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This is to certify that the

thesis entitled

OPTIMIZATION OF TUBE
HYDROFORMING PROCESS

presented by

Naeem Zafar

has been accepted towards fulfillment
of the requirements for

MS degree in _Mechanical Engineering

mgflm‘a

Major professor

Date Xz/f3l/02'

0-7639 MS U is an Affirmative Action/Equal Opportunity Institution

 

 

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DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 c:/CIRC/DateDue.p65—p. 15

 

OPTIMIZATION
OF TUBE HYDROFORMING
PROCESS
By

Naeem Zafar

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Mechanical Engineering

2002

ABSTRACT
OPTIMIZATION
OF TUBE HYDROFORMING
PROCESS
By

Naeem Zafar

This thesis presents an approach to optimize the tube hydroforming process using
a Genetic Algorithm (GA). A GA in combination with a structural finite element code is
used to optimize the internal hydraulic pressure and feed rate while satisfying the forming
limit diagram (FLD). Aside from the target of cost reduction, the industrial enterprises
are aiming for optimization of their products regarding weight as well as stability and
rigidity. This requires reevaluation of conventional design solutions, manufacturing
techniques and material selections in the search for alternative solutions. Such an
alternative with interesting technical and economical potential is hydroforming, a method
for manufacturing a wide range of complicated hollow components from tubular or sheet
blank material by means of water pressure. This method can achieve short development
times, reduced number of operation steps, high precision, undisturbed material structure
and increased uniform strength in the component.

This thesis presents the finite element simulation of the tube hydroforming
process and optimization of the loading paths, (i.e., internal pressure and axial feed

versus time).

ACKNOWLEDGEMENTS

I wish to express my gratitude to my co-advisors Dr. Farhang Pourboghrat and
Dr. Erik D. Goodman for all the support, insight, guidance and expertise offered
throughout my research. I would like also to thank ACD Associates specially Dr. David
J. Eby and Ranny Sidhu for their advice and assistance during this research.

Also, I would like to thank the computational Mechanics Research Group for their
support and assistance. Finally, I would like to thank my advisor Dr. Ronald C. Averill
for his incredible amount of guidance, understanding and foresight throughout the

duration of my thesis.

iii

TABLE OF CONTENTS

LIST OF TABLES ................................................................................ v

LIST OF FIGURES ............................................................................... vi

CHAPTER 1

INTRODUCTION ................................................................................ 1
1.1 Introduction ....................................................................... l
1.2 Problem Definition .............................................................. 2
1.3 Literature Review ....................................................................... 4

CHAPTER 2

DESCRIPTION OF TUBE HYDROFORMING PROCESS ................................... 9
2.1 Introduction ....................................................................... 9
2.2 The hydroforming process .................................................... 11
2.3 Forming forces and Pressures ................................................... 12
2.4 Tube/Die Contact ..................................................................... 14
2.5 Tube hydroforming systems ................................................... 14
2.6 Failure Modes ..................................................................... 20
2.7 Advantages and Disadvantages ................................................... 23

CHAPTER 3

TUBE HYDROFORMING SIMULATION AND OPTIMIZATION ............... 25
3.] Introduction ..................................................................... 25
3.2 Description of the geometry ................................................... 26
3.3 Simulated Tube ..................................................................... 31
3.4 Results before using optimization software ................................. 35
3.5 Experimental validation ............................................................ 41
3.6 Optimization using HEEDS ................................................... 45
3.7 Results after using optimization software ................................. 55

CHAPTER 4

SUMMARY AND CONCLUSIONS ................................................... 66

REFERENCES .............................................................................. 68

LIST OF TABLES

Table (3.1) Material properties of tube ................................................... 27
Table (3.2) True Stress - Plastic Strain data for Aluminum 6061-T6 [20] ...... 29
Table (3.3) FLD data for Aluminum 606l-T6 .......................................... 32
Table (3.4) Pressure and Feed versus time for tube hydroforming simulation ...... 36

Table (3.5) Internal Pressure and Feed versus Time (Generation=6, Evaluation=224). . .57

LIST OF FIGURES

Figure (1.1) - One-eighth model of the tube hydroforming optimization problem. ...... 3

Figure (2.1) - Applications in Automotive Industry [23] ................................. 10
Figure (2.2) — Industrial and Sanitary Applications [4] ................................. 10
Figure (2.3) - Tube hydroforming process [24] .......................................... 12
Figure (2.4) — Pressure and Forces in tube hydroforming [2] ................................. 13
Figure (2.5) - Low Pressure Forming System [3]. .......................................... 15
Figure (2.6) - Forming Comer Radius, Low Pressure Forming System [3]. ............... 16
Figure (2.7) - High Pressure Forming System [3]. .......................................... 17
Figure (2.8) — Forming Comer Radius, High Pressure Forming System [[3]. ...... 18
Figure (2.9) - Sequenced Pressure Forming System [3]. ........................ 19
Figure (2.10) - Forming Comer Radius, Sequenced Pressure Forming System [3]. ...... 19

Figure (2.11) - Failure cases that can occur during the hydroforming of rotationally

symmetrical workpiece shapes [4] ................................. 21
Figure (3.1) - Tooling setup of square expansion process ................................. 26
Figure (3.2) - Experimentally obtained stress-strain relationship [20] ............... 30
Figure (3.3) - Forming limit diagram for Aluminum 6061-T6 ........................ 33

Figure (3.4) - One eighth model of tube hydroforming simulation, meshed and unmeshed
....................................................................................... 35

Figure (3.5) - Internal Pressure and Feed versus Time relationship ........................ 36

vi

Figure (3.6) - Maximum radius obtained for the deformed tube before failure ...... 37

Figure (3.7) - Forming Limit Diagram for the deformed tube before failure ...... 38
Figure (3.8) - Variation in thickness for the deformed tube ................................. 39
Figure (3.9) - Von Mises stress Plot for the deformed tube ................................. 40
Figure (3.10) - Experimental setup for tube hydroforming [25] ........................ 41

Figure (3.11) - Forming Limit diagram for the simulated tube before failure, full view

................................................................................................. 42
Figure (3.12) - Cracks in experimental and simulated tube .................................. 43
Figure (3.13) — Simulated and experimentally tested tube, shape comparison ...... 44
Figure (3.14) - The structure of a simple Genetic Algorithm [19] ........................ 48
Figure (3.15) - Flow chart for simulated annealing [19] ................................. 55
Figure (3.16) - Internal Pressure and Feed verses Time Relationship (Gen=6,Evl=224)

................................................................................................ 58

Figure (3.17) - Maximum radius obtained before failure, after optimization using HEEDS

................................................................................................ 58
Figure (3.18) - Forming limit diagram for the deformed tube before failure, after
optimization ....................................................................................... 60
Figure (3.19) - Thickness plot for the deformed tube before failure after optimization

................................................................................................ 61
Figure (3.20) - Von Mises stress plot for the deformed tube before failure, after
optimization ....................................................................................... 62
Figure (3.21) - FLD plot for the deformed tube before failure, after optimization (full-
scale model) ....................................................................................... 63

vii

Figure (3.22) - Thickness plot for the deformed tube before failure, after optimization
(full-scale model) .............................................................................. 64
Figure (3.23) - Von Mises stress plot for the deformed tube before failure, after

- optimization (full-scale model) ............................................................ 65

viii

CHAPTER 1
INTRODUCTION

1.1 Introduction

Optimization problems are present in all disciplines and thus a wide range of
goals exists. For example, the goals of manufacturing optimization problems can be to
maximize product output, minimize material waste, minimize cycle time and maximize
equipment use, etc. These goals, of course, coincide with the ultimate goal of business
related optimization problems, which is to maximize profit. In the engineering field
uncountable optimization problems exit. Of these, manufacturing process parameters
optimization is the concern of the current study. In manufacturing process optimization
goals are similar to other optimization problems. Many automobile and aerospace
optimization problems are concerned with determining the design that minimizes weight
while maintaining structural integrity and function.

Various techniques are used to solve optimization problems. Conventional
techniques use gradient-based approaches, which have numerous limitations. Random
search techniques are also limited to a class of practical problems that have small search
spaces. A directed random search method such as a Genetic Algorithm (GA) is a multiple
search method that can be an effective optimization approach to a broad class of
problems. This technique is used to solve the process optimization problem in the current

research.

1.2 Problem Definition

With the advancement in computer controls and high-pressure hydraulic systems,
tube hydroforming (THF) process has become a viable method for mass production of
structural components. Modern presses have independent control of axial feeding,
internal pressure and counter pressure which allows the manufacturing of increasingly
complex parts using the tube hydroforming process.

