

2.9 Discussions

The optimal design of vehicle structure is a classical nonlinear optimization problem. According to

different targets, the nonlinearity can be in the constraint (when optimizing weight) or object function

(when optimizing the crashworthiness performance). This nonlinearity is due to the complex geometry

and its behavior under high speed impact loading, which is very hard to decompose and get the

analytical function. Metamodel method shows a good performance when trying to solve such

problems, which is also demonstrated in this subsystem. To ensure the accuracy of metamodel, a well-

designed DoE study is quite necessary to train the model with least losing of information in real

21

scenarios. And if possible, an automatic iteration of DoE and model training will be even better. A

derivative-free optimization/searching method then will find the local optimum over the metamodel.

The problem of converging into different local minimums is still very common over the metamodel,

especially when the real problem is highly nonlinear. Thus, a good global searching method is still

needed.

From above, we can see that metamodel-based method has several steps to conduct and is very costly

of time in sample preparation and of computation in metamodel generation. A popular research in this

area is trying to optimize the whole procedure: using the least samples to generate the best fitting

metalmodel. And in real engineering application, another important problem for such research method

is to setup the design space, which highly requires expert-knowledge in such domain.

22

3 METAMODEL-BASED HOT STAMPING OPTIMIZATION OF B-PILLAR --- Ying

Luo

There are two basic approaches on the design of EVs: conversion- design and purpose-built design.

Either an existing ICE vehicle is converted into an electric vehicle by replacing the propulsion system,

while keeping the general structure, or a completely new vehicle is designed to specifically fit the

purpose of using electric propulsion. Both methods and current supermini class models will be

introduced. Those ICE models designed with alternative propulsion taken in mind will be included in

the purpose-built section. The overview presents selected EV models‘ performance, body design and

choice of drive train equipment as well as their packaging and safety performance. To demonstrate the

difficulties and measures taken in the field of EV crash safety the selection is focussed on purpose-

built EVs that stand out for their performance, size, weight, design or safety concepts. EG-class M1

vehicles, which need to meet the standard safety regulations and demand a regular driving licence,

were preferred.

Additional models and details are summarized in table 1 in the appendix, which is neither claimed to

be complete in all details nor in the range of current models, due to the lack of published details on the

new models and the abundance of EVs currently introduced. It will, however, provide an overview

over current developments in mini EVs. Unfortunately there is only very few information released on

crash data of EVs.

3.1 Problem Statement

The automotive industries have been interested in light weighted components and the safety as well as

the quality of vehicles for years. With years of development in this industry people have found out that

both weight reduction and performance improvement can be achieved by optimizing structure design,

enhancing the properties of materials and improving the process of manufacturing. As a key

manufacturing process, stamping plays an important role in the entire manufacturing process

especially welding and the quality and performance of the vehicles. Additionally, stamping also

implies limitations and prospects of the structure geometry and material selection, contributing to the

design stages. It has been proved that properly control of the stamping process and carefully design of

the die geometry could enhance the material formability, allowing more design possibility. Moreover,

properly designed stamping could directly increase the accuracy of the final structure, and help to

modify the quality of the vehicles.

23

This optimal subsystem aims at decreasing the manufacturing cost of the automotive B-pillar by

minimizing the maximum punch force during stamping and maximize the formability of the blank at

the same time, taking consideration of the limits of material formability during the process, the trade-

off with the structure geometric design stage and industrial requirements. Moreover, the effects to

welding process are also considered in the entire optimization system.

Punch force, according to previews studies, has been proved to be affected by four main factors, which

namely are blank holder force, friction conditions and the punch stroke. Nevertheless, some other

factors such as the final thickness of the blank that is introduced to avoid crack at the edge of the

structure and minimize the total mass the B-pillar is also important.

Fig.1 B-pillar in vehicles: a) the outside and inside B-pillar; b) the simplified structure of B-pillar; c)

the dimensional parameters of simplified B-pillar

Fig.2 Hot stamping process in industry

As presented in the above figure, B-pillar of the vehicle is simplified as a U shape shown in Fig1.b,

most of the geometric characteristics of the structure shown in Fig1.a is given by the design stage. In

24

the stamping process, a rectangular blank with properly designed dimensions is assembled between a

die and two blank holders. A punch with designated geometries then moves down with a constant

speed to form the final shape. In order to reduce noises from non-considered factors, the manipulation

of the blank is assumed to be the same on the direction as the pillar goes. Therefore, only cross-section

of the sample is analyzed. During this process, four factors (i.e. the blank holder force, the coefficient

of friction, the punch stroke and the final thickness of the blank) are used as the four variables, and the

properties of the material as well as the initial dimensional factors are considered as parameter.

Fig.3 The setup of stamping process

25

3.2 Nomenclature

Objective Value

The mass of the B-pillar

Material Properties

Sy Yield Strength

St Tensile Strength

Y Young‘s Modulus

Poisson's ratio

E Elongation

K Strength coefficient

n Strength exponent

l Elongation of the material at stamped temperature

intermediate variable

td Designed thickness of B-pillar

to Initial thickness of the blank

tf Final thickness of B-pillar

h Stroke of the punch

hd Designed depth of B-pillar

R Designed radial of die

Designed angle 1 of B-pillar

Wo Original width of blank

26

Top width of B-pillar

2 Bottom width of B-pillar

Area of the each blank holder

Machine setting

Maximum punch force

Blank hold force

The coefficient of friction between the blank and die

Maximum acceptable friction coefficient

Minimum acceptable friction coefficient

Geometric Design Parameters (see Fig 5)

Length of lower component

Length of middle component

ℎ Length of higher component

Total width of component

Stamping width of component

Stamping Angle

Material Parameters

Sy Material Yield Strength

E Material Elongation

Strain of the Material

Stress of the Material

Density of material

Number of spot-welds

The failure stress of spot-weld

27

3.3 MATHEMATICAL MODEL

3.3.1 Objective function

The aim of the subsystem is to minimize the punch force during stamping.

ℎ 2 ℎ ℎ

The objective function can be achieved by performing simulation of stamping process accounting for

the different variables, and by carrying out the DOE (design of experiment), a series of punch force

depending on the change of variables are recorded.

3.3.2 Design Variables

The decision variables are selected based on previous study investigating factors that could affect

punch force most significantly, and some factors that have influence on the formability of metal.

Decision Variables:

Blank hold force in stamping

The punch coefficient of friction

h Punch stroke

Final thickness of B-pillar

3.3.3 Parameters

Material Properties:

28

=130MPa Yield Strength

= 270 MPa Tensile Strength

= 210 GPa Young‘s Modulus

0.3 Poisson's ratio

= 270 MPa Strength coefficient

= 0.17 Strength exponent

= 0.78 Maximum elongation at the stamping temperature

Other parameters

= 1.5 Safety factor

= 0.78 Coefficient of friction without lubrication

= 0.05 Minimum coefficient of friction

= 3mm The width of heat affect zone

Minimum blank force based on industry experience

29

Geometric factors

3.3.4 Constraints

All the variables and parameters that help to perform a simulation should result a stamping model

satisfying the material limits, the geometric limits, physical limits as well as industrial limits. At the

same time, the limits introduced by other subsystem should also be considered.

For the material properties requirement, and based on the simulation results, the maximum normal and

shear stress could be derivate and mustn‘t exceed the material tensile stress with accounting for the

safety factor.

(equation 3.1)

The geometry limits are caused by the geometric design of the structure as well as the welding process

requirements. Based on the geometric relationship, the upper limit of the punch stroke is determined.

2

2

2 ℎ

2 (equation 3.2)

ℎ

( )

ℎ (

)

= 2 mm Designed thickness of B-pillar

ℎ = 35 mm Designed depth of B-pillar

Designed angle of B-pillar

= 340 mm Original width of blank

= 140 mm Top width of B-pillar

2 = 60 mm Bottom width of B-pillar

= 60 mm2 Area of the each blank holder

30

In practice, the blank holder force is applied to avoid wrinkling during stamping. Therefore, the blank

holder force could not be too small. On the other hand, this limit is set to avoid damage and wear for

the stamping machine. Hence, the blank holder force should not be excessively large, which could lead

fracture of the material. The maximum blank holder force is therefore roughly calculated by

considering the surface shear stress raised by friction.

FB F ’ (Equation 3.3)

FB F’ - (Equation 3.4)

FBmin FB F (Equation 3.5)

Another important consideration of the stamping process is the formability of the material, which

could be approximated by the allowed minimum thickness after stamping. In order to avoid failure

during the stamping process, and based on the physical requirement that the blank volume should be

constant during the whole process, the lower and upper bound of the final thickness could be

determined:

(Equation 3.6)

(Equation 3.7)

Moreover, the coefficient of friction should be consistent to the industrial used value

(Equation 3.8)

(Equation 3.9)

31

3.3.5 Summary of Model

By simplifying the model and eliminating the obvious redundant equalities, the summary

model could be built.

Objective

Function: (ℎ ) 2 (Equation 3.10)

Subject to (Equation 3.11)

2

2

ℎ

(Equation 3.12)

ℎ

( )

ℎ (

)

(Equation 3.13)

(Equation 3.14)

(Equation 3.15)

(Equation 3.16)

(Equation 3.17)

(Equation 3.18)

(Equation 3.19)

(Equation 3.20)

(Equation 3..21)

32

3.4 Optimization Model Analysis

3.4.1 Metamodeling

As an effective approach for engineering design problems, metamodel-based optimization is widely

used in the problems with significantly complex relationships among variables, parameters, constrains

and objects. The typical step of processing metamodeling method is show as follows:

Fig 3.4 Steps of metamodeling

3.4.2 Design of Experiment and Simulation

In order to obtain simulation samples with variable values more entirely and randomly spread in the

design domain, Latin hypercube method was used to determine the sample values. The model was

built and simulated using Abaqus and the correlate values were reported when simulation finished.

