An interpolation-free ALE scheme for unsteady inviscid flows computations with large boundary displacements over three-dimensional adaptive grids

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Contents lists available atScienceDirect

Journal

of

Computational

Physics

www.elsevier.com/locate/jcp

An

interpolation-free

ALE

scheme

for

unsteady

inviscid

flows

computations

with

large

boundary

displacements

over

three-dimensional

adaptive

grids

B. Re

a

,

∗

_,

_{C. Dobrzynski}

b

,

c

_,

_{A. Guardone}

a

a_{DepartmentofAerospaceScienceandTechnology,PolitecnicodiMilano,20156Milano,Italy} b_{BordeauxINP,IMB,UMR5251,F-33400,Talence,France}

c_{InriaBordeauxSud-Ouest,TeamCARDAMOM,F-33405Talence,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received3October2016

Receivedinrevisedform12March2017 Accepted15March2017

Availableonline6April2017

Keywords:

Meshadaptation ALEscheme

Finitevolumediscretization Tetrahedralgrid

Eulerequations

AnovelstrategytosolvethefinitevolumediscretizationoftheunsteadyEulerequations within the Arbitrary Lagrangian–Eulerian framework over tetrahedral adaptive grids is proposed.The volume changesdue tolocal mesh adaptation are treatedas continuous deformationsofthe finitevolumesand they aretakenintoaccount byaddingfictitious numericalfluxestothe governingequation.Thispeculiarinterpretationenablestoavoid anyexplicit interpolation of the solution between different grids and to compute grid velocities so that the Geometric Conservation Law is automatically fulfilled also for connectivity changes. The solution on the new grid is obtained through standard ALE techniques,thus preservingthe underlyingschemeproperties,suchas conservativeness, stability and monotonicity. The adaptation procedure includes node insertion, node deletion,edge swappingand points relocationand it is exploited bothto enhancegrid qualityaftertheboundarymovementandtomodifythegridspacingtoincreasesolution accuracy.Thepresentedapproachisassessedbythree-dimensionalsimulationsofsteady andunsteadyflowfields.Thecapabilityofdealing withlargeboundarydisplacementsis demonstratedbycomputingtheflowaroundthetranslatinginfinite- andfinite-spanNACA 0012wingmovingthroughthedomainattheflightspeed.Theproposedadaptivescheme isappliedalsotothesimulationofapitchinginfinite-spanwing,wherethebi-dimensional characteroftheflowiswellreproduceddespitethethree-dimensionalunstructuredgrid. Finally, the scheme is exploited in a piston-induced shock-tube problem to take into accountsimultaneouslythelarge deformationofthe domainandthe shockwave. Inall tests,meshadaptationplaysacrucialrole.

©2017TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCC BYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

The numericalsimulation ofunsteady fluid dynamicsandcontinuum mechanics problemsoftenrequires todeal with large deformationsofthe computationaldomain and,atthe sametime, toprecisely representtheboundariesand/or the interfaces of multi-material systems. The capability to cope with both requirements is determined by the choice of the kinematicdescription,whichexpressestherelationshipbetweenthedeformingcontinuumandthecomputationalgrid.The

*

Correspondingauthor.

E-mailaddress:barbara.re@polimi.it(B. Re).

http://dx.doi.org/10.1016/j.jcp.2017.03.034

forre-zoningcanbefoundin[17,7]andreferencestherein.Forwhatconcerns there-map step,thesolutioninterpolation must be conservative, accurate andmust preserve monotonicity of the solution. A very effective approach that satisfies all thisrequirements is based on the integration of the volume swept by the cells during the re-zone phase, followed by a repairstep in which mass is re-distributed betweenneighboring cells to ensure local bound preservation [18–20]. OthertechniquesexploittheFlux-CorrectedTransportmethodtoenforcemonotonicity,bycombininglow-orderfluxeswith high-orderones[21,14,22].

Inaddition toalgorithms thatimplementseparately the threeALEphases, schemesthat donot performexplicitlythe re-mapphasebutembeditintothesolutionupdatearealsoavailable[2,23–25,15,26].Inthiscase,thegoverningequations are solveddirectlywithin theALEframework, bytakinginto accountthemotionof thecontrolvolumes. Alsohigh-order (morethansecond)schemeshavebeenproposed,asforexample[27,28].

AlthoughitcanbeobtainedquitestraightforwardlyfromthestandardEulerianformulation,theALEformulationincludes sometime-dependentgeometricquantitiesinvolvingpositionsandvelocitiesofthegridnodesthathavetobeproperly eval-uated.Tothisend,theGeometricConservationLaw(GCL)anditsdiscretecounterpart—theDiscreteGeometricConservation Law (DGCL) [3]—play a crucial role. This consistency condition states that a numerical scheme designed to enforce the governingequationsovera movinggridshouldpreserveauniformflow.TheGCLcanbeenforcedbyanappropriate com-putationofthegeometricquantities.FulfillingtheDGCLisasufficientconditiontobeatleastfirst-ordertime-accurate[29], althoughitisneitheranecessarynorasufficientconditionforhigh-orderaccuracy[23].Ontheotherhand,itsfulfillmentis anecessaryandsufficientconditionforpreservingnon-linearstability[5].Therefore,itisgenerallyacceptedthatsatisfying theDCGLimprovestimeaccuracy,numericalstabilityandavoidsspuriousoscillations[30,24].Anupdatedand comprehen-sive review ofthis subjectcan be found in [31]. Note that the scheme conservativeness is ofparamount importance in fluid–structure interaction andaeroelasticproblems,where violating theDGCL maylead toan erroneouscomputation of thefluttervelocity[3].

Mostof thestandard ALE schemes usefixed mesh topology,namely there-zoning phase aims atmodifying only the positionofthegridnodeswithoutchangingtheconnectivity.Thisconstraintresultsinalimitationonthedeformationthat thegridisabletotackle,becauselargedisplacementsofthenodesmayleadtoentangled,andthusinvalid,elements. More-over,sincethegridnodesaremoved fromonegridregiontoanother,onlyalimitedcontroloverthemeshresolutioncan be generallyachieved[32]. Toovercomethisdrawback, some adaptivere-zone approacheshavebeenproposed. Adaptive MeshRefinement within theALEframework hasbeenusedtoinsert(or delete)grid nodesinstructuredgrids by Ander-son et al.[33] andbyMorrell etal.[34],butbothapproachesare basedonan underlyingfixedconnectivity,becauseonly thecelloftheinitialgridcanbesub-divided[32].Inordertoavoidmeshentanglement,thegenerationofanewdualmesh ateach time step isproposed by Springel [35].This techniquesreliesupon the fact thatthe nodesof theVoronoimesh generatedfromamovingDelaunaytriangulationexperiencealwaysacontinuousdeformation,evenifthemesh-generating points move discontinuously. Alternatively, local mesh adaptation techniques [36–39] may be exploited to enhance grid qualityaftermeshdeformationandtoimprovesolutionaccuracyinmoving-boundaryproblems,asforinstancein[40–43]. Unfortunately,thisoperationrequiresanon-trivialinterpolationofthesolutionfromtheoldgridtothenewone,thatmay originateproblemsinenforcingconservativeness andmonotonicityandcomplicatetheimplementationofmulti-steptime integrationalgorithms[44].

The extensionof ALEschemesvariabletopology grids hasbeeninvestigated insteadin[32,45], wheretheedge swap-ping techniqueshasbeen exploitedtomodify thegrid withoutchangingthe numberofgrid nodes.Furthermore,several techniques,seeforinstance[46,47]for2D and[48–51] for3D,propose totreatgrid connectivitychangeswithin theALE frameworkthankstoanexplicitinterpolationofthesolutionfromtheoldtothenewgrid,whichhowevermayleadstothe samedrawbackshighlightedbefore.Analternativeapproach,availableonlyfor2Dproblemstoauthors’knowledge,consists intheso-calledspace–timemeshesthatallowtohandlelargeboundarymovementswithintheALEformulationbytreating the time asthe third spatial dimension [52,53].Finally, the recentwork of Barral andco-workers [54] proposes an ALE formulationformoving boundaryproblemsableto dealwithconnectivitychanges dueto meshadaptationthanks tothe interpolationschemedescribedin[55],which,however,isconservativeonlyifthevolumeofthedomaindoesnotchange, i.e.ifitsboundariesdonotmatchbetweenconsecutivetimestepsanerrorintheglobalmassconservationisintroduced.

fictitiouscontinuousdeformationsofthefinitevolumesassociatedtothenodesinvolvedinthemodification.Theadditional deformations dueto grid adaptation resultinadditional fictitiousALEfluxes, whichcan be easily taken intoaccount by standard ALE techniques, without requiring any explicit interpolation of the solution over the new grid. Admittedly, as observedin[56],theALEmappingisindeedequivalenttoaninterpolation,butitdoesnotrequireanyspecialtreatmentto guaranteetheconservativeness,themonotonicityandaccuracyofthescheme.Moreover,alsogeometricquantitiesinvolved inthisfictitiousdeformationareevaluatedsothattheDGCLisautomaticallyenforcedeveninpresenceoftopologychanges. Nospecialtreatmentsorinterpolationsarerequiredtoimplementmulti-steptime-integrationtechniques.

