A new strategy for the numerical modeling of a weld pool

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A new strategy for the numerical modeling of a weld pool

Yassine Saadlaoui, Eric Feulvarch, Alexandre Delache, Jean-Baptiste Leblond, Jean-Michel Bergheau

To cite this version:

Yassine Saadlaoui, Eric Feulvarch, Alexandre Delache, Jean-Baptiste Leblond, Jean-Michel Bergheau.

A new strategy for the numerical modeling of a weld pool. Comptes Rendus Mécanique, Elsevier,

2018, 346 (11), pp.999-1017. �10.1016/j.crme.2018.08.007�. �hal-01902116�

Contents lists available atScienceDirect

Comptes Rendus Mecanique

www.sciencedirect.com

Computationalmethodsinweldingandadditivemanufacturing/Simulationnumériquedesprocédésdesoudageetdefabricationadditive

A new strategy for the numerical modeling of a weld pool

Yassine Saadlaoui

^a

, Éric Feulvarch

^b,∗

, Alexandre Delache

^c

, Jean-Baptiste Leblond

^d

, Jean-Michel Bergheau

^b

aUniversitédeLyon,ENISE,UJM,LTDS,UMR5513CNRS,58,rueJean-Parot,42023Saint-Étiennecedex2,France bUniversitédeLyon,ENISE,LTDS,UMR5513CNRS,58,rueJean-Parot,42023Saint-Étiennecedex2,France cLMFAUMR5509CNRS,sitedeSaint-Étienne,UniversitédeLyon,UniversitéJean-MonnetdeSaint-Étienne,France

dUPMC,UniversitéParis-6,UMR7190,InstitutJean-Le-Rond-d’Alembert,Tour65-55,4,placeJussieu,75252Pariscedex05,France

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received6March2018 Accepted15May2018 Availableonline23August2018

Keywords:

Surfacetension Freesurface Finite-elementmethod Weldingprocess

Welding processes involve high temperatures and metallurgical and mechanical conse- quences that must be controlled. For this purpose, numerical simulations have been developed tostudythe effectsofthe processonthefinalstructure.Duringthe welding process, the material undergoes thermal cycles that can generate different physical phenomena,likephasechanges,microstructurechangesandresidualstressesand distor- tions.Buttheaccuratesimulationoftransienttemperaturedistributionsinthepartneeds tocarefullytakeaccountofthefluidflowintheweldpool.Theaimofthispaperisthus toproposeanewapproachforsuchasimulationtakingaccountofsurfacetensioneffects (includingboththe “curvature effect” andthe “Marangoni effect”),buoyancy forces and freesurfacemotion.

Theproposedapproachisvalidatedbytwonumericaltestsfromtheliterature:asloshing test andaplatesubjectedtoastaticheatsource.Then,the effects ofthe fluidflowon temperaturedistributionsarediscussedinahybridlaser/arcweldingexample.

©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Weldingisaveryoldassemblyprocessdatingfromthemetalage.Differenttechniqueshavebeendevelopedovertime:

assembliesbyheatingandhammeringatthebeginning,arcwelding,resistancewelding andmorerecently,electronbeam welding,andlaserwelding.Thelargeuseofweldingprocessesandtheneedtoimprovethemtoachieveabettercontrolof theweldingqualityledtothedevelopmentofseveralnumericalsimulationmethods,whichmayprovidevaluableassistance tostudytheeffectsofweldingonthefinalstructure.

Dependingon the objectivestargeted by thesimulation,different approacheshavebeen developed.Theseapproaches canbe classifiedintotwo groups,comprisingrespectivelythermo-mechanicalsimulations topredictthemetallurgicaland mechanicalconsequencesinducedbywelding,andsimulationsoftheprocessitselfincludingthefluidflowintheweldpool.

Industrialcompaniestook alargepartinthedevelopmentofthermo-mechanicalsimulations sincetheobjectivesofthese simulations are toestimate the residualstresses anddistortions[1–5], thus providingvery usefulinformationforfatigue lifetimepredictionsorweldingsequenceoptimization[6].Thethermalpartofthiskindofsimulationisgenerallysimplified

*

Correspondingauthor.

E-mailaddress:eric.feulvarch@enise.fr(É. Feulvarch).

https://doi.org/10.1016/j.crme.2018.08.007

1631-0721/©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense} (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Fig. 1.(a)Simplifiedschematicoftheweldingprinciple,(b)simplifiedschematicofthematerialstatesandfluidflowinthemoltenpoolduringthewelding process[23].

inordertoreducethecalculationtimesinceitisusedtosimulatelargestructures[7–9].Theheatinputisthenrepresented by anequivalent heatsource,theparametersofwhichmustbepreviouslyidentified onexperimentalheataffectedzones.

The thermo-mechanical computation is thereforeachieved intwo steps:a coupledthermal andmetallurgicalsimulation, followed bya mechanicalcomputation takingaccountofthetemperatures andmetallurgicalphasespreviously computed [10,11].

However,thistypeofsimulationisnottotallypredictiveasfarastemperaturedistributionsareconcerned.Theirpredic- tion indeedrequirescarefullysimulatingthefluidflow, whichplays an importantrole,particularlyonthemorphology of theweldpool.

Several authorshavethereforedeveloped numericalapproaches[12–22], theaim ofwhichis tosimulatethe physical phenomena involvedby theprocessitself(Fig.1).Among thesephenomena,thefluidflowinthemoltenpool,particularly relatedtothesurface tension,hasadirecteffectonthefieldtemperatureandweldpoolmorphology[24–26].Thisiswhy severalstudieswerecarriedouttomodelthesurfacetensioninthemoltenpool[27,28].Theeffectofthesurface tension on thefluid flowarisesfromtwofactors: first,surfacetensiongeneratesanormalforceontothefreesurface ofthefluid, proportionaltothecurvatureofthissurface(“curvatureeffect”),andseconditgenerates,throughatemperaturegradient,an additionaltangential forceontothefreesurface(“Marangonieffect”).Theclassicalimplementationmethodsofthesurface tension in finite-element codes consider these two effects in a naturalway. They includethe loads applied on the free surface ofthefluidamongtheexternalforcesoftheproblem[29,30].Inthistypeofapproach,theimplementationofthe tangential effectdoesnotraiseanyproblem.However,thenormaleffectisverydifficulttoincorporate,since itrequiresa directcalculationofthisforcethatdependsonthefreesurfacemotion[31,32].

