Malde Thesis

8/18/2019 Malde Thesis

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THERMOMECHANICAL MODELING AND OPTIMIZATION OFFRICTION STIR WELDING

A Thesis

Submitted to the Graduate Faculty of the

Louisiana State University andAgricultural and Mechanical College

in partial fulfillment of therequirements for the degree of

Master of Science in ndustrial !ngineering

in

The "epartment of Construction Management and ndustrial !ngineering

byManthan Malde

#$!$% &smania University% 'yderabad% ndia% ())*"ecember ())+


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ii

Acknowledgments

,ould li-e to e.press my sincere gratitude to "r$ T$ /arren Liao% my advisor% for his

invaluable guidance encouragement and support e.tended throughout the study$

am than-ful to the committee members% "r$ Muhammad /ahab and "r$ 0ius 1$ !gbelu for

giving their valuable time to serve in the e.amination committee and for their comments and

inputs in my ,or-$

,ould li-e to than- 2inay 3aghuram% 4en 4iong% Shivani "aftardar and 5ranthi 5umar

Charlapally for their valuable discussions on the topic$

This is a great opportunity for me to e.press my gratitude to my friends$ ,ould li-e to than- my

friends at LSU for their constant help and for ma-ing my stay at LSU memorable and en6oyable$

Than- you 5alyana% 3aghava% 3avi% 2arun% Sampath% Sameer% 0hani% Shilpa% 'emalatha%

Srila-shmi% Anuradha and Amit$ shall al,ays remember the good times had ,ith you all$

am also pleased to than- my friends 2en-at% Shailey% 5rishna and Sriram for their continuous

support and inspiration$

,ould li-e to than- my parents for their unconditional love% support and encouragement

throughout my life$ Last but certainly not the least% ,ould li-e to than- my brother and sister7

in7la, for their continuing concern and advice$ /ords cannot e.press ho, than-ful am to them

for everything$


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8

T !le o" Contents

Ac-no,ledgments$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

List of Tables $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

List of Figures $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

vi Abstract $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

vii

9$ ntroduction$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$9$9 #ac-ground $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$9$( Advantages and "isadvantages $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$9$8 3esearch &b6ective$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

($ Literature 3evie,$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$($9 &n /elding 3esidual Stress$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

($9$9 3esidual Stress Measurement $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$($( &n Modeling of Friction Stir /elding 0rocess$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

($($9 Thermal Modeling $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$($($( Thermomechanical Modeling $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

($8 &n &ptimi=ation of the 0rocess $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$($8$9 Use of Surrogate Models $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

8$ Methodology &vervie, $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

:$ Thermomechanical Model of FS/$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$:$9 Model "evelopment of Friction Stir /elding for 8):L Stainless Steel $$$$$$$$$$$$$$$$$$$$$$$$ 9+:$( Thermal Model $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

:$($9 Assumptions$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$:$($( Geometry$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$:$($8 !lements Used $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$:$($: Mesh "evelopment $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$:$($> Material 0roperties $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$:$($* #oundary Condition $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$:$($; 'eat Flu. nput $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

:$8 Mechanical Model $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$:$8$9 Assumptions$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$:$8$( !lements Used and Mesh "evelopment $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$:$8$8 0lasticity Model $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$:$8$: #oundary Conditions $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

:$: Simulation $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

>$ 2alidation of Thermomechanical Model of Friction Stir /elding $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$


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:

>$9$9 Temperature 3esponses $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$>$9$( Stress 3esponses $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$


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*$ 0arametric Study and Surrogate Models of FS/ 0rocess $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$*$9 "esign of !.periments $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

*$9$9 !ffect of Factors on Temperature "istribution and 3esidual Stress$$$$$$$$$$$$$$$$$$$$$$$ 8>*$( Surrogate Models of Friction Stir /elding$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

*$($9 "evelopment of Model for 3esponse ? Temperature $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

*$($( "evelopment of Model for 3esponse ? 3esidual Stress$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$*$8 !stimation of 0erformance of "eveloped Surrogate Models$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

;$ "etermining &ptimal FS/ 0arameters Using mproved 'armony Search Algorithm $$$$$$$ :(;$9 Formulation of &ptimi=ation 0roblem $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$;$( Solution Methodology Using 'armony Search Algorithm $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

;$($9 mproved 'armony Search Algorithm $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$;$($( 0seudo Code$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

;$8 &ptimi=ation 3esults for FS/ 0rocess $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$;$8$9 3esults for Model 9 and Model ( $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$;$8$( 3esults for Model 8 and Model : $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$


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L#st o" T !les

Table :$9 Thermal material properties of 8):L stainless steel $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Table *$9 0rocess parameters% range and design levels used $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Table *$( 3egression statistics of linear and nonlinear surrogate models $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Table ;$9 Comparison bet,een optimi=ation and musical performance B>+ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Table ;$( 0arameters used for 'SD in this study $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Table ;$8 &ptimi=ation results of Model 9 and Model ( ,ith T L# E 9))) and T U# E 98)) $$$$$$$$$$>)

Table ;$: &ptimi=ation results of Model 9 and Model ( ,ith T L# E 9)>) and T U# E 99>) $$$$$$$$$$>9

Table ;$> &ptimi=ation results of Model 9 and Model ( ,ith T L# E 99:) and T U# E 99>) $$$$$$$$$$>(

Table ;$* &ptimi=ation results of Model 8 and Model : ,ith constraints T L# E 9)))% T U# E98)) and 3 U# E 89) $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Table ;$; &ptimi=ation results of Model 8 and Model : ,ith constraints T L# E 9)>)% TU# E99>) and 3 U# E 89) $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Table


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L#st o" F#g$%es

Figure 9$9 Friction stir ,elding operation principle B: $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Figure 8$9 Methodology of model7based optimi=ation of FS/ process $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Figure :$9 Three dimensional thermal solid element S&L ";) B:+ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Figure :$( Three dimensional surface effect element SU3F9>( B:+ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Figure :$8 Schematic representation of boundary condition for thermal analysis $$$$$$$$$$$$$$$$$$$$$$$$(:

Figure :$: Temperature dependent mechanical properties of 8):L stainless steel B(< $$$$$$$$$$$$$$$$(;

Figure :$> #ilinear isotropic stress7strain model for 8):L stainless steel $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Figure :$* Flo,chart of sequentially coupled thermomechanical analysis $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Figure >$9 Comparison of temperature distribution along the transverse direction at ,eldingtime tE $( Temperature distribution on top surface of the ,or-piece at ,elding time% tE >)$:sec $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Figure >$8 2ariation of transient temperature 7 comparison of simulated results and resultsfrom hu and Chao s Model $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Figure >$: 2ariation of the longitudinal residual stress along the traverse direction at themiddle section of the ,or-piece$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Figure *$9 0lot of main effects for temperature $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Figure *$( 0lot of main effects for residual stress $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Figure *$8 2ariation of temperature on top surface of the ,or-piece at different ,eldingspeeds $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Figure ($:% HE)% E) for optimal parameters 'E : mmIs $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Figure ($:% HE) m E) for optimal parameters 'E ;;($+; /and SE ($89( mmIs $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Figure


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A!st% ct

This thesis research implemented an e.isting thermomechanical model of friction stir ,elding

process% and studied the surrogate model7based optimi=ation approach to obtain optimal process

parameters for the modeled friction stir ,elding process$ As an initial step% the

thermomechanical model developed by hu and Chao for friction stir ,elding of 8):L stainless

steel ,as replicated using A SHS$ The developed model ,as then used to conduct parametric

studies to understand the effect of various input parameters li-e total rate of heat input% ,elding

speed and clamping location on temperature distribution and residual stress in the ,or-piece$

/ith the data from the simulated model% linear and nonlinear surrogate models ,ere constructed

using regression analysis to relate the selected input process parameters ,ith response variables$

Constrained optimi=ation models ,ere formulated using surrogate models and optimi=ation of

process parameters for minimi=ing cost and ma.imi=ing throughput ,as carried out using

improved harmony search algorithm$ To handle the constraints% "eb s parameter7less penalty

method ,as used and implemented in the algorithm$

t is learned from this research that@ J9K heat input is mainly constrained by the lo,er bound of

the temperature for ma-ing good ,elds J(K the optimal ,elding speed must balance the loss of

heat input and the gain in productivity J8K clamping closer to the ,eld is better than a,ay from

the ,eld in terms of lo,ering the pea- residual stresses$ Moreover% the nonlinear surrogate

models resulted in a slightly better optimal solution than the linear models ,hen ,ide

temperature range ,as used$ 'o,ever% for tight temperature constraints% optimi=ation on linear

surrogate models produced better results$ The implemented improved harmony search algorithm

seems not able to converge to the best solution in every run$ evertheless% the non7converged

solution it found ,as very close to the best$

vii


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9

&' Int%od$ct#on

&'& ( ckg%o$nd

Friction Stir /elding JFS/K is a revolutionary solid state ,elding technique invented at The