A challenging aspect regarding the extension of the interpolation-free ALE scheme in [56,57] to unstructured three-dimensional meshesconcernsthevariablelocaltopology ofthetetrahedralgrids,whichrequiresoneto handlea variable number ofelementsinvolved forthelocalgridmodification (withtheexception ofthesplit ofasingle element infour). Consider forexample anode insertion by edgesplit.In atriangulargrid, onlytwo elements sharean edge,sowhen the edge is splitonly thesetwo elements haveto be modified. Conversely, intetrahedral grid,an edge can be sharedby an arbitrary number of elements, so an arbitrarynumber of finite volumes have to be included in the series of fictitious continuous deformationsusedtotreatconservativelytheconnectivitychangewithintheALEframework.Furthermore,the three-dimensional version introduced inthiswork iscapable to dealwitha broader suiteofadaptation procedures than theone describedin[56],includingnodeinsertionby theDelaunay triangulation.Thistechnique consistsinre-generating a local portionof the triangulation and newelements inserted in thisway show usually better qualities than the ones createdthroughedgesplit[58].Inordertoeffectivelydealwiththedifferentaspectsthatcharacterizemeshadaptationin threedimensions,asforinstancealsothemultipleoptionsthatareavailableforanedgeswapping,theexternalre-mesher Mmg3d[59]isexploitedinthepresentwork.

The paperis organized asfollows.First,in Section 2the finite-volumeedge-base ALEschemefordynamic grids with fixed connectivityispresentedtointroducetherelevantconcepts andthenotation.The evaluationofsomekeyquantities isexplainedinSubsections2.2and2.4.Section3describesthemodificationrequiredforits extensiontoadaptivegrid.In particular,Subsection3.1detailsthethree-stepsprocedurethatenablestheinterpretationofthegridconnectivitychanges asseriesoffictitiouscontinuousdeformationsforatetrahedralgrid.Then,abriefoutlineofthemeshadaptationtechniques used in the numerical experiments isgiven Section 4. Afterwards, the resultsobtained withthe proposed approachare presented in Section 5. Subsections 5.4 and5.3 illustrate the results for a flow around an infinite-span wing traveling throughthedomainattheflightvelocityforonechordandexperiencingapitchingmotion,respectively.InSubsection5.4, theexperimentofawingtranslatingthroughthedomainispresentedforafinite-spanwing.Subsections5.5and5.6show theresultsobtainedintestscharacterizedbylargeboundarydisplacements.Finally,theconclusionsofthepresentworkare drawninSection6.

2. Finite-volumeALEschemefordynamicmeshes

TheEulerequationsdescribethebehaviorofcompressibleinviscidfluidflowsandforsufficientlyhighReynoldsnumber flows aroundaerodynamicbodiesthey provideanaccuraterepresentation,exceptinthethin,boundarylayerregionsclose to solidwalls. IntheALEframeworkforacontrol volume

C(

t

)

∈ !

containedinthedomain

!(

t

)

andmoving atvelocity

v

(

x

,

t

)

,theEulerequationsread d dt

!

C(t) udx

+

"

∂C(t) [f

(

u

)

−

uv]

·

n ds

=

0

,

(1)

where x

∈ !

⊆ R

3 _{istheposition vector,t}

_{∈ R}

+ _{isthe time,u isthevector ofconservative variables,f}

₍

_u

₎

_{istheinviscid} flux and n

(

s

,

t

)

∈ R

3 is theoutward unit vector normalto the boundary

∂

_C

ofthe control volume

_C

,which is function

of theboundarycoordinates

∈ ∂!

⊆ R

2 _{andoftime.The vector ofconservativevariables u}

_{: !}

_{× R}

+

_{→ R}

5 _isdefinedas

u

= [

ρ

,

m

,

Et

]

T,with

ρ

thedensity,m themomentumdensityandEtthetotal(internalpluskinetic)energydensity,while

thefluxfunctionf isdefinedas

f(u

)

=

#

m

,

m

⊗

m

/

ρ

+ $(

u

)

I

3

,

$

Et

+ $(

u

)

%

m

/

ρ

&

T (2)

where

$(

u

)

= $(

ρ

,

m

,

Et

)

is the pressure function, which depends on the thermodynamic model, and

I

3 is the 3

×

3

identity matrix. Inthe presentwork, the polytropic idealgas modelis adopted todescribe the behavior ofthe fluid. To makethegoverningequations(1)complete,suitableinitialandboundaryconditionsmustbespecified[60].

As anticipated in Section 1, when solving the governing equations over moving grids, the fulfillment of the GCL is beneficial toavoid spurious oscillations andpreserve non-linearstability [3,5,30,24]. Tothis end, theALE schemehas to compute exactlythe trivialsolution ofauniformflowover amoving domain.ApplyingthisrequirementtoEquations (1)

Fig. 1. Cellinterfacedefinitionbetweentwotetrahedralelements.Theportionofthefinitevolumessurroundingthenodesiandkassociatedtothefour tetrahedralelementsdefinedbythegridnodesi,k,v1,v2,v3,v4areshown.Intheleftpicture,domaingridelementsareconsidered,whileintheright

onethefacesi–v2–k andi–k–v4lieontheboundary∂!.Gridedgesaredrawnbyblacklines.TheportionsofCiandCkarepaintedrespectivelyblue andgreen ,whiletheirboundaries∂Ci and∂Ckaredarker,i.e. and .Theredcolor identifiesthecellinterface∂Cik= ∂Ci∩ ∂Ck,whichis shownentirely.Fortheboundarycase(atright),thepatterns and indicate theboundaryinterfaces∂_{Ci∩ ∂!}and∂_{Ck∩ ∂!}.(Forinterpretationofthe referencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

whichcanbeviewedasasupplementaryconstrainttotheflowequations(1)thatthegridvelocityhastofulfillduringthe motion.

2.1. Finitevolumediscretizationwithinthenode-pairrepresentation

Thespatially-discretecounterpartoftheEulerequations(1)overadynamicunstructuredgridisobtainedherethrough thefinitevolumeformulation.Athoroughdescriptionofthisprocesscanbefoundin[57,56];hereonlythemainstepsare brieflyrecalledtointroducethenomenclatureusedinthefollowing.

The domain issplit into NV non-overlapping volumes anda node-centered approach is adopted.As shownin Fig. 1,

thecontrolvolume

C

i

(

t

)

thatsurroundsthenode i iscomposedbya portionofall gridelementssharingthe nodei.The

portionoftheboundarythatitshareswithitsneighboringvolumek,i.e.

∂

_C

ik

(

t

)

= ∂C

i

∩ ∂C

k,isusuallyreferred toascell

interfaceanditisassociatedtothenode-pairik,thatisthecoupleofinteractingnodesiandk [61].Then,givenageneral approximation

%

oftheintegralofthenumericalfluxacrossthisinterface,andasimilar

%

∂fortheeventualfluxacrossthe

portionof

∂

_C

i thatliesontheboundary,thenode-pairrepresentationoftheEulerequationsforeachfinitevolumereads

d

dt[Viui]

=

'

k∈Ki,̸=

%(

ui

,

uk

,

ν

ik

,

η

ˆ

ik

,

η

ik

)

+ %

∂

(

ui

,

ν

i

, ˆ

ξ

i

, ξ

i

)

∀

i

∈

K

(4)

where Vi isthe cell volume,ui isthe cell-averaged value ofthe solution,

K

isthe set ofall finite volumesand

K

i,̸= is thesubset ofthose finitevolumessharingaportion oftheir cellinterface withi,i.e.

K

i,̸=

= {

k

∈ K,

k

̸=

i

| ∂C

i

∩ ∂C

k

̸= ∅}

.

