Simplifi ed modelling of Fixtures in FE Welding Simulation

MASTER’S THESIS

OLOF WIIPPOLA

Simplifi ed modelling of Fixtures in FE Welding Simulation

MASTER OF SCIENCE PROGRAMME Engineering Physics

Luleå University of Technology

Department of Mathematics

in FE welding simulation

De ember12,2006

Olof Wiippola

olof.wiippolagmail. om

Master Thesis

Institute of mathemati s

Luleå University of Te hnology

Thegoalwiththismasterthesiswork,istondasimpliedwayofmodelling

the xture in FE welding simulation. To take the entire xture into the

welding model, is very CPU-time onsuming sin e the onta t between the

xture and the work pie e is di ult to simulate. The method usedtoday,

is based on xed degree of freedom boundary ondition. This method do

not in lude the stiness of the xture, the fri tion and the thermal ee ts

between the xture and the work pie e. In order to get a more a urate

modell, these things an not be negle ted. The method that is presented

in this master thesis report, uses already existent fun tions in the FEM

program MSC.Mar . The main idea is to use non-linear springs and non-

linear elasti foundation to model fri tion and support. Sin e the onta t

for ebetweenthextureandtheworkpie edire tlyae tsthefri tionfor e,

thedistribution fun tionsthat des ribes how a known onta t for e at one

point distributedthrough thewhole xture must be used. This fun tion is

onstru ted in two ways, the rst uses the entire model with the xture,

the se ond uses Bu kingham's pi theorem. The tests that where arried

out shows that the distribution fun tions gives a good results, but there

are several problems that must be solved before the methods an be used

in welding simulations. The subroutine used in welding simulation do not

supporttheneedednewstyletableinputwhi hisneededinordertouseedge

foundation as the fri tion for e. MSC.Mar has problem with intera tion

betweenthefa elmandtheedgefoundation. Theproblemwiththefa elm

andtheedgefoundationismostliklyabug,sin ethisproblemdonota ure

inthe newrelea e ofMSC.Mar r3.

Keywords: welding simulation, non-linear spring, elasti foundation, edge

foundation,fa e foundation, onta t

Thismaster thesisworkhasbeen arried out at Volvo AeroCorporation in

Trollhättan. Iwouldliketothankmysupervisors,Seniorle torBoKjellmert

at LTUand Torbjörn Kvist at Volvo Aero. A spe ialthanksto Dr. Henrik

Alberg for his support and advi e during my work at Volvo aero. I would

alsolike to thank thesta of9634 for theresupport.

OlofWiippola

Trollhättan, De ember12, 2006

Contents

1 Introdu tion 1

2 Problem statement 2

2.1 Ba kground . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Denition of theproblem . . . . . . . . . . . . . . . . . . . . 2

2.3 Problemsetup . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Methods 4 3.1 Denitionsoffun tionsinMSC.Mar . . . . . . . . . . . . . . 4

3.2 Modellingof onta t . . . . . . . . . . . . . . . . . . . . . . . 6

3.2.1 Shell onta t . . . . . . . . . . . . . . . . . . . . . . . 6

3.2.2 Dete tion of onta t . . . . . . . . . . . . . . . . . . . 6

3.2.3 Conta t toleran e. . . . . . . . . . . . . . . . . . . . . 6

3.2.4 Conta t between deformable bodies. . . . . . . . . . . 7

3.2.5 Penetration . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2.6 Heatuxdue to onta t between deformablebodies . 7 3.3 Modellingof fri tion andsupportwithout onta t . . . . . . . 9

3.3.1 Springs . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.3.2 Elasti foundation . . . . . . . . . . . . . . . . . . . . 9

3.3.3 Fa elm . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Modelling and solution 11 4.1 The models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.1.1 Thereferen emodel . . . . . . . . . . . . . . . . . . . 11

4.1.2 Thesimpliedmodel . . . . . . . . . . . . . . . . . . . 11

4.2 First step, fri tion test . . . . . . . . . . . . . . . . . . . . . . 12

4.2.1 Conta t for edistribution . . . . . . . . . . . . . . . . 13

4.2.2 How to usethedistributionfun tion . . . . . . . . . . 17

4.2.3 Implementation andresultofthersttest usingpoly- nomialtting . . . . . . . . . . . . . . . . . . . . . . . 18

4.2.4 ImplementationandresultofthersttestusingBu k- ingham'sPi theorem . . . . . . . . . . . . . . . . . . . 20

4.2.5 How to useresultsof therst test . . . . . . . . . . . 21

4.3 Se ondstep . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.4 Welding simulations . . . . . . . . . . . . . . . . . . . . . . . 24

