A domain decomposition method for quasi-incompressible formulation with discontinuous pressure fields

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A domain decomposition method for

quasi-incompressible formulation with discontinuous pressure fields

Pierre Gosselet, Christian Rey, Françoise Léné, Pascal Dasset

To cite this version:

Pierre Gosselet, Christian Rey, Françoise Léné, Pascal Dasset. A domain decomposition method for

quasi-incompressible formulation with discontinuous pressure fields. Revue Européenne des Éléments

Finis, HERMÈS / LAVOISIER, 2002, 11, pp.363–378. �hal-01224421�

formulations with dis ontinuous pressure eld

Appli ationtotheme hani alstudyofaexiblebearing

P. Gosselet

∗

, C. Rey

∗

, F. Léné

∗

and P. Dasset

∗∗

*LaboratoiredeModélisation etMé aniquedesStru tures,

FRE2505duCNRS,UPMC,8rueduCapitaineS ott,75015PARIS

**SNECMAMoteurs,33187LeHaillan

February 3,2020

Abstra t

We study the implementation of a domain de omposition method for stru tures with quasi-

in ompressible omponents. We hoseamixedformulationwherethepressureeldisdis ontinuous

on the interfa es between substru tures. We propose anextension of lassi al pre onditioners to

this lassofproblems. Thenumeri alsimulationoftheme hani albehaviouroftheexiblebearing

ofthe nozzle ofa solidpropellant booster is then ondu tedusing various Newton-Krylov parallel

approa hes. Wepresentthemainme hani alresultsand omparethenumeri alperforman eofthe

parallelapproa hestoasequentialapproa h.

Nousétudionslamiseenoeuvred'uneméthodededé ompositiondedomainepourstru turesà om-

posantsquasi-in ompressibles.Uneformulationmixteà hampdepressiondis ontinuauxinterfa es

entre sous-stru tures est retenue ; nous proposons, pour ette lasse de problèmes, une extension

despré onditionneurs lassiques. La miseenoeuvredelasimulationnumériquedu omportement

mé aniqued'unebutéeexibledetuyèredepropulseuràpropergolsolidepardiversesappro hespar-

allèlesitérativesdetypeNewton-Krylovestalorsproposée. Nousprésentonslesprin ipauxrésultats

mé aniquesainsiquelesperforman esnumériquesobtenuesparlesappro hesparallèlesretenueset

uneappro heséquentielle.

Keyword: Domainde ompositionmethod,Newton-Krylov,quasiin ompressibility,mixedformula-

tion

Mots- lésMéthodededé ompositiondedomaine,Newton-Krylov,quasiin ompressibilité, formu-

lationmixte

1 Introdu tion

Primalanddualdomainde ompositionmethods[6 ,3℄are amongtherstnon-overlappingdomain

de ompositionmethodsthathavedemonstratednumeri als alabilitywithrespe ttobothmeshand

subdomainsizes. They haveprovedtheire ien yonmanytypesof problemssu has se ondand

fourthorderlinear(stati anddynami )elasti ity,heterogeneousproblems...andtheyare urrently

extendedtootherproblemssu hasStokes'equation[1℄.

Inthispaperwefo usonthe omputationofquasi-in ompressibleelastomeri omponentsusing

aprimal domainde ompositionmethod. Numeri al simulation of the behaviour of su hmaterials

whi h main properties are the ability to handle large deformations, the non-linearbehaviour and

thequasiin ompressibility,requirestousemixeddispla ement-pressureniteelements[6, 2℄. More

pre iselywedeal withthe ase ofdis ontinuouspressureeld ontheinterfa e ofsubstru turesand

ontinuousor dis ontinuouspressure eld inside substru tures. Su h an approa h orresponds to

any substru turation whenthe pressure is dis ontinuous between niteelements, and to physi al

de ompositionbetweendierentpie eswhenthe ontinuityofpressureisensuredinsidesubstru tures

(e.g. interfa e between steel and elastomeror two dierent elastomeri pie es). However, as the

in ompressible/ ompressibleheterogeneity is nottaken intoa ount ina satisfyingway by urrent

pre onditioners, we extendthem to this lassof problems. Beside, be ause of non linearities, we

use anon-linear solverleading to the solutionto a sequen eof ill- onditioned linearsystemswith

both non invariant matrix and right-hand side. Various strategies to a elerate the solution to

su essivesystems[9 ,10 ,11 ℄havealreadybeendeveloped,andevaluated oupledwithadualdomain

de ompositionmethod. Weassesstheperforman e ofsu hKrylova eleration approa hes oupled

withtheprimaldomainde ompositionmethod.