Two methods have been applied to obtain the relationship between the objective function (and certain

constraints) and variables. Firstly the recorded data were analyzed by regression and therefore find out

the error level. By adjusting the range and order of the polynomial function, and the error was

minimized, and the form of the function was obtained. Since the relationship between the object and

Yes

No No

Yes

Optimization problem

definition

DoE analysis in the

feasible domain

FE simulation using

sampling points

Refine the

model by more

Construct the metamodel

Validate the model Perform the optimization

Check the convergence

Output

33

variables is not clear, all the variables were scaled to the same range to ensure that their influence

would not be affected by their value level. Secondly the determined polynomial function was analyzed

using stepwise algorithm to eliminate the less important ones. By combination of these two method,

the objective function was achieved:

2 ℎ

(rsq = 0.80)

(Equation 3.22)

Similarly, one constraint function, namely the maximum stress of the blank, was obtained without

scaling:

(rsq = 0.78)

(Equation 3.23)

Five randomly selected samples were selected and simulated to validate the polynomial functions

obtained.

Table 3.1 Validation form

Variables Sample_1 Sample_2 Sample_3 Sample_4 Sample_5

7.3 5.9 13.4 10.6 9.2

2.1 1.6 1.9 2.4 1.7

0.6 0.1 0.3 0.5 0.7

36.1 40.5 44.9 48.3 52.4

Error 19.8% 14.3% 15.7% 25.6% 20.1%

34

3.4.3 Dependent Analysis

Table 3.2 Variable dependent analysis

Variables 𝐟 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖 𝟗

𝑭𝑩𝑯𝑭

𝒕𝒇

𝛍

𝒉


The monotonicity analysis is performed as shown in Table. 10.

35

Table 3.3 Monotonicity analysis

Variables 𝐟 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖 𝟗

𝑭𝑩𝑯𝑭 + ? + + -- --

𝒕𝒇 + -- + -- --

𝛍 ? + + + -- --

𝒉 -- + + -- --

Activity

Active

for

&

Active

for

Active

for

Active

for

Active

for

Active

for

According to the analysis above, it is obvious that all variables correlating to different constraints have

at least one upper and one lower bound, implying a well boundedness of the problem. Meanwhile,

since some of the constraints has non-linear functions, there will be some extra boundedness.

3.4.5 Activeness Analysis

As discussed above, the problem is well bounded. Now it is necessary to check if all constraints are

active. Firstly from the monotonicity analysis table, is upper bounded by both and and

lower bounded by and . Based on the monotonicity principles, at least one of and are

active.

:

:

Obviously, that is a stronger boundary than , so that can be defined as inactive constraint.

By applying this method to check all constraints, the activeness could be determined and the analyze

results are show in the monotonicity table.

36

3.4.6 Numerical Results

This problem is to find a best trade-off between the minimizing the punch force and maximizing the

blank formability. Therefore, different optimization could be found based on different consideration of

the weight of each objective (i.e. minimizing punch force, maximizing the punch stroke and

minimizing the final thickness).

By considering the three objective as equally important, the optimization of the problem could be

found out by applying Fmincon-interior-point algorithm, which shows an optimized result with

different initial guess. The Following are four guesses of the initial guesses:

Scenario 1:

Initial guess Final value Value of the

objective after

optimization

Number of

interations 𝑭𝑩𝑯𝑭 7.04kN 9.41kN

0.52

45 𝒕𝒇 1.6mm 1.98mm

𝛍 0.64 0.33

𝒉 37.99mm 43.77mm

Scenario 2:


objective after

optimization

Number of

interations 14.16kN 9.42kN

0.51

34

1.8mm 1.97mm

0.17 0.32

45.83mm 43.78mm

37

Scenario 3:


objective after

optimization

Number of


0.52

55 2.33mm 1.98mm

0.06 0.34

52.53mm 43.67mm

Scenario 4:


objective after

optimization

Number of


0.52

40 1.48mm 1.98mm

0.43 0.32

49.94mm 43.77mm

Scenario 5:


objective after

optimization

Number of


0.52

39 2.17mm 1.98mm

0.53 0.31

39.7mm 43.79mm

38

3.4.7 Parameter Analysis

With all other parameters fixed and only change the weight factor of each sub-objective function, the

results changed slightly as shown below:


objective after

optimization

Number of

interations

12.67kN 8.77kN

0.61

29 2.17mm 2.01mm

0.53 0.38

39.7mm 41.72mm


objective after

optimization

Number of

interations

12.67kN 7.86kN

0.47

47 2.17mm 1.72mm

0.53 0.23

39.7mm 45.98mm

39

However, when to change the other parameters including such as the initial dimension of the blank,

the solution tends not valid, and the functions obtained by simulation and regression have been

significantly changed. For that reason, it can just conclude the this optimization solution is only valid

for these specific siduation.

3.5 System Trade-Off

The subsystem of stamping optimization is closely related to other subsystems especially the structure

optimization subsystem. The dimensional variables are bounded by the design requirements, and the

welding subsystem also implied some limits to the final thickness. Meanwhile, the initial thickness of

the blank is leaked to the material cost in the cost subsystem. Finally, the punch force is a limit for the

cost subsystem to estimate the tool life.

3.6 Problem Discussion

There are some problems can be observed in this problem. Firstly when the parameters linked to the

stamping process (such as the blank initial dimension) change, the entire optimization would fail.

Besides, the error in validation is not low enough. There are several possible reasons that can explain

the phenomenon. One of the most possible reasons is that simple polynomial function cannot express

relationships between the variables and punch force. For example, according to the previews studies,

with the change of the die dimension, the tendency of the change of the punch force would be different

with punch stroke increasing (i.e. in some experiments, the punch force would increase with the punch

stroke increasing, while in some other cases, there will be a decreasing when punch stroke reach a

certain value). Meanwhile, with theoretical mechanism analysis, there should be at least one

exponential relationship between the punch force and coefficient of friction. In conclusion, the

relationship between the variables and objective function are too complex to process a linear

regression analysis. On the other hand, the number of samples of this problem is possibly not enough

to present a right relationship between the variables and objective functions. In the future work, it is

recommended that more samples are simulated and other metamodeling-based method such as neural

net work could be used to find a more nonlinear relationship.

40

4 Optimization of Laser Welding Process

4.1 Background

Laser welding has been widely used in the manufacturing processes of vehicles due to its obvious

advantages. The focused laser beam possesses the high power density, the low and localized energy

input, the ability to be precisely controlled, to name only a few. These benefits of laser beam could

result in a high depth-to width ratio, a small heat-affected zone, minimal distortion and residual

stresses. Thus, various vehicles structures including B-pillar (Fig. 4-1) prefer to use laser welding [6,

7].

Fig. 4.1: Laser Beam welding of B-Pillar

In the case of B-pillars, outer and inner plates are jointed together using spot laser welding. This joint

has to possess enough shear strength in order to avoid fracture during the side collision. The joint

strength of the laser welds depends on the weld joining area [8], penetration and residual stress which

are largely determined by the properties of materials, the parameters of laser welding input parameters,

including output power, welding speed, focal position, shielding gas and position accuracy [9]. In

addition, the heat input and heat affect zone affect the welding quality due to their influence of causing

distortion during or after the welding process.

41

4.2 Subproblem Description

For simplified model, this individual study focuses on optimizing the laser welding variables such as

beam power, beam exposure time, incident angle of the beam (shown in Fig. 4-2), distance between

two adjacent spot welds and total number of spot welds in order to maximize the joint strength and to

keep the heat input at reasonable range.

The variables of laser power, exposure time and incident angle exert effect on the geometry of spot

welds, especially for beam width and depth of penetration. Those are highly likely to decide the

strength of single spot weld. At the same time, the strength of weld in B-pillar is the comprehensive

result of the strength of the total number of spot welds. Distance between two neighbouring spot

welds has direct relationship with the number of spot welds. Thus, the strength function can be

expressed with those variables.

(a)

(b)

Figure 4.2 Geometry of spot weld, B-pillar and properties of laser

The constraints can be obtained through the requirement of welding and physical phenomena. It will

be discussed in detailed in the following parts.

42

4.3 Nomenclature

S Safety factor

Shear Stress During Collision

[] Acceptable Shear Stress for Weld

F The Tensile Force During Collision

n Total Number of Spot Welds per Side

BW Bead Width

p Beam Power

Dp Depth of Penetration of The Weld

a Incident Angle

t Exposure Time

Absorption Coefficient

L Length of The B-Pillar

d Distance Between Two Adjacent Spot Welds

h Thickness of the Workpiece

Q Heat Input for Single Weld

Qmax Acceptable Heat Input for Single Weld

Ultimate Tensile Strength of the High Strength Steel

Safety Factor That Depends on the Potential Consequences of Failure

43

4.4 Mathematical Model

During side collision, the joint connecting inner and outer B-pillar primarily bears shear stress due to

the dissimilar plastic deformation between these two B-pillars. The resulting tremendous shear stress

may cause the failure of the weld. In order to reduce the possibility of failure of the joint, the joint

strength has to be strong enough to overcome the huge shear stress in the process of crash.