/elding nstitute JT/ K in 9++9 B9 $ The FS/ process operates belo, the solidus temperature

of the metals being 6oined and hence no melting ta-es place during the process$ This process is a

derivative of the conventional friction ,elding and is being used to produce continuous ,elded

seams for plate fabrication B( $ Since its invention in 9++9% continuous attempts have been made

by researchers to understand% use and improve this process$

Friction Stir /elding is a hot7shear 6oining process in ,hich a non7consumable% rotating tool plunges into a rigidly clamped ,or-piece and moves along the 6oint to be ,elded B8 $ The

cylindrical rotating tool used in FS/ has a profiled threaded or unthreaded probe of length less

than the ,eld depth% e.truding from the tool shoulder$ The operating principle of FS/ process is

presented in figure 9$9$

Figure 9$9 Friction stir ,elding operation principle B:


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The FS/ process is initiated by plunging of a rotating tool into the 6oint until the shoulder

contacts the top surface of the ,or-piece$ As the tool translates along the 6oint% heat is generated

by rubbing action of tool shoulder against the ,or-piece$ Additional heat is generated by visco7

plastic dissipation of mechanical energy at high strain rates due to interactions bet,een tool and

,or-piece B> $ The heat thus generated results in thermal softening of the material$ The thermally

softened material is contained at the underside by a bac-ing plate% at the sides by non7softened

parent material% and at the topside by pin force$ The softened material is then forced to flo, by

the translation of the tool from the front to the bac- of the pin ,here it cools% consolidates and

results in 6oint formation B* $FS/ process requires a tool of harder material than the ,or-piece material being ,elded B( $

0reviously% FS/ ,as used for soft ,or-piece materials li-e aluminum alloys% lead% =inc% and

magnesium$ 'o,ever% ,ith the development of tools made from refractory material li-e tungsten

and superabrasive materials li-e polycrystalline diamond J0C"K and polycrystalline cubic boron

nitride J0C# K% FS/ of high temperature materials ,as made possible B; $ As FS/ process is a

solid state process% it requires lo, heat input and it results in lo, distortion% no

macrosegregation% and a finely recrystallised microstructure$ For these reasons% FS/ has been

investigated for ,ide range of materials including high melting temperature materials such as

austenitic stainless steels B< $

The feasibility of FS/ for high melting temperature materials have been studied and reported$

Studies have sho,n the feasibility of FS/ in several steels and have reported that the

mechanical properties of friction stir ,elds are comparable to those of base material B


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&') Ad* nt ges nd D#s d* nt ges

The FS/ process has demonstrated a number of advantages over the conventional ,elding

process$ Some of the advantages of this thermomechanically energy efficient process are B( @

9$ The process temperatures in FS/ are much lo,er than the fusion techniques$ This results

in avoiding problems ,hich occur ,ith liquid phase% such as alloy segregation% porosity

and crac-ing$

($ The process can be easily automated as it is machine tool technology based$

8$ 'igh integrity similar and dissimilar ,elded 6oints are produced for an increasing range

of materials ? aluminum% =inc% lead% copper% magnesium% titanium and steel$

:$ 3eduction in production costs in further processing and finishing is possible as the

surface appearance of FS/ approaches to that of a rough machined surfaces$

>$ o filler material or shielding gas is required$

*$ The process produces lo,er levels of distortion in the ,or-piece compared to fusion

,elding$

;$ The FS/ process can be carried out in all positions ? vertical and overhead$ The process

can also be operated under,ater$

L479))% and corrosion resistant alloys such as A S 89*L and 8):L B98 $ The

FS/ process offers advantages in terms of productivity and cost$ Compared to conventional


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:

fusion ,elding processes such as arc and laser beam% FS/ is highly energy efficient and the

estimated reduction in energy usage is by *) to


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parameters% the ,elding process can be considered as a multi7input% multi7output process$ Thus

appropriate combinations of ,eld parameters have to be chosen to produce high quality ,elds

,ith minimum detrimental residual stresses and distortions B9* $ This thesis research focuses on

investigation of input parameters that control the formation of residual stresses in 8):L stainless

steel friction stir ,elds and on model7based optimi=ation of the process$

The main ob6ectives of this thesis are JiK to develop and validate a three dimensional

thermomechanical model of FS/ process and to predict the developed residual stresses% JiiK to

study the effects of various process parameters on ,eld temperature history and residual stresses

using the developed model% and JiiiK to optimi=e FS/ process ,ith model7based approach usinga traditional nonlinear optimi=ation procedure and improved 'armony Search Algorithm$

The rest of the thesis is organi=ed as follo,s@ Chapter ( revie,s related ,or-s on modeling and

optimi=ation of FS/ process$ n chapter 8 the methodology used for achieving the set ob6ectives

is described$ Chapter : outlines the computational approach used in the development of

thermomechanical model of FS/ process$ Chapter > deals ,ith validation of the developed

thermomechancial model$ Chapter * presents the design of e.periments and results from

parametric studies of the developed model$ Chapter * also discusses the development of

surrogate models for the t,o chosen responses% temperature and residual stress$ Chapter ;

presents the formulation of optimi=ation models and its solution using improved harmony search

algorithm$ 2alidation of optimi=ation results are presented in chapter


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)' L#te% t$%e Re*#ew

This section has been divided into three parts$ The first part outlines the techniques used for

measurement of residual stresses% the second part revie,s ,or-s related to thermomechanical

modeling of FS/ process% and the third part revie,s ,or-s related to optimi=ation of FS/

process$

)'& On Weld#ng Res#d$ l St%ess

/elding cycle often results in formation of residual stresses$ The residual stresses are the loc-ed7

in stresses left out in the ,or-piece after the ,elding process is completed$ The locali=ed heating

and non7uniform cooling during ,elding% results in a comple. distribution of the residual stresses

in the 6oint region along ,ith undesirable deformation or distortion of the ,elded structure B9; $

3esidual stress can be beneficial or harmful depending on its compressive or tensile nature$

Tensile residual stresses can cause crac- initiation B8 % reduce the performance or cause failure of

manufactured product B9< $ These tensile stresses may also increase the rate of damage by

fatigue% creep or environmental degradation$ &n the other hand% compressive stress can lead to

performance benefits B9+ $

)'&'& Res#d$ l St%ess Me s$%ement

!stimation of residual stresses is usually done using measurement techniques ? destructive and

non7destructive techniques B8% 9+ $

"estructive technique involve partial destruction such as drilling a hole% sectioning a layer etc$

and using speciali=ed strain gauge rosettes to measure strain relief in the material$ Some of the

common destructive methods include@

9$ 'ole7drilling method


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($ 3ing core technique

8$ #ending deflection method

:$ Sectioning method$

&n the other hand% in non7destructive techniques% measurement is carried out ,ithout destroying

the ,eld$ These techniques provide more accurate results than destructive techniques$ The most

commonly used techniques for non7destructive measurement include@

9$ 47rayI neutronI synchrotron diffraction

($ Ultrasonic technique

8$ Magnetic methods$

The diffraction techniques are based on using lattice spacing as strain gauge$ Ultrasonic

technique uses the variation of ultrasonic ,ave propagation in materials under the action of

mechanical stress% ,hile the magnetic methods rely on the interactions bet,een magneti=ation

and elastic strain in ferro7magnetic materials B9+ $

n recent years% ,ith the development of po,erful computing facilities% finite element analysis

methods have been applied to model the ,elding process and to estimate residual stresses$ Some

of the attempts to model FS/ process and estimate residual stresses are described in the

follo,ing section$

)') On Model#ng o" F%#ct#on St#% Weld#ng P%ocess

Friction Stir /elding ,as invented and e.perimented at The /elding nstitute% U5 in 9++9$

Since then% several e.perimental methods% numericalIanalytical and finite element methods have

been developed and studied by many researchers to understand the thermal and

thermomechanical interactions ta-ing place during FS/$ "espite significant advances in the

FS/ process% the comple. thermomechanical interactions ta-ing place have not been fully


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<

understood$ n order to predict the residual stress developed during friction stir ,elding%

thermomechanical models are studied$ n most cases% decoupled analysis ,as used to estimate

the residual stresses$ n a decoupled analysis% first pure thermal problem is solved and then the

calculated temperature fields are used as input to the mechanical models$

)')'& T,e%m l Model#ng

Understanding the heat generation and the temperature history during the FS/ process is the

first step to,ards understanding the thermomechanical interaction ta-ing place during the

,elding process$ The initial modeling approaches focused on appro.imate estimation of heat

generated during the FS/ process$ Gould and Feng B() developed a preliminary thermal model

to predict the temperatures of friction stir ,elds using the 3osenthal equations to describe a

moving heat source$ The heat input ,as described as a function of process parameters such as

tool rpm and force on tool$

Chao% Ni and Tang B(9 formulated a boundary value problem for tool and ,or-piece in order to

study the heat transfer in friction stir ,elding$ They determined the frictional heat flu. from the

measured transient temperature fields obtained in the finite element analyses$ n an attempt to

predict the flo, of material around the tool% Colegrove et al. B(( presented a finite element

based thermal model of FS/$ Their model included the bac-ing plate and the tool$ n their ,or-%

the heat input ,as fitted through iterative process for verification bet,een the modeled and

e.perimental values$

An input torque based thermal model for prediction of temperature in friction stir ,elds of Al7