Moreover,themetricquantities

η

ik,

ν

ik,

ξ

i,

ν

i havebeenintroduced[62].Theyaretheintegratednormal(

η

ik,

ξ

i)andthe

interface velocities(

ν

ik,

ν

i) oftheinterface

∂

_C

ikandoftheboundary interfaceassociated tothenode i,respectively, and

aredefinedas

η

ik

=

!

∂Cik nidx

,

ν

ik

=

!

∂Cik v

·

nidx

,

ξ

_i

₌

!

∂Ci∩∂! nidx

ν

i

=

!

∂Ci∩∂! v

·

nidx (5)

and,accordingtothestandardvectornotation,

η

ik

= |

η

ik

|

,

η

ˆ

ik

=

η

ik

/

η

ik,

ξ

i

= |ξ

i

|

and

ˆξ

i

= ξ

i

/ξ

i.Althoughitisnotexplicitly

writtentolightenthenotation,metricquantitiesdependontimeaswellasthecellinterfacesbecauseofgridpointsmotion. TheirexpressionsintermsofgridnodespositionaregiveninSubsection2.2.

Inthiswork,theTotalVariationDiminishing(TVD)approachisusedtoobtainedthehigh-resolutionexpressionforthe integrated numericaldomain fluxes[60,63]. The vanLeer limiter [64] is used to switch fromthe second-order centered approximation to the first-orderupwindscheme ofRoe [65] neardiscontinuities. Boundary conditions are enforced ina weakform,i.e.byevaluatingtheboundaryflux

%

∂inasuitableboundarystate[56].

Fig. 2. Metricsdefinitioninnode-pairrepresentationforanelementwithafaceontheboundary.Thesixtriangularfacetsdefiningthedomaincontribution oftheconsideredelementmtothe∂Ciaredepictedinblue .Eachinterfaceportionhasasverticesthebarycenteroftheelement ,ofaface andof anedge .Theredpattern highlightsthetriangularfacet∂Cm

ik,f,relatedtothefacei– j–k.Consideringthatthefacei–v– j liesontheboundaryandit

formstheboundaryelementm∂_{,thegreenpattern depictstheboundaryelementalportionoverwhichtheelementalcontributionξ}m∂

i,f∂ iscomputed, wheretheconsideredfacef∂_{istheedgei–v.Finally,thevolumeV}m

ik,f ofthesub-tetrahedronthathas∂Cmik,f asbaseandthenodeiasoppositevertexis

showninlightblue .(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Thepreviousequationcanbeautomaticallysatisfiedbysplittingthederivativeofthecellvolumeintothevolumesweptby the interface

∂

C

ikand

∂

C

i

∩ ∂!

,respectively,namely dV_dti(t)

=

(

_Ki,̸=dV_dtik

+

dV_dti,∂.Theso-calledInterfaceVelocities

Consis-tency(IVC)conditions[66]aresoobtained dVik

(

t

)

dt

=

ν

ik

(

t

)

∀

k

∈

K

i,̸= and

dVi,∂

(

t

)

dt

=

ν

i

(

t

)

if i

∈

K

∂

,

(7)

where

K

∂ isthesetofboundarynodes.Relations(7),givenaknowngridmotion,allowtocomputeavalueoftheinterface velocities that automatically satisfiesthe GCLconstraintand they complementtheSystem (4)made of5

×

NV Ordinary

DifferentialEquations.

2.2. Metriccomputationontetrahedralgrids

The expression forthemetric quantitiesandthecell volume inEquation(4) arenow obtainedintermsof thenodes positions fortetrahedral grids,onwhichthenode-pairscorrespond totheedges.FromFig. 1,itcould benoticedthat the generic cell interface

∂

C

ik iscomposed by differentcontributions pertaining to all grid elements that share theedge ik.

Considering only the elementalinterface contribution

∂

_C

m_ik fromthe element m, it iscomposed by two triangularfacets

∂

_C

m_ik,f stemmingfromthedifferentfaces f ofthetetrahedron m towhichthe edgeik belongs.The threenodesofthese

facets are the barycenters ofthe element xm, ofthe edge xik and of the face xf (Fig. 2). According to this partition of

the interface, the integrated normals are computedon each triangularfacets andthen summed together. The elemental contributiontotheintegratednormal

η

ikoftheelementmfortheface f canbethereforecomputedas

η

m_ik,f

= −

1

/

2

$

xf

−

xm

%

× (

xik

−

xm

) .

(8)

Similarly, the boundary elemental contribution is computed over the triangle formed by the boundary node i, the barycentersoftheboundaryelementm∂_{andofitsface f}∂_,namely

ξ

m_i,∂_f∂

= −

1

/

2

$

x_f∂

−

x_m∂

%

×

$

xi

−

xf∂

%

.

Themetricvectors

η

ikand

ξ

icanbethencomputedbysummingalltheelementalcontributions

η

ik

=

'

m∈εi∩εk

'

f∈Fik,m

η

m_ik,f and

ξ

_i

=

'

m∂_∈ε∂ i

'

f∈Fi,m∂

ξ

m_i,∂_f∂

,

(9)

where

ε

i and

ε

k are theset of theelements sharing the node i andk,respectively,

ε

∂i is the setof boundaryelements

sharing thenodesi,

F

ik,m (and

F

i,m∂)is thesetoffaces ofm (m∂)sharing theedgeik (nodei). Theintersection

ε

i

∩

ε

k

denotes the set ofall elements sharing the edge ik.When summing all contributions, particular care must be takento preserve thecorrectorientationofthemetricvectors,becauseEquation(8)assumesthatthecontribution

η

m

ik,f isoriented

Fig. 3. Volumesweptbytheelementalinterface∂Cikm,f,duringthetimeintervaltn≤t≤tn+1,astheresultofthedisplacementofthenodesfromtheir

positionsxn_toxn+1_{.Theredpattern highlightstheelementalinterface∂C}m

ik,f,whiletheredarea depictsthesweptvolume*Vmik,,nf.(Forinterpretation

ofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

AlsothevolumeVi ofthefinitevolume

C

i iscomputedbysummingelementalcontributions.Accordingtothe

subdivi-siondescribed above,eachtetrahedron issplitinto24sub-tetrahedra,eachhavingthetriangularfacet

∂

C

ikm,f asbaseand

thenodeiorkastheoppositevertex.Thus,thevolumeofthefinitevolume

_C

iiscomputedas

Vi

=

'

m∈εi

'

k∈Km i,̸=

'

f∈Fik,m V_ikm_,f

,

with V_ikm_,f

=

1 3

(

xm

×

xi

)

·

η

ik,f

,

(10)

where

K

mi,̸= arethethreeverticesofthetetrahedronmdifferentfromi.

Theproposednode-pairfinitevolumesdiscretizationisverystandardandstraightforwardtobeimplemented.Afterthe median-dualmeshisgeneratedandtheedgestructure computed,no informationaboutelements isrequiredby theflow solver[67,68].Notethattheedge-basedrepresentationheredescribedisindependentfromthetypeofelementoftheinitial gridanditisthereforeverysuitableforhybridgridsmadeofelementsofdifferenttypes[69].Moreover,Selmin[61] and SelminandFormaggia[62]provedthatforfixedsimplexgrids(triangularin2Dandtetrahedralin3D)thedescribedfinite volumediscretizationisequivalent toafiniteelementone withlinearelements,exceptadifferenttreatmentofboundary terms.

2.3. TimeintegrationoftheALEequations

Equations(4)and(6)areintegratedintimewiththeBackwardEulerscheme.Thefully-discretesystemofthegoverning ALEequationstherefore,reads

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

V_in+1un_i+1

₋

V_inun_i

_{= *}

t

- '

k∈Ki,̸=

%(

ui

,

uk

,

ν

ik

,

η

ˆ

ik

,

η

ik

)

n+1

+ %

∂

(

ui

,

ν

i

, ˆ

ξ

_i

, ξ

i

)

n+1

.

,

i

∈

K

*

Vn_ik+1

= *

t

ν

_ikn+1

,

k

∈

K

i,̸=

*

Vn_i,∂+1

= *

t

ν

_in+1

,

i

∈

K

∂ (11)

where

*

Vn_ik+1and

*

V_in_,∂+1 arethevolumesweptbytheinterfaces

∂

_C

ikand

∂

C

i

∩ ∂!

duringthetimestep

*

tn

=

tn+1

−

tn,

seeFig. 3.ExplicitexpressionsforthesweptvolumesaregiveninthefollowingSubsection2.4.