The emphasis on lightweight vehicle design also increases the requirements for
parts produced using this technology. While hydroforming is new to the forming scene, it
has quickly gained acceptance as a viable alternative to assembled stampings. Only a
decade after its large volume applications, hydroforming is no longer being used only as
a substitute for stampings. Many new vehicle platform designs are calling for the use of
hydroformed components as the primary design option.

The metal forming industry is using finite element analysis (FEA) process
simulation to develop successful tube hydroforming processes that would produce parts
without defects. FEA can be conducted to predict most of the defects in the deformation
zone such as buckling, wrinkling and bursting. These types of failure are caused by
excessive axial feed and insufficient internal pressure.

To avoid the defects in THF, the applied internal pressure must be high enough to
suppress buckling but not high enough to cause bursting. For this purpose FEA
simulations are used. However, in conventional process simulation procedures, the user
must estimate a pressure versus time, as input into the FEA programs. As a result many
simulations must be conducted to obtain the desired results. Therefore, there is a need to

develop a methodology to predict the loading paths (i.e., pressure and axial feeding

2

versus time) necessary to hydroform a tubular part for a given geometry and material. In
the current research a Genetic Algorithm is used to automatically optimize the process
parameters. This technique is expected to reduce the number of simulations necessary to
determine the “best” loading paths.

In the present work, a square shape die and a circular tube blank are considered.
Displacement and internal pressure curves sought during the optimization are such that
the circular tube can expand in to a square shape die to a maximum extent without
bursting, buckling, and wrinkling. Consider a point ‘P’ on the tube as shown in the Figure

(1.1). As the tube expands in the die, the distance ‘U’ traveled by the point increases.

 

 

Square
3 Die
YZ 3] \
Symmetry U
P
Circular
Tube Blank
XZ
Symmetry
X 77777

Figure (1.1) One-eighth model of the tube hydroforming optimization problem

Now the optimal value of the displacement and internal pressure is determined to
maximize the distance ‘U’ without severe thinning and while satisfying the forming limit

diagram (FLD). The FLD provides information about how much a particular metal can

be deformed before necking occurs. The principal strains for each element of the
hydroformed tube must lie under the major strain versus minor strain curve of forming
limit diagram to avoid bursting the tubular blank within the square die for the optimal

value of the loading path.

1.3 Literature Review

Olhoff and Taylor [6] present the basic concepts of structural optimization
emphasizing the fundamentals. They discuss the features and elements involved
optimization of both discrete and continuum structures, and also discuss the mathematical
formulation of such problems. According to [6], fundamental structural optimization
problems were of interest to the likes of Galileo (1638), J. Bernoulli (1687), Newton
(1687), Lagrange (1770), and later to Saint-Venant (1864), Maxwell (1869), and Levy
(1873).

Olhoff and Taylor [6] group design variables into six categories. Cross-sectional
design variables are the first group. These variables refer to the properties of structural
elements such as plate thickness and second moment of area, etc., which are typically
assumed to be continuous but may also be discrete. Topological and configurational
design variables describe the structural layout by specifying such things as the number
and location of structural members and joints in a design. Shape design variables are
variables, which are usually continuous and describe the structure shape. Material design
variables describe material properties and are generally discrete. The final category is the
support or loading design variables, which describe the loading and boundary conditions

on the structure. This type of variable can be either discrete or continuous.

An optimal design problem can be termed discrete or continuous depending on
the dependence of the design variables on the spatial coordinates of the problem. Design
problems such as truss structures are discrete whereas plates and beams are often
continuous.

In design problems certain constraints are imposed on all designs to help identify
feasibledesigns. Olhoff and Taylor [6] divide constraints into two categories; behavioral
and geometric constraints.

Behavioral constraints are, in general, non-linear constraints that can further be
classified as equality or inequality. Equality constraints are, for example, equations of
state and compatibility, which govern the structural response by placing limits on stress
or deflection.

Geometric constraints can also be classified as equality or inequality. Geometric
constraints place restrictions on the design variables due to manufacturing limitations or
required appearance of the design.

To determine the “fitness” or “goodness” of a design, an objective function is
used. The objective function is an expression given in terms of the design objectives,
which have a value for each design or each point in the design space. While single-
criteria optimization problems have been thoroughly studied, there are few realistic
problems that contain merely a single objective to optimize. Multi-criterion optimization
(MCO) problems involve the simultaneous optimization of several criteria and
consequently are more difficult to solve.

Olhoff and Taylor [6] mention three of the better-known methods for solving

MCO problems: weighted sum method, goal attainment or bound method, and the

constraint method. A brief discussion of these methods will be given next. For
explanation purposes c, (D) will represent the criteria for design D, and the intent is to
minimize with respect to D the vector {ch c2, C3, on}.

For the weighted sum method the problem is posed as:

min
D Eilwici (1.1)

Where “’1‘ 2 0 are weights specified by the designer based on the experience and

judgement of the designer. This method determines D, which minimizes the weighted
sum of all criteria.

In the constraint method the problem is given as:

min(ck)
subject to 6,- —big() (i=l,2,...n, i at k) (1.2)

Where the constraint bounds 17:20 are again specified by the designer.

The goal attainment or bound method defines the problem as:

nun(§)
D

subject to WiCi —5 S 0 (i=l,2,...n) (1.3)

This method can be used to solve a min-max problem where for example the criteria c,-

are identified with a maximum such as when the problem is stated as:

min max (wici )
(l .4)
max

In this situation the weighted sum and constraint methods cannot be used.

Stadler and Dauer [7] first present a brief historical review of the field of multi-
criteria optimization, tracing its origins to the area of economics. Stadler and Dauer [7]
discuss optimization as finding Edgeworth Pareto optimal points (EP-points), which are
named after the two economists Francis Ysidro Edgeworth and Vilfredo Pareto. EP-
points are optimal or compromise points in a multi-criterion optimization problem.

Stadler and Dauer [7] outlined four steps involved in the solution of a multi-
criterion optimization problem. First an appropriate mathematical model must be chosen,
second a design set must be selected, third preferences are determined to allow the best
design to be chosen, and finally the optimality concept or condition must be selected.

Stadler and Dauer [7] state the method of constraint approach is a good choice for
solving multi-criteria objective problems. This approach is good since nearly any
mathematical programming code can be used. Many standard approaches for analyzing
multi-objective linear programs are based on the simplex algorithm, which generates a set
of acceptable points from which the best design is chosen.

Finally Stadler and Dauer [7] outline examples of multi-criterion optimization
problems from a survey they conducted. Two simple examples mentioned were the
design of a simple arch structure, and of an elastic truss structure. More difficult and
interesting optimization problems analyzed by Professor R. Statnikov of the Research

Institute for Mechanical Engineering of the Russian Academy of Sciences in Moscow

were also briefly mentioned. Statnikov applied his optimization methods to airframe and
body design of the Russian space shuttle Buran. In this problem mass, structural rigidity,
rib spacing, and cross-section shape were the design variables. This optimization effort
gained a 300kg reduction in the tail section of the shuttle alone. Other work by Statnikov
included design of the Russian truck ZIL. This problem contained 12 criteria.

Another example mentioned in [7] was conducted by H. Eschenauer, a Professor
in the Laboratory for Structural Optimization at the University of Siegen, Germany. In
his work Eschenauer solves the general optimal design problem for laminated conical

shells with weight and deformations design criteria.

CHAPTER 2
DESCRIPTION OF TUBE HYDROFORMING PROCESS

2.2 Introduction

The hydroforming of tubes and hollow sections is by no means a new technique.
The process has been around for decades and was used in the ‘70s [1] for the manufacture
of designer sanitary appliances. The versatile technology available nowadays, with its
high degree of reliability particularly in the field of control technology, has paved the
way for numerous new hydroforming applications in various branches of industry in
recent years.

The possibility of manufacturing ready-made precision parts in a wide variety of
shapes and sizes which are both strong and lightweight means that this process will not
only make its way in fields of application normally reserved for sheet metal forming, but
also in fields which until now have been the exclusive domain of die casting and milling
technology. The increasing use of hydroformed parts in the automobile industry is an
impressive indicator of the importance of this promising technology, which is still very
much in its infancy. It is mainly used in the automobile industry particularly for the
manufacture of exhaust system components, engine cradle, radiator support, instruments
panel support, side rails (Figure 2.1) and for producing industrial and sanitary appliances

(Figure 2.2).

Roof Headers

    
 
 

Instrument Panel
Supports ‘

Radiator Supports

  
 

Engine ’ '

Cradles Upper Frame Rails

Figure (2.1) Applications in Automotive Industry [23].

 

Figure (2.2) Industrial and Sanitary Applications [4].