Allnumeri alassessmentsrelatetoavery hallengingindustrialproblem: thenumeri alsimula-

tionofsteel-elastomerstratiedstru tures. These stru turesarewidelyusedinaerospa e industry

revolvingma hines,blade-rotor onne tionsofheli opters,ro ket-nozzle onne tionsoftheArianeV

laun her. They maytake theformof aexiblesteel-elastomerstru turelo ated between thebody

andthenozzleofasolidpropellantbooster,ofwhi htheengineofthepowdera elerationstagesof

theArianeVlaun herisatypi alexample.

Thus, we present inse tion (2) the formulation of the problem and the generi algorithms to

a hieve the simulation. We give in se tion (3) the extension of traditionalpre onditioners to the

quasi-in ompressible aseandinse tion(4)aKrylova elerationte hniquetosolvethesequen eof

linearsystemsresultingfromthe linearization ofthenon-linearproblem. Inse tion (5)we present

theexiblebearing whi hsupportsthe assessments,and asso iated me hani al results; se tion (6)

sumsupnumeri alperforman e. Se tion(7) on ludesthisarti le.

2 Overview of the models and methods

2.1 Lagrangian formulation

We onsiderthe omputationoftheequilibriumpositionofabody

Ω

^madeupofaquasi-in ompressible hyperelasti materialundergoing largedeformation. We hoose alagrangianformulation where all

variablesaredenedinthereferen e onguration.Let

f

^{denotethebodyfor e,}

g

^{thesurfa etra tion}

imposedon

∂ g Ω

^,

u 0

^theimposeddispla ementonthe omplementarypartoftheboundary. Taking into a ount the in ompressibility leads to the introdu tion of an unknownpressure eld

p

^{. The}

resear hoftheequilibriumofthebody(deadloadingassumption)isequivalenttotheresear hofthe

saddlepointofthefollowinglagrangian:

(u, p) ∈ ({u 0 } + H) × P L(u, p) =

Z

Ω

W(F )dΩ + Z

Ω

p(h(J) − 1

2K p)dΩ − Z

Ω

f udΩ − Z

∂ _g Ω

gudS

⁽¹⁾

Theproblemthenreads:

Find

(u, p) ∈ ({u 0 } + H) × P / ∀(v, q) ∈ H × P, Z

Ω

∂W

∂F (Id + ∇u) : ∇vdΩ + Z

Ω

ph ^′ (J) ∂J

∂F : ∇vdΩ = Z

Ω

f vdΩ + Z

∂ _g Ω

gvdS

Z

Ω

(h(J) − 1

K p)qdΩ = 0

(2)

Where

H

^and

P

^{arethe spa esofadmissible} displa ement andpressure elds,

F

^{isthegradientof}

the deformation (

F = Id + ∇u

^,

J = det(F )

^),

K

^{is the} ompressibility modulus of the material.

Freeenergy

W(F )

^{anbe hosenfromdierentmodels([12 ,5℄). Forisotropi materials,it isoften}

writtenasafun tionofthe

C = F ^T F

^{tensorinvariants}

W(F ) = ¯ W(I 1 , I 2 , J)

^where

I 1 = Tr(C)

^and

I 2 = ¹ ₂ (Tr ² (C) − Tr(C ² ))

^{. Amongotherswe itethe}Mooney-Rivlinmodel:

W(I ¯ 1 , I 2 ) = C 10

2 (I 1 − 3) + C 01

2 (I 2 − 3)

Where

C 01

^and

C 10

^{are onstantsthat} hara terizethematerial. Fun tion

h(J)

^{analsobegivenby}

variousmodels,inthesimplest ase (linearmodel)

h(J) = (J − 1)

^.

Thenumeri alsolutiontothisvariationalproblemis lassi ally ondu tedusingtheniteelement

method.Subspa es

H

^and

P

^{arerepla edwithnitedimensionsubspa es}

H h ⊂ H

^and

P h ⊂ P

^{. Let}

usunderlinethat the onstru tionofmixedniteelementsmustinparti ular omplywith ompat-

ibility onditions (Ladyzenska-Babuska-Brezzi ondition[2℄) thus restri ting thepossible hoi esof

approximationspa es. However ommon hoi esfor

3

^{Dproblemsarethe}

Q 2 − P 1

^{hexahedralelement}

(

27

displa ement nodes,

4

^{pressure nodes)and the}

Q 2 − Q 1

^{hexahedral element (}

20

displa ement nodes,

8

^{pressurenodes).}

2.2 Newton-type algorithms

The problem arising from the nite element method is nonlinear, let us write it

F(x) = 0

^with

x = (u h , p h )