Severalmethodshavebeendevelopedinordertomodelthefreesurface duringthefinite-elementcomputation. These methods can be classified into two categories: the Lagrangian approaches(Arbitrary Lagrangian Eulerian: ALE) [33] and Eulerian approaches(volumeoffluid: VOF[34] and level-set [35]).The twoEulerian methodscan compute the“keyhole collapse” andhelp predictdefect formationsuch asporosity [36,37]. However, takingintoaccount thesolid deformation using these two methods is very difficult.For this, the ALE approach is often used during the numerical simulation of welding [38–40].It consistsincalculatingthe displacementoftheboundary nodesphysicallyandthatof thebulk nodes arbitrarily. Thismeans that the solutionsinside the domainare computed withan Eulerianformulation. Thus, the mesh movementismodifiedinordertoimprovethemeshquality.

The aimofthispaperisto proposeanewapproachforthesimulationoftheweld pooltakingaccountofthesurface tensioneffects(includingboththe“curvatureeffect”andthe“Marangonieffect”),buoyancyforces,andfreesurfacemotion.

Thissimulationwillbeusedinfurtherstudiestosimulatetheinteractionbetweenthefluidflowandthesoliddeformation.

Forthispurpose,theapproach developedbyLeblondetal.[41] tomodelthesurface tensionisused.Ithasbeenapplied uptonowto simulatesimpleacademicbenchmarks.Inthiswork,afirstapplicationtoanindustrialproblemisproposed (weldingprocess). Thismodeling ofsurface tension consistsinincorporating the normalandtangential effects ina very simple andefficientwaycompared toclassical methods. Inaddition,a pragmaticALEapproach is developedto simulate thefreesurfacemotion.Thechoiceofthisapproachisbasedontwocriteria:thesimplicityofitsimplementation,andthe possibilitytotakeintoaccountthesoliddeformationinasimpleway.

Thepaperisorganizedasfollows:

– insection2,themodelingofthethermal-fluidproblemispresented,itcontainsthreeparts:anillustrationofthefluid andthermalformulation,themodelingofthesurfacetension,andthemodelingofthefreesurface;

– insection3,avalidationofthemodelingofthesurfacetensionandfreesurfaceiscarriedoutusingnumericalvalidation tests;

– insection4,theresultsofthethermal-fluidsimulationoflaser/archybridweldingarepresented.

2. Modelingthethermal-fluidproblem 2.1. Strongformulation

Let usconsider a bounded fluid domain Ω ofboundary ∂Ω. The coupled heat transfer and fluid flow problems are governedbythefollowingthreeequations.

Massconservationequation

Thefluidissupposedthermallydilatableandmechanicallyincompressible:thevolumevariationisonlyduetotempera- turevariations,thestresseshavenoeffect.FromtheEulerianstrainrateD=¹₂(grad v+(grad v)),letusdefinetheEulerian mechanicalstrainratebytheformulation:

Dm

=

^D

+ ρ ˙

3

ρ

^{I (1)}

Themassconservationequationthenwrites:

divv

+ ρ ˙

ρ ⁼

^tr

⁽

^Dm

^{) =}

^{0 (2)}

In practice, the variations of

ρ

are only due to the thermal dilation effects. It follows that ρ^˙

ρ ∼= −βT˙, where β is the coefficientofvolumetricexpansion.

Energybalanceequation

ρ ∂

H

∂

t

+

^v

·

^gradH

−

^div

(λ

gradT

) −

^r

=

^{0 (3)}

Here

ρ

isthedensity, H,thespecificenthalpy,v,thefluidvelocity,λ,theheatconductivity,T,thetemperature,andrthe volumeheatsource.Theenthalpy Hisafunctionoftemperatureincludingboththesensibleheatandthelatentheatdue tophasechange,and

ρ

andλcanbetemperature-dependent.Theassociatedboundaryconditionsare:

T

=

^T0 on

∂Ω

T (4)

λ

gradT

·

ⁿ

=

^{q on}

∂Ω

q (5)

T0 istheknowntemperatureon∂ΩT andqatemperature-dependentheatfluxon∂Ωqwhoseoutsideunitnormalisn.

Momentumequations(Navier–Stokesequations)

−

^gradp

+

^{div s}

+

^fv

= ρ ∂

v

∂

t

+

^{grad v}

·

^v

(6) The fluid flow is supposed to be laminar. Here p is the fluid pressure, s, the stress deviator, and fv, the volume force.

Buoyancyforcescanbeconsideredusingthevolumeforceswhicharegivenbythefollowingequation:

f_v

= ρ

₀

1

− β(

T

−

^T0

)

g (7)

gbeingthegravitationalaccelerationand

ρ

0themassperunitvolumeatthereferencetemperatureT0.

Thefluidissupposed tobehaveasaNewtonianfluid.TherelationbetweentheEulerianmechanicalstrainrateD_m and thestressdeviatorscanthereforebewrittenas:

Fig. 2.P1+/P1 element.

s

=

²

μ

D_m (8)

where

μ

isthefluidviscosity.

Theassociatedboundaryconditionsare:

v

=

v^d on

∂Ω

v (9)

σ _·

n

= ϕ

^d on

∂Ω

_ϕ (10)

Here v^d and

ϕ

^dare theprescribedvelocityandsurface forceon∂Ωv and∂Ω_ϕ respectively,and

σ

=^s−^{pI istheCauchy} stresstensor.