2.2 The hydroforming process

In tube hydroforming, a straight or prebent tubular blank is placed in an
encapsulating die. The ends of the tube are sealed, and the inside is filled and pressurized
with a hydraulic fluid. At this point, forming pressure can be supplied in one of four ways
to achieve the final part shape:

1. By applying an axial force. The incompressibility of the fluid then expands the
tube to fit the shape of the die.

2. By increasing the internal pressure. As the internal pressure increases, it bulges
the tubular blank outward to fit the shape of the die.

3. By simultaneously applying an axial force and increasing internal pressure.

4. By applying an axial force first, and then, when the axial force fails to bulge the
tubular blank completely to the final shape, increasing the internal pressure to
complete the forming process.

Figure 2.3 illustrates the tube hydroforming process.

11

      
 

Move in horizontal
cylinders, control
water pressure &
guide counter holder

Open press
and eject

Figure (2.3) Tube hydroforming process [24].

2.3 Forming forces and Pressure

The principle of tube hydroforming requires the following three forces to

complete the process (see Figure 2.4):

1. The closing force, F,, is needed to close the tool.

2. The forming force, F“, acts mainly on the ends of the tube and introduces axial
compressive stresses into the tube wall while transporting the material into the
forming zone.

3. The die force, Fw, acts in a direction perpendicular to the tube wall and also on
elements branching off from the workpiece.

Initial internal pressure is generated by a pressure intensifier. During the process,
pressure is applied to the inner profile of the tube, nearly allowing a free flow of material.
When the internal pressure is low, the tube expands freely in die space without contact
with the die surface. When the pressure is high, the tube comes in contact with the die
surface and is deformed against the die profile. For this reason, the internal pressure and

axial flow of the material must be closely controlled.

 

 

 

 

 

 

 

 

    

 

‘”" @
irate V 1FS

Figure (2.4) Pressure and forces in tube hydroforming [2].

 

l3

2.4 Tube/Die contact

When the length of the tube in contact with the die is shorter, the required forming
pressure is lower. When the contact length increases, the required forming pressure also
increases [2].

When the ratio between the cross-sectional areas of the blank and the die is
smaller, the contact length increases at a faster rate, causing a quicker increase in the
forming pressure requirement [2]. Softer materials and longer tube blank lengths have the
same effect. A larger cross-sectional area of the die results in a larger required forming
pressure.

In simple terms, friction in the tube hydrofroming process can be described by the
master-slave concept of friction [2]. This considers friction for the contact areas between
the tool and the tubular blank. The coefficient of friction has a significant influence on

the axial ram force necessary for the process.

2.5 Tube Hydroforming Systems

Currently there are three different tube hydroforming systems, low pressure
forming, high pressure forming and pressure sequence or 2 pressure system. Each method
forms the tube using different mechanisms, which will be described, in the following

sections.

2.5.1 Low Pressure Forming

l4

Low pressure forming occurs when the internal forming pressure is less than the
pressure required to expand the tube along the corner radius of the final part. The internal
pressure required to yield the tube in this area can be detemiined using hoop stress
formulas. This technique can be used to form a tube with flat areas, large comer radii,
and simple cross sectional designs. Figure (2.5a) shows an example of a simple cross
section that can be formed by low pressure. However, difficulties such as pinching arise
with the complex shapes, such as the form in figure (2.5b). Figure (2.6) Shows the tube
comer in a die with a large comer radius. The original wall thickness is maintained in the
comer because the forming stress did not exceed the material yield strength.

Tube expansion during low-pressure hydroforming is possible, but it is limited to
cross-section with a large radius. Tube expansion at the ends is possible with end feeding,

but it must be done mechanically prior to hydroforming.

 

a. Simple form b. Complex Form

Figure (2.5) Low Pressure Forming System [3].

R=1UT

/////

-Hoop Stress at corner Radius
is below the Material Yield
Strength

~Cannot Form Sharp Corner
Radii

-Limited to Simple section
Shapes

\\\\\

 

Figure (2.6) Forming Corner Radius, Low Pressure System [3].

2.5.2 High Pressure Forming

High pressure forming uses high internal pressure so that the tube hoop stress at
the comer radius is higher than the material yield strength. In doing this, the tube fills the
corner radius areas of the die by local stretching or expansion. Figure (2.7) shows the
cross section of a tube in a complex shaped die. The die closes on the tube without
pinching because the tube is smaller than the die cavity. As the higher pressures force the
tube to the final shape, there is a reduction in tube wall thickness in the areas shown in

Figure (2.8).

 

 

 

Before Pressure Afler Pressure

Figure (2.7) High Pressure Forming System [3].

Since high pressure causes significant friction between the tube and the die, the
tube stretches significantly less in the flat areas than in the corner radius where material
thinning occurs. The tube material must have high enough elongation to withstand the
forming without bursting. Since press tonnage is proportional to forming pressure, and
higher pressures require larger press sizes, a limitation occurs for larger structural parts.

Tube expansion (beyond forming the comer radius) is possible during high
pressure hydroforming but it requires tube materials with high elongation and often times
in-process annealing. Tube expansion at the ends using axial end feed in the forming die

can also be employed.

Reduced Thickness

R=3T
_ i,
A -Radius formed When Internal

pressure Creates Hoop stress

I Above the Material Yield
Strength
/ -Tube Expansion and Reduced
P J

 

wall thickness in corners
-High Material Elongation
Required

-High Press Tonnage Required

 

.1 T

Figure (2.8) Forming Corner Radius, High Pressure Forming System [3].

2.5.3 Sequenced Forming Pressures

A pressure sequence system forms the corner radius by forcing the tube to flow
into the comer without stretching or expanding the tube to fill the die cavity. The comer
radius area is formed by exceeding the yield limit of the tube material in a bending mode,
rather than a tensile mode as done in high pressure systems. This is accomplished by
adding an additional pressure stage while the die is approaching the final closed position.
Figure (2.9) shows a cross section of a die and tube during the pressure stages.

Figure (2.10) shows the die comer and how the pressure forms the radius in the
tube by drawing the tube into the comer area. The free flow of metal in direction “F”
allows an outside comer radius to be formed equal to 3 times metal thickness. During the
first stage the pressures are low resulting in lower die friction, which is important for
allowing the metal to slide along the die surface and into the comer area. The final

formed tube is of uniform wall thickness around the circumference including the comer

l8

radii. After completion of the first stage a higher pressure is required to completely form

the sides of the tube.

 

Ist Pressure Stage 2nd Pressure Stage

Figure (2.9) Sequenced Pressure Forming System [3].

R=3T

-Meta| Flows into Corner Radius
Areas from Direction "F"
-Forming Pressure is Lowto
Minimize Die Friction

-Wall Thickness is Uniform
Around Circcumference

-The Following. Second Stage
Pressure is Higher to Set
Accuracy ofthe Final Form
-Required Forming Pressures
are Significantly Less Than High
Pressure system

r,\\:\\\

 

A

Figure (2.10) Forming Comer Radius, Sequenced Pressure Forming System [3].

Tube expansion during the hydroform process can be done during the second
pressure stage. If required, expansion of the tube using end feed in the hydroform die is

done as part of this stage. Tubes can also be mechanically expanded with end feed ahead

of the hydroform die if there is a local indent or other complex form near the end of the
final part.

Tube forming capability improvements have been developed using pressure
stages, or a pressure sequence to improve metal flow in the forming dies. This multiple
pressure sequence forms the tube into the complex shapes of the die cavity in a manner
that reduces the natural pinching and buckling tendencies of traditional hydroforming.
This process allows for the lower elongation values typical of aluminum and high

strength steels, and also eliminates the need for in process annealing [3].

2.6 Failure Modes

The following failure modes [4] can occur when tubes with a straight longitudinal
axis are being expanded as shown in Figure 2.11.
Buckling
Bursting
Wrinkling
Folding back

2.6.1 Buckling

There is a danger of buckling at the start of the process as a result of excessively
high axial force (feeding) acting on the starting tube. This is a function of the tube
parameters. The permitted buckling force at the start of the process can be estimated
theoretically [5,8]. The danger of buckling persists beyond the initial stage of the process

and lasts throughout the start-up phase. It is thus necessary to influence the process,

20

through the process control, in such a way that the reduction in free tube length is

coupled with a rapid increase in the section modulus of the tube cross-section [4].

   

    
 

:5} ' Tube pushed

Buckling . inwards

Upsetting of
the tube-wall

Figure (2.11) Failure cases that can occur during the hydroforming of rotationally
symmetrical workpiece shapes [4].