^{. The prin iple of Newton's algorithms is to build a sequen e of linear systems the}

solutions of whi h onverge to the solution to the non-linear problem. There are many versions

of Newton'salgorithms, the most widelyused is Newton-Raphson's. This last method onsists in

iteratively substitutingthesolutiontotheequation

F(x) = 0

^{foritsrstorderlimited}development around

x ^k

^:

F(x ^k ) + dF(x ^k )

dx (x ^k+1 − x ^k ) = 0

⁽³⁾

Newton-Raphson's method is known to onverge fast when properly initialized. A very ommon

extensionisthe in rementalalgorithmwhi h onsistsindening"stepsofloading"and ndingthe

step as an e ient initialization. The linear system arising from Newton-Raphson's linearization

reads:

 K uu K up

K up ^T K pp

  v ^k q ^k



=

 f u

f p



with

 v ^k = u ^k+1 − u ^k

q ^k = p ^k+1 − p ^k

⁽⁴⁾

 K uu (u ^k , p ^k ) 

ij = Z

Ω

 ∂ ² W

∂F ² : ∇Φ i



: ∇Φ j dΩ +

Z

Ω

p ^k h ^′′ (J)  ∂J

∂F : ∇Φ i

  ∂J

∂F : ∇Φ j

 dΩ +

Z

Ω

p ^k h ^′ (J)  ∂ ² J

∂F ² : ∇Φ i



: ∇Φ j dΩ



K up (u ^k , p ^k ) 

ib = Z

Ω

h ^′ (J)  ∂J

∂F : ∇Φ i

 Ψ b dΩ

 K pp (u ^k , p ^k ) 

ab = − 1

K Z

Ω

Ψ α Ψ β dΩ



f u (u ^k , p ^k ) 

i =

Z

Ω

f Φ i dΩ + Z

∂ g Ω

gΦ i dS − Z

Ω

∂W

∂F : ∇Φ i dΩ

− Z

Ω

p ^k h ^′ (J) ∂J

∂F : ∇Φ i dΩ

 f p (u ^k , p ^k ) 

a = −

Z

Ω

Ψ α



h(J) − 1 K p ^k

 dΩ

(5)

Wherefun tions

(Φ i )

^and

(Ψ α )

^{are the basis of the} displa ement and pressure elds. For amore ompletedes riptionofNewton'stypealgorithmforin ompressiblenon-linearelasti ity,readers an

referto[7 ℄.

Remark: Whenusing domainde ompositionmethods,be ause ofinsu ient Diri hlet's onditions

orinternalme hanisms,thestinessmatrixofsomesubstru turesmaybenotinvertible;the ompu-

tationofthekernelofthematrixisthenanimportantpoint. Asfarasweknow,therearenogeneral

resultswhi hindi ateapriorithe ompositionofthekernel. What anbedemonstratedforsubstru -

tureswithoutme hanismisthattherstsystemisthelinearizedelasti itysystem,thentheve tors

ofthekernelare(rigidbodydispla ements,zeropressure),forfollowingsystemsve tors omposedby

(admissibletranslations, zero pressure)alwaysbelong to thekernel. Wehave neverobservedother

kindsofnullspa e modes (translationsand rotationsfor therstsystem, onlytranslationsfor the

following systems). Sowe propose touse ageometri al omputationof therigidbody motionsfor

therstsystemandjustsuppresstherotationsforthefollowingsystems.

Duetotheinversibilityofthe

K pp

^{submatrix,pressurenodes anbeeliminatedfromtheresolution}

pro essusingaS hur ondensation. One ansolvethefollowingsystemforthedispla ementunknown

and omputepressureaspost-pro ess:

Kv ˜ ^k = ˜ f

^with

 K = K ˜ uu − K up K pp ⁻¹ K up ^T

f = f ˜ u − K up K pp ⁻¹ f p

(6)

Inthe asewheretherearenonodesontheinterelementboundary(dis ontinuouspressureeld)whi h

isthe ase ofthe

Q 2 − P 1

^{hexahedralelement,this} ondensationis usuallya hievedat theelement s aleataverylow ostsin ethe

(Ψ α )

^{fun tions anbe hosen}orthonormalandthen

K pp = − _K ¹ I d

^.

2.3 Primal domain de omposition method

We briey re all the primal domain de omposition method[6 ℄ ina generi ase, the next se tion

fo uses onitsextensionto mixed displa ement-pressure formulations. We onsider thedis retized

problem (4). Let us make a non-overlapping onform partition of dis retized domain

Ω

^into

N

subdomains

(Ω ^(s) ) 16s6N

^{,theinterfa e ofasubdomainis denedas}

Υ ^(s) = ∂Ω ^(s) \∂Ω

^{,the omplete}

interfa e

Υ

^{istheunionoftheinterfa esofall}substru tures.