2.2. Finiteelementformulation

Theweakformulationoftheenergybalanceequation aswellasthefiniteelementformulationofthethermalproblem arequiteclassical,anddetailscanbefoundin[42,43].Wethereforefocusonthefluidproblemformulationinthesequel.

Thefiniteelementformulationofthefluidproblemrestsonamixedvelocity–pressure(v,p)variationalformulation.

Theproblemisnowtofind(v,p)∈(Vad×Pad),∀^t∈ [⁰,T]^,suchas∀(v^∗,p^∗)∈(V_ad⁰ ×Pad):

−

Ω v^∗

ρ

∂

v

∂

t

+

^{grad v}

·

^v

dv

−

Ω

D^∗

:

²

μ

Dmdv

+

Ω div

v^∗

pdv

+

Ω

v^∗

·

^fvdv

+

∂Ωϕ

v^∗

· ϕ

^dds

=

⁰

Ω p^∗

divv

+ ρ ˙ ρ

dv

=

0

(11)

withtheinitialconditionv(x,t=⁰)=^v0(x). InEqs. (11),D^∗=¹₂(grad v^∗+(grad v^∗))and:

Vad

= {

^v

∈

^H1

(Ω) |

^v

=

^vdon

∂Ω

v

}

V_ad⁰

= {

^v

∈

^H1

(Ω) |

^v

=

^0on

∂Ω

v

}

Pad

= {

^p

∈

^L2

(Ω) }

P1+/P1 tetrahedralelements(Fig.2)areusedforthediscretizationofEqs. (11).

Velocityapproximation

The velocityfieldisdecomposedintoapartvs calculatedfromthevelocitiesofthevertexnodesgatheredinthearray V_s,usingtheshapefunctionsN_s oftheclassicalfour-node tetrahedron[42],plusanotherpartv_b calculatedfromavector _bofadditionaldegreesoffreedomofthebubble(internal)nodeandtheassociatedshapefunctionN_b:

v

=

^vs

+

^vb (12)

with

v_s

=

^Ns

.

V_s

=

4

n=1

N_snV_sn v_b

=

N_b

.

_b

(13)

withNb=^minn=^1to4Nsn.

Notethat b doesnot representthevelocity ofthecentroidoftheelementbecausetheshape functionsofthevertex nodesNsdonotvanishatthispoint.

The virtual velocity v^∗ isdecomposedin thesame way.Notethat the “bubblevelocity” vb vanishes oneach element boundary∂Ω^e (bubblenodeprinciple)andthereforeatanypointof∂Ω.

Pressureapproximation

ThepressureiscalculatedfromthevaluesofthevertexnodesPs:

p

=

^Ns

.

Ps

=

4

n=1

Ns_nPs_n (14)

andsoisalsothetemperature.

The Eulerianmechanical strain rateD_m,aswell asits virtual counterpart D^∗_m, are then decomposedin thefollowing form:

Dm

=

Dms

+

Dm_b

D_ms

=

^Ds

+ ρ ˙

3

ρ

^I

⁼

1 2

grad v_s

+ (

grad v_s

)

+ ρ ˙

3

ρ

^I

D_mb

=

^Db

=

¹ 2

grad v_b

+ (

grad v_b

)

(15)

Equations(11) with(8),(12) and(15) canthereforeberewritten,foranyv^∗_s,v^∗_b,andp^∗:

−

Ω v^∗_s

ρ

∂

v

∂

t

+

grad v

.

v

dv

−

Ω

D^∗_s

:

2

μ

Dmsdv

−

Ω

D^∗_s

:

2

μ

Dm_bdv

+

Ω div

v^∗_s

pdv

+

Ω

v^∗_s

·

^fvdv

+

∂Ωϕ

v^∗_s

· ϕ

^dds

=

⁰

−

Ω v^∗_b

ρ

∂

v

∂

t

+

^{grad v}

.

v

dv

−

Ω

D^∗_b

:

²

μ

Dm_sdv

−

Ω

D^∗_b

:

²

μ

Dm_bdv

+

Ω div

v^∗_b

pdv

+

Ω

v^∗_b

·

^fvdv

=

⁰

Ω p^∗

div

(

v_s

+

^vb

) + ρ ˙ ρ

dv

=

⁰

(16)

Ds andD^∗_s areconstantover eachelementΩ^e astheyinvolvethederivativesoftheshape functionsoftheclassicalfour- node tetrahedron. Assuming that ρ^˙

ρ and

μ

are constant over each element (in practice ρ^˙

ρ and

μ

are calculated at the centroidofthetetrahedron),Dms isalsoconstantovereach element.Since v_b vanisheson theelementboundary∂Ω^e,it followsthat:

Ω^e

D^∗_s

:

²

μ

Dm_bdv

=

²

μ

D^∗_s

:

Ω^e

Dm_bdv

= μ

D^∗_s

:

Ω^e

grad v_b

+ (

grad v_b

)

dv

= μ

D^∗_s

:

∂Ω^e

n

⊗

^vb

+

^vb

⊗

^{n ds}

=

⁰

(17)

Inthesameway,onecanshowthat:

Ω^e

D^∗_b

:

²

μ

D_msdv

=

^{0 (18)}

Animplicitbackwardschemefor^∂_∂^v_t andalinearizationoftheconvectiontermareusedforthetimeintegrationofEqs. (16).

Itisalsosupposedthatthebubblenodedoesnotcontributetotheacceleration(justthevertexnodesaretakenintoaccount fortheinterpolationofacceleration).