2.6.2 Bursting

There is a danger of bursting once medium-level expansion has been attained
(d1/d0 > 1.4) as a result of an excessively high internal pressure, where do and (11
represents the diameter of the tube before and after expansion. The bursting process is
triggered by local constriction of the tube wall; the start of constriction is a major
function of the starting wall thickness. These processes generally have a characteristic
intermediate form of tube wall (in the form of a bulge towards the outside) associated

with them [9]. The development of this intermediate form can be influenced via the

21

process control; to avoid the danger of bursting it must be ensured that the tube wall is

lying against the die by the start of constriction at the latest.

2.6.3 Wrinkling

Wrinkles that are symmetrical to the longitudinal axis can be eliminated by
increasing the internal pressure in the final phase of the expansion process [4]. Additional
wrinkles can, however, occur in the center of the workpiece, even in long expansion dies,
as a result of excessively high axial force (feeding). The formation of these wrinkles can
be avoided through an appropriate form of process control. It is also possible to estimate

the danger of wrinkling by theoretical means, employing simplifying assumption-[10].

2.6.4 Folding back

The danger of the workpiece folding back is conditioned both by the design of the
die and by the expansion geometry and the tube parameters [11]. Folding back occurs
when tubes are expanded in dies where tube wall material is forced into the die from tube
holders. If this is done with thin walled tubes (so/d0 < 0.05), where so and do are thickness
and diameter of the tube before expansion, then there is a danger of the tube folding back
in areas of large-scale expansion (d1/d0 > 1.8). In the case of tubes with thicker walls
(so/d0 > 0.1), by contrast, there is a danger of the tube wall being compressed in the
holder at lower expansion ratios already if the force required to feed further material into

the expansion die is greater than the upset force. Figure 2.11 shows the transition from

22

the holder to the expansion die, by way of example. The tube becomes compressed here

and if further tube wall material is fed in from holder, material segregation can occur.

2.7 Advantages and Disadvantages

The advantages of the tube hydrofroming process are numerous. New and
different materials can be used to produce parts. Lightweight materials with high
strength-to-mass ratios, high-strength steels, and laminates can be used. Inexpensive,
nonalloyed steels can also be substituted since, because of deformation, the tube material
strain hardens and achieve higher strength than that of the initial tube.

Tube hydroforming allows complex part design because it can achieve deeper
draws, varying thickness, varying cross sections, and severe shapes. These specific
benefits are most important to part design:

1. Part consolidation means a single part replaces an assembly. We can reduce the
number of parts using this technology.

2. Weight savings are significant as compared to conventionally formed parts (up to 40
to 50 percent). The savings are gained from the production of hollow parts, which
have an inherently higher strength/weight ratio.

3. The stronger parts are due to a continuous weld versus the conventional spot welds.
Strength is also increased by the inherent work hardening of the process. This allows
for the use of thinner and less costly materials [12].

4. Surface and dimensional quality of part is improved. By providing a uniform strain
distribution, the process has better shape control, reduced spring back, and less

surface distortion. Outer dimensions are kept to exact tolerances with high precision.

23

In low volume production, the percentage of the final product cost accounted for by the
tooling is significantly higher than in higher volume production [13]. The development
and lead times are shorter than those for conventional methods of production, the
prototype tooling costs less, and the material changes do not require new tooling.

The tube hydroforming process increases the formability of a material, which
allows a reduction in the number of forming steps for a product, in turn reducing the
number of tooling sets required.

The elimination of subsequent spot welding operations from the forming process
creates significant savings in manufacturing costs. The capabilities of the process
eliminate many secondary operations for additional savings.

This process also has drawbacks. In general:

1. The tube hydroforming process has a slow cycle time and, therefore, a low
production rate. This limits the process to low-volume production or requires
additional investment in equipment and tooling.

2. The dies in the tube hydroforming process require a high level of surface polishing.
The frictional forces caused when the sheet blank contacts the die must be
minimized as much as possible. Additionally, the surface quality of the die directly

affects the surface quality of the final part’s outer surface.

24

CHAPTER 3
TUBE HYDROFORMING SIMULATION AND OPTIMIZATION

3.1 Introduction

It is very cost-effective to make the right decisions early in a design process.
Computer simulations are important as a tool to support these decisions. Nowadays, most
components are simulated and their characteristics are studied. If their behavior appears
to be non-satisfactory changes are made. The simulations are based on nominal data such
as geometry, material parameters, residual stresses, etc.

The vehicle industry has established the technique of simulating the
manufacturing process [14]. By dealing with sheet metal forming, the technique is highly
developed and reliable enough to eliminate costly tools for test pressing. In other
industries such as injection moulding and forging the technique was a tool used only in
the academic world but today different software and packages are developed rapidly for
commercial use.

Among the most common applications today is tool design and process
optimization for the process industry. For a process like hydroforming this is of special
significance. Designers don’t have any designer’s handbooks or deep knowledge in this
area due to the relative immaturity of the process.

Various commercial Finite Element Analysis (FEA) software packages such as
PAM-STAMP and LS-DYNA are capable of simulating the tube hydroforming
processes. A successful tube hydroforming process requires proper combination of part
design, material selection, and application of internal pressure and axial feeding. By

using FEA methods, process parameters can be determined and then optimized using

25

optimization software before manufacturing the dies and starting die try—out and process
development. The use of automated design optimization software to optimize the process
parameters in a hydroformed part has never been attempted to date. This is the goal of the

present work.

3.2 Description of the geometry

Figure (3.1) illustrates isometric and front views of the tube hydroforming set up
for the square expansion process. The major components of the equipment include two
die halves with square cavities in the central area, and a circular tube blank.

The central section of the die cavity has square dimensions of 50.8 mm by 50.8
mm with comer radii 6.35 mm The length of the cavity is about 50.8 mm Circular tubes
with diameter 50.8 mm and length 200 mm are used in the experiments and the
simulations. The long sections on both sides of the central section are included to

investigate frictional and axial displacement issues.

Circular
‘/— Tube Blank

       
  

Square
section 3, ,

Die hart/2.x

 

Figure (3.1) Tooling setup of square expansion process.

26

3.2.1 Tube Material

The tubes are made of Aluminum 6061 with T6 and T4 condition and wall

thickness of 1.00 mm. Properties of this material are listed in Table 3.1.

Table 3.1 Material properties of the tube

 

 

 

 

Modulus of Density Poisson’s Yield Tensile
Material n-value elasticity ratio Strength Strength
(GPa) (Kg/m3) (MPa) (MPa)
24:??? 0.0717 69 2700 0.33 235 282.77
g‘é‘grlnlfi‘l’m 0.1748 69 2700 0.33 118 216.47

 

 

 

 

 

 

 

 

3.2.2 Strain Hardening Exponent

The strain-hardening exponent (n-value) as listed in Table 3.1 is calculated using

the Considere criterion [15].

63—: = (3.1)
and using the power law relationship
0' = kg" (32)
we have
99. = nan-1 = 0' = k8" (3.3)

de
From Equation 3.3, the following relationship between the maximum strain and n can be

determined

27

3 = 1 (3.4)
8

The true stress at the maximum load is also known from [15]

n
Su = 43) (3.5)

e
Where e = 2.7183 is the base of the natural logarithm.

Using Equations 3.2 and 3.5 and assuming that 0' = 0'}, when 8 = 0.002, it is possible to

obtain the following equation for calculating n.

€n(0'y)— €n(Su)
" = €n(0.002)+€n(e)—€n(n) (3‘6)

 

The Equation 3.6 can be solved for n, iteratively, for known values of the ultimate tensile

strength, 5“, and the yield stress, 0),, of the material. From the uniaxial tensile test for
6061-T6 Aluminum we get a value of 5,, = 282.77 MPa at the ultimate load. Using the
0.2% offset condition we also calculate the yield stress to be 0'), = 235 MPa.

By substituting for these values into equation 3.6, we get the following n-value
n = 0.0717
The strength coefficient (k) was then calculated from Equation 3.2.

k = 366.93 Mpa

3.2.3 Stress-Strain relationship

Table 3.2 shows the true stress and plastic strain data for Aluminum 6061-T6
obtained through tensile test of the material [20]. Based on this data Figure 3.2 shows the

true stress plastic strain relationship.