The joint strength has direct relationship with the bearing capacity. Thus, the primary objective of this

subsystem is to maximize the bearing capacity under the reasonable heat input and the diminutive heat

affected zone.

Figure 4.3 Force Analysis in B Pillar

4.4.1 Objective Function

Even through the actual stress distribution over the joint is very compressive, an estimate of the value

of stress in the joints can be done under the assumptions that stress distribution is uniform along the

welds and residual stress of the joints is ignored. This assumption makes sense due to the fact that the

distribution of residual stress is pretty complex and difficult to be calculated; more importantly,

residual stress could not be obvious if some measures are taken during the process of design and

manufacturing [10].

The shear stress across the weld is depends on the force F, the across area of the weld, and the total

number of spot welds. Here, we just use the to represent the shear stress over the weld, and

this term will be discussed in detail in the metamodel.

The relationship between and is given by:

(Equation 4.1)

44

The following acceptable stresses can be assumed for the strength of welded joints for static load [11].

Actually during the side collision, the load changed dramatically, while [τ] can be estimated by using

the coefficient 3.5 in the denominator.

(Equation 4.2)

By the nature of safety factor, , this means, has to be larger than so that the weld

would not be damaged. Usually, the value of S is larger, the strength of the weld is stronger. Thus, the

objective function of this subsystem can be represented by equation (7.3):

(Equation 4.3)

4.4.2 Constraints

The depth of penetration depends on laser power p, exposure time t and incident angle [12]. The exact

form of will be explored in the metamodel. In order to produce an acceptable weld, the

depth of penetration should not be less than the thickness of single plate:

ℎ (Equation 4.4)

The heat input has direct relationship with the beam power, beam exposure time and the absorption

coefficient. It can be calculated by:

(Equation 4.5)

The large heat input would cause distortion of B pillar and enlarge the heat-affected zone, which has

adverse effect on the strength of spot weld. Here, we limit to 1.5 kJ. So, another constraint is

given by:

45

(Equation 4.6)

The total number of spot weld and distance between them is limited by the length of B pillar:

(Equation 4.7)

(Equation 4.8)

The welding operations are performed using 2 kW continuous wave ND:YAG laser. Limited by the

capacity of the laser equipment, the constraint for the power is given by:

(Equation 4.9)

Welding quality depends on the beam exposure time to a large extent. Too short beam exposure time

leads to the insufficient weld; on the other hand, with too long beam exposure time, the welding bead

will be damaged resulting in the poor quality of spot bead. Here, the range for beam exposure time

remains the same as that of the reference [12]:

(Equation 4.10)

The incident angle of the beam also has effect on the weld geometry. Welding with straight beam ( 90°

beam incident angle) may damage the optics by back reflection of the beam. So the scope of beam

incident angle is kept the same as that of the reference [12]:

(Equation 4.11)

In addition, the distance between spot weld affects the strength of B pillar. When the distance is larger

than 70 mm, the strength of B pillar reduces obviously. At the same time, this distance is restricted by

the design of B pillar, this is 24 mm. So the range of distance is given by:

(Equation 4.12)

46

4.4.3 Metamodel

Linear and quadratic polynomial equations for predicting the depth penetration and the bead width

were developed. A stepwise method in Matlab was used to fit the second order polynomial equation

(4.13) in the reference [12] data and to identify the relevant model terms.

∑ ∑ (Equation 4.13)

4.4.3.1

The stepwise method is to build the quadratic form for the function of depth of penetration (DP) with

respect to exposure time (t), beam power (p) and incident angle (a). The figure below is results for

stepwise and the detailed code is attached as Appendix. From the results of stepwise (figure 4.4), it can

be found that the significance for the terms of t, a, t2 and a2 is worst and can be removed. Regression

method is utilized to find out the coefficient of the rest terms. Finally, the equation is shown as

equation (4.14):

2

(Equation 4.14)

47

Figure 4.4 the results of stepwise

To verify the equation, the curve of the function is plotted for each parameter, and the dots in the

figure below are original data.

Figure 4.5 Depth of penetration against beam exposure time

Figure 4.6 Depth of penetration against beam power

48

Figure 4.7 Depth of penetration against incident angle.

From the figure 5-7, the curve is in good agreement with the original data, so the metamodle is

acceptable.

49

4.4.3.2

As for the function of bead width with the variables of beam power, incident angle and beam exposure

time, the stepwise method can be also use to identify the significant terms, a and p×t, and then the

equation (4.15) are obtained using the regression method:

(Equation 4.15)

Verification has been taken and the curves are in good agreement with the original data (figure 8-10).

Figure 4.8 Bead width against beam exposure time

The highest tensile force is parameter from design system. Here F=20 kN, but more investigation is

required to verify this value.

50

Figure 4.9 Bead width against beam power

Figure 4.10 Bead width against incident angle

51

2

(Equation 4.16)

2

(Equation 7.17)

F is the parameter which comes from structure subsystem.

Finally, verification is taken with the help of Matlab. The correlation coefficient R2 is close to 1 from

table 1 for both DP (p,t,a) and BW(p,t,a) , thus the regression equation is significant.

Table 4.1 Statistics for metamodel

Function R2 statistic F value Prob > F error variance

DP (p,t,a) 0.9921 526.62 <0.0001 0.0009

BW(p,t,a) 0.9302 159.90 <0.0001 0.0034

4.4.4 Model summary

First of all, the variables, parameters and constants that are used in this model is summarized in table

2.

In addition, after eliminating the intermediate variables, the mathematical model can be constructed

table 3.

52

Table 4.2 Summary for variables, parameters and constants

Elements Name Value Unit

Variables p kW

t s

a Rad

n per side

d mm

Parameters F 20,000 N

h 2 mm

L 1075 mm

Constants 1444 MPa

0.693

53

Table 3 Summary for model

2 (Equation 4.18)

S.t.

2

(Equation 4.19)

2 (Equation 4.20)

(Equation 4.21)

(Equation 4.22)

(Equation 4.23)

(Equation 4.24)

(Equation 4.25)

(Equation 4.26)

(Equation 4.27)

(Equation 4.28)

(Equation 4.29)

2 (Equation 4.30)

54

Param

eters

4.4.5 FDT Analysis

Table 4.4 FDT Analysis

f 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖 𝟗 𝟐

p ▲ ▲ ▲ ▲ ▲

t ▲ ▲ ▲ ▲ ▲

a ▲ ▲ ▲ ▲

n ▲ ▲ ▲

d ▲ ▲ ▲

F ▲

h ▲

L ▲


Table 4.5 Monotonicity Analysis

f 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖 𝟗 𝟐

p N/A N/A + + -

t N/A - + + -

a N/A - + -

n - + -

d + + -

Variab

les

55

From the monotonicity analysis above, f decreases with respect to n, and only g3(+) wrt n, so g3 is

active. For nonobjective variable d, g3 and g11are increasing constraints and g12 is decreasing

constraint. So the variable d is bounded both blow and above. For the variables p, t and a, the function

do not show obvious monotonicity. Yet these variables are bounded both below by at least one

nonincreasing semiactive constraint and above by a least one non-decreasing semiactive constraint.

Thus, the problem is well-bounded.

Except the objective function, the constraint g1 are nonlinear, the other constraints behave linearly

with single variable, and thus the figures of the behaviour of those constraints with respect to single

variable are ignored. And the figures of the behaviour of g1 are similar to figure 5-7, so those figures

will not be presented. Below are function behaviours with respect to only one variable while all others

are fixed.

Figure 4.11 Function vs incident angle

56

Figure 4.12 Function vs beam power

Figure 4.13 Function vs exposure time

From figure 11-13, function decreases with respect to incident angle, beam power, and exposure time

with other fixed variables even through the function is second order equation.

57

4.5 Optimization Study

4.5.1 Process Description

The optimization process will calculate the optimum values of design variables, namely n, p, t, a, and

d. The minimum search function in MATLAB, fmincon.m, is employed to find the optimum. The tool

fmincon has four algorithm options: 'interior-point', 'sqp', 'active-set', 'trust-region-reflective'. All of

those are tried to check the optimum results. The overall process flow for optimization is shown in

figure.

Figure 4.14 Process flow

Different initial values were tested to check whether the optimal values found by fmincon converge to

the same value. Additionally, four kinds of algorithm in fimincon.m were applied to verify the

optimum mutually.

58

4.5.2 Algorithm Compare

For the same parameter, four kinds of algorithm in fimincon.m are tried from the same initial points.