Intheprevioussystem,allquantitiesattimetn _{areknown,aswellasallthegrid-dependentquantitiessincegridmotion}

isknown.Thus,thetwolastequationsarenotcoupledtotheEulerequations.However,thecoupledformofthissystemis preferredheretohighlighttheexistenceoftheconsistencyconstraints,seeEquation(7),ontheinterfacevelocities.

Apseudo-timestepmethodisused tosolvethenon-linearSystem(11) [70].At eachpseudostep,amodified Newton methodisusedtosolvethelinearsystem,inwhich,accordingtothedefect-correctionapproach,theexactJacobian ofthe integratedfluxfunctionisreplacedbyanapproximatedone[71].

2.4. Interfacevelocitycomputationontetrahedralgrids

The last two equations inSystem (11)are the so-called DiscreteGeometric Conservation Laws (DGCL) [5],which are obtained by integrating in time the GCL condition (3). The DGCL states that the variation in volume in a certain time interval should balance the volume swept by its boundary during the same interval. In System (11), this constraint is matchedby computingeach interfacevelocity,

ν

ik and

ν

i,interms ofthevolumeswept bythe correspondingportion of

The swept volumesarecomputedby recasting tothe sameinterface partition introduced inSubsection2.2.Thus, the volumesweptbyadomaininterfaceis

*

Vik

=

tn+1

!

tn

!

∂Cik v

(

t

)

·

ni

(

t

)

dx dt

=

'

m∈εi∩εk

'

f∈Fik,m

*

Vm_ik,f

,

where

*

V_ikm_,f isthevolumesweptby

∂

_C

m_ik,f,asshowninFig. 3.Assumingaconstantvelocityofthegridpointsduringthe

timestep,itcanbeevaluatedas

*

Vm_ik,,n_f

=

v n m

+

vnf

+

vnik 3

·

tn+1

!

tn

η

m_ik,f

(

t

)

dt (12) where vn

j

= (

xnj+1

−

xnj

)/*

tn arethe velocitiesof thebarycenters ofthe element xm,ofthe face xf andof theedge xik,

for j

=

m

,

f

,

ik,respectively.Consideringalinearfunctionforthepointposition,i.e.xj

(

t

)

=

xnj

+

vnj

(

t

−

tn

)

,theintegrated

normalportion

η

m ik,f

(

t

)

results

η

m_ik,f

(

t

)

=

η

n_ik

+

t

−

tn

*

tn

/

−

3

η

n_ik

+

4

η

n+ 1 2 ik

−

η

nik+1

0

+

2

1

_t

−

tn

*

tn

2

2

/

η

n_ik

−

2

η

n+ 1 2 ik

+

η

nik+1

0

(13)

wheretheindices f andmareomittedintherightsidetosimplifythenotationand

η

n+12

ik isdefinedas

η

n+ 1 2 ik

=

1

/

4

(

η

nik

+

η

nik+1

)

−

1

/

8

-(

xn_f

−

xn_m

)

× (

x_ikn+1

−

xn_m+1

)

+ (

xn_f+1

−

xn_m+1

)

× (

xn_ik

−

xn_m

)

.

.

(14) Finally,integratingintimeandsubstitutingtheexpressionsforthevelocitiesofthepoints,theequationforthevolume sweptbytheinterface

∂

C

mik,f duringthetimestep

*

tn isobtained:

*

Vm_ik,,n_f

=

1 18

(

x n+1 m

−

xnm

+

xnf+1

−

xnf

+

xnik+1

−

xnik

)

· (

η

mik,,nf

+

η

mik,,nf+1

+

4

η

m,n+1 2 ik,f

) .

(15)

Withasimilarprocedure,thefollowingexpressionforthevolumesweptbyaboundaryinterfaceisobtained

*

Vm_i,f∂∂,n

=

1 18

$

xn_m+∂1

−

x n m∂

+

xn_f+∂1

−

x n f∂

+

xn_i+1

−

xni

%

·

$

ξ

m_i∂,n ,f∂

+ ξ

m∂_,n+1 i,f∂

+

4

ξ

m∂_,n+1 2 i,f∂

%

,

(16) with

ξ

n+ 1 2 i,f∂

=

1

/

4

(ξ

ni

+ ξ

ni+1

)

±

1

/

8

-(

xn_m

−

xn_f

)

× (

x_in+1

−

xn_f+1

)

+ (

xn_m+1

−

xn_f+1

)

× (

xn_i

−

xn_f

)

.

wherethesign

±

dependson

whetheriisthefirst(

−

)orsecond(

+

)nodeofthenode-paircorrespondingtotheface f∂_.

3. ConservativeALEschemeforadaptivemeshes

IntheprevioussectiontheIVCconditions,solvedalongwiththeEulerequationsinSystem(11),wereshownto guaran-teethefulfillmentoftheDGCLondynamicgridswithfixedconnectivity.Moreprecisely,thechangeinpositionandshape ofthefinitevolumescanbeeasilytakenintoaccountbycomputingtheinterfacevelocitiesasthesumofthevolumeswept by theinterfacesduringthegriddeformation.Thistechniqueisnowextendedtoadaptivegridswithvariableconnectivity bymeansofthethree-stepsprocedureproposedbyGuardoneandco-workersforbi-dimensionalproblems[57,56],that al-lowstocomputethesweptvolumealsoinpresenceoftopologychanges.Thiscontribution,labeled

*

V_ikA tohighlightthatit

originatesfromaconnectivitymodification,isthensummedtotheoneduetosimpledeformation,labeled

*

VD

ik.Thenthe

IVC-compliantinterfacevelocitiesarecomputedfromthelasttworelationsofSystem(11)considering

*

Vik

= *

VikD

+ *

VikA.

Athoroughdescriptionofthisprocedurefortriangulargridsintwodimensionscanbefoundin[57,56],whilehereits ap-plicationtotetrahedralgridsisdetailed,followedbythefinalsystemsofALEgoverningequationsforadaptivegrids.

3.1. Continuousinterpretationoftopologymodifications

The fundamentalideaofthethree-steps procedureistodescribethegridmodification asasequenceoffictitious con-tinuousdeformationsforwhichthevolumessweptbycellinterfacescanbe computedfromEquations(15)and(16).This can beaccomplishedbythefollowingthree-steps procedure,inwhicha fictitioustime

ζ

isusedto betterdistinguishthe

differentphases,withtheinitialconfigurationat

ζ

=

0 andthefinaloneat

ζ

=

1.

Fig. 4. Exemplary three-stepsprocedure applied to the splittingof the edge ik. The portions of the finite volumes pertaining to the polyhedron

i–k–v1–v2–v3–v4 andassociatedtothenodesi,kand jareshown,respectively,withblue ,green ,andred (thedarkercolorsreferto

theboundariesofthefinitevolumes).Theprocedurebeginswiththecollapsephase.Theconnectivitychange,showninthethirdandfourthpictures, occursatζ₌0.5 whenthesurfaceoftheinvolvedinterfaceshasnullarea.Then,theexpansiontakesplace:thenodesi,k,v1,v2,v3,v4returntotheir

originalpositionsandthenewfinitevolumeCjexpandstoreachthefinalconfiguration(atζ₌1).For0< ζ <1,thegreylinesshowthegridconnectivity intheoriginal/finalconfigurationasareference.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversion ofthisarticle.)

Fig. 5. Localcharacterofconnectivitymodifications,exampleforanedgesplitin2D.Theinitialandthefinalconfigurationforthesplitoftheedgeikare shownoverlapped.Theboundariesofthefinitevolumesassociatedtoiandkaredrawnwithdifferentlinestodisplaythemodificationsduetotheedge split.Thepolyhedronformedbytheunionoftheinvolvedelementsconsistsinthequadrilaterali–k–v1–v2.Outsidethisregion,therearenodifferences

betweentheinitialandthefinalconfiguration.

2. Connectivitychange,

ζ

=

0

.

5:whenallinvolvedelementsreachnullvolumes,nodesmaybeinsertedordeleted.

3. Expansion,0

.

5

< ζ <

1:allelementsstillactive(i.e.notdeletedat

ζ

=

0

.

5)expandtothefinalconfiguration.