Using lassi alnotation(

i

^{standsforinternaldegreeoffreedom,}

b

^{forboundarydegreeoffreedom),}

thestiness matrixofthe

s ^th

^{subdomainreads:}

K ^(s) = K _ii ^(s) K ^(s) _ib K _bi ^(s) K ^(s) _bb

!

(7)

Theprimalapproa hsimply onsistsineliminatinginternaldegreesoffreedomfromthe omplete

problemwhi h anbedoneindependentlyonea hsubstru ture onstru tingthelo alprimalS hur

omplement

S ₁ ^(s)

^{. Let}

u

^bethedispla ementeldoftheinterfa edegreesoffreedom,theproblemto solvethenreads:

Su = b

^with



 

 

 

  S = P

s

B ^(s) S ₁ ^(s) B ^(s)T b = P

s

B ^(s) b ^(s) S ^(s) ₁ = K _bb ^(s) − K _bi ^(s) K _ii ^{(s) −1} K _ib ^(s)

b ^(s) = f _b ^(s) − K _bi ^(s) K _ii ^{(s) −1} f _i ^(s)

(8)

The

B ^(s)

^{matrixproje tsthelo alinterfa e}

Υ ^(s)

^{ontheglobalinterfa e}

Υ

^{. Fortheprimalapproa h,}

oppositeto the lassi al dualmethod(FETI), rosspoints(pointsshared by morethan

2

^{substru -}

tures)arenotrepeatedwhendes ribing

Υ

^.

Duetotheexisten eofe ientpre onditioners,system(8)issolvedusingaKryloviterativesolver

(Conjugategradient,GMRes

. . .

^{)whi hiswellsuitedtotheparallel}ar hite tureofmodern omput- ers. TheNeumann pre onditioner onsists inapproximating the inverse of the sumof lo al S hur

omplementsbythesumoftheinverseoflo alS hur omplements.Let

M ⁻¹

^bethepre onditioner:

M ⁻¹ = P

s

D ^(s) B ^(s) S ₂ ^(s) B ^(s)T D ^(s) S ^(s) ₂ = S ₁ ^{(s) +} = β ^(s) K ^{(s) +} β ^(s)T

(9)

β ^(s) = (0 i Id b )

^{extra ts fromve tors dened on thesubdomain}

Ω ^(s)

^{their tra eon their interfa e}

Υ ^(s)

^.

K ^(s)+

^{is a}pseudo-inverseofmatrix

K ^(s)

^,

S ^(s) ₂

^{is thelo aldual S hur} omplement.

D ^(s)

^isa

diagonal s alingmatrix(

P D ^(s) = Id Υ

^{). Whendealing with}homogeneousstru tures,

D ^(s)

^anbe

hosenequaltotheinverseofthemultipli ityofea hdegreeoffreedom. Forheterogeneousstru tures

[13 ℄,s alinghas to provideinformation aboutthe dieren eof stinessbetween subdomains,most

oftenthisitemofinformationisextra tedfromthediagonalofthe

K ^(s) _bb

^matrix:

D ^(s) _i = (B ^(s) Diag(K ^(s) _bb )B ^(s)T ) i

( P

k

B ^(k) Diag(K _bb ^(k) )B ^(k)T ) i

(10)

Tobe omes alablewithrespe tto thenumberofsubstru tures,theprimal approa hequipped

withtheNeumannpre onditionerhas tobeenri hedwitha oarseproblem. Theidea isto ensure

thatve torsthat aremultipliedby generalizedinversematri es

K ^(s)+

^{belongto theimageof}

K ^(s)

^.

Thismethodisreportedasbalan ingmethod[8℄be auseitsme hani alinterpretationistoensurethe

equilibriumofea hsubstru turefa eup torigid-bodyloadings. Noting

r

^theresidual

(r = b − Su)

^,

pre onditioning onsistsin omputing

M ⁻¹ r

^.

M ⁻¹ r = P

s

D ^(s) B ^(s) β ^(s) K ^{(s) +} β ^{(s) T} B ^(s)T D ^(s) r

∀s β ^(s)T B ^(s)T D ^(s) r ∈ Im(K ^s )

⇔ ∀s R ^(s)T β ^{(s) T} B ^(s)T D ^(s) r = 0

^with

Span(R ^(s) ) = Ker(K ^(s) )

⇔ ∀s (D ^(s) B ^(s) β ^(s) R ^(s) ) ^T r = 0

⇔ G ^T r = 0

^with

G = . . . D ^(s) B ^(s) β ^(s) R ^(s) . . .