The results are show below Table (3-5):

%%%Shown in the Matlab command window %%%%%%%%%%%%%%%%%%%%%

Local minimum found that satisfies the constraints. Optimization completed because

the objective function is non-decreasing in feasible directions, to within the default value of

the function tolerance, and constraints are satisfied to within the default value of the constraint

tolerance.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Table 4.5 Results of h=2mm using 'interior-point' algorithm

Group 1 Group 2 Group 3 Group 4

Var. Initial Out Initial Out Initial Out Initial Out

n 20 45.8 10 45.8 30 45.8 53 45.8

p 0.5 1.77 0.75 1.77 0.83 1.75 0.62 1.70

t 0.5 1.23 0.5 1.22 1.2 1.24 0.93 1.28

a 1.31 1.48 1.33 1.48 1.45 1.48 1.37 1.48

d 30 24 30 24 53 24 25 24

Function -4.6231 -4.6231 -4.6231 -4.6231

Iterations 61 31 25 18

59

Table 4.6 Results of h=2mm using 'sqp' algorithm



n 20 45.8 10 45.8 30 45.8 53 45.8

p 0.5 2 0.75 1.44 0.83 1.44 0.62 1.71

t 0.5 1.08 0.5 1.50 1.2 1.50 0.93 1.27

a 1.31 1.48 1.33 1.48 1.45 1.48 1.37 1.48

d 30 24 30 24 53 24 25 24

Function -4.6231 -4.6231 -4.6231 -4.6231


Table 4.7 Results of h=2mm using 'active-set' algorithm



n 20 45.8 10 45.8 30 45.8 53 45.8

p 0.5 1.75 0.75 1.44 0.83 1.45 0.62 1.45

t 0.5 1.24 0.5 1.50 1.2 1.50 0.93 1.49

a 1.31 1.48 1.33 1.48 1.45 1.48 1.37 1.48

d 30 24 30 24 53 24 25 24

Function -4.6231 -4.6231 -4.6231 -4.6231


60

'trust-region-reflective'

The following results would show on the screen:

―Warning: The default trust-region-reflective algorithm does not solve problems with the constraints

you have specified. FMINCON will use the active-set algorithm instead.‖ This means that 'trust-

region-reflective' failed.

Table 4.8 Results compare for different algorithm

Algorithm Initial point x1 x2 x3 x4

'interior-point' Iterations 61 31 25 18

'sqp' 14 14 11 5

'active-set' 19 15 12 8

Compare the results above, from different initial points and through different optimization algorithm,

the output of variables of p and t are different while both the other output variables and final optimized

value function are the same. This is because for different initial guesses, the constraint g2 is active

leading to the product of p and t is constant. t the same time, the function is affected by ―p×t‖ by

coincidence. So the value of function does not change. Furthermore, the total number for iterations in

'sqp' is the least among the algorithm. Thus, the algorithm 'sqp' is selected to complete the following

parametric studies.

Corresponding to baseline [ 40 1.8 1.2 1.4 30]T

, function value f0= -3.5. Improvement is defined by

|f1- f0/f0|

Table 4.9 Results compare for improvement


Element Initial Out Initial Out Initial Out Initial Out

Function value f1 -4.6 -4.6 -4.6 -4.6

Improvement 30% 30% 30% 30%

61

4.5.3 Parametric Studies

Table 4.10 Results of h=2.5mm, F=25 kN using 'sqp' algorithm



n 20 45.8 10 45.8 30 45.8 53 45.8

p 0.5 1.95 0.75 2 0.83 1.52 0.62 1.47

t 0.5 1.11 0.5 1.08 1.2 1.42 0.93 1.48

a 1.31 1.48 1.33 1.48 1.45 1.48 1.37 1.48

d 30 24 30 24 53 24 25 24

Fval -3.70 -3.70 -3.70 -3.70


Table 4.11 Results of h=2.2mm, F=30 kN using 'sqp' algorithm 'sqp'



n 20 45.8 10 45.8 30 45.8 53 45.8

p 0.5 1.53 0.75 1.80 0.83 1.65 0.62 1.45

t 0.5 1.41 0.5 1.20 1.2 1.32 0.93 1.49

a 1.31 1.48 1.33 1.48 1.45 1.48 1.37 1.48

d 30 24 30 24 53 24 25 24

Fval -3.0758 -3.0758 -3.0758 -3.0758


62

From the table above, it can be found that the optimized objective function value changed with

different parameters of thickness and force. In addition, for the same input initial variables, the output

variables of p and t also change with different parameters. Finally, the iterations vary a little with

different parameters for the same input initial guess for variables.

When the function is optimized, the variable d reach the low boundary, while the variable a reach the

upper boundary. Also, the other variables n, p, t are the interior solution. From the table, it can be

found that g2, g3, g9 and g12 are active. This result agrees with the monotonicity analysis that g3 is

active.

4.5.4 Discussion of results

The link between this subsystem and the other subsystems are shown below:

Figure 1.15 Interaction between Laser welding optimization and other subsystems

It is possible that the ideal h and d for design optimization and optimization of stamping process can

not been processed by laser welding. Or there is high possibility that acceptable beam power and

exposure time would cause high cost for optimization of manufacturing system, which causes

conflicts. Thus, it is admirable to construct tradeoff between these subsystems.

Some basic rule to optimize laser welding process can be summarized below:

The distance between two neighboring spot welds d should get the value in its low boundary.

The total spot welds per side n can be obtained by transforming the active geometry constraint

g3 into equality.

The incident angle a should be chosen as its upper boundary.

The beam power or exposure time is constrained by the heat input constraint.

63

5 COST OPTIMIZATION OF B-PILLAR PRODUCTION

5.1 Problem Statement

With the rapid development of automotive industry, market competition becomes more and more

serious. Vehicle companies need to reduce products costs, improving products quality and shorten

manufacturing time. The general determining factors in deciding on vehicle production process is

usually cost, that is, to choose a process that produces the required quality at the relatively lowest

possible cost. In this part, we focus on cost reducing of the vehicle B-pillars in order to occupy more

market share. It means that the objective of this subsystem is to minimize the cost of producing B-

pillar.

Production is a vital aspect to decrease the cost in the whole designing, manufacturing and assembly

process. Producing B-pillar mainly contains two processes which are stamping and laser welding. The

costs of this process are specific to some elements of the work. Traditionally, these costs are regarded

as consisting of those for labor, operating and machine. Therefore optimization of labor cost, operating

cost, equipment cost and so on can improve productivity and reduce cost.

Figure 5.1 Automotive manufacturing system

64

5.2 Nomenclature

In this section, we present some parameters and variables, those are used to mathematics

model.

5.2.1 Parameters:

Table 5.1 Parameters

Raw material length of B-pillar 1075mm

Raw material width of B-pillar 340mm

Punch length 1100mm

Punch width 160mm

Punch thickness 70mm

Die length 1100mm

Die width 340mm

Die thickness 70mm

Blank holder length 1100mm

Blank holder width 60mm

Blank holder thickness 20mm

Punch material (Carbon tool steel) cost $900/ton

Die material (Carbon tool steel) cost $900/ton

Blank holder material (Carbon tool steel) cost $900/ton

B-pillar raw material (High strength steel) cost $900/ton

Labor hourly cost $15/ hour

Welding machine cost $10,000

Welding machine lifetime 9 years

65

Hot stamping machine cost $50,000

Hot stamping machine lifetime 10 years

Hot stamping machine power 39 kW

Heating furnace cost $15,200

Heating furnace lifetime 10 years

Heating furnace power 500 kW

Robotics cost $22,000

Robotics lifetime 10 years

Robotics power 3 kW

Hot stamping cooling speed 20s/ available steel plate

Working time each year 360 days

Working time each day 24 hours

Electric charge [10] $0.073/kWh

Density of steel

Die lifetime [11] 11000 times

Total number of spot weld per side 45

Argon cost [12] $0.1/L

Argon usage 5L/min

Absorption coefficient 0.693

Maximum power of spot laser welding machine 2.0kW

Minimum power of spot laser welding machine 0.5kW

Maximum one spot laser welding time 1.5s

Minimum one spot laser welding time 0.5s

Maximum strokes per minute of stamping machine 25 strokes/min

66

Minimum strokes per minute of stamping machine 0

Maximum energy of spot laser welding machine 1.5kJ

Maximum height of B-pillar 45mm

Minimum height of B-pillar 35mm

Maximum raw material thickness of B-pillar 3.0mm

Minimum raw material thickness of B-pillar 1.0mm

5.2.2 Variables

Table 5.2 Variables

SPM Strokes per minute of stamping machine(strokes/min)

h Height of B-pillar(mm)

P Welding power (kW)

t Beam exposure time (s)

H Raw material thickness of B-pillar(mm)

67

5.3 Mathematical Model

Designers typically use some models and simulation tools to finish functional analysis of the design.

According to the current production, we can build a model to simplify the optimization problem.

We simplified the manufacturing system model as dedicated line, that is, there are one Welding

machine and one stamping machine manufacturing. In this manufacturing line, there are some B-

pillars being produced. The model is not to produce a specific production schedule for each of the

individual machines in the machine group. Rather, it is intended to offer a realistic functional form for

the manufacturing costs that are used in the objective function.

The specific steps to optimize cost are as follows. Firstly, I build a model to optimize the production

process in spite of other designing and manufacturing optimization such as welding and stamping

optimization. Secondly, I take parameters and results of other optimization into account to adjust the

optimization result of this part. Thirdly, we can obtain a relatively better whole designing,

manufacturing and assembly optimization result of B-pillar via receiving information, having feedback

to each other and changing some parameters.