*)*97T* alloy ,as developed by 5hand-ar et al B(8 $ n their model% the heat generated by tool

rotation and linear traverse of shoulder and pin% has been correlated ,ith actual machine po,er

input$ This estimated heat ,as applied as a moving heat to obtain the temperature distribution


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across the ,eld$

The above mentioned models did not include the tool penetration and pulling out phase$ Song

and 5ovacevic B(: proposed a coupled heat transfer model of both the tool and the ,or-piece

for FS/ to include the tool penetration and pulling out phase$ A moving coordinate ,as adopted

to reduce the difficulty of modeling the heat generation due to the movement of the tool pin$ The

finite difference method ,as used for solving the control equations and the results obtained ,ere

in good agreement ,ith the e.perimental results$

2ilaca et al $ B(> developed an analytical thermal model for simulation of friction stir ,elding

process$ The model included simulation of the asymmetric heat field under the tool shoulder

resulting from viscous and interfacial friction dissipation$ The analytical model also considered

the influence of hot and cold FS/ conditions into the heat flo, around the tool$

The focus of all the thermal models ,as to understand the process of heat generation and to

predict the temperature distribution in the ,or-piece and tool$ A thermal model forms the basis

for the development of mechanical and microstructural models$

)')') T,e%momec, n#c l Model#ng

n order to estimate residual stress and distortions in ,or-piece resulting from ,elding process%

thermomechanical models ,ere developed and studied$ &ne of the first thermomechanical

models for FS/ ,as studied by Chao and Ni B(* $ A decoupled heat transfer and a subsequent

thermomechanical analysis for Al *)*97T* ,as used in their study$ 'eat generated from friction

bet,een tool shoulder and ,or-piece ,as implemented as the heat input$ The empirical equation

for calculating the heat input to the ,or-piece is given by equation J($9K$

q r =3Qr33

for ri ≤ r ≤ r oJ($9K

(OJro P ri K


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,here q r is the rate of heat input% and are the radii of the shoulder and the nib of the pin�� tool% and� is the total rate of heat input to the ,or-piece e.pressed as sho,n in equation J($(K$

22

� =

�� (�� + �� + �� )

45(�� + �� )

J($(K

,here% is the tool rotational speed% Q is the frictional coefficient% and Q is the do,n,ard force$�The total heat input and heat transfer coefficient ,ere estimated by fitting the measured

temperature data ,ith the analytical model by a trial and error approach$ The temperatures thus

obtained from the analysis ,ere used to determine the residual stress retained in the friction stir

,elds$ The ma.imum residual stresses ,ere reported to be 8) of the yield strength of the

material$

Chen and 5ovacevic B(; proposed a three dimensional finite element analysis model to study

the thermal history and thermomechanical process in butt ,elding of aluminum alloy *)*97T*$

The model incorporated the mechanical reaction of the tool and thermomechanical processes of

the ,elded material$ The friction bet,een the material% the probe and the shoulder ,as included

in the heat source$ 47ray diffraction technique ,as used to measure the residual stresses

developed in the plate and the measured results ,ere used to validate the efficiency of the

proposed model$ From the study% it ,as reported that fi.turing release to the ,elded plates

affected the stress distribution of the ,eld$

hu and Chao B(< presented three7dimensional nonlinear thermal and thermomechanical

simulations using finite element analysis code ?/!L"S M on 8):L stainless steel friction stir

,elded plates$ nitially% a heat transfer problem ,as formulated as a standard boundary value

problem and ,as solved using the inverse analysis approach$ The total heat input and heat

transfer coefficient ,ere estimated by fitting the measured temperature data ,ith the analytical

model$ Later% the transient temperature outputs from the first stage ,ere used to determine


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residual stresses in the ,elded plates using a three7dimensional elastic plastic thermomechanical

model$ Convection and radiation ,ere assumed to be responsible for heat loss to the ambient on

the surface$ Their model provided good match bet,een e.perimental and predicted results$ They

reported that the residual stress in the ,elds after fi.ture release decreased significantly as

compared to those before fi.ture release$ They also reported that about >) of the total

mechanical energy developed by FS/ machine ,as utili=ed in raising the temperature of the

,or-piece$

Soundarara6an et al. B(+ developed a finite element thermomechanical model ,ith mechanical

tool loading considering a uniform value for contact conductance and used for predicting the

stress at ,or-piece and bac-ing plate interface$ The non7uniform contact conductance ,ere

defined from pressure distribution contours and used in predicting the temperatures in the

thermal model$ The thermomechanical model ,as then used in predicting the developed stresses$

5hand-ar et al $ B8) developed coupled finite element models to predict residual stress in AA7

()(:% AA7*)*9 and SS 8):L friction stir ,elds$ n their models% the temperature history

predicted by the thermal model ,as sequentially coupled to a mechanical model to assess the

residual thermal stresses developed during the ,elding$ t ,as found that clamping constraints

and their locations had significant locali=ed effects on the stress components in the unaffected

base metal beyond the heat7affected =one$

Feng et al. B89 presented a more detailed thermal7metallurgical7mechanical model to study the

microstructure changes and their effects on residual stress distribution in friction stir ,eld of Al*)*97T*$ n their approach% the first stage involved a transient nonlinear heat flo, analysis to

determine the temperature distribution$ The frictional heating in the thin layer near the interface

,as treated as a surface heat generation term% %�

,hich ,as estimated by the equation J($8K$ 2ημFωr for Rpin ≤ r ≤ R shJ($8K60(R sh − R pin

)


20/87

,here Q is the do,n,ard force% is the rotational speed% Q is the process efficiency% Q is the�

interpretive coefficient of friction% and 3pin and 3sh the radii of the pin and the shoulder respectively$ n the second stage% using the temperature history from the thermal model as input%

the metallurgical calculations ,ere performed in the mechanical analysis as a part of material

constitutive definition subroutine$ t ,as reported that residual stresses had strong dependence on

the ,elding speed$

Li et al $ B8( presented a semicoupled thermomechanical finite element model containing both

thermal load and mechanical load$ Their model included an autoadapting heat source in the

thermal model and fi.tures ,ere included in the mechanical model$ They reported that in the

case of ()(:7T* alloy% stresses at the retreating side of the ,eld ,ere smaller than those at the

advancing side$

#astier et al. B88 used computational fluid dynamics pac-age to estimate the material flo, and

temperature field in ;)>) aluminum alloy$ They used the results to estimate residual state

induced in friction stir ,elding process based on elasto7viscoplastic constitutive la,$ They also

reported from the parametric study that the ,elding speed and rotational speed had influence on

the level of residual stresses and distortions developed during ,elding$

Some researchers conducted e.perimental studies to investigate the effect of process parameters

on the residual stresses$ 0eel et al $ B8: investigated the microstructure% mechanical properties

and residual stress as a function of ,elding speed for AA>)


21/87

98

decreases% ,hile the pea- longitudinal stress increases$

Staron et al $ B8> conducted e.perimental study on residual stress states in FS/ 6oints in *$8 and

8$( mm thic- AA()(: sheets that had been ,elded under mechanical tensioning$ They ,ere

successful in reducing the tensile residual stress in the ,eld =one by induction of large

compressive stresses through mechanical tensioning$

"attoma et al $ B8* evaluated the residual stress fields in similar and dissimilar 6oints in ()(:7T8

and *)* 6oints made by FS/$ The influence of process parameters on the

,eld quality ,as assessed by Analysis of 2ariance JA &2AK methods using the e.perimental

results$ A complete t,o factor factorial e.periment% ,ith three replicates ,as performed by the

authors$


22/87

9:

Meng et al $ B8< used a multi7targeted optimi=ation ,ith constraint based on genetic algorithm

for optimi=ation of stir head dimensions$ The ob6ective function employed ,as an analytically

derived mathematical model relating heat input coefficient ,ith tool parameters$ The goal of

optimi=ation ,as to determine the shoulder diameter and pin diameter of the stirring tool for

ma.imi=ing the tensile strength of the friction stir ,elds of aluminum7lithium alloy$

n addition to design of e.periment techniques% some evolutionary algorithms ,ere utili=ed for

optimi=ation of FS/$ Fratini and Corona B8+ investigated FS/ lap 6oint resistance optimi=ation

using gradient techniques$ They combined the gradient technique and the finite difference

method to determine the optimal rotating speed and ,elding speed in order to ma.imi=e the 6ointstrength per unit length$

andan et al. B:) used genetic algorithm to determine four process parameters by minimi=ing

the difference bet,een the numerical model and e.periments$ The process parameters included

variable friction coefficient% the e.tent of stic-ing% the heat transfer coefficient% and the e.tent of

viscous dissipation converted into heat$ These selected parameters ,ere optimi=ed by a genetic

algorithm using a limited volume of measured temperatures at several monitoring locations

during FS/ of dissimilar aluminum alloys AA 9()) and AA *)*9$

Use of Artificial neural net,or- JA K ,as proposed by &-uyucu et al $ B:9 to obtain

correlation bet,een FS/ parameters and mechanical properties of aluminum plates$ Their

attempt ,as to correlate the parameters rather than to optimi=e them$ The input parameters ,ere