(11)

This ondition isimposedusing aproperinitialization(

u 0 = G(G ^T SG)G ^T b

^{) and aproje tor}

P = Id − G(G ^T SG) ⁻¹ G ^T S

^;thepre onditionerthenreads

P M ⁻¹

^.

3 Extension of primal domain de omposition method to

quasi-in ompressible material with dis ontinuous pressure

eld

Inthispaperwedealwiththe aseofdis ontinuouspressureeldsattheinterfa eofsubstru tures.

Thepressureeld insidesubstru turesmay beeither ontinuous(e.g. hexahedral

Q 2 − Q 1

^{) ornot}

(e.g. hexahedral

Q 2 − P 1

^{). Inthe aseof ontinuouspressureeldinside}substru ture,su hamodel orrespondsto physi alde ompositions between stu k pie es(whatevertheirmaterialmaybe,e.g.

interfa ebetweensteelandelastomerortwodierentelastomersortwodierentpie esofthesame

elastomer). Hen eallpressuredegreesoffreedomare onsideredinternal.

Forthefollowingequations

i

^and

b

^{standforinternalandboundary}displa ementdegreeoffreedom,

p

^{forpressure degreeoffreedom. Sin epressureis onsideredasaninternaleld, the} ondensation shownin(6) anberealizedatthesubstru tures ale(ifnotyetrealizedattheelements ale)without

modifyingtheglobalproblem. Theinterfa eproblemthenreads:

Su = ˜b ˜

^with



 



 

 S = ˜ P

s

B ^(s) S ˜ ₁ ^(s) B ^(s)T ˜b = P

s

B ^(s) ˜b ^(s) S ˜ ₁ ^(s) = ˜ K _bb ^(s) − ˜ K _bi ^(s) ( ˜ K _ii ^(s) ) ⁻¹ K ˜ _ib ^(s)

˜b ^(s) = ˜ f _b ^(s) − ˜ K _bi ^(s) ( ˜ K _ii ^(s) ) ⁻¹ f ˜ _i ^(s)

(12)

Theexpressionofthematri esandve torsaboveistheexpansionofequation(6):

K ˜ _ii ^(s) K ˜ _ib ^(s) K ˜ _bi ^(s) K ˜ _bb ^(s)

!

= K _ii ^(s) − K _ip ^(s) K pp ^(s)

−1 K _pi ^(s) K _ib ^(s) − K _ip ^(s) K pp ^(s)

−1 K _pb ^(s) K _bi ^(s) − K _bp ^(s) K pp ^(s)

−1 K _pi ^(s) K _bb ^(s) − K _bp ^(s) K pp ^(s)

−1 K _pb ^(s)

!

f ˜ _i ^(s) f ˜ _b ^(s)

!

= f _i ^(s) − K _ip ^(s) K pp ⁻¹ f p ^(s)

f _b ^(s) − K _bp ^(s) K pp ⁻¹ f p ^(s)

!

(13)

Noteaboveall thattheresultingstinesss aling

D ˜ ^(s)

^{isbuiltfromthediagonal}

K ˜ ^(s) _bb

^,thatisto

sayfromthediagonalofthematrix

(K _bb ^(s) − K _bp ^(s) K pp ^(s)

−1 K _pb ^(s) )

^.

Howeverifwedonot ondensethepressure,wehave:

K ^(s) =









K _ii ^(s) K _ip ^(s) K _pi ^(s) K pp ^(s)

! K _ib ^(s) K _pb ^(s)

!



K _bi ^(s) K _bp ^(s) 

K _bb ^(s)









(14)

Su = b

^with



 

 

 

 



 

 

 

 



S = P

s

B ^(s) S ₁ ^(s) B ^(s)T b = P

s

B ^(s) b ^(s)

S ₁ ^(s) = K _bb ^(s) − 

K _bi ^(s) K _bp ^(s)  K _ii ^(s) K _ip ^(s) K _pi ^(s) K pp ^(s)

! −1

K _ib ^(s) K _pb ^(s)

!

b ^(s) = f _b ^(s) − 

K _bi ^(s) K _bp ^(s)  K _ii ^(s) K _ip ^(s) K _pi ^(s) K pp ^(s)

! −1

f _i ^(s) f p ^(s)

!

(15)

Notethats alingmatrix

D ^(s)

^{asso iatedtothe}non- ondensedpressureproblemisthendire tly builtfromthe

K _bb ^(s)

^matrix.