28

Table 3.2 True Stress Plastic Strain data for Aluminum 6061-T6 [20]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Plastic Strain True Stress (MPa)
.000000000E+00 .235000000E+O3
.553623000E-02 .239000000E+03
.848696000E-02 .242400000E+03
.l 14594000E-01 .244300000E+03
. 144225000E—01 .246850000E+O3
.173916000E-01 .248980000E+03
.203617000E-Ol .251040000E+03
.233313000E-01 .253140000E+03
.263004000E-01 .255270000E+03
.292706000E-01 .257330000E+03
.322416000E-01 .259330000E+03
.352130000E-01 .261300000E+O3
.38 1 880000E-01 .263030000E+03
.41 1 63 8000E-01 .264700000E+03
.441406000E-01 .266300000E+O3
.471213000E-01 .26763000OE+03
.500984000E-01 .269210000E+03
.530804000E-01 .270450000E+03
.560617000E-01 .271740000E+03
.590464000E-01 .272800000E+03
.620317000E—01 .273810000E+03
.650174000E-01 .274800000E+O3
.680030000E-01 .275790000E+03
.709904000E-01 .276660000E+03
.739794000E-01 .277420000E+03
.769701000E-01 .278060000E+03

 

 

.799604000E-01

 

.278730000E+03

 

29

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.8295 12000E-01 .279370000E+03
.859442000E-01 .279850000E+03
.889372000E-01 .280330000E+03
.919307000E-01 .280780000E+03
.949248000E-01 .281 190000E+03
.979207000E-01 .281470000E+03
. 100917000E+00 .281760000E+03
. 103912000E+00 .282070000E+03
. 106909000E+00 .282290000E+O3
. 109906000E+00 .282460000E+03
.1 12905000E+00 .282580000E+03
. l 15903000E+00 .282700000E+03
. 1 18902000E+OO .282770000E+O3
. 1 36902000E+00 .282770000E+03

 

300 1

./

250

d N
or O
O O
L

True Stress (MPa)

50‘

True Stress-Plastic Strain Relationship

 

 

 

T j

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Plastic Strain (mmimm)

Figure (3.2) Experimentally obtained stress-strain relationship [20].

30

0.14

 

.l -_..__.... _._- 2.- a-

'D
.o
(D

 

The tool geometry has been created in I-DEAS master series [21] and imported
into DYNAFORM-PC [22] on an IGES file. DYNAFORM-PC [22] is a pre- and post-
processor for LS—DYNA [17]. Other geometries like tube blank and punch has in a
similar way been created directly in I-DEAS and imported from an IGES file.

The commercial program LS-DYNA from Livermore Software Technology
Corporation has been used to analyze the hydroforming process. The program is an
explicit FEA program and solves nonlinear problems. In comparison to an implicit FEA
program the explicit is much faster due to the advantage of avoiding matrix inversions

and it also requires less data storage [16].

3.3 Simulated Tube

The tube has been modeled with shell elements. In most explicit codes, shells are
assumed to be composed of a number of laminae, with each lamina satisfying the plane
stress assumption. One constitutive evaluation is performed for each lamina, yielding a
potentially complex through-thickness stress distribution. Numerical integration with one
integration point in each lamina is used to compute the resultant forces and moments on
the shell. In LS-DYNA, this integration may use from two to five pre-defined points
through the thickness, or any number of points in a user-defined integration rule. Fewer
points yield sufficient accuracy in regions of predominantly membrane behavior, while
more points improve accuracy when bending effects and plasticity are combined [18]. In
this case the Belytschko-Tsay [17] element with five integration points through the

thickness has been used.

31

The tools for this simulation are modeled as rigid. By modeling the tooling as
rigid in the finite element model, a fine mesh may be used to accurately resolve the
details of tooling geometry (such as radii of curvature) without significantly increasing
the cost of the analysis. Elements using the rigid material don’t affect the time step, and
therefore rigid element sizes may be selected solely on the basis of accurately
representing the tooling shape.

The material model used is a transversely anisotropic elastic plastic model. This is
material 37 in LS-DYNA [17].

The formability of tube material is often evaluated from strain analysis through
the Forming Limit Diagram (FLD), which represents the relationship between the
limiting major and minor principal strains in the plane of the strained tube.

Table 3.3 shows the FLD data for n =0.0717 and thickness of 1.00 mm for
Aluminum 6061-T6. Figure 3.3 shows the Forming Limit Diagram based on FLD data
listed in Table 3.3 for the same material.

Table 3.3 FLD data for Aluminum 6061-T6

 

 

 

 

 

 

 

 

 

 

 

Minor Strain (mm/mm) Major Strain (mm/mm)
-.494296300E+00 .700827800E+00
-.446287100E+00 .642684000E+00
-.400477600E+00 .585192700E+00
-.356674900E+00 .528683500E+00
-.314710700E+00 .473534200E+00
-.274436800E+00 .420171 100E+OO
-.235722300E+00 .369068400E+00
-. 198450900E+OO .320743700E+00
-.162518900E+00 .275750900E+00

 

 

 

32

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-.127833400E+00 .234667600E+00
—.943 106800E-01 . 198078800E+00
-.618754000E-01 .166556200E+00
-.304592100E-01 .140633600E+OO
.000000000E-t-00 . 120780400E+00
.295588000E-01 .146086300E+00
.582689100E-01 .166843000E+00
.861777000E-01 .183938500E+00
. 1 133287OOE+00 .198065200E+00
.139761900E+00 .209770100E+00
.165514400E+00 .219489900E-l-00
. 190620400E+00 .227575800E+00
.2151 11400E+OO .234312500E-l-00

 

Maior Strain-Minor Strain Relationship

on On.)
,-

 

 

Major Strain (%)

vr."

70.00% -

       

80.00% d

50.00:: .

10.00% -

 

n Mil

 

 

-G0.00%

-20.00% 40.00%

Minor Strain (%)

-50.00% 40.00% -30.00%

33

0.00%

10.00% 20.00%

Figure (3.3) Forming Limit Diagram for Aluminum 6061—T6.

30.00%

 

The tube material is assumed to be uniform throughout the thickness. The friction
coefficient is also assumed to be a constant value of 0.03 throughout [2]. One-eighth part
of tube is considered in the FEM model (due to existing symmetries) to reduce the
simulation time as shown in Figure 3.4.

The mesh has been defined with adaptive characteristics. This improves the
curvatures in the small radii of the tool where it is needed the most instead of defining
small elements all over the tube.

In order to avoid properly account for blank/tool contact in the hydroforming
process, the blank is modeled as a slave surface and tooling as the master surface. The
two contact surfaces are individually described by a collection of contact segments,
which are simply element faces on the contact surface. For each slave node, the contact
algorithm first searches through the master segments to determine if any penetration has
occurred. If a slave node has penetrated a master segment, then the penalty method is
used to compute a restoring force, which is applied to the slave node to return it to the
master surface, and an equal and opposite force is applied to the master segment [18].
Once this process is completed, the slave and master designations are interchanged and

the algorithm is applied again.

34

m
“HE
x
g;

m ’-
g,
33‘
its
El

12
a:

 

Figure (3.4) One-eighth model of tube hydroforming simulation, meshed and unmeshed.

3.4 Results before using optimization software

A number of iterations have been done using LS-DYNA for different pressure and
feed profiles to find the maximum expansion of tube within the square die without
bursting the tube and satisfying the Forming Limit Diagram. One of the best pressure and
feed profile data is listed in Table 3.4 below and Figure 3.5 shows their relation with

respect to time. These were obtained through a manual optimization process.

35

Table 3.4 Pressure and feed verses time for tube hydroforming simulation

 

 

 

 

 

 

 

 

 

 

 

 

Time (Sec) Pressure (Mpa) Feed (mm)
0.00 0.00 0.00
0.005 9.5 0.5
0.010 9.5 3.5
0.015 10.5 3.5
0.020 10.5 6.5
0.025 11.5 6.5
14 ~ ..

 

‘ ‘ Pressure

 

 

 

 

 

'5‘ .
'o -
CD
0)
u.
"o 8 .
C
(1)
2t
:5, 5 i X/F { . -
9 Displacement
S / curve
0‘)
9
0. 4 .
. 2%. I
2 ..
,..n’
0 A r I r r r i
0 0 E05 0.01 0.015 0.02 0 025 0.03

Time (Secs)

Figure (3.5) Internal Pressure and Feed versus Time Relationship.
At time step 0.01448 sec, internal pressure is about 10.3 Mpa and feed is about

3.5 mm. The maximum expansion of tube before failure at above pressure and feed rate is

illustrated in Figure 3.6. Figure 3.6 shows that the tube expands up to 14.8 mm radius. A

36

slight increase in pressure above 10.3 Mpa results in the bursting of the tube at the given

 

 

 

 

 

 

 

 

 

feed rate.
2< 25.4 :4
a 7?
Y2 Symmetry
25.4R
14.8R B
J:
Yl
XZSymmetry v All d' . .
2...? n r I rmensrons in mm

Figure (3.6) Maximum radius obtained for the deformed tube before failure.

Principal strains for the deformed tube lie within the Forming Limit Diagram as

shown in Figure 3.7. Figure 3.8 shows the thickness plot and Figure 3.9 shows the von

Mises stress plot for the tube before failure.