,eld speed and tool rotational speed ,hile the output parameters included mechanical propertiessuch as tensile strength% elongation% hardness of ,eld metal and hardness of heat affected =one$

The obtained model ,as used to calculate mechanical properties of ,elded Al plates as a

function of ,eld speed and rotational speed$


23/87

Conventional parametric design of e.perimental approach is cumbersome and requires large

number of e.perimental trials$ Statistical techniques are often used to reduce the number of

e.periments conducted$ La-shminarayanan et al. B:( used one such statistical technique -no,n

as Taguchi technique to determine the effect of three process parameters% i$e$ tool rotational

speed% traverse speed% and a.ial force on the tensile strength of friction stir ,elded 3"!7:)

aluminum alloy$ 1ayaraman et al $ B:8 used a similar technique to find the effect of three process

parameters on the tensile strength of friction stir ,elded A89+ aluminum alloy$ n both these

studies% the authors performed Analysis of 2ariance JA &2AK to identify statistically significant

process parameters$

)'+'& 0se o" S$%%og te Models

Some techniques such as surrogate model or meta7model based optimi=ation have been used in

optimi=ation$ The benefit of using surrogates in optimi=ation is that a fast appro.imate model

instead of a computationally e.pensive model can be used to speed up the optimi=ation process

B:: $

!langovan et al $ B:> developed a mathematical model using response surface method J3SMK to

develop relationship bet,een four process parameters and tensile strength for AA*)*9$ The

process parameters included tool rotational speed% ,elding speed% a.ial force and the tool pin

profile$ A similar study ,as carried out by #abu et al $ B:* but on a different aluminum alloy

AA((9+$ #oth the studies used 'oo-e and 1eeves search algorithm to achieve ma.imum tensile

strength$ #oth the studies reported close match bet,een the optimi=ed values and the

e.perimentally determined values$

More recently% Liao and "aftardar B:; proposed a model7based approach for optimi=ation of

FS/ process for AA(9+>7T


24/87

relate three process parameters such as heat input% ,elding speed and shoulder diameter ,ith

ma.imum temperature at selected location$ Further% a constrained optimi=ation model ,as

formulated and solved using five population7based metaheuristics to find the optimal solutions$

The performance of different metaheuristics ,as evaluated and it ,as reported that differential

evolution technique had the best performance$


25/87

+' Met,odolog1 O*e%*#ew

To accomplish the research ob6ectives set forth for this study% a methodology ,as developed$

The methodology ,as essentially a model7based approach for optimi=ation of FS/ process$ The

first tas- ,as to develop and validate a thermomechanical model of FS/ process in

consideration of various published papers as discussed in literature revie,$ The model chosen for

this tas- ,as the thermomechanical model developed by hu and Chao B(< for FS/ of 8):L

stainless steel$ The thermomechanical model ,as developed using commercial finite element

analysis program A SHS R 799$)$ n order to validate the developed model% the output of the

model ,as correlated ,ith the published results$ &nce developed% the thermomechanical model

,as used to simulate the process$ The model ,as then e.trapolated to perform parametric studies

in order to investigate effects of various process parameters on temperature distribution and

residual stress in the ,or-piece$

The ne.t step ,as to construct surrogate models using the data generated by the

thermomechancial model$ Linear and nonlinear surrogate models ,ere constructed to relate

process parameters ,ith responses% i$e$% temperature and residual stress measured at selected

location$ The performance of the developed surrogate models ,as estimated using several

statistical measures$ n the ne.t step% constrained optimi=ation models ,ere formulated ,ith goal

of ma.imi=ing throughput and minimi=ing manufacturing costs$ The optimi=ation models ,ere

solved using a traditional nonlinear optimi=ation procedure and a population7based

metaheuristics% improved harmony search algorithm$ Finally% the optimal results ,ere validated

by simulation using A SHS R $ Figure 8$9 presents an overall methodology of surrogate model7

based optimi=ation of friction stir ,elding process$


26/87

FSW Model7 Thermomechanical

modeling using A SHS R

Model 2 l#d t#on7 Correlation of model output

,ith results from B(< $

Des#gn o" E3.e%#ments nd . % met%#c st$d17 dentify input factors and output responses7 "etermine significant factor effects

Selected output responses7 temperature andresidual stress

Const%$ct#on o" s$%%og te models7 Linear and nonlinear models for

temperature and residual stress7 !stimate model performance using

statistical measures

Fo%m$l t#on o" o.t#m#/ t#on models7 "efine model ob6ectives7 "efine constraints

O.t#m#/ t#on to "#nd o.t#m l .%ocess. % mete%s7 Solve using mproved 'armony Search

Algorithm7 Solve using traditional nonlinear

optimi=ation

2 l#d t#on o" o.t#m l %es$lts7 2alidate solution ,ith finite element analysis

Figure 8$9 Methodology of model7based optimi=ation of FS/ process


27/87

4' T,e%momec, n#c l Model o" FSW

4'& Model De*elo.ment o" F%#ct#on St#% Weld#ng "o% +54L St #nless Steel

The Finite !lement Method JF!MK offers a ,ay to solve comple. continuum problems by

subdividing it into a series of simple interrelated problems$ F!M is most commonly used in

numerical analysis for obtaining appro.imate solutions to ,ide variety of engineering problems

B:< $ n the present study% a commercial general purpose finite element program A SHS R 99$)

,as used for numerical simulation of friction stir ,elding process$

The A SHS R program has many finite element analysis capabilities% ranging from simple%

linear% static analysis to a comple. nonlinear% transient dynamic analysis B:+ $ The thermal and

mechanical responses of the material during friction stir ,elding process are investigated by

finite element simulations$ n this study% a sequentially coupled thermomechanical model is

developed for analysis$ First% a nonlinear% transient three7dimensional heat transfer model is

developed to determine the temperature fields$ Later% the temperature fields are used as input for

a nonlinear% rate independent% three7dimensional structural model in order to predict the

distortions and the residual stresses$ The finite element models are parametrically built using

A0"L JA SHS 0arametric "esign LanguageK provided by A SHSR

B:+ $ The models are then

validated by comparing the results ,ith established numerical data$

4') T,e%m l Model

The purpose of the thermal model is to calculate the transient temperature fields developed in the,or-piece during friction stir ,elding$ n the thermal analysis% the transient temperature field �

,hich is a function of time and the spatial coordinates J % % K% is estimated by the three� � � �dimensional nonlinear heat transfer equation J:$9K$

�(�

��2

�(

�

+��

2 �( �+��2

��+ ��= ��

��


28/87

J:$9K

,here is the coefficient of thermal conductivity% is the internal heat source rate% is the� ��

mass7specific heat capacity% and is the density of the materials B() $ The heat transfer model�developed for the thermal analysis is described in the follo,ing section$

4')'& Ass$m.t#ons

A number of assumptions have been made in developing the finite element thermal model% ,hich

include@

• /or-piece material is isotropic and homogeneous$

• o melting occurs during the ,elding process$

• Thermal boundary conditions are symmetrical across the ,eld centerline$

• 'eat transfer from the ,or-piece to the clamp is negligible$

4')') Geomet%1

n the numerical model% only half of the ,elded plate is modeled as the ,eld line is the

symmetric line$ Symmetric condition is used to reduce the simulation time$ The ,or-piece has

dimensions of )$8):< m . )$9)9* m . )$))89< m$

4')'+ Elements 0sed

n the present thermal analysis% the ,or-piece is meshed using a bric- element called S&L ";)$

This element has a three7dimension thermal conduction capability and can be used for a three7

dimensional% steady7state or transient thermal analysis B:+ $ The element is defined by eight

nodes ,ith temperature as single degree of freedom at each node and by the orthotropic material

properties$ 'eat flu.es or convections Jbut not bothK can be input as surface loads at the element


29/87

faces as sho,n by the circled numbers on the element geometry in Figure :$9$ An advantage of

using this element is that% the element can be replaced by an equivalent structural element for the

structural analysis$

Figure :$9 Three dimensional thermal solid element S&L ";) B:+

As S&L ";) cannot apply heat flu. and convection at the same time% a three7dimensional

thermal7surface7effect element ,as used$ For applying convection on the ,or-piece surface%

SU3F9>( ,as used overlaying it onto faces of the base elements made by S&L ";)$ The

convections ,ere applied as a surface load by choosing 5!H&0T J


30/87

Figure :$( Three dimensional surface effect element SU3F9>( B:+

4')'6 M te%# l P%o.e%t#es

Thermal properties of the material such as thermal conductivity% specific heat% and density are

temperature dependent$ An accurate estimation of temperatures is critical in FS/ process

because the stresses and strains developed in the ,eld are temperature dependent$ Therefore%

temperature dependent thermal properties of 8):L steel are used in finite element model$