4.4.1 Welding simulation setup . . . . . . . . . . . . . . . . 24

4.4.2 Therst welding simulation . . . . . . . . . . . . . . . 24

4.4.3 Fa elm togetherwithelasti foundation . . . . . . . 24

5 Con lusions 26 5.1 Errors inthesimpliedmodel . . . . . . . . . . . . . . . . . . 26

5.2 Re ommendations . . . . . . . . . . . . . . . . . . . . . . . . 27

A Full size plots a B Element type 75 f B.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f B.2 Bilinearinterpolation . . . . . . . . . . . . . . . . . . . . . . . f C Element type 7 g D Matlab ode h D.1 Distributionfun tion onstru tionusingBu kingham'sPiThe- orem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . h D.2 Distribution fun tion onstru tion usingnumeri al polyt . . i

Chapter 1

Introdu tion

Welding ofdetailsto theaeroindustryisavery omplextask. There areno

roomfor errors during the welding pro ess. That is why, itis important to

know as mu h as possible about how the material is going to be deformed

duringthewelding pro ess. Thedeformationandstressinthematerial dur-

ingandafter welding areof interest. To make physi al testsofdeformation

and stress during the development step of a new details is both ost and

time onsuming. That iswhy,simulation toolsare oftenused. Onemethod

that is often used is the Finite Element Method (FEM). FEM is a numer-

i al method of solving mathemati al problems. In this Master Thesis, the

possibilityof analternative way ofxture representation isinvestigated.

Chapter 2

Problem statement

This hapter givestheba kground anddenitionof theproblemon whi hthe

master thesis Simplied modelling of xtures in FE welding simulation is

based.

2.1 Ba kground

Inthe aerospa eindustries, itisvery important tohave goodknowledge of

all problemsthat mayo urduring themanufa turing pro ess. To identify

theseproblemsbyexperimentisbothtimeand ost onsuming. Thatiswhy,

simulationtoolsare used.

In welding simulations, deformations and stresses are often interesting.

The purpose of the welding simulations is to nd optimal manufa turing

pro essesand manufa turing on epts. Theweldingpro essesarevery om-

pli ated and CPU-time onsuming to model. E ient methods of xture

simpli ationdoesnot exist at Volvo AeroCorporation today. If thewhole

xture would be a part of the simulation model, the CPU-time that would

be requiredinorder to getgoodsimulation results wouldin rease to alevel

thatisimpossible towork with. ThatiswhyVolvoAeroCorporationtoday

doesnotin lude thextureintothelargeweldingmodels. Insteadxdegree

offreedom(D.O.F)atthe boundariesareused. Thesesimpli ationsdonot

in lude the ee ts ofthe xture su h asthestiness of the xture, fri tion

andthermal ee tsbetween thexture and thework pie e.

2.2 Denition of the problem

As mention earlier, the fri tion between the xture and welding obje t is

today negle ted inthe welding simulations. The goal of this master thesis

workistodevelopamethodthatin ludesthestinessofthexture,fri tion

and thermal ee ts between the xture and the work pie e into the FE

welding simulations, without signi antly in reasing the CPU-time. The

xture and plate on wit h thesimulations arebased on areshown ingure

2.1and 2.2.

2.3 Problem setup

Thexture in2.1isthe referen emodel, onwhi hthesimpli ationsareto

be made. Figure 2.2shows theplate thatis to bewelded. The thi kness of

theplate inthe gure 2.2is

1.65mm

^{. Asshown in gure 2.1, the left part}

ofthe plate isxed inthe lamped area. The right part oftheplate,on the

otherhand, an move upfrom thexture. The lamping for e of theplate,

omes fromtheboltfor eand the weight oftheupperpartofthe xture.

Fixture top

Plate Fixture

300 mm

300 mm Bolts

60mm

Figure2.1: Fixturesetup

000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000

111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111

00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000

11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 0000000

0000000 0000000 0000000 0000000 0000000 0000000

1111111 1111111 1111111 1111111 1111111 1111111 1111111

00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000

11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 0000 0000 1111 1111

0 1 0 100

11 0000 0000 1111 1111 0000

0000 00 1111 1111 11

Clamp area

Plate 100 mm

190 mm Weld path

Figure 2.2: Plate

Chapter 3

Methods

In this hapter, the methods that are going to be used in order to onstru t

a simpli ation ofthe xture problemin FEwelding simulation are going to

be presented.