UsingEqs. (17) and(18),Eqs. (16) canthereforeberewrittenatthefinalinstantofanytime-step[t,t+t],foranyv^∗_s, v^∗_b,andp^∗:

−

Ω v^∗_s

ρ

vs

−

^vts

t

+

^grad

(

vs

+

^vb

) ·

^vt

dv

−

Ω

D^∗_s

:

²

μ

Dmsdv

+

Ω div

v^∗_s

pdv

+

Ω

v^∗_s

·

^fvdv

+

∂Ωϕ

v^∗_s

· ϕ

^dds

=

⁰

−

Ω v^∗_b

ρ

v_s

−

^vt_s

t

+

^grad

(

v_s

+

^vb

) ·

^vt

dv

−

Ω

D^∗_b

:

²

μ

D_mbdv

+

Ω div

v^∗_b

pdv

+

Ω

v^∗_b

·

^fvdv

=

⁰

Ω p^∗

div

(

vs

+

vb

) + ρ ˙ ρ

dv

=

0

(19)

whereallthequantitiesareconsideredattimet+t,exceptforv^t,whichrepresentsthevelocityattimet.

UsingnowtheVoigtnotationfortheEulerianstrainrateanddefining,from(13) and(15),Bs andBbmatricessuchthat D_s=^Bs·^VsandD_b=^Bb·_b,Eqs. (19) leadtothefollowingexpressionsofthenodalresidualvectors:

T_s

= −

Ω

N_s

·

^vs

−

^vts

t

ρ

dv

−

Ω

N_s

· ρ

grad

(

v_s

+

^vb

) ·

^vtdv

−

Ω

B_s

·

²

μ

D_msdv

+

Ω

(

grad N_s

)

pdv

+

∂Ωϕ

N_s

· ϕ

^dds

+

Ω

N_s

·

^fvdv

=

⁰

T_b

= −

Ω

N_b

·

^vs

−

v^t_s

t

ρ

dv

−

Ω

N_b

· ρ

grad

(

v_s

+

^vb

) ·

^vtdv

−

Ω

B_b

·

²

μ

Dm_bdv

+

Ω

(

grad N_b

)

pdv

+

Ω

N_b

·

^fvdv

=

⁰

T_p

=

Ω N_s

div

(

v_s

+

^vb

) + ρ ˙ ρ

dv

=

⁰

(20)

Note that thesecond equation ofsystem(20) defines,at theelement level,thefollowing linearrelationbetween Vs,_b andPs:

K_bs

·

^Vs

+

^Kbb

·

b

+

^Kbp

·

^Ps

−

^Fb

=

^{0 (21)}

with:

K_bs

=

Ω

N_b

· ρ

grad N_s

·

^vtdv

+

Ω N_b

· ρ

tN_sdv K_bb

=

Ω

B_b

·

²

μ

B_bdv

+

Ω

N_b

· ρ

grad N_b

·

^vtdv

K_bp

= −

Ω

(

grad N_b

)

·

Nsdx

F_b

=

Ω

N_b

·

^fvdv

+

Ω N_b

·

^vts

t

ρ

dv

(22)

Equations(20) aresolvedusingaNewton–Raphsonmethod.Aftereliminationof_bfromEq. (21),ateachiterationcorrec- tions δVs andδPsarecalculatedthroughthefollowingequations:

Kss

−

^Ksb

·

^K−_bb¹

·

^Kbs Ksp

−

^Ksb

·

^K−_bb¹

·

^Kbp

K_ps

−

^Kpb

·

^K−_bb¹

·

^Kbs

−

^Kpb

·

^K−_bb¹

·

^Kbp

δ

V_s

δ

P_s

=

T_s

T_p

(23) with:

K_sb

=

Ω

N_s

· ρ

grad N_b

.

v^tdv

Kss

=

Ω

B_s

·

²

μ

Bsdv

+

Ω

N_s

· ρ

grad Ns

·

^vtdv

+

Ω N_s

· ρ

tgrad Nsdv K_sp

= −

Ω

(

grad N_s

)

p

·

^Nsdx

(24)

K_ps

=

^Ksp

K_pb

=

^K_bp ⁽²⁵⁾

The processis repeatedaslongas^Ts>

ε

s and^Tp>

ε

p,where

ε

s and

ε

p are precisionthresholds prescribed bythe user.

2.3. Modelingthesurfacetensioneffects

Oneofthemaineffectstobeaccountedforinweldpoolsimulationsisrelatedtosurfacetension.Surfacetensioninduces apressurenormalto thefreesurface ofthe fluid,proportionaltothecurvatureofthissurface,anditsgradient generates a tangential surfaceforce (Marangoni effect).As farastheimplementation ofthesurface tensioninfinite-element codes [44,45] isconcerned,thesetwoeffectsaretakenintoaccountinanaturalwaythroughtheboundaryconditionsasdefined inEq. (7) byincludingtheloadsappliedonthefreesurfaceofthefluidamongtheexternalforcesoftheproblem.Insuch anapproach,iftheMarangonieffectcanbeintroducedinastraightforwardway,theimplementationofthecurvatureeffect is farmore complicated, asit requires the preliminarycalculation of themean curvature (traceof thecurvature tensor) ofthefreesurface [31,32].Butthepositionofthefreesurface isgivenbythefluid motion,andthecurvaturecalculation requiresthe evaluationofthe spatial secondderivatives ofthedisplacement field. Thiscalculation canbe achievedquite accuratelyusingC¹elements,buttheseelementsareveryheavyandtime-consuming.WhenclassicalC⁰elementsareused, anotherwayconsistsincalculatingthe divergenceoftheunit normaltothe freesurface, andthespatial firstderivatives ofthe shapefunctions[46,47].Butthe calculationofthe unit normal,generallyobtainedby averaging thenormalofthe surfaceelementsconnectedtoeachnodeconsidered,caninducesignificanterrorsintheresults.