The thermal material properties of 8):L stainless steel are tabulated in Table :$9$The thermal

property values are obtained from B(9 % and for higher temperatures the values are linearly

e.trapolated$

Table :$9 Thermal material properties of 8):L stainless steel

TemperatureJ℃) ) ()) :)) *)) *8 >+8 >+8

Thermal Conductivity� � 9:$( 9*$+ ()$* (8$8 (;$< (;$<

"ensity�

;9< ;:)* ;:)*


31/87

n order to define the temperature dependent properties% combination of M0T!M0 and

M0"ATA commands ,as used$ M0T!M0 ,as used to define a series of temperatures% and later

M0"ATA ,as used to define corresponding material property values$

4')'7 (o$nd %1 Cond#t#on

#oundary condition for FS/ thermal model ,ere specified as surface loads through A SHS R

codes$ Assumptions ,ere made for various boundary conditions based on data collected from

various published research papers B(( $

Convective and radiative heat losses to the ambient occurs across all free surfaces of the

,or-piece and conduction losses occur from the ,or-piece bottom surface to the bac-ing plate$

To consider convection and radiation on all ,or-piece surfaces e.cept for the bottom% the heat

loss is calculated by equation J:$(K$��

E Q P ) D J : P : K J:$(K��

,here is absolute temperature of the ,or-piece% ) is the ambient temperature% Q is the� �

convection coefficient% Q is the emissivity of the plate surfaces% and E >$*; . 9)79( ( is� �� ℃

the Stefan7#olt=mann constant$ n the current model% a typical value of Q ,as ta-en to be 9)

( using an ambient temperature of 8)) 5 and Q ,as ta-en to be )$9; for 8):L steel$�� ℃n order to account for the conductive heat loss through the bottom surface of ,eld plates% a high

overall heat transfer coefficient has been assumed$ This assumption is based on the previous

studies B(9% (< $ The heat loss ,as modeled appro.imately by using heat flu. loss by convection

given by equation J:$8K$�� = �� − �0

J:$8K

,here is a fictitious convection coefficient$ "ue to the comple.ity involved in estimating the��

0


32/87

contact condition bet,een the sheet and the bac-ing plate% the value of had to be estimate��


33/87

(:

d by assuming different values through reverse analysis approach$ n this study% the optimi=edvalue of �� ,as found to be 9)) � ��

2℃ $ Figure :$8 sho,s the schematicrepresentation of

boundary conditions that ,ere used for thermal analysis

Figure :$8 Schematic representation of boundary condition for thermal analysis

4')'8 He t Fl$3 In.$t

'eat is produced in the friction stir ,elding process due to the friction bet,een the tool shoulder

and ,or-piece interface and due to the plastic deformation of the ,eld metal near the pin$ The

heat generated by the plastic deformation of ,eld metal near the pin is of negligible magnitude

and is difficult to quantify B(9% >8% >: $ 'ence% it ,as neglected in this study$ Therefore in this

model% the heat generated by friction bet,een the ,or-piece and tool shoulder is the only source

of heat generation$

The total heat input in ,atts for this model is calculated through Chao et al$ B(9 equation and�


34/87

is applied as amoving heat

flu.$ The totalheat input is�

given byequation J:$:K$

22

� =�� (�� + ��

�� + �� )45(�� + �� )

J:$:K

,here is the tool rotational speed% Q is the frictional coefficient% Q is the do,n,ard force% and�

and are the radii of the shoulder and the nib of the pin tool$��

The rate of heat input to the ,or-piece q r is assumed to be a.is7symmetric and linearlydistributed in the radial direction B(9 and is calculated by equation J:$>K$

q r =

3Qr33

for ri ≤ r ≤ r oJ:$>K

2π(r o − r i)

n the present simulation% the heat flu. q r obtained from the equation J:$>K is applied assurface load using tabular boundary condition$ The movement of FS/ tool is implemented by

creating a local cylindrical coordinate system and calculating heat load at each node at each

instantaneous time step$

The dimensions for tool and values for other parameters used in this study ,ere obtained from

hu and Chao B(< for correlation to the published research data$ The tool shoulder diameter

used in this study ,as 9+$)> mm% ,hile the pin diameter ,as assumed as =ero$ The assumption

,as made based on findings from 3ussell and Sheercliff B>> that the heat generated at the pin of

the tool is in the order of ( of total heat and hence negligible$ Fitted values of and ,ere� ��

used in this study$ For the verification of the model% values of heat input E ;*) ,atts and E� ��

9)) ( for 8)) rpm ,ere used$�� ℃

4'+ Mec, n#c l Model


35/87

The second step in the thermomechanical analysis is development of the mechanical model$ The

temperature distributions obtained from the thermal analysis are used as input to the mechanical

model$ This model is used to estimate the ,eld induced residual stresses$ The mechanical model

developed for the analysis is described in this section$


36/87

4'+'& Ass$m.t#ons

The follo,ing assumptions have been made in developing the structural model@

• "eformation occurs symmetrically along the ,eld line% so only half of the ,or-piece is

modeled$

• The plate material is homogeneous$

• The effect of creep is neglected because there is no cyclic thermal load involved$

4'+') Elements 0sed nd Mes, De*elo.ment

A structural element defined by eight nodes Ji$e$% S&L "9K having three degrees of freedom at

each node is used for the modeling of plate$ This element supports plasticity% hyperelasticity%

stress stiffening% creep% large deflection% and large strain capabilities B:+ $

n the present analysis% the heat transfer model containing the equivalent thermal element

S&L ";) is replaced by S&L "9 by s,itching the element type from thermal to structural

using the command !TC'G$ The advantage of using this element type is that the temperatures

obtained from thermal step can be applied as element body loads at the nodes$ The geometry%

node locations% and the coordinate of this element are equal to those of S&L ";) element$ An

identical mesh pattern generated for the thermal analysis is used in the structural analysis$

4'+'+ Pl st#c#t1 Model

0lastic behavior involved in friction stir ,elding process begins ,hen the induced stress e.ceeds

the yield point of the material$ The plasticity is characteri=ed by nonlinear relationship bet,eenstress and strain$ The plasticity model is defined by three essential principles ? a yield criterion% a

flo, rule and a hardening rule B:+ $ A yield criterion determines the stress level at ,hich

yielding is initiated% a flo, rule relates the applied stress increments to the resulting plastic strain


37/87

increments once plastic flo, has begun% and a hardening rule describes the change in the yield

criteria as a function of plastic strains B:+%>* $

n the present thermomechanical analysis% the incremental theory of plasticity is employed$ The

plastic deformation of the material is assumed to obey von Misses yield criterion% the associated

flo, rule and the ,or- hardening rule$ This assumption is made based on the assumption made

by hu and Chao B(< in their study$ Accordingly% a bilinear isotropic hardening model J# S&K%

provided by A SHS R soft,are is used$ A # S& model incorporates von Mises yield criteria% and

associated flo, rules coupled ,ith isotropic ,or- hardening rule$ n the model% the stress7strain

behavior is described by bilinear stress7strain curves$ Figure :$: presents the yield stress%

Houng s modulus and thermal e.pansion coefficient of 8):L stainless steel at various

temperatures$ A constant plastic modulus of ($< G0a is used in all calculations to consider the

effect of strain hardening on the residual stresses$ Figure :$> sho,s the stress7strain behavior of

bilinear isotropic material used in the analysis$

Figure :$: Temperature dependent mechanical properties of 8):L stainless steel B(


38/87

Figure :$> #ilinear isotropic stress7strain model for 8):L stainless steel

4'+'4 (o$nd %1 Cond#t#ons

n the present analysis% sequentially coupled finite element analysis is carried out$ The

temperature histories obtained from thermal analysis are applied as body loads in the mechanical

analysis$ The forces from the thermal e.pansion of the ,or-piece material are the only forces

considered in this analysis$

The follo,ing boundary conditions are utili=ed for the mechanical analysis@

• The ,or-piece is constrained of vertical motion at the bottom surface$

• The ,or-piece is fi.ed through clamping by 8):$< mm long L7shaped steel strip J(>$:

mm . (>$: mm . *$8> mmK on each plate at a distance >)$< mm from the ,eld center$

Totally rigid boundary conditions are applied at these clamping locations$ The clamping

constraints are released after the ,eld cools do,n to room temperature$

• There are no displacements along the symmetric surface$


39/87

4'4 S#m$l t#on

The thermomechanical modeling ,as carried out in t,o stages$ Transient thermal analysis is the

first stage follo,ed by nonlinear transient structural analysis in the second stage$ Figure :$*

illustrates the flo, diagram of the method used for the finite element analysis$ Since the problem

involves nonlinear analysis% full e,ton73aphson option ,as used to solve the nonlinear

equations$

Figure :$* Flo,chart of sequentially coupled thermomechanical analysis


40/87

6' 2 l#d t#on o" T,e%momec, n#c l Model o" F%#ct#on St#% Weld#ng

For validating the thermomechanical model developed using A SHS R % it ,as essential to

correlate the developed model ,ith the published results$ For this purpose% the developed

thermomechanical model ,as verified ,ith numerical results obtained by hu and Chao B(< $