5.3.1 Constraints

At first, we choose one kind of spot laser welding machine, its power range is from 0.5kW to 2kW, so

the constraint for the power is given by:

P and

That is and (Equation 5.1)

This time of this machine which finish one spot laser weld is from 0.5s to 1.5s, so the

constraint for the spot welding time is given by:

t and (Equation 5.2)

That is, and (Equation 5.3)

68

We choose one kind of stamping machine, its strokes per minutes can adjust from 0 to 25, so

the constraint f or the stroke time is given by:

and

That is, and (Equation 5.4)

The absorption coefficient is 69.3% of the spot laser welding machine, the maximum energy

of the welding system is 1.5KJ, so the constraint for the energy is given by:

* Pt

That is, 0.693 * Pt (Equation 5.5)

The stamping is ahead of laser welding, the production rate of stamping must be higher than

the production rate of laser welding, otherwise, the dedicated line will bring the blockage. So

the constraint for production rate is given by:

60/(t* )/2 SPM/2

That is, 60/(t*45)/2-SPM/2 (Equation 5.6)

According to the reality, the range of height of B-pillar is always from 35mm to 45mm in the

engineering. So the constraint for height of B-pillar is given by:

and

That is, (Equation 5.7)

According to the reality, the range of raw material thickness of B-pillar is always from 1.0mm

to 3.0mm in the engineering. So the constraint for raw material thickness of B-pillar is given

by:

69

and


The cooling is ahead of hot stamping. After the raw material heated in the heating furnace, the

material would be cooled and then go to the hot stamping. The production rate of cooling

must be higher than the production rate of hot stamping, otherwise, the dedicated line will

bring the blockage. So the constraint for production rate is given by:


5.3.2 Modeling

B-pillar production mainly includes two processes which are hot stamping and spot laser welding. The

total cost in the manufacturing contains raw material cost, hot stamping cost and spot laser welding

cost.

Figure 5.2 The component of total cost

70

5.3.2.1 The cost in the stamping process,

The stamping cost includes machine cost, electrical cost, labor cost, die cost, heating furnace cost,

robotics cost.

Figure 5.3 The component of stamping cost

One stroke of the stamping machine forms an available steel plate, two pieces of available

steel plate constitute a B-pillar.

Machine cost = 2* Hot stamping machine cost / Hot stamping machine lifetime / Working

time each year/ Working time each day/ 60minutes/ Strokes per minute of stamping machine =

2* 2 50,000/ 10 / 360 / 24 / 60 / SPM=

/ B-pillar

(Equation 5.10)

Electrical cost = 2* Hot stamping machine power * Electric charge/ 60 minutes/

SPM+2*Heating furnace power* Electric charge/ 60 minutes/ SPM +2*4* Robotics power *

Electric charge/ 60 minutes/ Strokes per minute of stamping machine = =2 / 60 / SPM +

/ 60 / SPM + 2 / 60 / SPM =

B-pillar

(Equation 5.11)

Stamping

Cost

Machine cost

Electrical cost

Labor cost

Die cost

Robotics cost

Heating furnace

cost

71

Labor cost = 2*2* Labor hourly cost/ 60 minutes/ Strokes per minute of stamping machine =

= 2 15 / 60 / SPM=

/ B-pillar (Equation 5.12)

Die cost = 2*[2* Blank holder length* Blank holder width* Blank holder thickness* Density

of steel* ton/kg *Blank holder material (Carbon tool steel) cost + Die length * Die width

*Die thickness*Density of steel* ton/kg *Die material (Carbon tool steel) cost +Punch

length*Punch width*(70- Height of B-pillar) *Density of steel* ton/kg *Punch material

(Carbon tool steel) cost] / Die lifetime = ton/kg ton/kg ton/kg ]/

= = $[0.036+2.26 / B-pillar

(Equation 5.13)

Heating furnace cost=2* Heating furnace cost/ Heating furnace lifetime/ Working time each

year/ Working time each day/ 60minutes/ Strokes per minute of stamping machine =2

= 2 15,200/ 10 / 360 / 24 / 60 / SPM=

/ B-pillar

(Equation 5.14)

Robotics cost=2*4* Robotics cost / Robotics lifetime/ Working time each year/ Working time

each day/ 60minutes/ Strokes per minute of stamping machine =

= 2 22,000/ 10 / 360 / 24 / 60 / SPM=

/ B-pillar

(Equation 5.15)

5.3.2.2 The cost in the welding process

The spot laser welding cost includes machine cost, electrical cost, labor cost, argon cost, robotics cost.

72

Figure 5.4 The component of welding cost

(Total number of spot weld per side*t) is equal to the beam exposure time per side in a B-pillar.

Double beam exposure time per side is the beam exposure time in a B-pillar.

Machine cost = 2* Welding machine cost / Welding machine lifetime / Working time each

year/ Working time each day/ 60minutes/[60/( Beam exposure time * Total number of spot

weld per side)] = = 2 10,000/ 9 / 360 / 24 / 60 /

[60/(t*45)] = / B-pillar

(Equation 5.16)

Electrical cost = 2* Welding power * Electric charge/ 60 minutes/ [60/( Beam exposure time

* Total number of spot weld per side)] + 2* 2* Robotics power * Electric charge/ 60 minutes/

[60/( Beam exposure time * Total number of spot weld per side)]= =2 P* / 60 / [60/(t*45)] + 2 3 /

60 / [60/(t*45)] = B-pillar

(Equation 5.17)

Labor cost = 2* Labor hourly cost/ 60 minutes/ [60/( Beam exposure time * Total number of

spot weld per side)]= = 2 15 / 60 / [60/(t*45)]= / B-pillar

Argon cost = 2*Argon cost* Argon usage/[60/( Beam exposure time * Total number of spot

weld per side)]= = 2 0.1 / B-pillar

(Equation 5.18)

Welding

Cost

Robotics cost

Machine cost

Electrical cost

Labor cost

Argon cost

73

Robotics cost=2*2* Robotics cost / Robotics lifetime/ Working time each year/ Working time

each day/ 60minutes/ [60/( Beam exposure time * Total number of spot weld per side)]= = 2 $22,000/ 10 years/ 360 days/ 24 hours/ 60

minutes/ [60/(t*45)]= / B-pillar

(Equation 5.20)

5.3.2.3 Material cost

Material cost =2* Raw material length of B-pillar * Raw material width of B-pillar * Raw

material thickness of B-pillar * Density of steel* Raw material (High strength steel) cost of B-

pillar =

=

/B-pillar

(Equation 5.21)

5.3.2.4 Model Sumary

Min f = Machine cost in the stamping process + Electrical cost in the stamping process + Labor cost

in the stamping process + Die cost in the stamping process + Heating furnace cost in the stamping

process Robotics cost in the stamping process + Machine cost in the welding process+ Electrical

cost in the welding process + Labor cost in the welding process + Argon cost in the welding process

+ Robotics cost in the welding process + Material cost

74

That is

Min f= 𝟓 𝟖𝟑 𝟑 + 𝟐 𝟒

+𝟐 𝟐𝟔 𝟒 𝟕 𝟓 𝟔 𝟑𝟔]

B-pillar

(Equation 5.22)

Subject to

(Equation 5.23)

2 (Equation 5.24)

(Equation 5.25)

(Equation 5.26)

(Equation 5.27)

(Equation 5.28)

0.693*Pt-1.5 (Equation 5.29)

0.67- 0.5*SPM*t (Equation 5.30)

(Equation 5.31)

(Equation 5.32)

(Equation 5.33)

2 (Equation 5.34)

(Equation 5.35)

75

5.4 Optimization Model Analysis


A monotonicity analysis with respect to the design variables are performed for checking whether the

problem is well constrained. The monotonicities of all the variables with respect to the objective

function and constraints are shown in the table below.

Table 4 Monotonicity Analysis

SPM h P t H

f - - + + +

- Active for variable P

2 +

- Active for variable t

+

-

+

+ +

- -

-

+ Active for variable h

- Active for variable H

2 +

+ Active for variable SPM

76

When other variables are fixed, and only SPM can change. Objective function decreases in a variable

SPM. Constraint and Constraint increase in a variable SPM.

, 0, so

When other variables are fixed, and only h can change. Objective function decreases in a variable h.

Constraint increases in a variable h. , so

When other variables are fixed, and only P can change. Objective function increases in a variable P.

Constraint decreases in a variable h. , so

When other variables are fixed, and only t can change. Objective function increases in a variable t.

Constraint and Constraint decrease in a variable t. 0.67-

0.5*SPM*t , when SPM is from 0 to 3, and can not be judged the relationship, so

When other variables are fixed, and only H can change. Objective function increases in a variable H.

Constraint decreases in a variable H. , so

77

5.5 Numerical Optimization

%Shown in Matlab command window%%%%%%%%%%%%%%%%%%%%%%%%%%

Local minimum possible. Constraints satisfied.

fmincon stopped because the predicted change in the objective function is less than the default

value of the function tolerance and constraints are satisfied to within the default value of the

constraint tolerance.

<stopping criteria details>

Active inequalities (to within options.TolCon = 1e-006):

lower upper ineqlin ineqnonlin

1 3 2

2

5

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

There are four algorithms in fmincon.m. I used one of algorithm, that is, 'interior-point' algorithm. I

tried this algorithm from different initial points. The results are shown below:

78

Table 5.5 Optimization results for various starting points

Test 1 Test 2 Test 3 Test 4


p 0.5 0.5002 0.9 0.5024 0.65 0.5001 1.2 0.5004

t 0.5 0.5 1.3 0.5 0.75 0.5 0.6 0.5

SPM 1 3 2 3 1 3 3 3

h 35 44.9996 45 44.9902 37 44.9996 40 44.9982

H 1 1 2 1 3 1 3 1

Fval 8.1794 6.5771 7.8988 6.5771 18.7869 6.5771 17.0141 6.5771


From the results above, the 'interior-point' algorithm, we can get the local minimum which is

$6.5771/B-pillar. From the initial points and out points in different situations, we can obtain

improvement over starting points. Then, different parameters are tried to optimize the function again.