Bothproblems(whetherthepressureis ondensedornot)areequal:

S = S ˜

^,

˜b = b

^{. Itisthen}

abnormalthats alingmatri es shoulddier

D ˜ ^(s) 6= D ^(s)

^{. Infa tthe} ondensation ofthe pressure nodesleadstoanoverestimationofthe stiness;we thenpropose twodierents alingswhi hwork

newhether the materials are ompressible or not. Therst one is built from the

K _bb ^(s)

^diagonal

(before ondensation). Sin eobtainingthis informationmaynotbe easywhenusingelement-s ale

ondensation,weproposease onds aling, simplerbutevenbetter,whi hisbasedontheshearing

modulus

µ

^{ofthedierentmaterials}

D ^(s) _j = ^µ

(s) P j

k µ ^(k) _j

.

Table (1)summarizes the performan e of the dierent s alings for the industrialstru ture de-

s ribedse tion(5). Theelementusedisanhexahedra

Q 2 −P 1

⁽

27

displa ementnodes,

4

^{internalpres-}

surenodes). Thenews alingsshowtheire ien y,theyevenmanagetoa hievebetterresultsthan

the omputationof thehomogeneousstru ture. Theee tof theperturbation (

−K _bp ^(s) K pp ^(s)

−1 K _pb ^(s)

⁾

introdu ed by the ondensation anbeobservedonthe usualstiness s aling: the perturbation is

biggerforthese ondsystemthenitrequiresmu hmoreiterationsto onverge.

De omposition Typeofs aling Numberofiterations

Firstsystem Se ondsystem

6a-1r(6pro .) Topologi al

290

^>

1000

6a-1r(6pro .) Usualstiness

( ˜ D) 120 726

6a-1r(6pro .) Stinessbefore ondensation

48 44

6a-1r(6pro .) Shearingmodulus

43 39

6a-1r(6pro .) Homogeneousstru ture

93 116

Table1: A tionofthes aling-mixedelement

Q 2 − P 1

Remark: Of ourse,thesameanalysis anbe ondu tedfromthedualdomainde ompositionmethod

(FETIalgorithm). Newduals alings anbedenedonthebasisofthesameprin iple,theyproved

similare ien y.

4 Krylov a eleration strategy: GIRKS

The ontextofthestudyistheresolutionofasu essionoflinearsystems,letus onsiderthesolving

ofthe

(k + 1) ^th

^system

S ^k+1 u ^k+1 = b ^k+1

^{,theaimofthefollowingstrategyistoreusethe}information generatedduringtheresolutionofprevioussystemstosolvethe urrentsystem. Theresolutionofa

linearsystemwithaKryloviterativesolverleadsto the onstru tionofatleastonebasis

W ^k+1

^of

theKrylovsubspa efor whi hthe

Γ ^k+1 = W ^k+1T S ^k+1 W ^k+1

^{matrixiseasily}invertible. Inthe ase ofaConjugateGradient,notethat

W ^k+1

^{isthenthesetofresear hdire tionsand}

Γ ^k+1

^adiagonal

matrix.

The GIRKSalgorithm is ageneralization of augmentedKrylov subspa e methods for multiple

right hand sides [14 ℄ to the ase of non-invariant matri es (multiple left hand sides). It has two

distin ta tions,rstaninitializationIRKSanda orre tionofthepre onditionerGKC.

TheIRKSalgorithm(IterativereuseofKrylovsubspa es[11 ℄)isbaseduponaniterativeapproa h

makingitpossibletoevaluateatlow ostarelevantinitializationofalinearsystemwithrespe tto

previously generated Krylov subspa es. On e the initialization stage is omplete, the algorithm

is subje t to a restarting pro edure whi h an be onsidered as a Conjugate Gradient algorithm

augmentedwiththeKrylovsubspa egeneratedduringtheinitializationstage. TheGKC(Generalized

pre onditioningproblem.

Figure (4) gives the omplete algorithm of proje ted pre onditioned onjugate gradient with

GIRKSa eleration.

5 Study of the exible bearing

5.1 Des ription

Theorientationofthenozzleofaboosterisa hievedwithaexiblebearing. Thisbearingisastratied

stru turewiththinspheri al steeland elastomerlayers,it is maintainedbytwometalli supports.

Theexiblebearingwestudy(g. 2)wasproposedbySNECMAMoteurs,itwasdesignedtoletthe

bu klingofsteellayersappear. Thisbu klingwasobservedwhenperforminganexperimentalstudy

ofthesolidpropellant boosterofAriane5ro ket.