Fig. 4showstheapplicationofthethree-stepsproceduretoanedgesplit,whichconsistsintheinsertionofanewnode alongagridedge.WithreferencetoFig. 4,considerthattheedgebetweennode i andk hastobesplittoinsertthenew node j.Thismodificationinvolvesalltheelementsthatshareinitiallytheedgeik,thatarethefourelementsshowninthe picture,becausethefinitevolumesassociatetonodesi,k,v1,v2, v3,v4 havetobemodifiedtogeneratethenewvolume

associatedwiththenewnode.Oncetheinvolvedelements havebeenidentified,thecollapsephase canbecarriedout.At thispoint,thenewnodeisinsertedandconnectedwiththeexistingonestogeneratenewtetrahedra.Finally,theelements expandtowardsthefinalconfiguration,whichcomprehendsthenewfinitevolumeassociatedtothenode j.

Thecollapseandtheexpansionphasearesimplycontinuousdeformations,thereforethesweptvolumescanbecomputed fromEquations(15)and(16).Conversely,sincetheconnectivitychangeoccurswhiletheelementshavenullvolume—and the interfaces have nullarea—no volume is swept by any interface inthis phase. Thus, the connectivitychange has no effects interms ofinterface velocitiesor DGCLsatisfaction andit doesnot generateany numericalfluxes. The complete contributiontothesweptvolumeduetoadaptation

*

VA

ikisgivenbythesumofthevolumesweptduringthecollapseand

theexpansionphases.Itisimportanttonoticethelocalcharacterofthemodification:theexternalfacesofthepolyhedron formedby theunionofthe involvedelements remainunchanged,see Fig. 5fora2D example.Indeed,sincenovariation occursinthegridelementsoutsidethepolyhedron,novariationoccursinthefinitevolumeseither.Then,thevolumeswept bytheinterfaceslocatedoutsidethepolyhedronduringthewholeprocedureisnull,becausetheinterfacessweepthesame volumebutwithoppositesignsduringthecollapseandtheexpansionsteps.

Thethree-stepsprocedureisgeneralanditcanbeexploitedtocomputetheIVC-compliantinterfacevelocitiesincaseof differentlocalmeshadaptationtechniques,likenodeinsertion,nodedeletionandedgeswap.Clearly,theelementsinvolved aredifferentdependingonthegridmodification.Table 1describestheelementsinvolvedandthecollapsepointforallthe localadaptationtechniquesusedinthepresentwork.

Table 1

Three-stepsproceduredetailsfortheconsideredlocalmeshadaptationtechniques.Foreachtechnique adoptedinthepresentwork,theelementscomposingthepolyhedronthatdelimitsthefinitevolumes influencedbythegridmodificationareindicated,alongwiththechosencollapsepoints.

Adaptation Polyhedron made by Collapse point

Edge split Elements sharing the edge New node

Element split Element to split New node

Delaunay insertion Elements in the cavity New node

Edge collapse Elements sharing the node to be deleted Node to be deleted

Fig. 6. Three-stepsprocedureappliedtothecollapseoftheedgeij.Theportionsofthefinitevolumespertainingtothepolyhedroni–k–v1–v2–v3–v4and

associatedtothenodesi,kand jareshown,respectively,withblue ,green ,andred (thedarkercolorsrefertotheboundariesofthefinite volumes).Atζ₌0.5 (thirdandfourthpictures),thenode jiscollapsedoverthenodei.Theelementsthatsharetheedgeijaredeleted,whileforthe otherelementsthenode jissubstitutedbythenodei.Finally,onlytheelementsthatarenotdeletedareexpandedtoreachthefinalconfiguration(at

ζ=1).Thegreylinesshowthegridconnectivityintheoriginal/finalconfiguration,for0< ζ <1.(Forinterpretationofthereferencestocolorinthisfigure

legend,thereaderisreferredtothewebversionofthisarticle.)

topology.Indeed,theseelementsare deletedat

ζ

₌

0

.

5 andtheydonot takeparttotheexpansion phase.Alsothefinite volumeconnectedtothenode j isdeletedat

ζ

=

0

.

5.Nevertheless,itshouldbenoticedthatduringthecollapsephasethe interfacesof

_C

jsweepanon-nullvolume,whichgeneratesnon-nullinterfacevelocities.Thisfacthastobe carefullytaken

intoaccountintheflowsolver,asdescribedinnextsubsection.

3.2. ALEschemewithvariabletopologies

In this subsection, System (11) is extended to adaptive grids with variable connectivity. Before presenting the final systemofequations,thereader’sattentionisdrawntosomepeculiaritiesofthepresentapproach.Adetaileddescriptionof thewholeprocedurecanbefoundin[57,56,72].

Afirstremarkconcernsthetimedependenceofthegridconnectivity.Duringeachtimestep,avariablenumberofgrid nodescanbeaddedordeleted.Consequently,boththesetofgridnodesandthesetofthefinitevolumeschangeintime. Inthefollowing,asuperscriptisaddedtotheirsymbolstoindicatethetimesteportimeintervalatwhichtheyshouldbe considered.Forinstance,

K

n+1 denotesthesetofnodesthatcomposethegridattimeleveltn+1_.

Second, whenanewnodeisaddedtothetriangulation,anequationforthebalanceoftheconservativevariablesover the newfinite volume hastobe added to thegoverning equationssystemandadditional IVCconditionsare required to computethenewinterfacevelocities.TheseequationsdonotdifferfromtheonesinSystem(11),howevertheyaresimpler sinceifthenodei isinsertedbetweenthetimesteptn _andtn+1_,thevolumeV

i andthevolumesweptbyitsinterfaceare

nullforalltimestepspreviousthantn+1_.

Third, special care must be taken in dealing with the treatment of the deleted interfaces. When the grid edge ik is deleted, during thethree-steps procedure theassociated interface

∂

_C

ik is collapsedto reacha nullarea at

ζ

=

0

.

5; from

then on,it is considered as deleted.However, during the collapse phase it sweeps a non-null volume, whichhas to be includedintothecomputation oftheassociatedinterfacevelocity

ν

ik eveniftheedgeikisnolongerpresentinthemesh.

Thus, supposing that thismodificationoccurs during theadaptation phase performedbetweentn _andtn+1_{,theinterface}

velocity is notnullalsofor thenext time step,i.e.

ν

n+1

ik

̸=

0. Asa consequence,theassociated numericalflux isnot-null

and it is included in the system of governing equations by adding the term

%

n_ik+1

= %(

un_i+1

,

un_k+1

,

ν

_ikn+1

,

η

ˆ

n_ik+1

,

0

)

where

η

n_ik+1

=

0 becauseofthe nullarea of integrationwhile thenormal unit vector isdefined as

η

ˆ

nik+1

=

limt→tn+1

η

ˆ

ik

(

t

)

[57].

Furthermore,theassociatedIVCconditionshavetobeenforceduntiltheinterfacevelocityisidenticallynull.Asexplained by Isola et al.[56],thefact thattheinterface velocitydoesnot necessarilybecomenullwhiletheinterface collapsesand isremovedfromthemeshisaconsequenceofthedifferentialnatureoftheGCLconstraintanditdoesnotdependonthe timeintegrationscheme.

⎪

⎪

⎪

⎪

⎪

⎩

*

V_ik

= *

t

ν

_ik

,

k

∈

K

i,̸=

*

V_in_,∂+1

= *

t

ν

_in+1

,

i

∈

K

[∂n,n+1]

where

K

[in,̸=,n+1)

= {

k

∈ K,

k

∈ K

/

ni,+̸=1

|

ν

ikn+1

̸=

0

}

isthesetoffinitevolumesthathavesharedaportionoftheirboundarywith

C

i intheintervalfromtn totn+1 butthatdonotshareanyportionsattn+1,while

K

[i,n̸=,n+1]

= K

i[,n̸=,n+1)

∪ K

ni,+̸=1 includesall finitevolumesthatshareorhavesharedtheirboundarywith

C

i.

Thefirst andthesecond equationsofSystem(17)expressthebalanceof conservativevariablesforthe finitevolumes associatedtoallnodesofthetriangulationattimetn+1 _{andtothenodesremovedbetweent}n _andtn+1_{,respectively.The}

computationofthesolutiononnodesremovedduringthepreviousstepsisessentialtothecorrectevaluationofthesolution onthe existingnodes.Indeed,ifthe removed node j was connected totheexisting node i byan edge thatis nolonger presentinthegrid,theflux

%

n_ij+1isrequiredtocomputetheactualsolutionatnodei,butitsevaluationrequiresthevalue

ofun_j+1,whichiscomputedviathesecondequationofSystem(17).Then,sincetheinterfacevelocity

ν

ij remainsdifferent

fromzeroonlyforthestepnexttoitsdeletion,fort

>

tn+1 theequationforthenode j isdiscardedfromthesystemand

thesolutionuj isnotcomputed.