From the table below, it can be found that the function value changed for different parameters while

the iteration does not change too much.

79

Table 5.6 Results of 'interior-point' algorithm using n=30



p 0.5 0.5002 0.9 0.5024 0.65 0.5001 1.2 0.5004

t 0.5 0.5 1.3 0.5 0.75 0.5 0.6 0.5

SPM 1 3 2 3 1 3 3 3

h 35 44.9996 45 44.9902 37 44.9996 40 44.9982

H 1 1 2 1 3 1 3 1

Fval 6.4756 6.4756 6.4756 6.4756


Table 5.7 Results of 'interior-point' algorithm using n=40



p 0.5 0.5002 0.9 0.5024 0.65 0.5001 1.2 0.5004

t 0.5 0.5 1.3 0.5 0.75 0.5 0.6 0.5

SPM 1 3 2 3 1 3 3 3

h 35 44.9996 45 44.9902 37 44.9996 40 44.9982

H 1 1 2 1 3 1 3 1

Fval 6.5233 6.5233 6.5233 6.5233


From the table above, we can find when we change the total number of spot weld per side

from 45 to 30 and 40, the local minimum will decrease. It means that the cost which produces

one B-pillar will decrease. And the iteration does not change too much.

80

5.6 Discussion of Results

From the parameters study results, the lowest cost of manufacturing is obtained when the

beam power, stamping height and exposure time reach their lower boundary while the SPM

and the thickness of plate get their values on upper boundary. It makes sense that when beam

power and exposure time decrease, energy cost for laser welding reduces. The productivity

would increase with stamping height decreasing and SPM increasing. Thus the cost of

manufacturing process is the lowest at the above minimizer just for this subsystem.

Figure 5.5 Relation between manufacturing subsystem and the other subsystems

The low cost requirement of manufacturing subsystem optimization required that small geometry for

design, high speed for stamping process, and low power and exposure time for laser welding. While

these requirement maybe cause trouble in other subsystem. So, tradeoff is needed to construct the

complete project.

81

6 System Integration Study

The optimization study at a subsystem level has been done and valuable insights have been gained.

The objective of this project is to optimize the B pillar as a system based on the subsystems. The major

tasks in a system integration study are:

(1) Identifying the system level interactions

(2) Integrate the subsystem models to create the complete system model.

(3) Formulating and solving the optimization problem

Each of these will be discussed in the following sections.

6.1 System Interactions

There are four subsystems for the design and manufacturing processes optimization of a car‘s B-

pillars: Lightweight design, optimization of stamping process, optimization of laser welding process

and optimization of manufacturing system. The system level interactions is pictured as shown below

Figure 6.1 Interactions between Subsystems

Each subsystem has its own requirement, and when the subsystems are integrated into the system, a

number of conflicts occur. The conflicts between different subsystems are summarized below:

82

Table 8.1 Conflicts between different subsystems

6.2 Problem Formulation

Once the subsystem models are integrated, the optimization problem for the complete system has to be

defined. There are two objectives for the system: one is to reduce the cost to build lightweight B-pillar

structure, and the other is to reduce the structure weight. This is multiobjective problem. The ultimate

objective function can be obtained by assigning subjective weights to each objective and summing up

these two objectives multiplied by their corresponding weight.

The overall system problem can be formulated as follows

ℎ 2 ℎ

Subjected to the following two kinds of constraints:

Total mass

Peak acceleration

Maximum intrusion

Spot-welding safety factor

Spot-welding quality

83

Stamping punch force

Stamping elongation

Manufacturing cost

Spot-welding geometry

The specific constraints are shown as Matlab code in the appendix.

Final Set of variables are summarized as

Table 6.1 Variables summary

Description Symbol

laser power p

exposure time during laser spot-welding T

frequency of stamping SPM

stamping depth h

thickness of raw steel panel t0

thickness of lower component panel t1

thickness of upper component panel t2

84

Objects of other subsystem problems now become the constraint function after reference object values

are defined as the design boundary, which are shown in following table. Besides, the system

problem also subject to those constraints functions that subsystems already have.

Table 6.2 Constraints functions summary

Total mass ∑

Peak acceleration ℎ

Maximum intrusion ℎ

Spot-welding safety factor

2

Spot-welding quality

2

Stamping quality ℎ 2 ℎ ,

where

6.3 Optimization Approaches

In the system level, Matlab function: fmincon is still applied to find out the optimum solution. 5 times

of running are conducted, no solution is obtained. The fmincon reports no feasible design space with

all these constraints provided. Then the design space is enlarged by loose the bounds, and another 5

rounds are tried. Only one solution is obtained, which is shown as follows:

Table 6.3 Optimization result summary

Description Initial Output

laser power 0.7 1.6

exposure time 1.3 0.7

frequency of stamping 1.6 2.6

ℎ stamping depth 44 48

thickness of raw steel panel 1.9 1.0

thickness of lower component panel 1.6 1.0

thickness of upper component panel 2.3 1.0

Function value 5.2 2.9

Improvement 44%

Iteration 94

85

From the result we can tell that although the improvement of object function is significant, some of the

design variables in this result already exceed over the initial feasible area, like ℎ, the stamping depth

which is initially defined between 35-45mm . This means the variables are chosen from the design area

that metamodel has no real information but only regression trend, thus the result is quite suspicious for

their ability to reflect the real situation. The validation of this result needs experiments in real scenario,

which are only accessible for subsystem 1 in our problem, and its result, though shows good match,

cannot be enough to prove the whole result for system optimization.

6.4 Dicussion

The system integration in optimization problems is always a very challenging task. To ensure that the

design space of each subsystem can overlap with others, expert knowledge in not just sub disciplines

but also system level is required to well define the problem before optimization. Besides, there are 4

nonlinear constraints in our system level problem that use non-analytical model to simulate the real

result, which may cause significant discontinuity of the final design space, and make the optimization

algorithm difficult to converge. Heuristic method may be needed to deal with such situation. And

again, a efficient heuristic method still requires expert knowledge to reduce the ―guessing‖ time to

initial the optimization iteration.

86

7 REFERENCES

[1] Pan, F., Zhu, P., & Zhang, Y. (2010). Metamodel-based lightweight design of B-pillar with TWB

structure via support vector regression. Computers & Structures, 88(1-2), 36-44. Elsevier Ltd.

doi:10.1016/j.compstruc.2009.07.008

[2] Qin Y., Kong X., & Luo, W. (2011). Finite element model updating of airplane wing based on

Gaussian radial bassis function response surface. Jounral of Beijing University of Aeronautics

and Astronautics, 37(11), 1465-1470. 2011.11

[3] Firat, Mehmet, Osman H. Mete, Umit Kocabicak, and Murat Ozsoy. "Stamping Process Design

Using FEA in Conjunction with Orthogonal Regression." Finite Elements in Analysis and Design

46.11 (2010): 992-1000.

[4] Z. Marchiniak, J.L.Duncan, S.J.Hu, ―Mechanics of Sheet Metal Forming‖, 2002;

[5] Peng, L., P. Hu, X. Lai, D. Mei, and J. Ni. "Investigation of Micro/meso Sheet Soft Punch

Stamping Process – Simulation and Experiments." Materials & Design 30.3 (2009): 783-90.

[6] http://www.seas.virginia.edu/research/lam/pdfs/speaker%20presentations/Havrilla-

UofV_Laser%20Based%20Manufacturing%20in%20the%20Automotive%20Industry.pdf

[7] C.M. Sonsino, M. Kueppers, M. Eibl, G. Zhang, Fatigue strength of laser beam welded thin steel

structures under multiaxial loading, International Journal of Fatigue, Volume 28, Issues 5–6,

May–June 2006, Pages 657-662.

[8] Y.S. Yang, S.H. Lee, A study on the joining strength of laser spot welding for automotive

applications, Journal of Materials Processing Technology, Volume 94, Issues 2–3, 29 September

1999, Pages 151-156.

[9] Q. Huang, J. Hagstroem, H. Skoog, G. Kullberg, Effect of laser parameter variation on sheet

metal welding, Int. J. Join. Mater. 3 (3) (1991) 79–88.

[10] Minggang, ji et al., Design of Machinery, Higher Education Press, 2006

[11] Phillips, Arthur L ,, and American Welding Society.,. Welding Handbook. New York,

1957.

[12] Siva Shanmugam, N., G. Buvanashekaran, and K. Sankaranarayanasamy. "Some Studies on Weld

Bead Geometries for Laser Spot Welding Process Using Finite Element Analysis." Materials

http://www.seas.virginia.edu/research/lam/pdfs/speaker%20presentations/Havrilla-UofV_Laser%20Based%20Manufacturing%20in%20the%20Automotive%20Industry.pdf

http://www.seas.virginia.edu/research/lam/pdfs/speaker%20presentations/Havrilla-UofV_Laser%20Based%20Manufacturing%20in%20the%20Automotive%20Industry.pdf

87

& Design 34 (2012): 412-26.