Thestru tureis lampedononeexternalring,aradialdispla ementimposedononepointatthe

bottomofthe nozzlemodelsthe turningloading(5 degrees),a

4

^{MPapressure duetothe gasesis}

imposedatthetopoftheexiblebearing(g. 3). Theresolutionis ondu tedintwosteps: rstthe

turningproblemissolvedusingtwononlinearin rements(

10

^{linearsystems),thenthe} ompression problem is solved with

4

^{nonlinear in rements (}

37

^{linear systems) or}

10

^{nonlinear in rements(}

48

linearsystems)whetherwewantthebu klingtoappearornot.

Resultingfrom theidenti ationofthematerials,simple onstitutive lawswere hosen. Steelis

denedusingaSaint-VenantKir homodel(Youngmodulus

E = 2.10 ⁵

^{MPa,Poisson's oe ient}

ν = 0.3

^{). Nearly}in ompressibleelastomerisdenedusingaMooney-Rivlin elasti potential(

C 10 = 0.2

^MPa,

C 01 = 0.

^MPa,

K = 2000

^MPa).

Many di ulties arisewhen arryingout thenumeri alsimulation ofthis exiblebearing,rst

non-linearitiesdueto thelargestrains,theinstabilitiesandthebehaviouroftheelastomer,se ond,

thehighheterogeneities(

5

^{degreesofmagnitudeseparatetheshearingmoduli)andlastthelargeand}

massiveaspe tofthis

3

^{Dproblem. Thesimulationis ondu tedusinga}

Q 2

^{hexahedralniteelement}

(

20

displa ementnodes)forsteelanda

Q 2 − Q 1

^{hexahedralniteelement(}

20

displa ementnodes,

8

pressurenodes)forelastomer,whi hleadsto

75900

^{degreesoffreedom.}

5.2 Me hani al results

A ordingtoexperimentalresults,thebu klingis omputedundera

3

^{MPapressure. Bu kling auses}

bifur ationsofthedispla ementofsomepoints(g. 5,6).

5.3 De ompositions used for the parallel simulation

Thegeometryofthe stru tureis axisymmetri (whilethe loadingisnot). Substru turesare hand-

made de omposing eitherthe axial se tion or the rotation. Thenomen latureof a de omposition

reads

N

^a-

M

^rwhere

N

^{standsforthenumberof}substru turesintheaxialse tion,

M

^forthenumber

ofsubstru turesintherotation.

Inthe aseofrotation-de omposedsubstru tures,pressureisdis ontinuousattheelastomer/elastomer

interfa eand ontinuousinsidesubstru tures.However,me hani alresultsareidenti alwhateverthe

de omposition.

6 Numeri al results

All omputationspresentedherewererealizedontheSGIORIGIN

2000

^{ofthePledeCal ulParis}

Sud. We ompareperforman elevelsforthebu klingandnon-bu klingproblems,of lassi alprimal

approa h, GIRKS-primal approa h and dire t sequential approa h. The dire t sequential solver

requires

8667

^{stosolveonelinearsystem.}

6.1 Non-bu kling problem

As said before, this loading history leads to the omputation of

48

^{linear systems with dierent}

matri esand right handsides. Thedire tsequentialapproa his ompletedin

115

^h

30

^{min. Parallel}

performan eresultsaregivenintable(2).

As anbeseen,performan elevelsstronglydependonthe hoi eofthede ompositionthoughthe

numberofsubdomainsisalmost onstant. Twofa torsjustifythesevariations,rsttheheterogeneity

ofthe interfa e (de ompositions giving bestresults ontain onlymono-materialsubstru tureswith

dierent materialsfa ing, while lesse ientde ompositionspossessmulti-materialsinterfa eswith

samematerialsfa ing),se ondtheaspe tratioofsubstru tures(duetothelowersparsityofmatri es,

massivesubstru turesinvolvelonger omputationtimeformatrixmanipulation).

k

^{linearsystems}

(S ^q u ^q = b ^q ) q=1,...,k

^weresolved

forthe

(q) ^th

^{linearsystem,wenote}

W ^q = {w ^q ₀ , . . . , w ^q _{r q−1} }

^{setofresear hdire tions}

r ^q = dim(W ^q )

Γ ^q = W ^{q T} S ^q W ^q

^{(diagonalmatrix)}

s ^q

ponderationterm(mostoften1)

Solution to the (k + 1) ^th system 1. IRKS approach

1.1 Initialization ˆ

u 0 = G(G ^T SG) ⁻¹ G ^T b + P χ ˆ

r 0 = b − S ˆ u 0

1.2 Iterations i = 0, . . . , p /ˆ z p = 0 ˆ

z i = P

"

k

P

q=1

W ^q Γ ^{q −1} W ^{q T}

# ˆ r i

ˆ

w i = ˆ z i + ⁱ⁻¹ P

j=0

ˆ

γ ij w ˆ j ( ˆ w 0 = ˆ z 0 ) ˆ γ ij = − _{( ˆ} ^(ˆ _w ^{z i} _j ^,S _,S ^{w ˆ} _{w ˆ} ^j _j ⁾ ₎ ˆ

x i+1 = ˆ x i + ˆ α i w ˆ i

ˆ

r i+1 = ˆ r i − ˆ α i S ˆ w i

    ˆ

α i = _{( ˆ w} ^{(ˆ r i ,ˆ z i )}

i ,S w ˆ _i )