The model used for validation had dimension of 8):$< mm . 9)9$* mm . 8$9< mm of 8):L

stainless steel material$ The tool shoulder diameter ,as 9+$)> mm and the tool pin diameter ,as

*$8> mm$ The tool rotational speed ,as 8)) rpm and the applied do,n,ard force ,as 89$9 5 $

The ,elding ,as assumed to start at a location *$: mm a,ay from the edge of the ,or-piece and

stop after translation of (;+$: mm along the ,eld line ,ith a velocity of 9$*+8 mmIs$

t ,as difficult to predict the values for the convective heat transfer coefficient at bottom surface

and the total rate of heat input$ hu and Chao B(< conducted inverse analysis to fit the values of

these t,o uncertain parameters ,ith ma.imum temperature measured during FS/ e.periments$

To correlate the model to e.isting numerical data% fitted values of and are used in this� ��

study$ A convection coefficient of 9)) ( ,as applied at the bottom surface of the�� ℃

,or-piece$ The heat input of ;*) ,as applied as a moving heat flu. along the ,eld line$�

Additionally% a convection coefficient of 9) ( ,as applied at all the surfaces e.cept the�� ℃ bottom surface$

6'&'& Tem.e% t$%e Res.onses

Measurement of temperature ,as made by hu and Chao B(< through the use of 8* gauge 57

type thermocouples embedded at nine locations on the top and bottom surface along the

transverse section of the ,or-piece$ The graph in figure >$9 sho,s the comparison of

instantaneous e.perimental and simulation results for top surface of ,or-piece$ The ,or-piece

temperature ,ere measured and calculated along the traverse direction of ,eld line at tE


41/87

89

seconds% i$e$% at a distance of 9>($: mm from the edge of the ,or-piece$ From the figure >$9% it is

seen that the highest temperature during the ,elding is distributed ,ithin the shoulder region and

has the value bet,een +)) and 99>) $ This range is lo,er than the melting temperature of 8):L℃stainless steel$

Si !"#$ion r%s!"$s

&000

'h! #n h#o*s $%s$ #$#

+00

'h! #n h#o*s F,-r%s!"$s

600

400

200

0

0 20 40 600 &00Distance from the weld

line (mm)

Figure >$9 Comparison of temperature distribution along the transverse direction at ,elding timetE $( sho,s the temperature distribution on the top surface of the ,or-piece measured at

,elding time tE >)$: sec$ Figure >$8 sho,s the variation in temperature ,ith respect to time at

location J4E9>($:% HE9($;% E)K of the ,or-piece for both the results obtained by hu and Chao

B(< and by the model developed in this study$ The overall trend of the predicted temperature

profile is similar to that obtained by hu and Chao B(< % thus verifying the validity of the model

developed in this study$

T e m p e r a

t u r e

( 0 C )


42/87

Figure >$( Temperature distribution on top surface of the ,or-piece at ,elding time% tE >)$: sec

600

500

400

Si !"#$ion R%s!"$s

'h! #n h#o*s-o %"

300

200

&00

050 &00 &50

200 250Flow Time(seconds)

Figure >$8 2ariation of transient temperature 7 comparison of simulated results and results fromhu and Chao s Model

T e m p e r a

t u r e

( ⁰ C )


43/87

6'&') St%ess Res.onses

The temperature fields obtained from the thermal model are used as input for the mechanical

simulation for calculation of residual stresses$ The primary residual stresses in FS/ ,ere

observed in the longitudinal direction$ Therefore% only longitudinal stresses ,ere considered in

this study$ Figure >$: sho,s the comparison of results from hu and Chao s model B(< and

simulation results of longitudinal residual stresses for the top surface$ The residual stresses ,ere

measured along traverse direction at a distance of 9>( mm from the end of the ,or-piece$

Fi.ture release ,as modeled in order to estimate the effect of clamping$ t ,as observed that the

residual stress in the ,elds decreased significantly after the fi.ture release$ The overall trend of the developed model for prediction of residual stress is similar to that of hu and Chao B(< % thus

verifying the validity of the model developed in this study$

600

500

400

300

200

Si !"#$ion ./% or% r%"%#s%

Si !"#$ion. 1 $%r r%"%#s%

'h! #n h#os o %" ./% or% r%"%#s%

'h! #n h#o*s -o %". 1 $%r r%"%#s%

&000

.&000 20 40 60 0 &00&20

.200 Distance from the weld line (mm) at x=152 mm

Figure >$: 2ariation of the longitudinal residual stress along the traverse direction at the middlesection of the ,or-piece

x x

( ! p a

)


44/87

7' P % met%#c St$d1 nd S$%%og te Models o" FSW P%ocess

n order to conduct parametric investigation of FS/ process% design of e.periment methodology

is implemented in this study$ "esign of e.periment J"o!K technique is used to optimi=e the

number of e.periments required to determine the effects of various factors affecting the response

of the system B>* $ "o! helps to eliminate the need for e.tensive e.perimental analysis and in

turn reduces the computational time and cost$ The follo,ing sections describe the details of "o!

and development of surrogate models for FS/ process$

7'& Des#gn o" E3.e%#ments

Thermal and thermomechanical models developed in the chapter : are used as base models for

carrying out parametric studies$ An e.periment in this study ,ould refer to a distinct numerical

simulation run for a given set of input parameters$ The first step in "o! is to identify important

independent input factors and response variables$ The response variables selected for this study

are ma.imum temperature JTK and residual stress J3K$ #oth these selected responses are recorded

at a selected location i$e$ 4E 9>($: mm% HE ) mm% and E ) mm$ The process parameters heatinput J'K and ,elding speed JSK are chosen as input variables affecting the response variable

temperature JTK% ,hile the parameters '% S and clamping location JCK are chosen variables

affecting the response residual stress J3K$ The ne.t step is to identify the range and the specific

levels at ,hich selected factors have to be varied$ Table *$9 lists the process parameters% their

range and selected levels used in this study for response variables T and 3$

The final step in the parametric design is to perform the required number of e.perimental runs

and analy=e the significant factor effects$ The total number of e.perimental runs to be conducted

is identified from the total number of factors and the number of levels selected$ Table A$9 in

appendi. A depicts the design matri. for response variable T used in screening design for


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parametric study$ Table A$( in appendi. A depicts the design matri. for the other selectedresponse% residual stress J3K$ The observations ,hich e.ceeded 9:>) ℃% the melting point of 8):L stainless steel% ,ere omitted from design matri. ,hen formulating surrogate models$

Table *$9 0rocess parameters% range and design levels used

Res.onse P%ocessP % mete%s 0n#ts R ngeLe*el

&Le*el

)Le*el

+Le*el

4Le*el

6

TemperatureJTK

/eld Speed J S K mmIsec )$>7($>: )$> )$ 9$)) 9$*+ ($>:

'eat nput J H K ,att >))7+;) >)) *)) ;*) +;) 7

3esidualStress J3K

/eld Speed J S K mmIsec )$>7($>: )$> )$ 9$)) 9$*+ ($>:

'eat nput J H K ,att >))7+;) >)) *)) ;*) +;) 7Clamping

location J C K mm >)$(7;*$( >)$( ;*$( 7 7 7

7'&'& E""ect o" F cto%s on Tem.e% t$%e D#st%#!$t#on nd Res#d$ l St%ess

Figures *$9 and *$( depict the plots of main effects for temperature and residual stress%

respectively$ These plots help to assess the effect of each factor graphically$ The figures *$9 and

*$( sho, that heat input factor has a significant effect on both temperature and residual stress

and a direct proportionality can be seen bet,een the heat input factor and the responses$

Temperature decreases ,ith increasing ,elding speed$ Figure *$8 sho,s the variation of temperature on top surface of the ,or-piece for ,elding speeds )$>) mmIs to ($>: mmIs at

constant heat input of *)) /$ The pea- temperature tends to increase as the ,elding speed is

reduced$ &n the other hand% it is observed residual stress first increases ,ith increase in ,elding

speed and then tends to slightly decrease at higher ,elding speeds$

The clamping location also has a significant effect on the residual stress$ t is observed from

figure *$( that if the clamp location is nearer to the ,eld% lo,er residual stresses are developed$

As the clamp location moves further a,ay from the ,eld line% level of residual stress increases$


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!ain "ffects #lot forTemperature

#$#-%#ns

&300

&200

&&00

&000

00

500 600

60 0

0 50

0 +5

& 00

& 6 2 54

Figure *$9 0lot of main effects for temperature

!ain "ffects #lot for $esidual%tress

#$#-%#ns

360

340

320

300

500 600 60 0 0 50

05

& 00

& 6 2 54

360340

320

300

50 2 6 2

Figure *$( 0lot of main effects for residual stress

%#$ 8%"