3.1 Denitions of fun tions in MSC.Mar

•

^{MSC.Mar : FEMprogram thatisused inthismaster thesis.}

•

^{Table driven input: Almostall inputs into MSC.Mar an be table}

driven. Thatmeansthattheinputvalue(referen evalue)ismultiplied

bythevalueofthetable. Thetable anbeafun tionofseveraldierent

variables, su h as time, displa ement, urrent oordinates and so on.

Forexample,ifaboundary onditionofthetypepointloadisusedwith

an input value in the

x − direction = 1

^{. That means that the point}

loadis 1 inx-dire tion. Ifthe table 3.1is usedas an inputtable, the

pointloadwillbethepointloadreferen e value(whi hisoftensetto1)

multipliedbythe multipli ation value at ea h timestep. That means

thatthe pointload will hange it'svaluewithtimea ording totable

3.1 and gure 3.1. For example after 1 timeunit, the multipli ation

value is 50 a ording to table 3.1 and gure 3.1, this multiplied with

the referen evalue1 givesthe point load50N.

Table 3.1: Exampleof atable for thetabledriven input

Time Multipli ation value

0 0

1 50

2 150

3 300

4 500

5 650

0 3

Time Multiplication

value

100

0 300

200

1 2 4

400 500 600

5

Figure3.1: The gureshows thetable plotfrom thetable driven input

Table 3.2: The point load at ea h time step with the point load referen e

vaule setto 1

Time Point load

0 0

1 50

2 150

3 300

4 500

5 650

3.2 Modelling of onta t

Theanalyzeof onta tis omplex,due tothefa tthatthepositionandthe

motionof the onta t bodymust be determined with a very high a ura y

to avoid thatthe onta tbodies move throughea h other.

There aretwo typesof onta t bodies inMSC.MARC,

•

^{Deformable onta t bodyis asetof elementsthat a tlikea body.}

Thedeformable onta t body an bein onta twithotherdeformable

bodiesor ridged bodies.

•

^{Ridged onta t body is a set of urves or surfa es that a ts as a}

body. Theridged bodydoesn't deform.

3.2.1 Shell onta t

Shell elements an bein onta t withridged bodies anddeformable bodies.

Anode ofashellelementissaidto be in onta twithasegement,whenthe

position of the shellelement node

±

^{halfthethi knessof theshell element,}

normalto the shellarewithin anothersegment.

3.2.2 Dete tion of onta t

Ea h potential onta t node is he ked, if near onta t is possible with a

segment. In3-D,fa esofelementsarepossible onta tsegment. MSC.Mar

usesanear onta talgorithmtodetermineif onta tnodesareneara onta t

segment. Ifanodeisinnear onta t, amoresophisti atedalgorithmisused

to dete t onta t.

3.2.3 Conta t toleran e

Itisunlikely,thatexa t onta t o urs,sin e thepositionofa nodeisa nu-

meri alvalueandtherearealwaysroundo-andtrun ationerrorsinvolved.

That iswhy onta t toleran eis needed. Conta t toleran eis themeasure-

mentofwhen onta t anbesaidtoo ur. Bydefault,the onta ttoleran e

issetto 25%ofthe shellthi kness. Ifavalueofthe onta ttoleran e istoo

large,it will ausetoo manynodesto bein onta t. Iftoo smallvalues are

hosen, it willlead to more omputation time.

3.2.4 Conta t between deformable bodies

When a 3-D deformable onta t body is reated, the program reates a

surfa e to outline the boundary of the onta t body. When a node of a

deformable onta t body omes in onta twitha segment,a tieis reated.

3.2.5 Penetration

To avoid penetration between elements, MSC.Mar has three pro edures

to dete t and avoid penetration. The default and theused pro edure, uses

in rementsplitting. Inthispro edure,thetimestepwhi h ausespenetration

∆t

^{isdevidedintotwo}subin rements. Intherstsubin rement,anode

n

1 ^is

not onstrainedandinthese ondsubin rementthenodeis onstrained,then

MSC.Mar has to nd thetimewhen onta t rst o urs, seeg. 3.2. The

timestep

∆t

_a^and

∆t

_b ^{is hoosenby}linearizingthedispla ementin rement.