[13] Aderoba, A., A generalized cost-estimation model for job shops. International Journal of

Production Econonomics, 1997

[14] http://www.net114.com/100376571p.html

[15] http://www.net114.com/101148101p.html

[16] http://zhidao.baidu.com/question/108800894.html

[17] http://www.alibaba.com/showroom/laser-welding-machine-price.html

[18] http://www.wjw.cn/product/MBR110226095213734898/PRO110411033834609919.xhtml

[19] http://www.machine35.com/product/trade/2-13.html

[20] http://zhidao.baidu.com/question/179273445.html

[21] http://www.jdzj.com/products/2011-8-4/7509176-1.html

[22] http://blog.sina.com.cn/s/blog_62f5246d0100m7af.html

[23] http://wenku.baidu.com/view/d20623e3524de518964b7dd8.html

[24] http://www.hilarion.com/igd.html

[25] http://wenku.baidu.com/view/83040809bb68a98271fefaf9.html

[26] http://woodhead.metapress.com

[27] http://www.alibaba.com/product-gs/518148409/press_used_metal_heating_furnace.html

[28] http://www.alibaba.com/product-gs/472508927/High_speed_Running_AMD_255_series.html

http://zhidao.baidu.com/question/108800894.html

http://www.alibaba.com/showroom/laser-welding-machine-price.html

http://www.wjw.cn/product/MBR110226095213734898/PRO110411033834609919.xhtml

http://www.machine35.com/product/trade/2-13.html

http://zhidao.baidu.com/question/179273445.html

http://www.jdzj.com/products/2011-8-4/7509176-1.html

http://blog.sina.com.cn/s/blog_62f5246d0100m7af.html

http://wenku.baidu.com/view/d20623e3524de518964b7dd8.html

http://wenku.baidu.com/view/83040809bb68a98271fefaf9.html

http://woodhead.metapress.com/

88

8 Apprendix

8.1 MATLAB Code for the structure optimization

% ===========================================================

% NeuralTrain is to train radical based neural network

% ===========================================================

function[net] = NeuralTrain(p)

% X = n x q matrix of q vectors with n dimensional continuous variables

% read training set from DoE spreadsheet

p = 4;

X0_in = csvread('DoE_sheet.csv');

% transpose the DoE matrix, for that the spreadsheet is in q x n formation

X0 = X0_in(1:25*p,:)';

% normalize design variables

lgt = size(X0,2);

h_0 = 35.0 * ones(1,lgt);

89

t1_0 = 1.0 * ones(1,lgt);

t2_0 = 1.0 * ones(1,lgt);

v_lower = 0.2 * ones(1,lgt);

% normalize training set

% X0_Norm(1,:) = v_lower + 0.6 * (X0(1,:) - h_0)/10;

% X0_Norm(2,:) = v_lower + 0.6 * (X0(2,:) - t1_0)/1.4;

% X0_Norm(3,:) = v_lower + 0.6 * (X0(3,:) - t2_0)/1.4;

% X_0 = X0_Norm;

X_0 = X0;

% read the FE simulation result of training set

D_0 = csvread('Intrusion.csv');

A_0 = csvread('Acceleration.csv');

q = numel(D_0);

Con_0 = zeros(2,q);

90

for i = 1:q

intr = D_0(i);

acc = A_0(i);

Con_0(:,i) = [intr,acc]';

end

size(Con_0);

size(X_0);

[onet tr] = newrb(X_0,Con_0,0.01,1.5,50,5);

% X_0 = n x q matrix of q normalized input vectors

% Con_0 = s x q matrix of q constraint value vectors

% mean squared error goal = 0.01

% spread of radial basis function = 2.5

% maximum num of neurons = 50

save metamodel.mat onet;

end

91

% ===========================================================

% Metafun is to generate the metamodel based on neural network

% ===========================================================

function[Con_X] = Metafun(X)

% X = n x q matrix of q vectors with n dimensional continuous variables

% normalize design variables

lgt = size(X,2);

h_0 = 35.0 * ones(1,lgt);

t1_0 = 1.0 * ones(1,lgt);

t2_0 = 1.0 * ones(1,lgt);

v_lower = 0.2*ones(lgt,1);

% normalize testing set

% X_Norm(1,:) = v_lower + 0.6 * (X(1,:) - h_0)/10;

% X_Norm(2,:) = v_lower + 0.6 * (X(1,:) - t1_0)/1.4;

% X_Norm(3,:) = v_lower + 0.6 * (X(1,:) - t2_0)/1.4;

92

% X_1 = X_Norm;

X_1 = X;

% load the neural network trained by traing set.

load metamodel.mat;

q = size(X,2);

Con_X = zeros(2,q);

for i = 1:q

Con_X(:,i) = sim(onet,X_1(:,i));

end

end

% ===========================================================

% Visualization provides the 3-D scatters for monotonicity study

% ===========================================================

X_1 = csvread('DoE_sheet.csv');

93

x1 = X_1(:,1);

x2 = X_1(:,2);

x3 = X_1(:,3);

n = 25;

xx1 = linspace(min(x1),max(x1),n);



[X1,X2,X3] = meshgrid(xx1,xx2,xx3);

H = reshape (X1,[n^3,1]);

Tl = reshape (X2,[n^3,1]);

Tu = reshape (X3,[n^3,1]);

%X = [35,1.4,1.4;40,1.2,1.6;44,1.1,1.7];

X = [H,Tl,Tu];

for i = 1:size(X,1)

94

[C] = Metafun(X(i,:)');

Mass(i,1) = Wgt(X(i,:)');

Intr(i,1) = C(1);

Acc(i,1) = C(2);

end

Mass3 = reshape (Mass,[n,n,n]);

Intr3 = reshape (Intr,[n,n,n]);

Acc3 = reshape (Acc ,[n,n,n]);

hmodel = scatter3(X1(:),X2(:),X3(:),5,Mass3(:),'filled');

hold on

hdata = scatter3(x1,x2,x3,'ko','filled');

axis tight

xlabel(xn(1,:))

ylabel(xn(2,:))

95

zlabel(xn(3,:))

hbar = colorbar;

ylabel(hbar,yn);

title('{\bf Quadratic Response Surface Model}')

legend(hdata,'Data','Location','NE')

% ===========================================================

% LatinDoE is to generate the initial points for optimization

% ===========================================================

function [D_1] = LatinDoE(i,j)

X = lhsdesign(i,j);

% i levels

% j variables

h = 35+10*X(:,1);

t_l = 1.0+1.4*X(:,2);

t_2 = 1.0+1.4*X(:,3);

96

D_1 = [h,t_l,t_2]';

end

% ===========================================================

% mycon defines the nonlinear constraints of optimization problem

% ===========================================================

function [c,ceq] = mycon(x)

[Con] = Metafun(x);

Intr = Con(1);

Acc = Con(2);

% baseline model: t1 = 1.8, t2 = 2.4, h = 40;

IntrBaseline = 70.0000;

AccBaseline = 5.5000;

c(1) = Intr - IntrBaseline ;

c(2) = Acc - AccBaseline;

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c(3) = 50 - Intr;

c(4) = 3 - Acc;

ceq = [];

end

% ===========================================================

% Wgt defines the object of lightweight design

% ===========================================================

function [w] = Wgt(X)

% X = n * 1 , the vector of n dimensional continuous variables

l = [3.750E-04, 3.887e-04, 4.026e-04];

u = [1.233E-03, 1.278e-03, 1.324e-03];

h = [35,40,45];

c_1 = (l(2)-l(1))/(h(2)-h(1));

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c_2 = (l(3)-l(2))/(h(3)-h(2));

c_3 = (l(3)-l(1))/(h(3)-h(1));

c_l = (c_1+c_2+c_3)/3;

c_4 = (u(2)-u(1))/(h(2)-h(1));

c_5 = (u(3)-u(2))/(h(3)-h(2));

c_6 = (u(3)-u(1))/(h(3)-h(1));

c_u = (c_4+c_5+c_6)/3;

h_r = X(1,1);

t_1 = X(2,1);

t_2 = X(3,1);

w_l = (c_l*(h_r-35)+3.750E-04) * t_1;

w_u = (c_u*(h_r-35)+1.233E-03) * t_2;

w = w_l + w_u;

end

99

% ===========================================================

% OptimizeFun uses the fmincon to find the optimal solution

% ===========================================================

function [] = OptimizeFun()

A = [];

b = [];

Aeq = []; beq = [];

lb = [35;1.0;1.0];

ub = [45;2.4;2.4];

X = cell(5,1);

x_Star = cell(5,6);

for i = 1:5

n = 3;

q = 6;

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x = LatinDoE(q,n);

X{i,1} = x;

for j = 1:q

x0 = x(:,q);

[xval,fval,exitflag,output] = fmincon(@Wgt,x0,A,b,Aeq,beq,lb,ub,@mycon);

Mass_Star(i,j) = fval;

x_Star{i,j} = xval;

Con = Metafun(xval);

Intru_Star(i,j) = Con(1);

Acc_Star(i,j) = Con(2);

Iteration(i,j) = output;

end

end

101

8.2 MATLAB Code for the stamping optimization

DoE

function [D_1] = LatinDoE (n)

% n value levels

p = 4; % p design variables

X = lhsdesign(n,p);

h = 35 + 20*X(:,1);

t = 1.4 + 1.0*X(:,2);

u = 0.05 + 0.73*X(:,3);

F = 85 + 165*X(:,4);

D_1 = [h,t,u,F];

end

102

Scaled objective function

clear all;

M = csvread('vv.csv');

hn=M(:,1);

tn=M(:,2);

un=M(:,3);

Fn=M(:,4);

Fpn=M(:,5);