37

FORMING LIMIT DIAGRAM
STEP 12 TIME = 144759975002
BOTTOM

 

 

 

 

 

 

Figure (3.7) Forming Limit Diagram for the deformed tube before failure.

38

STEP 12 TIME = 144759975002
THICKNESS

1.019542 . ,
1.010148 ‘
1.0000300 1
0.991354
0.981958 W .
0.972582
0.983187 ',_ -{
0.953771 ’
0.944375
0.934979 j;
0.925583 7 "

 

Figure (3.8) Variation in thickness for the deformed tube.

39

STEP 12 TIME = 14459975002
MAX_VONMISES

 

Figure (3.9) Von Mises Stress plot for the deformed tube.

40

319.54901 1

309559521

299.770530

289.881 439

279992249 .(

270103058 , i

250.213857 i

250324661
2411435471
2311546280

1%

I. . L‘
5‘ ;-.
. . w

220 - 657089 E: z. .

 

3.5 Experimental Validation

To validate the simulation results, Aluminum 6061 with T6 condition tube has
been tested experimentally in a Michigan State University research lab using the tube

hydroforming press as shown in Figure (3.10).

Typical UniPress Controller Application
for Hydrofonning

Clamp Actuators iii

   

   
  

- ”Feed Ram 2

Feed Ram 1

Figure (3.10) Experimental set up for tube hydroforming [25].

Figure (3.11) shows the Forming Limit Diagram for the simulated tube without
rupture at time 0.01447599 sec and at a pressure of 10.3 Mpa with feed of 3.5 mm and
Figure (3.12) shows rupture in the simulated and experimentally tested tube as the
pressure increases with the same feed rate. Figure (3.13) shows the shape comparison

between the simulated and experimentally tested tube.

41

    

Forming Limit Diagram
STEP 12 TIME = 1.4475997E-002

BOTTOM
1.0]—

 

0.8
0.6

0.4""\.

 

 

 

 

Figure (3.11) Forming limit diagram for the simulated tube before failure, full view.

42

 

43

Fonnlng lelt Dlagram
STEP 13 TIME = 157914255002

BOTTOM
1.0

 

0.0
0.5 ‘\
0.4 \..

0.2\»-..-_ -

 

01] watt... '

 

 

Figure (3.12) Cracks in experimental and simulated tubes.

 

 

Figure (3.13) Simulated and experimentally tested tube shape comparison.

Now the optimization software HEEDS (Hierarchical Evolutionary Engineering
Design System) [19] is used to find the optimal pressure and feed rate profiles for the

maximum expansion of the tube within a square die before rupture.

3.6 Optimization using HEEDS

The optimization problem in the current research is classified for HEEDS as type
one, which compromises between global and local search techniques. For this problem
classification type, HEEDS performs combinations of evolutionary, gradient based, and
design of experiments search heuristics [19]. A brief description of search techniques

used is presented in the section below.

3.6.1 Evolutionary Search

HEEDS [19] employs a Genetic Algorithm (GA) to perform evolutionary search.
A GA is a powerful technique for search and optimization problems, and is particularly
useful when the design space is large and complex. The main problem with using a
simple GA is the potentially large number of design evaluations required to obtain a set
of satisfactory solutions. HEEDS reduces the number of evaluations required to obtain a
set of satisfactory solutions by hierarchically decomposing a problem with multiple
agents that represent the problem in various ways, while combining efficient local search
methods (automated design of experiments, nonlinear sequential quadratic programming,

and simulated annealing).

45

A GA is a search procedure based on the mechanics of natural selection. Specific
knowledge is embedded in a chromosome (or design vector), which represents a possible
design with a set of values of all the design variables. The number of choices per design
variable determines the fidelity (or resolution) of each design variable.

A GA creates and destroys designs during a process that involves decoding each
chromosome, evaluating its satisfaction of constraints and its performance relative to the
objectives, and then allowing a simulated “natural selection” to determine which designs
are eliminated and which survive to generate other, derivative designs. Designs that
perform well (relative to constraints and objectives) have a higher probability of
surviving to influence future designs (their “offspring”). During reproduction, the two
genetic operators commonly modeled that produce new chromosomes (or design vectors)

are called crossover and mutation.

3.6.1.1 Crossover

The crossover operation (sometimes also called “recombination”) forms a new
solution by combining parts of two existing solutions. A high crossover rate (fraction of
population replaced by crossover during one generation of reproduction) will produce
many new designs in each generation, but will also have a high probability of disrupting
(and potentially losing, at least temporarily) higher—performance designs already found,
and requires more evaluations of constraints and objectives in each generation (which are

typically the most costly computing operation in the entire problem).

3.6.1.2 Mutation

46

Mutation is a reproduction operation that produces a new solution from a single
existing solution, through any of several ways. Mutation can change the value of one
design variable, or of many simultaneously, and can change them in uniform random
ways, or distributed normally, for example, around the current values of the design
variables. Mutation helps maintain diversity and reduces the possibility of premature
convergence (the tendency of a set of solutions to come to closely resemble each other,
thereby making it difficult for crossover to generate solutions that differ very much from
the current set).

A set of co-existing designs defines a population, while successive populations
are termed generations. That is, each period during which a set of existing solutions are
evaluated, then used with natural selection, crossover, and mutation to generate a new set
of solutions, is called a generation. A large population typically contains more genetic
diversity (i.e., more different values of design variables), that typically improves the
ultimate results of the GA search. However, the more new individuals created in each
generation, the more computer time must be spent evaluating the constraints and
objectives of the new individuals, so there is a tradeoff that must be made.

Figure 3.14 displays the structure of a simple GA. The simple GA begins by
creating a single initial population, wherein chromosomes (vectors of design variable
values) are randomly created. At this point the performance (constraint satisfaction and
objective values of each design is evaluated. Biased by the evaluations obtained, the GA
uses unary (mutation) and binary (crossover) operators on these designs to create another
population. This population probabilistically maintains the previously high performing

designs while discarding poorly performing designs. New population members are

47

evaluated, and then additional rounds of generation and selection are performed. This is
repeated until satisfactory solution(s) are obtained. Incorporating these processes in a
computer algorithm produces an algorithm that solves problems in a manner reminiscent

of natural evolution.

 

Create the Initial
Pooulation

1

Evaluate Fitness

 

 

 

 

 

 

 

I

Create the Next
Generation
With Genetic Operators,
Selection .

 

 

 

 

   

 

Convergence?

Yes ¢

 

Optimal Solutions

 

 

 

Figure (3.14) The structure of a simple Genetic Algorithm [19].

3.6.1.3 Representation of Design Variables within a Chromosome

‘6 9,

A chromosome of p design variables (x,) can be represented in vector form as:
{x1, X2, X3, ....., xp). Each design variable represents a field (or locus) on the chromosome

(or design vector). The number of choices per design variable will define its resolution, or

48

fidelity. If a coarse representation of a continuous design variable is sought, then the field
size of the continuous design variable should have a small number of choices between the
lower and upper bounds of the design variable. A large number of choices between the
lower and upper bounds of a continuous design variable can be defined to refine the
representation of the continuous design variable. Each design variable (field) on the
chromosome can have an independent field size. If a design variable is truly discrete in
nature, then the field size of the discrete design variable should be set at the number of

choices in the discrete set.

3.6.1.4 Initial Population Generation

First, a GA typically performs random initialization, then evaluation of the initial
population. The GA stochastically creates a population of chromosomes composed of a
randomly chosen design variable at each field. A performance measure (or fitness) is
assigned to each design (or individual) based on the degree to which it satisfies the

problem’s constraints and extremizes its objectives.

3.6.1.5 Selection

The next process the GA must perform is selection. The selection mechanism for
a GA is simply the process that favors the selection of high-performance individuals to be
placed in the “mating” pool of individuals to undergo crossover, mutation, and/or
unaltered survival to the next generation. The performance measure (or fitness function)

provides a means of comparing individuals and the selection process determines how the

49

individuals are compared. Selection mechanisms determine which chromosomes are
placed in the mating pool. The mating pool is comprised of the individuals from which
the next generation will be created. Selection is the GA search catalyst that works by

mimicking natural selection.

3.6.1.6 Definition of Performance

The “goodness” of each design is represented with a single scalar value called the
performance measure. The performance measure is a composite of a number of
subsidiary measures, a set of objectives (each of which may be targeted for maximization
or minimization) and a set of constraints, for which violations are to be minimized. The
constraints enter into the performance according to the penalty method, which gives them
no influence so long as they are satisfied, but gives them increasing importance to
whatever extent they fail to be satisfied. Within any single agent, to evaluate the
performance measure (or fitness) of a design, the objective and constraint functions are

normalized, weighted, and aggregated:

Nobjs , , 2 Ncons ._ . -_ _ 2
P: 2 Rli-ffimzi & — 2 C mil—fa—tiiwzi i5— (3.7)
1:] Inil Inil i=1 Itil Itil

Where P is the performance measure, Nobjs is the number of objectives, and Ncons is the

number of constraints. R1,- is a constant used to linearly reward a design’s performance

due to extremizing of the ith objective function. R2; is a constant used to quadratically

reward a design’s performance due to extremizing of the ith objective function. The

50

ith objective function ( foi ) is normalized by the absolute value of ni. P1,- is a constant

th

used to linearly penalize a design’s performance for violating the i constraint function.