Finally, the last two rows of System(17) expressthe IVC conditions that allow to compute the cell volume changes duetomesh deformationandadaptation sothat theDGCL ismatched.Thanks tothe three-stepsprocedure describedin Subsection3.1,thesolutioncanbeconservativelycomputedwithintheALEframeworkeveniftopologychangesoccurdue togridadaptation.

3.3. Multiplemodificationsinvolvingthesamenode-pair

Duringasingle timestep diversetopology modificationsmayaffectthesameinterface.Theproposed approachenable tocopewiththissituationbysummingthecontributionsofallthecollapseandexpansionphasesexperiencedbythe in-terface.Althoughthischoiceisnotmandatory,itcanbeimplementedinanextremelystraightforwardwayinthenode-pair representationdetailedinSection2.Indeed,aftereachlocalmodification,alooponallinvolvedinterfacesisperformedand thevolume sweptby each oneis computedandaddedtothe correspondingnode-pair. At theendofthe timelevel,the sweptvolumeduetoadaptationissummedtotheoneduetodeformationandtoobtainthetotalamount

*

V_ikn+1.

Finally, particularcare is devoted to the update ofthe node-pair spatial discretizationafter a mesh modification. Al-thoughinatetrahedralfinite-volumedual-meshdiscretizationnode-pairsarenaturallyassociatedtogridedges,theadaptive schemerequiresonetoincludealsosome node-pairsthatareassociatedtodeletededges.Inotherwords,twogridnodes mayinteractwitheach othereven ifthey arenot connectedby anedge. Tothisend, whenthetopology re-construction requiresthecreationofa newnode-pair,itshould becheckedifinthesetofthe“deleted-but-still-active” node-pairs al-readyexiststheoneconnectingthesenodes.Ifthiscase,thenode-pairisrestoredandthevolumesweptbytheassociated interfaceistakenintoconsideration.

4. Overviewofthecomputationalprocedureforadaptivesimulations

TheproposedconservativeALEschemeforadaptivegridshasbeenimplementedinthe FlowMesh solverthatiscurrently underdevelopmentattheDepartmentofAerospaceScienceandTechnologyofPolitecnicodiMilano.Thisnumericaltoolis

ableto performunsteadythree-dimensionalsimulations overmoving-boundary problems,evenwhenthe domain

bound-aries experience large deformations[72]. Thissection gives an overviewof thedifferent techniquesandof thecomplete procedureusedtocarriedoutsuchsimulations.Additionaldetailscanbefoundin[72,73].

Fig. 7. Computationalprocedureforadaptiveunsteadyproblems.Atthebeginningofthetimestep,themeshdeformationstrategyisappliedtoupdatethe gridKn_tothegridKn+_{thatcomplieswiththeboundarydisplacement.Apredictionphaseisthenperformedtocomputethesolutionattime}tn+1_over

thisnewgrid.Meshadaptionisthencarriedoutandthetargetgridspacingisbuiltaccordingtothepredictedsolutionu(tn+1_,Kn+_{).Finally,thesolution} atthetimeleveltn+1_{overtheadaptedgridK}n+1_{iscomputed.Theredcolorhighlightsthenewoutputofeachcomputationalstep.(Forinterpretationof}

thereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

boundaryisredistributedamonginternalgridelements:thelargestelementsaccountforthemajorpartwhilethesmallest onesmovealmostrigidly,avoidingsopossibleentanglement.

If the above procedure does not succeed in producinga displacement field that results in a valid mesh,i.e. without poor-quality or negative-volume elements, the target displacement is subdividedin more intervalsuntil a valid mesh is obtainedthroughelasticanalogyorauser-definedmaximumnumberofsubdivisionsisreached.Intheformercasethe re-maining portionofthedisplacementiscarriedout,whileinthelattercasemeshoptimizationisperformedtorestoregrid quality [73].Toefficientlymodifythree-dimensionalgrids, thecapabilitiesofferedby theautomatictetrahedralre-mesher Mmg3d areexploited [59,42].Thislibraryis partofthe wideropen-source software Mmg which offers differenttoolsfor surface andvolumemeshadaptation,optimizationandgeneration.Inadditiontoaseriesoflocalgridmodificationswhich can beperformedbothontheinteriorandonthesurfaceofthegrid, Mmg3d implementsalsothegradationcontrol tech-nique [76] to limit thevariation insize among adjacentgrid elementandthe validity ofthe boundaryrepresentation is guaranteed by theHausdorffcriterion [77].Adetaileddescriptionofall thetechniquesexploitedin Mmg3d can be found in [78,42].The localoperators exploitedinthiswork arequite standard andincludethenode insertion throughelement split,edgesplitorDelaunaytriangulation,thenodedeletionthroughedgecollapse,theedgeswappingandthebarycentric regularization,seeTable 1.

Remark: itisimportanttohighlightthattheapplicationoftheconservativeALEschemehereproposedisnotlimitedto theseoperators,butitcanbeextendedtoalllocalmodificationsthatcanbedescribedbythethree-stepsprocedure.

During mesh optimization, each local modification aims at improving the quality of the involved elements which is definedas

Qm

=

α

Vm

/

$ (

6_i=1

ℓ

2_i

%

3/2

,

(18)

where Vm isthevolumeoftheelementm,

ℓ

i istheedgelengthand

α

isaconstantparameterintroducedtogive Qm

=

1

forregulartetrahedra,towhichcorrespondsthemaximumquality.

In thiswork, the aim ofremeshing is not limitedto the enhancement ofgrid quality after meshdeformation, but it is exploitedalsotoincrease solutionaccuracyandtodetect allrelevant flowfeatures whoseexactlocation isnot known a prioriorchangeintime [37,36].Thekey pointinthisprocess isthedefinitionofasuitable criterionable toefficiently identify where grid modifications are required and to prescribe a proper size map. Since the main scope of the work concerns the assessment of the proposed conservative ALEscheme over adaptive grids, local criteriabased on solution variations areadoptedbecauseoftheirsimplicityandeasy implementation.Theunderlyingideaofthiskindofadaptation criteria isthat thelargest errors occur wherethe solution exhibitsthe largest gradients, thereforethe solution accuracy can be increasedby gathering meshpoints in theseregions butit is not negatively affectedby a coarsening ofthe grid regions where the solution exhibits a smooth behavior. Thus, a local error indicator can be obtained from the first- or second-order derivatives ofarepresentative solutionvariable, asforinstancedensity,pressure, Machnumber, oroftheir combinationandthegridspacingusedastargetduringmeshadaptationcanbebuiltaccordingtoit.Morespecifically,the grid spacingisreducedwheretheerrorestimatorexceedsa refinementthresholdanditisenlargedwhere theestimator is belowacoarseningthreshold.Theseuser-defined thresholdsare prescribed intermsofthestandard deviationandthe meanoftheerrorestimatoroverthedomain.Onlytheedgesizeisprescribedandnoinformationabouttheelementshape ororientation isgivensincesimpleisotropicadaptationisperformed. Meshcoarseningplaysan importantroleespecially in unsteadysimulations where itavoids an excessivelyincrease inthe computational burdenthat maybe causedby the insertionofagreatnumberofnewnodes[79].

During the mesh adaptation process, the re-mesher sends to the flow solver the information concerning each local adaptation immediately after it is performed, so that the swept volume

*

VA of all involved interfaces can be update.

Ad-hoccallbackfunctions,similartotheonesproposedin[43],areusedtothisaim.Suchasanimplementationrequiresa stronglinkbetweenthesolverandthere-mesher,butavoidstostorethehistoryofthegridadaptationprocess.

Fig. 8. Atleft:geometryofthewholecomputationaldomainfortheinfinite-spanNACA0012wingtests.Atright:enlargementoftherectangularportion ofthedomainhighlightedatleft,thesurfacemeshescorrespondingtothewallatz₌0 andtothewingareshownandthewingedgesaredrawninred. (Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Fig. 9. Gridconvergencestudiesforthetransonicflowaroundthesteadyinfinite-spanwingtestcase.TheabsolutedeviationsoftheliftcoefficientCLwith

respecttoreferencevalueC∗

Lareshownversusthenumberofgridnodes.InadditiontotheresultsonthefixedgridsdescribedinTable 2,threeseriesof

adaptivesteadycomputationsaredisplayed.Todrawthedashedlinescorrespondingtofirst- andsecond-orderaccuracy,therelationbetweenthenumber ofnodesandthegridspacinghareretrievedfromgridswithhw=0.05 and0.025 inTable 2.

elasticanalogyfails.Theso-calledPredictionstepconsistsinthecomputationofthesolutionoverthemeshthat hasbeen

already deformed to comply with the boundary motion. Thanks to the prediction phase, mesh adaptation is driven by

criteriacomputedover asolutionupdatedon thenewmeshconfiguration,ratherthantheone atthe previoustime step. TheinclusionofthePredictionstepisfoundtoallowforlargertimestepswhilepreservingthesameaccuracyasitwillbe showninthefollowingSection5.Thus,thecompleteprocedurecanbesummarizedasfollows

I. Meshdeformation:thegrid

_K

n _isupdatedto

K

n+tocomplywiththeboundarypositionattn+1_.