Inthisstudy,we usetheefficientapproachsuggestedin[27,44,45] forthecurvatureeffectandgeneralizedin[41] for both the curvature andMarangoni effects. The approach is basedon the equivalence betweenthe effects of the surface tensionandthoseofafictitiousstressedmembranebondedtothefreesurface.Thesurfacetensionisthereforetakeninto accountthroughasurfacehydrostaticstateofstress

τ

=

γ

0 0

γ

,where

γ

isthesurfacetension,appliedtomembrane-type (skin)elements (Fig.3)thatdonothaveanymaterialproperties[41].Thetransmissionofthesurfacetensioneffectsfrom theskinelementstothe3Dfluidelementsiscarriedoutnaturallythroughthecommonnodesofthe2Dand3Delements.

Theweak formulationofthefluidproblemisnowmodified by addingthe term−

∂ΩfreeD^∗_s:

τ

ds,where∂Ω_free isthe freesurfaceofthefluid,intheleft-handsideofthefirstofEq. (19),whichisnowwritten:

−

Ω v^∗_s

ρ

v_s

−

^vt_s

t

+

^grad

(

v_s

+

^vb

) ·

^vt

dv

−

Ω

D^∗_s

:

²

μ

D_msdv

+

Ω div

v^∗_s

pdv

−

∂Ωfree

D^∗_s

: τ

ds

+

Ω

v^∗_s

·

fvdv

+

∂Ωϕ

v^∗_s

· ϕ

^dds

=

0

(26)

The proposedapproach doesnotrequirecalculating themeancurvatureofthe freesurface, whichmakes it efficientand accurate.Inaddition,itispossibletoaccountforbothnormalandtangentialeffectsinanaturalandsimplewaybysimply consideringtemperature-dependentvaluesof

γ

.Itissufficienttoconsiderthecorrectvalueofthesurfacetensionateach Gausspointofthemembraneelements.Thisallowsustoavoidthenumericalproblemsofclassicalapproaches.

2.4. Modelingthefreesurfaceevolution

Totake the free surface evolution into account, differentmethods havebeen proposed. Combined withfullyEulerian formulations,thevolumeoffluid[34] orlevelset[35] techniquesaregenerallyused.Alternatively,theArbitraryLagrangian–

Eulerian(ALE)approachconsistsinimposingthefreesurfacemotiontothemesh[38–40].Butinordertokeepagoodmesh quality, thenodesinside theweldpool mustinturnbe repositioned, thusdefining ameshmotionwhich mustbe taken intoaccountinthefiniteelementformulation.

The algorithm used in the presentwork is summarized in Fig. 4.The first stepconsists in recovering the mesh and the results for a given time step of the thermal-fluid computation. Here, a strong coupling between fluid andthermal

Fig. 3.Mesh containing skin elements (red 1D elements with sky blue nodes).

Fig. 4.Proposed algorithm of the ALE approach.

computations isused.Then thestatus ofeachnode (freesurface, fluid, solid)isdefined. Thefree-surfacenodes arethen displaced inordertofollowthefreesurface motion.The fluidnodesinsidethemoltenpoolare finallyrepositioned using anelasticcalculationwherethesolidnodesremainfixed.

TheelementshavingameantemperatureabovethemeltingtemperatureT_liqareconsideredasfluidelements,theothers as solid elements.For each node,depending on its positionandthe statusof theelements it isconnected to,four-node statusescanbedefined:

– solidnode(rednode):connectedtoatleastonesolidelement;

– solid/free-surfacenode(purplenode)–solidnodeconnectedtooneoftheskinelementsdefiningthefreesurface;

– fluidnode(greennode):connectedonlytofluidelements;

– fluid/free-surfacenode(skybluenode):fluidnodeconnectedtooneoftheskinelementsdefiningthefreesurface.

Thenormalatafree-surfacenodeisdefinedasthemeanvalueofthenormaloftheskinelementscontainingit(Fig.5).

The normalvelocitiesandthedisplacements ofthefree-surfacenodesare thencalculated usingthefollowingexpres- sions:

v_nk

(

t

+

t

) =

n_k

·

^vk

(

t

+

t

)

·

ⁿk (27)

Fig. 5.Definition of the normal at a node of the free surface and projection of material velocity.

Fig. 6.The mesh before updating the node positions.

Fig. 7.The mesh after updating the node positions.

un_k

(

t

+

t

) =

^un_k

(

t

) +

^vn_k

(

t

+

t

) ·

t (28)

Here v_nk and u_nk are respectivelythe normalvelocity andthe normaldisplacementof thefree-surfacenode k,v_k isthe velocityofthisnode,andn_kisitsnormal(Fig.6).

Once the displacements of the free-surface nodeshave been calculated, differentmethods can be used to reposition theinternalfluidnodes.Forexample,Heuzéetal.[48] usedabarycentricreplacement.Inthepresentapproach,anelastic computation is carried out inorder to findthe arbitrary displacements ofthe fluid nodes.In thiscomputation, the dis- placementsofthesolidnodesandthefree-surfacenodesareprescribed.Finally,themeshisupdatedusingthecomputed displacements(Fig.7).

Allthesestepsareperformedateachtimestepofthethermal-fluidcomputation.

Remark. For the sake of simplicity, the mesh velocity has not been included in the presentationof the finite element formulationabove.Butofcourse,themeshvelocityistakenintoaccountinalltheadvectiontermssuchasgrad(vs+^vb)·^vt inEq. (19),whichisnowwrittengrad(v_s+^vb)·(v^t−^vt_mesh),wherev^t_meshisthemeshvelocity.

Fig. 8.Schematization and mesh of the sloshing problem.