P2,- is a constant used to quadratically penalize a design’s performance due to the

th

violation of the current constraint function. The i constraint function( fci )is normalized

by the absolute value of its target ti. If the constraint is satisfied, C is set to zero; if the

constraint is violated, C is set to unity.

3.6.2 Nonlinear sequential Quadratic Programming

Nonlinear Sequential Quadratic Programming (NLSQP) is a gradient-based
search technique that efficiently solves an explicit set of optimization problems. NLSQP
is used to solve deterministic, unimodal, convex, first-order differentiable objective and
constraint functions. NLSQP will quickly converge to the global optimum if the objective
and constraint functions are deterministic, have first-order continuity, and satisfy
convexity criteria. NLSQP will quickly converge to a local optimum if the objective and
constraint functions are deterministic, have first-order continuity, but fail to satisfy
convexity criteria.

NLSQP is a single point search technique that applies gradients of the objective
and constraint functions (with respect to continuous design variables) to define a search
direction to improve the performance of the current design. Since most simulation
packages do not compute analytical gradients of objective and constraint function, the

calculations of gradients are approximated through numerical differentiation. The

51

numerical differentiation is implemented within HEEDS with a forward differencing
scheme.

3.6.3 Automated Design of Experiments

HEEDS does type-zero Design of Experiments (DOE) by reusing previously
computed evaluations to construct response surfaces and then using nonlinear sequential
programming to search, from an initial design, for designs projected to have improved

performance.

3.6.3.1 Type-Zero Design of Experiments

HEEDS conduct type-zero DOE about an initial “starting point” design by: l)
collecting a training set of previously evaluated designs based on simple normalized
distance criteria measured relative to the starting point, 2) fitting linear and quadratic
surfaces (in a least squares sense) to the training set, to approximate the various
constraint and objective functions in the vicinity of the starting point design, 3) applying
NSQLP using the approximated objective and constraint functions with their least
squares polynomial surfaces, and 4) automatically updating the training set for the next
DOE.

The training set for type-zero DOE is based on a simple normalized absolute
distance metric, and is collected from all previous design evaluations of identical

representation and performance type:

 

(3.8)

Where N is the number of design variables, X ,- is the starting point’s ith design variable,
Tiis the ith design variable of design under consideration to be included in the training

set, and Ri is the resolution of the ith design variable. The minimum distance is zero (the

starting point) and the maximum distance is unity. All designs that have an absolute
normalized distance less than or equal to a prescribed maximum distance and greater than
or equal to a prescribed minimum distance will be included in the training set of the

DOE.

3.6.4 Simulated Annealing

Simulated annealing (SA) is based on the statistical mechanics of annealing
solids. To understand how such an approach can be used as an optimization tool, one
must consider how to coerce a solid material into a low energy state. Annealing is a
process typically applied to solid materials to force the atomic structure of the material
into a highly ordered state. If a material is brittle in nature, its atomic structure is not in an
ordered state. If the material is heated to a high temperature, the atomic structure shakes
violently, allowing for alternate, more ordered, energetically favorable, atomic states. In
an annealing process, a material is heated to a high temperature and slowly cooled. If the
material is cooled suddenly, the atomic order is locked into a random unstable state. If the
material is cooled slowly, the atomic structure tends to fall into relatively stable
configurations for each temperature. As the temperature is slowly dropped, the material

state will eventually stabilize into a highly ordered state.

53

Implemented as an algorithm, SA begins with an initial design, a starting
temperature, an ending temperature, a total number of temperatures to consider, and the
number of iterations per temperature. As coarsely depicted in Figure (3.15), SA begins
with an initial design at a specified starting temperature. Then, the initial design is
subjected to a specified number of random perturbations based on the starting
temperature. If the staring temperature is close to unity, the perturbations are large,
yielding a broad search relative to the initial design. If the starting temperature is small,
the perturbations are small, yielding a localized search about the initial design. If a
perturbation locates a better design, the initial design is updated with the improved
design. Once the number of iterations at that temperature is exceeded, the temperature is
decreased and the perturbation process is iterated. This process of decreasing the
temperature and iterating repeats for the total number of temperatures. The ending
temperature should be small, and will result in more localized search than the starting

temperature.

54

 

 

 

Starting design

1

a Updated temperature

__.[ Design perturbation

Evaluate current design

i Yes

Too many evaluations? Q

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l Yes
Update
Improved design? ‘__’ current
design
iteration+l
Nn

 

 

Too many iterations?

 

 

 

 

Yes
Temperature+l 1 Yes

Too many temperatures? a

Figure (3.15) Flow chart for simulated annealing [19].

 

 

 

 

 

 

 

3.7 Results after using optimization software

HEEDS is used to optimize the process parameters; the recommended eight
procedural steps for HEEDS input are followed. A dynamic link library was written to

perform the objective and inequality constraint function evaluations. This problem does

55

not require design criteria template files to be created, so the first step to ‘Define a
Baseline Design’ is not required. For this problem, the objective function will be
maximized while satisfying the constraint on the forming limit diagram. This is done in
the second step that is to ‘Define the Design Criteria’. The design criteria are specified in
a Performance file with an id of l. The third step is ‘Define the Design Variables’.
HEEDS requires the definitions of static parameters of all possible design variables to be
declared in the ‘Definition.in’ file. For this problem, we do not need to define this file
because we use user—defined variables. These variables are declared in the
‘Representationl.in’ file. In the fourth step ‘Represent the Design Variables’, a collection
of design variables to be varied by an agent is defined in a ‘Representationl.in’ file. The
fifth step is ‘Define the Agent Topology’. A single agent is applied to this problem. The
‘Topology. in’ file defined all of the agents in the HEEDS project. Each agent is assigned
a representation id, a performance id and the number of neighbors for the agent.

The search parameters file is input on the HEEDS command line. This is done in
the sixth step, ‘Define the Search Parameters’. The seventh step is, ‘ Execute the
Optimization Run’ in which HEEDS executes the problem. HEEDS terminates after
evaluating the specified number of cycles. Upon termination, the designs found in the
HEEDS00.dat and HEEDSOO.plot files can be recovered, and reevaluated through post-
processing. This is done in the last eighth step ‘Post—Process the Solution(s)’.

HEEDS gets the strain components information as output from LS-Dyna and
calculates the principal strains, compares calculated principal strains with the principal
strains of the forming limit diagram for the tube material Aluminum 6061 with T6. After

satisfying the forming limit diagram, maximize the tube expansion within a square die

56

without rupture and then find the optimal value for the design variables i.e. pressure
versus time and feed versus time in step wise linear fashion. At the end of generation 6
and evaluation 224, the best design is found for the process parameters. Table (3.5)
shows the optimized pressure and feed rate data and Figure (3.16) illustrates the
respective profiles for the process parameters. Figure (3.17) represents the maximum

expansion of the tube in a square die without failure, after optimization.

Table (3.5) Internal Pressure and Feed versus Time
(Generation=6, Evaluation=224)

 

 

 

 

 

 

 

 

 

Time (sec.) Feed (mm) Internal Pressure (Mpa)
0.000 0.00 0.00
0.005 4.75 9.00
0.010 4.75 10.13
0.015 6.50 12.40
0.020 6.50 21.08
0.025 6.50 22.09

 

 

 

57

Pr-Iun IIPd I FIII (mm)

 

0 BIDS 0 01 0.015 0.02 0.025 BIB
Tim. ha

Figure (3.16) Internal Pressure and Feed versus Time Relationship (Gen=6, Evl=224).

 

a. A
Cd 0"

 

 

[If]!

   

YZ Symmetry

XZ Symmetry

X III/l

 

All dimensions in mm

 

 

Figure (3.17) Maximum radius obtained for the deformed tube before failure, after
optimization by HEEDS.

58

Figure (3.18) represents one-eighth model of the forming limit diagram for the
deformed tube. Thickness plot and von Mises stress plot are shown in Figure (3.19) and
Figure (3.20) respectively. The full-scale model for the deformed tube is shown in Figure
(3.21), Figure (3.22) and Figure (3.23), which illustrate the forming limit diagram,

thickness and von Mises plot respectively.