II. Predictionphase:thesolutionatthenewtimestep,overthegrid

_K

n+_,iscomputed.

III. Meshadaptation ofthegrid

K

n+ onthe basis ofthenewsolution u

(

tn+1

_,

_K

n+

₎

_{andconsistent updateofthe spatial} discretization.

IV. Computationofthesolutionun+1 overthegrid

K

n+1,usingasinitialguessthesolutionu

(

tn+1

,

K

n+

)

.

Table 2

Fixedgridsusedintheconvergencestudyforthetransonicflowaroundthesteady infinite-spanwingtestcase.Alinearvariationinthegridspacingisobtainedduring themeshgenerationbyimposingthevaluehwoverthewingsurfaceandthevalue

h_∞atfar-field;topreservethecorrectgeometryofthewingsurface,asmallervalue

hLE isimposednearthetrailingedge.Theratiobetweenthesethreeparametersis

keptconstantamongallgrids. Wing surface hw Far-field h_∞ Leading edge hLE Grid nodes Nv 0.02 0.8 0.002 127 367 0.025 1.0 0.0025 71 315 0.03 1.2 0.003 44 548 0.035 1.4 0.0035 33 172 0.04 1.6 0.004 21 952 0.05 2.0 0.005 12 211 0.06 2.4 0.006 9 267

Fig. 10. GridandMachcontourplotsofthetransonicflowaroundthesteadyinfinite-spanwingatM∞=0.755 andα∞=0.016◦_{overthewallatz=0} andoverthewingsurface.Sincethegridelementsoverthewingsurfaceareverysmall,tomaketheleftpictureclearer,onlythewingedgesaredrawn, inred.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Table 3

QualityimprovementatdifferenttimestepsofthetestEnhanceforthetranslatinginfinite-spanwingwith*t1,

correspondingtoaCourantnumberof2.Theaverage,theworstandthebestvaluebeforeandaftermesh opti-mizationarereported.Remindthatthebestqualityis1 andcorrespondstoregulartetrahedra.

Performed displacement

Before adaptation After adaptation

Average Worst Best Average Worst Best

4.8 % 0.695289 0.000083 0.999565 0.771129 0.041053 0.999369 10.4 % 0.644737 0.000024 0.998924 0.751961 0.104121 0.999074 16.0 % 0.629724 0.000013 0.997824 0.738972 0.108341 0.997708 25.6 % 0.580335 0.000018 0.998159 0.729795 0.083266 0.999420 33.6 % 0.607000 0.000005 0.998135 0.720783 0.078378 0.998894 41.6 % 0.619678 0.000023 0.998515 0.716963 0.079011 0.998515 44.8 % 0.657054 0.026241 0.998617 0.715037 0.083704 0.998617 56.8 % 0.563106 0.000005 0.998932 0.712779 0.081304 0.998932 64.8 % 0.614716 0.000012 0.998932 0.705929 0.057188 0.998932 76.8 % 0.580232 0.000001 0.998908 0.704456 0.078917 0.998606 86.4 % 0.595999 0.000029 0.998498 0.701020 0.074342 0.998498 94.4 % 0.608164 0.000010 0.998451 0.699856 0.084938 0.998451 5. Results

Fig. 11. Gridsandpressurecontourlinesoverthewallatz₌0 andoverthewingsurfaceatthebeginning,at33%,at66%andattheendofonechord displacementfortheinfinite-spanwinginthetestEnhance,wheremeshadaptationisexploitedonlywhenelasticanalogyfails.Thetestisperformedwith thetimestep*t1correspondingtoaCourantnumberof2.Forgridplots,thewingisonlysketchedbydrawinginredtheedgesbecausegridelements

overthewingsurfacearetoosmalltobeclearlyrepresentedinthepicture.ThebluelinelabeledX0

TLindicatestheinitialpositionofthetrailingedge.(For

interpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

theflightvelocity.Theflowfieldaround anoscillatinginfinite-spanNACA0012wingisalsoinvestigatedinSubsection5.3

andthethree-dimensionalresultsarecomparedwithreferenceones.Then,steadyandunsteadysimulationsareperformed alsoforafinite-spanNACA0012wing,amodelwhichisofprimary importanceinrotor-craftsCFDsinceitcan represent thetipofarotorblade.Moreover,differentlyfromtheinfinite-spanwingcase,thisflowfieldisfullythree-dimensional,so itisavaluabletesttoassessthecapabilityoftheproposedscheme.

Finally,inSubsections5.5and5.6,twotestscharacterizedbylargeboundarydisplacementsarepresented,i.ethe infinite-spanNACA0012wingtravelingfor20chordsandapiston-inducedshock-tubeprobleminwhichthedomainexperiencesa reductionofmorethanthehalfoftheinitialsize.

Inallnumericalexperimentspresentedinthissection,airismodeledasaconstant-specific-heatsidealgaswithspecific heatratio

γ

=

1

.

4.Theflow equationshavebeenmadedimensionlessby choosingthefollowingreferencevalues: Pref

=

Fig. 12. Gridsandpressurecontourlinesoverthewallatz₌0 andoverthewingsurfaceatthebeginning,at33%,at66%andattheendofonechord displacementfortheinfinite-spanwinginthetestAdapt,wheremeshadaptationisperformedeverytimestep.Thetestisperformedwiththetimestep

*t1correspondingtoaCourantnumberof2.Forgridplots,thewingisonlysketchedbydrawinginredtheedgesbecausegridelementsoverthewing

surfacearetoosmalltobeclearlyrepresentedinthepicture.ThebluelinelabeledX0

TLindicatestheinitialpositionofthetrailingedge.InSupplementary

Material,MoviesS1andS2showthevariationofthegridandofthepressureduringthissimulation,respectively.(Forinterpretationofthereferencesto colorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

5.1. Steadysimulationofinfinite-spanNACA0012wing

The computationaldomainbuilttomodeltheflow fieldarounda so-calledinfinite-spanNACA0012wingisshownin

Fig. 8.Thewingspanisequaltothechordcandthefar-fieldconsistsinacylindricalsurfacewitharadiusof12c,centered at the leading edge.For an infinite-spanwing, the lateral planes at z

=

0 and z

=

c are treatedas walls, which, under the considered inviscid approximation, correspond to symmetry boundary conditions. Therefore, the flow is intrinsically bi-dimensional.Afree-streamflowatMachnumberM∞

=

0

.

755 andincidence

α

∞

=

0

.

016◦ _{isimposed,alongwithunity} dimensionlesspressureanddensity.

A convergence study on fixed grids is performed by comparing the error on the liftcoefficient obtainedin different simulations.First,areferencevalueiscomputedonafinegridcomposedby622 693nodes,gatheredmainlynearthewing surface whereagridspacingbetween0

.

001 and0

.

01 isenforced.Itshouldbepointedoutthatsomeother finegridshave

Fig. 13. Liftcoefficientandintegralvalueofthedensityoverthedomainfortheinfinite-spanNACA0012winginthelaboratoryframe.Theresultsobtained byexploitingonlyedge-swapping(Swap),meshadaptationonlywhenelasticanalogyfails(Enhance)andeverytimestep(Adapt)arecompared.Timesteps

*t1and*t2correspondtoaCourantnumberof2and5,respectively.Intheleftpicture,thevalueoftheintegraldensity,labeledM,isscaledwithrespect

tothesteadyvalueM0.

Fig. 14. Absolutevariationoftheintegralvalueofthedensityoverthedomainfortheinfinite-spanNACA0012winginthelaboratoryframe,intheAdapt

testsperformedbyenforcingnocoarseningonthefar-fieldboundarytoavoidvariationsinthevolumeofthedomain,whichleadtovariationsinthe integraldensity.