%normalize

h = 0.1+0.8*(hn - min(hn)*ones(size(hn,1),1))/(max(hn)-min(hn));

t = 0.1+0.8*(tn - min(tn)*ones(size(tn,1),1))/(max(tn)-min(tn));

u = 0.1+0.8*(un - min(un)*ones(size(un,1),1))/(max(un)-min(un));

F = 0.1+0.8*(Fn - min(Fn)*ones(size(Fn,1),1))/(max(Fn)-min(Fn));

Fp = 0.1+0.8*(Fpn - min(Fpn)*ones(size(Fpn,1),1))/(max(Fpn)-min(Fpn));

x=[h t u F h.^2 t.^2 u.^2 F.^2 (h.*t) (h.*u) (t.*u) (t.*F) (u.*F) (h.*F)];

103

X=[ones(50,1),x];

%stepwise(X,Fp);

XF=[ones(50,1) h t u F h.^2 t.^2 u.^2 F.^2 (h.*t) (h.*u) (t.*u) (t.*F)

(u.*F) (h.*F)];

XF1= [ones(50,1) u F u.^2 (h.*t) (u.*F)];

[b,bint,r,rint,stats]=regress(Fp,XF1);

Constraint function

clear all;

M = csvread('ss.csv');

h=M(:,1);

t=M(:,2);

u=M(:,3);

F=M(:,4);

s=M(:,5);

x=[h t u F h.^2 t.^2 u.^2 F.^2 (h.*t) (h.*u) (t.*u) (t.*F) (u.*F) (h.*F)];

104

X=[ones(50,1),x];

%stepwise(X,s);

XF=[ones(50,1) h t u F h.^2 t.^2 u.^2 F.^2 (h.*t) (h.*u) (t.*u) (t.*F)

(u.*F) (h.*F)];

Xs=[ones(50,1) h t u F h.^2 t.^2 u.^2 F.^2 (h.*t) (h.*u) (t.*u) (t.*F)

(u.*F) (h.*F)];

Xs1= [ones(50,1) h t u F t.*F];

[b,bint,r,rint,stats]=regress(s,Xs1);

Fmincon

function f=objfun(x)

%x = [39.7,2.18,0.53,211.1];

M = csvread('vv.csv');

hn = M(:,1);

tn = M(:,2);

un = M(:,3);

Fn = M(:,4);

105

h = 0.1+0.8*(hn - min(hn)*ones(size(hn,1),1))/(max(hn)-min(hn));

t = 0.1+0.8*(tn - min(tn)*ones(size(tn,1),1))/(max(tn)-min(tn));

u = 0.1+0.8*(un - min(un)*ones(size(un,1),1))/(max(un)-min(un));

F = 0.1+0.8*(Fn - min(Fn)*ones(size(Fn,1),1))/(max(Fn)-min(Fn));

x = [h t u F];

W = csvread('obj.csv');

w_1 = W(:,1);

x_1(1) = 1;

x_1(2,1) = x(1);

x_1(3,1) = x(2);

x_1(4,1) = x(3);

x_1(5,1) = x(4);

x_1(6,1) = x(1)^2;

x_1(7,1) = x(2)^2;

x_1(8,1) = x(3)^2;

106

x_1(9,1) = x(4)^2;

x_1(10:15,1) = [x(1)*x(2);

x(1)*x(3);

x(2)*x(3);

x(2)*x(4);

x(3)*x(4);

x(1)*x(4)];

h_1 = [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ];

h_2 = [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ];

a = w_1'*x_1;

b = h_1*x_1;

c = h_2*x_1;

f = a-b+c;

107

end

function [c,ceq] = mycon(x)

% x = [h,t,u,F]

%x = [1,2,3,4];

C = csvread('constrain.csv');

c_1 = C(:,1);

x_1(1) = 1;

x_1(2,1) = x(1);

x_1(3,1) = x(2);

x_1(4,1) = x(3);

x_1(5,1) = x(4);

x_1(6,1) = x(1)^2;

x_1(7,1) = x(2)^2;

x_1(8,1) = x(3)^2;

x_1(9,1) = x(4)^2;

108

x_1(10:15,1) = [x(1)*x(2);

x(1)*x(3);

x(2)*x(3);

x(2)*x(4);

x(3)*x(4);

x(1)*x(4)];

c(1) = c_1'* x_1-270;

W = csvread('obj.csv');

w_1 = W(:,1);

c(2) = -w_1'*x_1;

ceq= [];

end

clear all;

x0= [36,1.6,0.64,117.26];

109

lb= [35,1.4,0.05,83.3];

ub= [55,2.4,0.78,250];

options = optimset('Algorithm', 'active-set');

[x,fval,exitflag,output]=

fmincon(@objfun,x0,[],[],[],[],lb,ub,@mycon,options);

110

8.3 MATLAB Code for the welding optimization

Appendix I

Before input the following code, the data ‗Depth‘ has to be uploaded into the Matlab.

Matlab code for regression the function DP:

clc;

clear all;

p=depth(:,1);

t=depth(:,2);

a=depth(:,3);

DP=depth(:,4);

x=[p t a (p.*t) (p.*a) (a.*t) (p.^2) (t.^2) (a.^2)];

X=[ones(27,1),x];

stepwise(X,DP);

X1=[ ones(27,1) p (p.*t) (p.*a) (a.*t) (p.^2)];

[b,bint,r,rint,stats]=regress(DP,X1);

Matlab code for regression the function BW:

clc;

111

clear all;

p=depth(:,1);

t=depth(:,2);

a=depth(:,3);

BW=depth(:,5);

x=[p t a (p.*t) (p.*a) (a.*t) (p.^2) (t.^2) (a.^2)];

X=[ones(27,1),x];

stepwise(X,BW);

Variables have been created in the current workspace.

>> X2=[ones(27,1) a (p.*t)];

[b,bint,r,rint,stats]=regress(BW,X2);

112

Step 1: Write a file objfun.m.


f=-0.0245*x(1)*((-1.1716+1.4939*x(4)+0.4549*x(2)*x(3))^2);

Step 2: Write a file confuneq.m for the nonlinear constraints.

function [c, ceq]= confun(x)

c = [1.1618+3.3717*x(2)-0.2715*x(2)*x(3)-1.3096*x(2)*x(4)-0.1223*x(4)*x(3)-

1.6284*(x(2)^2);

0.693*x(2)*x(3)-1.5;

(x(1)-1)*x(5)-1075];

ceq = [];

Step 3: Invoke constrained optimization routine.

clc;

clear all;

x0= [20,0.5,0.5,1.31,30]; % Make a starting guess at the solution

lb= [0,0.5,0.5,1.309,24];

ub= [Inf,2,1.5,1.484,70];

options = optimset('Algorithm','active-set');% Line a

[x,fval,exitflag,output]=

fmincon(@objfun,x0,[],[],[],[],lb,ub,@confun,options);

113

For the other fmincon Algorithms, they can be simply done by replace 'active-set' in line

a with 'sqp', 'active-set', 'trust-region-reflective' .

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Optimization h=2.5mm F=25KN


f=-0.0196*x(1)*((-1.1716+1.4939*x(4)+0.4549*x(2)*x(3))^2);


c = [1.6618+3.3717*x(2)-0.2715*x(2)*x(3)-1.3096*x(2)*x(4)-0.1223*x(4)*x(3)-

1.6284*(x(2)^2);

0.693*x(2)*x(3)-1.5;

(x(1)-1)*x(5)-1075];

ceq = [];

clc;

clear all;

x0= [53,0.62,0.93,1.37,25];

114

%x0= [20,0.5,0.5,1.31,30];

%x0= [10,0.75,0.5,1.33,30];

%x0= [30,0.83,1.2,1.45,53];

lb= [0,0.5,0.5,1.309,24];

ub= [Inf,2,1.5,1.484,70];

options = optimset('Algorithm', 'sqp');

[x,fval,exitflag,output]= fmincon(@objfun,x0,[],[],[],[],lb,ub,@confun,options);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Optimization h=2.2mm F=30KN


f=-0.0163*x(1)*((-1.1716+1.4939*x(4)+0.4549*x(2)*x(3))^2);


c = [1.3618+3.3717*x(2)-0.2715*x(2)*x(3)-1.3096*x(2)*x(4)-0.1223*x(4)*x(3)-

1.6284*(x(2)^2);

0.693*x(2)*x(3)-1.5;

115

(x(1)-1)*x(5)-1075];

ceq = [];

clc;

clear all;

x0= [53,0.62,0.93,1.37,25];

%x0= [20,0.5,0.5,1.31,30];

%x0= [10,0.75,0.5,1.33,30];

%x0= [30,0.83,1.2,1.45,53];

lb= [0,0.5,0.5,1.309,24];

ub= [Inf,2,1.5,1.484,70];

options = optimset('Algorithm', 'sqp');


116

8.4 MATLAB Code for the cost optimization

When the number of spot weld =40

Objective function


f=(1.15+0.00183*x(1))*x(2)+2.4/x(3)+2.26e-4*(70-x(4))+5.16*x(5)+0.036;

Nonlinear constraint


c = [0.693*x(1)*x(2)-1.5;

0.67-0.5*x(2)*x(3)];

ceq = [];

Optimization method

clc;

clear all;

x0= [0.5,0.5,1,35,1];

%x0= [0.65,0.75,1,37,3];

%x0= [0.9,0.13,2,45,2];

%x0= [1.2,0.6,3,40,3];

117

lb= [0.5,0.5,0, 35,1];

ub= [2,1.5,3,45, 3];

options = optimset('Algorithm', 'active-set' );


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