59

Formlng leIt diagram
STEP 20 TIME = 2500027551112
BOTTOM

1 0

 

0.0. \
0.5‘ x

q \

u

0.4 '\ s \

0.2:

 

 

 

0.0-

-0.5 -0.3 -0.1

 

 

Figure (3.18) Forming Limit Diagram for the deformed tube before failure, after
optimization.

STEP 20 TIME = 250012765002
THICKNESS

   

.
km?

‘ ‘ n :91"? “
\Mé‘ffl‘.“

 
 

   
  

 

1‘ ‘
4 1i
i- lists-la.
. ‘ \tfiiflwoflkw .1' .1
“.4 ,1 1‘11".“ ,1“-
“1 ,4 -".0"-111‘.\-‘,v‘.0'. '
' y ‘10», 11‘ p- ‘4‘“ I“.

.
”Mi-r a u 1
“01,110 ..0- .1110

in m 5““!

\“m 1| 3
«My»
‘1‘“

    
  
      
 

1.070432
1.053612
1.036793
1.019974
1.003155
0.986336
0.969516
0.952697
0.935878
0.919059
0.902239

Figure (3.19) Thickness plot for the deformed tube before failure, after optimization.

61

 

STEP 20 TIME = 250002765002
MAX_VONMISES

   

325.475577
308.759450 V
291 .043243 '
273.327025
255510825
237.894508

   
      
  
     

 

4 .
«'3‘ .0‘ 'o“ o

“s. 7
'5’: ' ‘5 . 220.170001 ‘
“‘:““¢:‘i"“lfl“i“‘i‘ 202.452173
0 ‘ “31'3“?“ 15.1.31,
a”, 11.1.1.1:- 184.745955 ,
157.029739
149.313522

 

Figure (3.20) Von mises stress plot for the deformed tube before failure, after
optimization.

62

Forming Limit Diagram
STEP 20 TIME = 250002755002

BOTTOM
1.0 ..

 

0.8
0.5
0.4

 

 

 

 

Figure (3.21) FLD plot for the deformed tube before failure, after optimization (full-scale
model).

63

STEP 20 TIME = 250002765002
THICKNESS

 

1.070432 I
1.053512 =22.

1.035793 3'
1.019974

1.003155 .24
0.985335
0.959515
0.952597
0.935078
0.919059
0.902239

kits! :1 .
r_. 54 2 .
“Ill-I ”:01.

12
8

‘3

t; ;::':-‘

Figure (3.22) Thickness plot for the deformed tube before failure, after optimization (full-
scale model).

STEP 20 TIME = 250002765002
MAX_VONMISES

 

325.475577

308.759450 -
291.043243 i
273.327025

255.510825 ,.
237.894508 4
220.178391 ffi
202.452173 E]
184.745955 ,2.
157.029739 4
149.313522 U

Figure (3.23) Von Mises stress plot for the deformed tube before failure after
optimization (full-scale model).

65

CHAPTER 4

SUMMARY AND CONCLUSIONS

This thesis presents an approach to optimize the tube hydroforming process and
consists of three major parts. A broad range of optimization topics was first reviewed
which included past research involving a broad class in the optimization of engineering
systems. The second section presented the tube hydroforming process and related theories
in detail. The third section contained the simulation and optimization of tube
hydroforming process. In the same section results before and after using HEEDS were
discussed and experimental validation was presented.

Hydroforming is an emerging technology that meshes well with the automotive
industry’s drive to make more efficient material use. The application of this concept to
structural parts leads to tubular box sections replacing stamped assemblies. Successful
tube hydroforming process requires proper combination of part design, material selection,
and application of internal pressure and axial feeding. By using finite element analysis
methods, process parameters can be determined by manual optimization before
manufacturing the dies and process development. The use of automated design
optimization software to optimize the process parameters in a hydroformed part has never
been attempted to date. This is the goal of the present work.

By using the optimal value of process parameters the tube expansion within the
square die without failure is about 64% more as compared to manual optimization. That
much expansion without failure was not possible with manual optimization of process

parameters for the tube made of Aluminum 6061 with T6 condition that has very low

66

value of strain hardening exponent. Only 8-10% reduction in thickness was noticed as

shown in thickness plot in Figure (3.22).

67

REFERENCES

68

10.

11.

LIST OF REFERENCES

Bemd Viehweger, Esslingen/Germany, “ Hydroforming applications in industry:
Qualified equipment ensures economic use”, conference, Innovation in
Hydrofroming Technology, 1996, Nashville, Tennessee.

Keith Brewster, Kevin Sutter, Mustafa Ahmetoglu, and Taylan Altan, “A Review
of Current Processes and Technology”, conference, Innovation in Hydrofroming
Technology, 1996, Nashville, Tennessee.

Murray Mason P. Eng, “Tube Hydroforming: Advancements Using Sequenced
Forming Pressures,” conference, Innovation in Hydrofroming Technology, 1996,
Nashville, Tennessee.

F. Dohmann, Ch. Hartl, “Tube hydroforming — research and practical
application”, Journal of Materials Processing Technology 7] (1997) 174-186.

F. Dohmann, P. Bieeling, Werkzeugparameter und ProzeBdaten beim
aufweitenden Innenhochdruckumformen, Umformtechnik 26 (1) (1992) 23-31.

Olhoff, N., Taylor. J.E., 1983, “On Structural Optimization”, Journal of applied
Mechanics, Vol. 50, pp. 1139-1151.

Stadler, W., and Dauer, J., 1992, “Multicriteria Optimization in Engineering: A
Tutorial and Survey”, Structural Optimization: Status and Promise, Progress in
Astronautics and Aeronautics, vol. 150, AIAA Inc., pp. 209-249.

A. Béhm, Numerische Simulation von Verfahren der Innenhochdruckum-

formung unter besonderer Beriicksichtigung des Aufweitens im geschlossenen
Gesenk. Dissertation, UniversitatGH Paderbom, 1993.

F. Klaas, Aufweitstauchen von Rohren durch Innenhochdruckumformen.
Fortschrittsbericht VDI, Reihe 2: Fertigungstechnik Nr. 42, Dusseldorf, and VDI-
Verlag, 1987.

W. J. Sauer et al., Free bulge forming of tubes under internal pressure and axial
compression, in: 6‘h North American Metal working Research Conference
Proceedings, SME, Conference University of Florida, Gainsville 16-19 April
1978, pp. 228-235.

F. Dohmann, Ch. Hartl, Liquid-bulge-forming as a flexible production method, J.
Mater. Process. Technol. 45 (1994) 37700382.

69

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

F. Klass, U. Lficke, and K. Kaehler, “Development of the Internal High-Pressure
Forming Process,” SAE, International Congress and Exposition, Detroit,
Michigan, 1993.

H. Amino, K. Nakamura, and T. Nakagawa, “Counter-Pressure Deep drawing and
Its Application in the Forming of Automobile Parts,” Journal of Materials
Processing Technology, volume 23 (1990), pp. 243-265.

Ingvar Rask and Dag Thuvesen, “Simulation of an engineering process — tube
hydroforming” IVF — The Swedish Institute of Production Engineering Research.

Hosford, W. F. and Caddell, R. M., Metal Forming - Mechanics and Metallurgy,
1983, Prentice-Hall, Inc.

Brénnberg N., “Computational Aspect on Simulation of Sheet Metal forming”,
Department of Mechanical engineering, Lin6kping Institute of Technology,
Sweden, 1994.

Hallquist J.O., LS-DYNA Theoretical Manual, Livermore Software technology
Corporation, Livermore, 1998.

Robert G. Whirley and Bruce E. Engelmann and Robert W. Logan, “Some
Aspects of Sheet Forming Simulation using Explicit Finite Element Techniques”,
CED-Vol. SIAMD Vol. 156, Numerical Methods for Simulation of Industrial
Metal Forming Processes ASME 1992. '

HEEDS Getting Started Manual by Applied Computational Design Associates,
Inc. MI, 48864, USA.

General motors material testing laboratory, Raj k Mishra, raj.k.mishrla@gm.com

 

I-DEAS Master Series, Theoretical Manual, Structural Dynamics Research
Corporation, Milford, Ohio 45150.

Dynaform—PC, Theoretical Manual, Engineering Technology Associates, Inc.
Troy, MI 48043.

VARI-FORM, “The ABSs of Hydroforming”, Technical Journal, Chesterfield, MI
4805 1 .

F. —U. Leitloff and W. Schafer, “Hydroforming Applications in the Automotive
Industry”, conference, innovations in Hydroforming Technology, Tennessee,

1996.

IT C Interlaken Technology Corporation, “Hydroforming Press Brochure”, Eden
Prairie, MN 55344 USA.

70