! e a n

%#$ 8 %"

! e a n

"# pin9 :o;#$ ion


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&400

&200

&000

00

600

400

200

00 20 40 60 0 &00 &20

Distance from the weldline (mm)

Figure *$8 2ariation of temperature on top surface of the ,or-piece at different ,elding speeds

7') S$%%og te Models o" F%#ct#on St#% Weld#ng

A surrogate provides fast appro.imations of the system response and it can be used for

optimi=ation studies B>; $ A surrogate can be used to model the design ob6ectives or model theconstraints$ n this study% surrogate models are constructed to establish relationship bet,een the

process parameters and the output responses$

A surrogate model for any given set of data can be modeled using linear or nonlinear regression%

neural net,or-s% response surface appro.imations% support vector regression% etc$ B:; $ n this

study% linear and nonlinear regression methods are used to construct surrogate models and later

their performances are evaluated$

7')'& De*elo.ment o" Model "o% Res.onse 9 Tem.e% t$%e

Multiple regression analysis ,as used to establish relationship bet,een the selected input process

parameters and the thermal response variable$ 'eat input J H K and ,elding speed J S K are the

T e m p e r a

t u r e

( 0 C )

0 50


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8<

selected input process parameters for the response temperature J T K$ The simulated data obtained

in table A$9 in appendi. A% is used for setting up surrogate models$

Minitab 9>% commercial statistical soft,are capable of data analysis% ,as used to compute the

regression constants for multi7linear regression model$ The fitted linear regression model for

temperature is given by equation J*$9K$

E *:; D 9(8; P *+> J*$9K� ∗� ∗�The results of multiple linear regression analysis are included in appendi. #$

Additionally% nonlinear regression models ,ere also setup using the simulated data obtained in

table A$9 in appendi. A$ The nonlinear regression analysis ,as carried out using "ataFit version

+$)% statistical soft,are capable of curve fitting and nonlinear regression analyses$ The fitted

nonlinear regression model for temperature obtained from "ataFit is given by equation J*$(K$

E 9


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Additionally% nonlinear regression model ,as also setup using the simulated data obtained in

table A$( in appendi. A$ The nonlinear regression analysis ,as carried out using "ataFit version

+$)$ The fitted nonlinear regression model for residual stress obtained from "ataFit is given by

equation J*$:K$

E e.pJ)$8:


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the predicted models$ is estimated by the follo,ing equation J*$>K$��

E ( D ln ( J KI D 9 J*$>K� �∗ � �� ,here n is the number of observations% and - is the number of parameters in the model$

The models to be compared are ran-ed according to their AIC and the model ,ith thelo,est �� is selected as the best fit model$

:$ The ad6usted coefficient of determination (�

Li-e ��% the ad6usted coefficient of determination is parameter independent and isusedas a measure to find the optimal regression model$ A higher value of �2 indicates

better fit$The values of �2 %��%�� and �2

,ere used to determine the goodness7of7fit of the surrogate

models$ Table *$( sho,s the regression statistics of linear and nonlinear surrogate models

developed for estimating temperature and residual stress$ From table *$(% it is seen that in case of surrogate models for temperature% the values of �2 and�2

are higher and the values of��

and are lo,er for nonlinear model ,hen compared to those of linear model$ This indicates��that the nonlinear model given by equation J*$(K fits the data better than the linear model given

by equation J*$9K$

Table *$( 3egression statistics of linear and nonlinear surrogate models

Res.onse2 %# !le

Reg%ess#onModel

E:$ t#onN$m!e% k

� ��

TemperatureLinear J*$9K 8 )$+;;( 9+:*+ 9*>$); )$+;8;

onlinear J*$(K 8 )$++$>8 )$+


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regression model had higher ( and (� � values and lo,er and values compared to th��


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elinear model% indicating nonlinear linear model given by equation J*$:K has better fit than linear

model given by equation J*$8K$ Thus the best models for estimating the responses% ,or-piece

temperature and residual stress ,ere nonlinear regression models$


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8' Dete%m#n#ng O.t#m l FSW P % mete%s 0s#ng Im.%o*ed H %mon1 Se %c,Algo%#t,m

The follo,ing section describes the model7based approach using mproved 'armony Search

J 'SK algorithm applied to the FS/ process in this study$

8'& Fo%m$l t#on o" O.t#m#/ t#on P%o!lem

The main goal of this research is to develop an optimi=ation strategy to determine process

parameters ,hich are able to optimi=e the ,eld quality$ The search for optimum is based on

ma.imi=ing the throughput and minimi=ing the manufacturing costs$ Therefore% the optimi=ation

problem is formulated as follo,s B:; @

M 3#m#/e Throughput

M#n#m#/e Cost

Sub6ect to%

JiK Maintaining good ,eld quality

JiiK The upper and lo,er limits of the process parameters

The production throughput for a ,elding process could be measured in terms of the length of

,eld completed% ,hich in turn relates to the ,elding speed$ Therefore% ma.imi=ing the

throughput for the process can be interpreted in terms of ma.imi=ing the ,elding speed$ The

costs relating to ,elding process include the cost of equipment% labor cost% and cost relating to

energy input$ 'o,ever% considering that equipment cost and labor cost are fi.ed for the process%

cost relating to energy input forms the dominant cost component$ Further% the ,eld qualities are

the result of thermomechanical history during ,elding and these ,eld quality constraints can be

equated ,ith constraints on temperature and residual stress$ Additional practical constraints are

applied from the bounds of process parameter values B:;% >: $


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n this research% the t,o conflicting ob6ectives i$e$ ma.imi=ing speed and minimi=ing cost are

handled by combining them into single ob6ective function ,ith equal ,eights applied to each of

the t,o ob6ectives$ The t,o ob6ective functions have different units of measurement$ To offset

the magnitude difference bet,een them% the process variables are normali=ed by dividing ,ith

the ma.imum value$

The optimi=ation models% formulated based on thermal model% have the follo,ing form@

Minimi=e P� �

Sub6ected to@ V V��

V V��

V V��

,here% is the temperature% is the heat input% and is the ,elding speed% L# and U# stands� � �for lo,er and upper bounds$

T,o optimi=ation models are formulated using the surrogate models developed for estimating

temperature$ These t,o models differ primarily on the equations for T@ called Model 9 if linear

equation J*$9K is used% Model ( if nonlinear equation J*$(K is used instead% for easy reference

later$

To avoid optimi=ation solutions that may e.ceed the desired residual stress limit% the

optimi=ation problem is modified by imposing additional constraints on residual stress and

clamping location$ The optimi=ation models have the follo,ing form@

Minimi=e P� �

Sub6ected to@ V V V V��

V V V V��


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V ,here is the location of the clamp from the ,eld centerline and is the residual stress$� ��

T,o optimi=ation models are formulated based on the above consideration$ These t,o modelsdiffer primarily on the equation for � and �@ called Model 8 if linear equation J*$9K for �and

linear equation J*$8K for is used% Model : if nonlinear equation J*$(K for T and nonlinear �

equation J*$:K for 3 used instead% for easy reference later$

8') Sol$t#on Met,odolog1 0s#ng H %mon1 Se %c, Algo%#t,m

Metaheuristics are high level heuristic algorithms ,idely used for solving optimi=ation problems$

n general% population7based metaheuristics such as ant colony optimi=ation% genetic algorithm%

harmony search% particle s,arm optimi=ation etc$ are more effective for constrained function

optimi=ation problems than single7point search metaheuristics li-e simulated annealing% tabu

search% iterated local search etc$ B:; $ 'armony Search algorithm J'SK% a population7based

metaheuristics is selected for this study because the optimi=ation problem formulated for friction

stir ,elding process is a constrained function optimi=ation$ 'S algorithm is inspired from the

musical process of searching for a pleasing harmony and has been successfully applied to various

optimi=ation problems B>< $

'S algorithm ,as proposed by Geem et al$ B>+ in ())9$ Unli-e ant colony optimi=ation and

particle s,arm optimi=ation ,hich are inspired from natureInatural phenomenon% harmony

search algorithm is inspired from an artificial phenomenon found in musical performance$ The

process of musicians in a musical performance to produce fantastic harmony pleasing to hear has

been compared to the process of optimi=ation in order to find the best solution$ The music from

combined instruments is 6udged by aesthetic standards% 6ust as the optimal solution is estimated

by ob6ective function$ Table ;$9 sho,s the comparison bet,een optimi=ation process and

musical performance$


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Table ;$9 Comparison bet,een optimi=ation and musical performance B>+

Com. %#son F cto% O.t#m#/ t#on P%ocess Pe%"o%m nce P%ocess

#est state Global optimum Fantastic harmony

!stimated by &b6ective function Aesthetic standard!stimated ,ith 2alues of variables 0itches of the instruments

0rocess unit !ach iteration !ach practice

8')'& Im.%o*ed H %mon1 Se %c, Algo%#t,m

The improved harmony search J 'SK algorithm developed by Mahdavi et al$ B*) is implemented

for optimi=ation process in this study$ An important consideration in the application of

optimi=ation methods is ho, the algorithm handles the constraints relating to the problem B>< $

n this study% the constraints are handled using the parameter7less penalty approach proposed by