Fig. 15. ErroronthefulfillmentoftheDGCLconditionandonthevolumeclosureinthetestAdaptoftheinfinite-spanNACA0012winginthelaboratory frame,withtwodifferenttimesteps.ThesetwoquantitiesarecomputedaccordingtoDefinitions(19).

inTable 2areusedtoperformsteadycomputations.TheerrorswithrespecttothereferenceliftcoefficientC∗

L aredisplayed

inFig. 9andanorderofconvergencebetweenoneandtwocanbeobserved,asexpected.

Fig. 16. Detailsofthecomputationalgridonplanez=0 andoverthewingsurfaceatdifferenttimesofthepitchinginfinite-spanwing.Thelastperiodof thesimulationwith*t1isshown.InSupplementaryMaterial,MovieS3displaysthegridduringthelastperiodofthissimulation.

Fig. 17. Mach contour plot on plane z=cat different times of the pitching infinite-span wing. The last period of the simulation with*t1is shown.

Inthe cyclelabeledB,fouradaptationsteps areperformedstartingfromthegridcharacterizedbyhw

=

0

.

003 inTable 2.

Fig. 18. Machcontourplotandpressureiso-lineseveryquarterofthelastperiodofthesimulationofthepitchinginfinite-spanwingwith*t1.Theiso-lines

Fig. 19. Variationofthenumberofgridnodesduringthewholesimulationofthepitchinginfinite-spanwingwith*t1.Twopeakscanbeobservedin

eachperiod,duetothetwomaximumintensitiesoftheshockontheupperandthelowerwingsurface.

Fig. 20. Liftcoefficientversusangleofattackfortheinfinite-spanpitchingwing.Theresultsobtainedinthelastperiodofthesimulationswithtwotime stepsareshownalongwiththeonesobtainedwithoutthepredictionstep.Moreoverthe2dresultsandtheonesprovidedby[81]areshown.

Fig. 21. Pressure contour plot on plane z=0 and over the wing for the steady simulation of the finite-span NACA 0012 wing. 5.2. Unsteadysimulationofinfinite-spanNACA0012winginthelaboratoryframe

Fig. 22. Liftcoefficientforthefinite-spanNACA0012winginthelaboratoryframe.Theresultsobtainedbyexploitingonlyedge-swapping(Swap)and adaptationateachtimestep(Adapt)arecompared.

Theinitialsolutionisobtainedfromthesteadyonebysubtractingthefree-streamvelocity.Adisplacementofonechord in thenegative x-direction at M∞

=

0

.

755 isimposed tothe wing during the simulation.The whole simulation time is

dividedinto125timesteps

*

t1,resultinginaCourantnumberof2ontheminimumgridedge,thatishmin

=

0

.

004.

BeforeintroducingtheresultsoftheadaptiveALEscheme,itshouldbeconsideredthatitwasnotpossibletosolvethis problem,i.e.toaccomplishthe wholedisplacement,onafixedconnectivitygrid,evenwithatime steptentimessmaller than

*

t1.

Inthefirsttest,labeledasEnhance,meshoptimizationisexploitedonlyduringthosetimestepsinwhichelasticanalogy failsinfollowingthe motionofthewing. Fig. 11showsthe gridsandthepressure contourlinesatthe beginningofthe simulationandwhenonethird,twothirdsandthewholedisplacementarecarriedout.Theschemeprovestoabletodeal withthelargemovementofthewingbutleadstoacoarseningofthegridandtoaslightdeteriorationofthesolution.The onlyaimoftheoptimizationinthistestistheenhancementofmeshqualityaccordingtoDefinition(18).Indeed,asTable 3

shows, theaverage,the worstandthebest elementquality areimproved thanksto there-meshing.However, becauseof mesh deformation atfixed connectivity, smaller elements, initially gathered inthe crucial regions of thegrid, are prone tobecome badlyshapedandconsequentlyto collapse.Thisdegradation occursmainly inthe regionadjacenttothewing leading andtrailingedge, wherethe averagesize ofthe gridelements at theendof thedisplacement islarger thanthe oneontheinitialgrid.Asimilartest,labeledinthefollowingSwap,isperformedinwhichmeshqualityisrestoredonlyby meansofedgeswapping,thusnonodeinsertionsordeletionsareexploited.

Then, the complete adaptive computational procedure (Adapt test) is adopted, using an error estimator base on the Hessianofthepressureandofthevorticity.AsitcanbenoticedfromFig. 12,theadaptiveprocessbasedalsoonthe solution leads toarefinementnearthetrailingandleading edge,andthe adaptivesolutions showlessoscillationsinthepressure field withrespecttothetest Enhance.Thedifferencesinthe solution atdiversetime levels arenegligible andthe steady natureoftheproblemiswellpreserveddespitetheunsteadysimulation.

Finally,Fig. 13showsaquantitativecomparisonoftheresultsobtainedintheunsteadysimulations Enhance,Adaptand

Swap,withtwodifferentvalues oftimestep:thealreadyintroduced

*

t1and

*

t2,whichcorrespondstoaCourantnumber

of5.Sub-Fig. 13(a)displaysthevariationsoftheliftcoefficient,whichremainlimited—approximatelywithin

±

0

.

0015 with

respecttothesteadycomputedvalueof0

.

00327—in theEnhanceandevenlessintheAdaptsimulations,wherethanksto

the meshadaptation thegrid spacingis modified toimprove solutionaccuracy. Conversely, inthe Swap teststheresults degenerateinthesecondhalfbecausetheedge-swappingtechniquealoneisnotabletoadaptthegridinaproperway.No significantdifferencesbetweendifferenttimestepsareobserved,especiallyintheadaptivetests.

Sub-Fig. 13(b)displaysthevariationintimeoftheintegralvalueofthedensityoverthedomain,computedbya trape-zoidalruleoneachfinitevolume.Alsointhiscase,thevariationsaroundtheinitialvalueareverylimited,belowthe0.25%. TheinitialincreasethatcanbeobservedintheAdapttestsisduetotheadaptationthatoccursonfar-field,whichis repre-sentedintheinitialgridasacircle.Sinceanaccuraterepresentationofthisboundaryisnotrequiredduringthissimulation, theadaptationparameterforthecorrespondingHausdorffcriterionisrelaxedand,accordingtothesmoothbehaviorofthe solutioninthisregion,thedeletionofsomenodesonthisboundaryleadtoaslightlymodificationofthedomainvolume. Thisde-refinementoccursinthistestandnotinthesteadyone,sinceinunsteadytestsmoreimportancehastobeplaced on thecoarseningprocess to prevent an excessivelyincrease inthe gridnodes[79].However, to assessmoreclearly the conservativenessoftheproposed approach,bothAdaptsimulationsare performedalsonotallowingcoarseningonthe far-field.Inthiswaytheintegralvalueofthedensityoverthedomainisnotaffectedby thevariationinthevolumedomain andremainsapproximatelyconstant(variationlessthan0.001%)duringthesimulation,asitcanbeevincedinFig. 14.

Finally,toassessthefulfillmentoftheDGCLcondition, theerrorcommittedateachtimestepinsatisfyingthediscrete counterpartofEquation(6)isreportedinFig. 15,alongwiththeerrorontheclosureofthedomain.Asexpected, thanks tothecomputationoftheinterfacevelocitiesdescribedinSubsection2.4,theseerrorscanbeconsiderednull,assessingthe capabilityof theschemeto automaticallyfulfilled theDGCL.For thesake ofcompleteness,the errorquantitiesshown in

Fig. 23. Gridsandpressurecontourlinesoverthewallatz₌0 andoverthewingsurfaceatthebeginning,at33%,at66%andattheendofonechord displacementforthefinite-spanwing.Forgridplots,thewingisonlysketchedbydrawinginredtheedges.ThebluelinelabeledX0

TLindicatestheinitial

positionofthetrailingedge.InSupplementaryMaterial,MoviesS5andS6showthevariationofthegridandofthepressureduringthissimulation, respectively.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

ErrDGCL

=

3

4

4

4

5

'

i∈Kn+1

⎛

⎝−

V n+1 i

−

Vin

*

t

+

'

k∈Ki,̸=

ν

_ikn+1

+

ν

n_i+1

⎞

⎠

2

:

!

n+1

,

(19a) Err!

=

3

4

4

4

4

5

⎛

⎜

⎝−

!

n+1

_{− !}

n

*

t

+

'

i∈Kn∂+1

ν

n_i+1

⎞

⎟

⎠

2

:

!

n+1

,

(19b)