3. Validationtests 3.1. Sloshingproblem

The sloshingproblemwasdevelopedbyRamaswamyetal.[49] tovalidate anALEformulation.Itconsistsinfollowing the oscillations of a liquid ina container. Sloshing was initially considered to be dueonly to gravity. This problemwas generalizedbyother authorsbyaccountingformorephysicalphenomena.Forexample,Popinetetal.[50] havesimulated in2Dtheoscillationsproducedbythesurfacetension.Theyhavecomparedtheirnumericalresultswiththeanalyticresults givenbyProsperettietal.[51].TheproblemwasconsideredlaterbyEl-Sayedetal.[25] usinga3Dmesh.

Inthiswork, thesloshingproblemwas usedtovalidatethe ALEapproachofthefree surfaceandthenormaleffectof thesurfacetension.Theaimofthenumericalmodelistofollowtheoscillationsofasingle liquidinacontainerthathasa squaregeometry.Initially,thefreesurfaceoftheliquidhasasinusoidalshapegivenbythefollowingequation:

H

(

x

) =

¹

+

^a0sin

π (

0

.

5

−

^x

)

(29) wherea0=⁰.01 istheinitialamplitude.Inthisequationalldistancesareexpressedinm.

Withregardtotheboundaryconditions,thebottomsurfaceisclampedandthenormalvelocitiesaresuchthatv·ⁿ=⁰ on the side walls. The free surface is thetop one. The mesh (Fig. 8) contains 4223nodesand 19,736 linear tetrahedral elements. Thefreesurface ismeshed with1002Dtriangular“skin”elements (redelements inFig. 8)toapplythesurface tension.The liquidpropertiesare: relative density

ρ

₌1 kg·^{m−3,kinematicviscosity}

η

₌0.01 m²·^{s−1,andgravity g}= 1m·^{s−2 (artificialvalue).}

Afirstsimulationiscarriedouttofollowthesloshingofaliquidundertheeffectofitssoleweight(gravityeffect).The resultsofthissimulationandtheanalyticalsolution[51] areprovidedbyFig.9.

Regarding the other simulation,the sloshingof the liquid is generated by its weight as well asthe surface tension.

Values ofthesurface tensionbetween0.001and1 N·^{m−1 areused.The resultsshow}proportionalitybetweenthevalue of the surface tensionand the intensityof liquid sloshing(Fig. 10). The comparisonbetween the numericalresults and the analytical solution shows a good correlation:the maximal error is lessthan 0.006%. We note that the value of this erroris verysimilarto thosefoundinthe literature.Forexample,Rabieretal.[32] havefound amaximal errorequalto 0.005%.However,theymeshed withadiscretizationtwicefinerthanours.Inadditiontothat,theyusedatimestepequal to0.0025 s,thatis ¹₈ ofthevalueusedinoursimulation(0.1 s).

3.2. Benchmarkproblem

TheBenchmarkconsistsofsomenumericalsimulationofthespotwelding“GTWA”process(Fig.11).Itwasdevelopedin a2DaxisymmetricoptionbyGirardetal.[52],ina3DoptionbyHamideetal.[53] andEl-Sayedetal.[25] tovalidatethe tangentialeffectofthesurfacetension.Inthiswork,apurelythermal-fluidcomputationofthisproblemwascarriedouton the quarterofa disk(Fig.11).A viscositydependingonthe temperatureallowsseparatingthefluidandsolid zones.The solidisthusschematizedasafluidwithahighviscosity(

μ

=^106kg·^m−1·^s−1).Themeshincludes68,130lineartetrahedral elements,762skinelementsand13,617nodes.Thelengthoftheelementsedgesis30 μminthefluidzoneandmorethan 100 μminthesolidzone.

The material used issteel, supposed to be isotropic, homogeneous, Newtonian, thermally dilatable,and mechanically incompressible.ItspropertiesaregiveninTable1.

Fig. 9.The displacement of the free surface under the effect of the liquid weight.

Fig. 10.The displacement of the free surface under the effect of the liquid weight and the surface tension;γ=⁰.001 N·^m−1andγ=^{1 N}·^m−1.

Fig. 11.Schematization [52] and mesh of the benchmark problem.

Withregardtotheboundaryconditions,thenodevelocitiesofthefreesurfaceandsymmetricboundariesaresuchthat v·ⁿ=^{0.Theheatfluximposedontheuppersurfaceofthediskisthesumofthreeterms:}

∅

total

= ∅

source

− ∅

convection

− ∅

radiation (30)

where:

∅

source

(

x

) =

^P 2

π

R²₀exp

−

^R2 2R²₀

(31)

Table 1

Characteristicsofthematerial.

Solidus temperature TS=^{1696 K}

Liquidus temperature TL=^{1740 K} Relative density (solid) ρ=^{7500 kg}·^m−3 Specific heat (solid) c=^{602 J}·^K−1·^kg−1 Thermal conductivity (solid) λ=^{24 W}·^m−1·^K−1 Relative density (liquid) ρ=^{6350 kg}·^m−3 Specific heat (liquid) c=^{695 J}·^K−1·^kg−1 Thermal conductivity (liquid) λ=^{20 W}·^m−1·^K−1 Dynamic viscosity μ=².5·^10−3kg·^m−1·^s−1 Coefficient of expansion β=10⁻⁴K⁻¹

Fig. 12.Temperature evolution with positive derivative of surface tension,^∂γ_∂T >0.

withR0=³·^{10−3m and P}=^{790 W.}

∅

radiation

= σ

0

T⁴

−

^T0⁴

(32)

with

σ

0 theStefan–Boltzmannconstant, theemissivityequalto0.5,andT0=^{300 K.}

∅

convection

=

^h

(

T

−

^T0

)

(33)

withhthetransfercoefficientequalto15 W·^m−2·^K−1.

Thelateraledgesofthediskareadiabatic,andalineartemperaturedecreasefrom800to400 Kisenforcedonitslower side,fromthecentertotheedge(Fig.11).

Duringthissimulation,onlythetangentialeffectofthesurfacetensionistakenintoaccount,withoutthenormaleffect.