"eb B*9 $ n "eb s approach% ,hen comparing t,o solutions% the constraints are handled using

the follo,ing clauses B*9 @

9$ /hen t,o feasible solutions are compared% the one ,ith better ob6ective value is chosen$

($ /hen a feasible and an infeasible solution are compared% a feasible solution ,ins over an

infeasible solution$

8$ /hen t,o infeasible solutions are compared% the one ,ith smaller constraint violation is

chosen$

8')') Pse$do Code

The pseudo code of the implemented improved harmony search algorithm% called 'SD% is given

belo,$

Step 9@ nitiali=e the problem and algorithm parameters

The optimi=ation problem is formulated as minimi=ing the ob6ective function and the


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:*

design variable bounds are defined$ The algorithm parameters are initiali=ed at this stage$

The parameters include the number of solution vectors in the harmony memory i$e$ the

harmony memory si=e J HMS K% harmony memory considering rate J HMCR K% ma.imum

and minimum pitch ad6usting rate J PARmax, PARmin K% ma.imum and minimum band,idth

Jbwmax , bw minK% and the number of function evaluations or stopping criterion J NI K$

Step (@ nitiali=e the harmony memory

'armony memory J'MK is initiali=ed ,ith randomly generated harmonies ,hich are

,ithin the acceptable design upper and lo,er bounds B UB, LB $ The infeasible solutions

are not eliminated but are handled by using "eb s strategy$

Step 8@ mprovise a ne, harmony

A ne, harmony vector is generated from 'M based on three rules JiK memory

consideration% JiiK pitch ad6ustment and JiiiK randomi=ation B*( $ The memory

consideration ensures that the design variable values are chosen from 'S memory ,hile

the randomi=ation step ensures random selection of a harmony vector$ 0itch ad6ustment

ensures that an ad6acent value from initial 'M is chosen$ This is implemented as follo,s@

/hile generation JgnK V a$ Update the pitch ad6usting rate J0A3K ,ith each generation for fine7tuning of

optimi=ed solution vectors% according to equation J*K in Mahdavi et$al$ B*) % ,hich is

denoted by equation J;$9K$

�� =

�� +

(�� −

�� ) � � J;$9K��

,here JQQ K is the pitch ad6usting rate for each generation��(�( ((((((((( b$ Update the band,idth Jb,K ,ith each generation for fine7tuning of optimi=ed solution


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:;

vectors% according to equation J;K in Mahdavi et$al$ B*) % ,hich is denoted by


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equation J;$(K$

�� =

��

"n� � � � �

QQ��

J;$(K

,here JQQ K is the band,idth for each generation$��(�( (((((((((Fo% each decision variable

i$ Construct a ne, harmony vector either by choosing each decision variable

from any specified 'M range based on probability 'MC3 Jmemory

considerationK or choosing a totally random harmony value from the feasiblerange ,ith probability of J97'MC3K Jrandom selectionK$

ii$ Chec- if a rand W 0A3% ,ith rand being a uniformly distributed random value

B)% 9 % and determine ,hether each component of the ne, harmony vector ∈obtained from memory consideration should be pitch7ad6usted$ Construct a

ne, harmony vector by updating the variables ,hich have to be pitch7ad6usted by �� > �(−&?&)% ,here �(−&?&)is a uniformlydistributedrandom value bet,een J79% 9K$

End "o%

Step :@ Update harmony memory

Update the harmony memory by replacing the ,orst one in the memory ,ith the ne,

one% if the ne, one improves it$ This is handled by ran-ing the solutions in archive by

first giving preference to feasible solutions over infeasible ones% then ran-ing feasible

solutions ,ith respect to their ob6ective values% and finally ran-ing infeasible solutions in

ascending order of constraint violation$

Step >@ Update the best solution and increment gn by one$


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End w,#le

Step *@ &utput the result of optimal solution and its ob6ective value$

The 'SD algorithm used in the present study differs from 'S in the follo,ing areas$ First of all

the 'SD uses "eb s strategy to handle constraints and it allo,s the use of 'M members that

violate the constraints$ This implementation thus avoids the e.haustive trial and error process of

generating a harmony memory ,ith each of its members satisfying all the constraints$

Additionally% 'SD calculates and stores constraint violation information associated ,ith each

harmony vector$ Further% 'SD differs from 'S in the ,ay the solutions are ran-ed and the best

solution is selected$

Ma6or parameters associated ,ith the 'SD algorithm include harmony memory si=e% HMS %

ma.imum number of function evaluations% NI % harmony memory considering rate% HMCR %

ma.imum and minimum pitch ad6usting rate% PARmax and PARmin% and ma.imum and minimum

band,idth% bwmax and bwmin$ The table ;$( lists the values fi.ed for the parameters used for this

study$

Table ;$( 0arameters used for 'SD in this study

P % mete%2 l$e

Model & nd Model ) Model + nd Model 4

HMS () 8)

NI 9))))) 9>))))

HMCR )$+ )$+

PARmax )$++ )$++

PARmin )$:> )$:>

bwmax : :

bwmin )$))))9 )$))))9


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8'+ O.t#m#/ t#on Res$lts "o% FSW P%ocess

The 'SD algorithm ,as applied to solve the optimi=ation model formulated in the previous

section for the targeted friction stir ,elding process$ "ue to the stochastic nature of the 'SD

algorithm% for each case% 8) independent runs ,ere made to produce sufficient statistical data$

The best% median% and ,orst results of ob6ective values and C0U time attained in 8) runs ,ere

recorded$ The solutions for the best results ,ere also recorded$ Additionally% for comparison% the

optimi=ation problems ,ere solved using the function fminc n available in MATLA#

&ptimi=ation toolbo.$ The function fminc n implements sequential quadratic programming

algorithm to find the constraint minimum of a scalar function of several variables starting at aninitial estimate B*8 $ The 'SD optimi=ation method ,as implemented in MATLA#$ All the

programs ,ere run on a ($** G'= ntel 0entium7" processor ,ith ( G# of random access

memory$

8'+'& Res$lts "o% Model & nd Model )

Model 9 uses the fitted linear regression equation of T% i$e$ equation J*$9K% ,hile the Model (

uses the fitted nonlinear equation of T% i$e$ equation J*$(K$ These optimi=ation models ,ere

solved using a ,ider bound of T% i$e$ E 9))) and E 98))$ The bounds for the other ��

process variables ,ere set at the lo,est and highest simulated values% i$e$ E >))% E +;)%��

E )$> and E ($>:$ These temperature range and bound values should be set in�� consideration of material properties and practical e.perimental constraints$ The melting point of

8):L stainless steel is about 9:>) $ To enable this study% the bound values selected are a rough℃guess around -no,n good temperature value belo, the melting range$

Table ;$8 summari=es the results obtained by 'SD and fminc n function for both the Model 9

and Model ($ The results indicate that@


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JiK The best ob6ective value of 7)$9>(;:� �mmIs ,as obtained for Model 9 using both algorithms$ /hile% a lo,er ob6ective value i$e$

7)$9**:+> and optimal solution E: mmIs ,as obtained for Model� �($ The result seems to indicate that optimi=ation on Model ( leads to a better solution$

JiiK The function fminc n found the best solution in all 8) runs$ &n the other hand% 'SD

algorithm ,as not able to converge to the best solution in many runs$ evertheless% the

average solution found in all runs is very close to the best$

Table ;$8 &ptimi=ation results of Model 9 and Model ( ,ith T L# E 9))) and T U# E 98))

(1 IHS; (1 fminconModel & Model ) Model & Model )

&b6ective 2alue

#est 7)$9>(; 7)$9>(;

Median 7)$9>(; 7)$9>(;

/orst 7)$9>(;(;

#est Solution'eat nput : ($>: ($>:

C0U Time#est 9*$)+8;>) 9*$:)*(>) )$)9>*(> )$)89(>

Median 9*$9;+* )$)*(>

/orst 9+$: 9*$+*) )$)*(> )$)+8*>)

umber of runs found the best solution (8 (> 8) 8)

To investigate the effect of narro,ing the range of temperature% Model 9 and Model ( are againsolved using a narro,er range of T% i$e$ ��E 9)>) and ��E 99>)$ Table ;$: summari=estheresults obtained by 'SD and fminc n function for both the Model 9 and Model ($ The results

indicate that@

JiK The best ob6ective value of 7)$99(8*+ ,ith optimal solution E:� �mmIs ,as obtained for Model 9 using both algorithms$ /hile% a lo,er ob6ective value i$e$


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7)$99;< and optimal solution E>$*;< / and E($>: mmIs ,as obtained for Model� �($ The result seems to indicate that optimi=ation on Model ( again leads to a better

solution$

JiiK The function fminc n found the best solution in all 8) runs$ &n the other hand% 'SD

algorithm seems not able to converge to the best solution i

Malde Thesis

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