∆t

a

= a a + b ∆t

∆t

b

= b

a + b ∆t

^(3.1)

Thatmeans that

∆t

_a ^{isthe timewhen onta trst o urs.}

3.2.6 Heat ux due to onta t between deformable bodies

a

b

Figure 3.2: Illustration that shows how the in rement splitting pro edure

works

When deformable bodies are reated, heat ux lms are automati ally

reatedat allthe boundaries. TheuxesusedinMSC.Mar dependsonthe

distan e

d

^{betweenthe onta tbodies. Therearetwodierentdistan esthat}

are used to al ulate the heat ux. The rst is the onta t distan e

d

_con^,

whenthebodiesaresaidtobein onta ta ordingtothe onta ttoleran e.

The se ond is the near onta t distan e

d

_near^{, when the distan e between}

elements is smalleror equal to the smallest element. For distan e

d < d

_con

theheatuxisdened as[4℄

q = H

T C

(T

2

− T

1

)

^(3.2)

where

q = heatux

H

_{T C} ^{= lm oe ient}

T

1 ^{= thesurfa e}temperature

T

2 ^{= the}interpolated nodaltemperature at onta t lo ation on the onta ted body.

Inthe near onta t ase, theheatuxis dened by

q = H

CV

(T

2

− T

1

) + H

N C

(T

2

− T

1

)

^{BN C}

+ σǫf T

_A2⁴

− T

_A1⁴

+

^(3.3)

+

 H

_CT



1 − d d

_near



+ H

BL

 d d

_near



(T

²

− T

1

)

where

q = heatux

H

_CV ^{= onve tion oe ient for neareld behaviour}

H

_{N C} ^{= natural onve tion for neareldbehaviour}

B

CN ^{= exponent asso iated withnatural onve tion}

σ

⁼ Stefan-Boltzman onstant

ǫ

^{= emissivity}

H

BL ^{= distan e dependent heattransfer oe ient.}

If the distan e

d > d

near^{, the heat onve tion to the} environment is dened by

q = H

CT V E

(T

²

− T

_{SIN K}

) + σǫf T

_A2⁴

− T

_A1⁴

(3.4)

where

q = Heatux

H

_{CT V E} ^{= heattransfer oe ient to the}environment

T

SIN K ⁼ environment sinktemperature.

Iftheneardistan e

d

_near^{isnotdenedinthethermalpartofthe onta t}

option,and

d > d

contact^{,eq.3.4is still valid.}

3.3 Modelling of fri tion and support without on-

ta t

There aretwopossibleways tomodelfri tionand supportwithout onta t.

The rst is the spring option, the se ond is the elasti foundation option.

Bothoptions an bemodelled withboth linearand non-linear properties.

3.3.1 Springs

The spring option an be used in both me hani al and thermal analysis.

During oupled thermo-me hani al analysis, thethermal part of thespring

a tsasalink. Thethermallinkis ontrolledwithaheattransfer oe ient.

The springs inMSC.MARC an be dened in three ways. All these spring

options need a beginning and a end node. It is important to remember to

apply xed boundary ondition to the begin node, in other ase the spring

will justoatinspa e.

•

^{FixeddegreeoffreedomThespringfor eisonlya tiveinonedegree}

of freedom between the beginning and the end node that denes the

spring.

•

^{To groundThespring for eis dire tedto origo.}

•

^{True dire tion The spring for e isa tive inthedire tion of theline}

between the two nodeswhi hdene thespring.

Thespring for e

F

_s ^{isdened by}

F

s

= k(U

1

− U

2

)

^(3.5)

where

k

^{is the spring stinessand}

U

^isthe displa ement.

3.3.2 Elasti foundation

Elasti foundations workslike springs, butthey don't operate on nodeslike

springs.

Edge foundation

Theedgefoundations,asthenamesays,operateontheedgesoftheelement.

Thefoundation for eis dened by

F

_f

= k (U

1

− U

2

)

^(3.6)

where

F

_f ^{isthe foundation for eand}

k

^{is the foundation stiness.}

U

1

− U

2

isthe displa ement of the edge.

Fa e foundation

Thefa efoundationworksliketheedgefoundation,butoperatesonthefa es

ofthe elementsinsteadof ontheedges. Thefoundation for eisdened like

eq. 3.6, but then

U

1

− U

2 ^isthedispla ement ofthefa esof theelement.

3.3.3 Fa e lm

In order to simulate the heat transfer to the xture and the environment,

thefun tionfa e lm an beused. Fa elmwaspresentedinse tion3.2.6.