Twovalues ofthederivative ofthesurface tension, ^∂γ

∂T =^{10−4 N}·^m−1·^{K−1 and ∂}_∂^γ_T = −^{10−4 N}·^m−1·^{K−1, areimposed on} the skinelements ofthe upperface ofthedisk. The firstthree secondsof theprocess are simulated,andthe time step is 0.1s during theformationoftheweldpool (thermalcomputation)and2·^{10−4 s after theactivationofthefluid flow} (thermal-fluidcomputation).

A comparisonbetweenthe Benchmarkandournumericalresultsiscarriedout. Itconsistsinexamining thefollowing quantities:

– temperatureontheradiusoftheupperfaceT(z=⁰). – radialvelocity vr(z=⁰).

Figs. 12and13showrespectivelytheevolutionofthetemperaturefield andthevelocityvector withapositivederivative of the surface tension with respect to temperature. Figs. 14 and 15 illustrate the caseof a negative derivative. A good correlation betweenourresultsandthebenchmark resultsisfoundin bothcases.Withregard tothebenchmark results, weusetheaveragevalueoftheresultsgivenbyfivecomputation codes[25,53].Amaximalerrorlessthan2%isfoundfor thetemperature. Regardingthevelocity, amaximalerrorof10%isfound.Themagnitudesofthetemperatureandvelocity arealsocheckedonthesymmetryaxisandhereagain,agoodcorrelationwithprevioussimulationsisfound.

Fig. 13.Velocity evolution with positive derivative of surface tension,^∂γ_∂T>0.

Fig. 14.Temperature evolution with negative derivative of surface tension, ^∂γ_∂T<0.

Fig. 15.Velocity evolution with negative derivative of surface tension, ^∂γ_∂T<0.

Fig. 16.Finite-element model: mesh.

4. Numericalsimulationofthehybridweldingprocess 4.1. FEmodel

In thissection,a 3D thermal-fluidsimulationofhybridlaser/arc welding ispresentedusingthe SYSWELD^® code [54].

It takesintoaccount thefluid flowthrough thesurface tensionandthebuoyancy.Both equivalent doubleellipsoidaland Gaussian surfaceheatsources[55] areappliedintheY directiontoametalblock 30 mminlength,10 mminwidth,and 20 mm inheight. Themodelhasa symmetryabouttheY Z plane(Fig. 16).Itisnotedthatthemeshistakensufficiently largeinorderthatthemoltenpooldoesnotreachtheboundaryofthedomaincomputation.Thismakesitpossibletoavoid theproblemsrelatedtothedefinitionofthenormalatnodesofthemeshboundaries.Onlyhalfofthestructureismodeled, andasingle-passhybridweldisconsideredwithaweldingspeedequaltov_welding=^6mm·^{s−1.Themesh(Fig.16)includes} 77,603 lineartetrahedralelements(P1+/P1),10,550lineartriangularelements (skinelements),and14,922 nodes.Inthe fluid zone, the elementsize is 500μm,and itcan reach 5000μmin the solidzone. The multiphasic ferritic steelgrade S355J2G3isused.Allthedatapertainingtothismaterialaregivenin[25].

Tostudytheeffectofthefluidflowonthetemperaturefieldandtheweldpoolmorphology,twosimulationsarecarried out. The first simulationis a thermal-fluid computation (with fluid flow). Here, the heating time is 1.4 s, including 1s for thecreationof themolten pool(thermal computation withstaticheatsources andwithoutfluid flow), and0.4s for the thermal-fluidsimulation (with fluid flow)with the motion of heat sources. The time step is equal to 0.1s for the creationofthemoltenpooland2.5·^{10−4 s aftertheactivationofthefluidflow.A}temperature-dependentsurfacetension

γ

=

γ

L+

α

(T−^TL) with

γ

L=⁰.15N·^{m−1, T}L=^1500◦C,and

α

=^{10−4 N}·^m−1·^{k−1 is consideredto includeboth the} normalandMarangonieffects.Thesecondsimulationisapurelythermalsimulation(withoutfluidflow)during1.4s.

In both simulations, heat exchangesdue toconvection andradiation on the boundarywith theexternal medium are takenintoaccount:

∅ = ∅

convection

+ ∅

radiation (34)

with

∅

convection

=

^h

(

T

−

^T0

)

(35)

∅

radiation

= σ

₀

T⁴

−

^T0⁴

(36) whereh=^40W·^m−2·^{K−1isthetransfercoefficient,T}0=^{293 Ktheroom}temperature,

σ

0 theStefan–Boltzmannconstant, and =⁰.7 designatestheemissivity.Theseconditionsareenforcedonallmeshfacesexceptforthesymmetryplane(Y Z), whereaninsulatingconditionisimposed.

4.2. Resultsanddiscussion

The thermal-fluidsimulation gives the distribution of the thermal field and fluid flow during the welding processes.

In thissimulation, the solid isschematized asa fluid witha high dynamic viscosity

μ

₌10⁶ kg·^m−1·^{s−1.The fluid is} supposed to be Newtonian, with a dynamic viscosity

μ

=².5·^{10−3 kg}·^m−1·^{s−1. The fluid flow is taken intoaccount} throughthesurfacetension(normalandtangentialeffects)andthebuoyancy.Thefluidflowisactivatedatt=^1s,thetime necessarytoformasufficientquantityoffluidintheweldpool.

Inthissection,acomparisonoftheresultsofthepurely thermalandthethermal-fluidsimulationsare given(Fig. 17).

Then, some illustrations oftheresults ofthe thermal-fluidsimulation aregiven(Figs. 18–21). Fig. 17illustrates thefield

Fig. 17.Temperature field att=¹.12 s (YZplane).

Fig. 18.Temperature field, and velocity vector of the fluid flow att=¹.02 s (YZplane).

temperatureforpurelythermalandthermal-fluidcomputationsatt=¹.12 s.Figs.18and19showrespectivelythetemper- aturefieldandthevelocityfieldofthefluidflowattwoinstants(t=¹.02 s,t=¹.13 s),thefirstwhentheheatsourcesare fixed,andthesecond duringthemotionoftheheat sources.Figs.20and21illustratetheevolutionofthefreesurfacein theweldpoolatthesametimes(t=¹.02 s,t=¹.13 s).Here,thedeformationofthefreesurfaceisinducedbythesurface tensionandthebuoyancyinthemoltenpool.

Theeffectofthefluidflowonthetemperaturefieldisverysignificant(Fig.17).Withregardtothethermal-fluidresults, we noteasignificant temperaturedrop.Thisisexplainedby thepositivederivative ofthesurfacetensionwithrespectto temperaturewhichgeneratesa penetratingweldpool,bringingthecoldmaterialfromtheedgetowardsthecenterofthe weldpool.

Duringthethermal-fluidsimulation,anaveragevelocityapproximatelyequalto0.3m·^{s−1 isfound.Thisvalueisconsis-} tentwithliteraturevalues[56,57],whereasamaximalvalueofvelocityequalto0.6m·^{s−1 isnotedwhentheheatsources} begintomove.Thisincreaseofvelocityislinkedtothehigh-temperaturegradient.Indeed,thegradientofthesurfaceten- sionisproportionaltothetemperaturegradient:therefore,whenthetemperaturegradientincreases,thevelocityincreases.

Also, thehigher ^∂γ

∂T, thehigher theMarangoni numberand thevelocity ofthe fluid flow. Thetwo effects ofthe surface tensionwith apositive ^∂γ

∂T are clearly visible inFigs. 18and19.These figures show that thefluid flow movesfromthe edgeofthemoltenzone toitscenter.Thus,a penetratingvortexisproduced.Thisvortexisresponsibleforthe changein theweldpoolmorphology.Tables 2and3showtheweldpoolmorphologyatdifferenttimesofthe“purely”thermaland thermal-fluidcomputations.Maximal gaps equalto15% forthelength, 9.6%forthe heightand1.6% forthewidthofthe weldpoolarefound.Apositive ^∂γ

∂T givesadeeperandnarrowerweldpool.Thiseffectofthefluidflowonthetemperature

Fig. 19.Temperature field, and velocity vector of the fluid flow att=¹.13 s (YZplane).

Fig. 20.Vertical displacement (Uz) of the free surface att=¹.02 s (YZplane).

fieldandtheweldpoolmorphologycaninfluencethemicrostructureandthemechanicalcomputations.Duringasimulation takingintoaccount onlythebuoyancyeffect(without thetensionsurface effects), amaximal valueequalto 0.03 m·s⁻¹ isfound.Therefore,itisabouttentimessmallerthanthatgeneratedbythesurfacetension.Thisshowsthatthefluidflow linkedtothesurfacetensionspeciallythetangentialeffect“Marangonieffect”isdominantinthiskindofprocess.Regarding thefreesurface,themaximalvalueoftheverticaldisplacementis0.5 mm,whentheheatsourcesarefixed(Fig.20),andit reaches0.8 mmduringthemotionofthesources(Fig.21).Thesevaluesaresimilartotheliteratureresults[18,58].

Fig. 21.Vertical displacement (Uz) of the free surface att=¹.13 s (YZplane).

Table 2

Weldpoolmorphologyatt=¹.05 s.

[mm] Thermal-fluid simulation Purely thermal simulation Gap

Length 10.85 10.88 0.3%

Width 4.78 4.86 1.6%

Height 2 1.93 3.5%

Table 3

Weldpoolmorphologyatt=¹.4 s.

[mm] Thermal-fluid simulation Purely thermal simulation Gap

Length 13.7 16.14 15%

Width 4.78 4.86 1.6%

Height 3 2.71 9.6%

5. Conclusion

Inordertostudy theinteractionbetweenthefluidflow andthesolid deformationduring thenumericalsimulationof theweldingprocess,a newtransient formulationforthenumericalmodeling ofaweldpool wasdeveloped.Itconsistsin consideringthe fluidflowinthe weldpoolthroughthe surfacetensionandthebuoyancy.Inthiscontext,anewmethod was usedto simulatethesurface tension.Itincorporates boththe normalforce exertedonto thefree surface (“curvature effect”)andthetangential effect(“Marangonieffect”)inaverysimpleandefficientwaycomparedwithclassicalmethods.

However,takingintoaccountthenormaleffectofthesurfacetensionstill requiresthesimulationofthefreesurface.That isthereasonwhya newALEapproachwas developedformodeling theevolution ofthefree surface.Thismethodallows takingintoaccountthesoliddeformationinasimpleway,whichisnotthecaseforEulerianmethods.

This work was validated using two numerical examples. The first one is the sloshing test; it consists in simulating the oscillationsofa liquidin acontainer, andit allowed validatingthenormal effectofthe surface tensionandtheALE approach.Thesecondoneisthebenchmarkproblem,itrepresentsthenumericalsimulationofspotwelding“GTWA”,and itallowedvalidatingthetangentialeffectofthesurfacetension.

After validation, the 3D thermal-fluid simulation ofa hybrid laser/arc was carried out. It gives good resultsin terms of the velocity ofthe fluid flow andthe evolution of the free surface compared to the literature results. Tounderstand theeffectofthefluid flowduring thesimulation ofhybridwelding,acomparisonofresultsbetweenpurely thermaland thermal-fluidsimulationswas performed.Itshowsthesignificantinfluenceofthefluidflowonthetemperaturefieldand the morphology of the weld pool. Forexample, we note a significant temperaturedrop in thethermal-fluid simulation.

Thismaybeexplainedbythepositivederivative ofthesurfacetensionwithrespecttotemperature,whichbringsthecold