1.3 End of IRKS V = {w 0 , . . . , w _p−1 }

Λ = V ^T SV

^{(diagonalmatrix)}

Q = Id − V Λ ⁻¹ V ^T S

^{proje tionmatrix}

2 . Conjugate Gradient with GKC

2 .1 Initialization x 0 = ˆ x p+1

r 0 = ˆ r p+1

2.2 Iterations i = 0, . . . , s y i = M ⁻¹ r i

Successive corrections q = 1, . . . , k

˜

r ^q _i = s q W ^q Γ ^{q −1} W ^{q T} r i − W ^q Γ ^{q −1} W ^{q T} S ^q y _i ^q−1 y ^q _i = y ^q−1 _i + ˜ r _i ^q

z i = QP y ^k _i w i = z i + ⁱ⁻¹ P

j=0

γ ij w j (w 0 = z 0 ) x i+1 = x i + α i w i

r i+1 = r i − α i Sw i

   

γ ij = − _(w ^{(z i} _j ^,Sw _,Sw ⁱ _j ⁾ ₎ α i = _(w ^(r _i ⁱ _,Sw ^{,z i )} _{i )}

Figure1: Algorithm: GIRKSwithProje tedPre onditionedConjugateGradient

stru ture

The turning stiness (g. 4) de reases

when the pressure inside the booster in-

reases, it anevenbe ome negative(the

exible bearing is then driving). This

evolution, aused by the displa ement of

pie es, is properly simulated during the

omputation.

1000 1200 1400 1600 1800

0 0.5 1 1.5 2 2.5 3 3.5 4

Turning stiffness (Nm/deg)

Pressure (MPa)

Buckling case Non buckling case

Figure4: Turningstiness

Problem Aver. CPUtime(s) /sys It. nb Gain

De omposition Method Fa torization Total Aver/sys Seq./Par.

17a-1r(17pro .) Primal 14.5 140.7 164 61.6

17a-1r(17pro .) GIRKS 14.5 78 50 111.1

6a-3r(18pro .) Primal 22.4 412.8 398 21

6a-3r(18pro .) GIRKS 22.4 256.7 183 33.7

3a-6r(18pro .) Primal 144 939.2 362 9.2

3a-6r(18pro .) GIRKS 144 400.9 120 21.6

Table2: Numeri alperforman eParallel/Sequential(non-bu kling)

TheKrylova elerationstrategyleadstosigni antspeed-up.TheCPUtime,thankstotheuse

ofGIRKS,in reasesfrom

38%

^to

58%

^{. GIRKSenablestosolveupto}

111

^{timesfasterthenon-linear}

problemthanthesequentialapproa husingonly

17

pro essors.

Figures(9)and(10)respe tivelyshowtheevolutionoftheaveragenumberof onjugategradient

iterationsandtheasso iatedaverageCPUtimetosolveea hlinearsystem. Duetotheverylow ost

ofGIRKS,iterationsandCPUtimegraphsarequite similar. As anbeseen,thea tionofGIRKS

grows as the nonlinear system number in reases due to the in reasing size of the stored Krylov

subspa es. In the ourseof the non-linearresolution, it may o ur that the information stored in

Krylovsubspa esbe omesnon-relevantandleadstoaperturbationleadingtostagnation. Thelinear

resolutionisthenrestartedwithdeletionofthesta kofKrylovsubspa es. Therestartingpro edure

anbeobservedongure(9)whentwopointsareasso iatedtothesamelinearsystem.

6.2 Bu kling problem

Thisloadinghistoryleadstothe omputationof

37

^{linearsystemswithdierentmatri esandright}

hand sides. The dire tsequential approa h is ompletedin

89

^{h. The} performan e results of the parallelapproa hesaregivenintable(3). Forthebu klingproblem,GIRKSisnotase ientasfor

thepreviousproblem,butitstillhasapositiveimpa t. Thebestresultisthenaresolution

67

^times

fasterthanthesequentialapproa husingonly

17

pro essors.

7 Con lusion

Inthispaperwe onsideredtheresolutionofhighlyheterogeneousstru turesinvolvingquasi-in ompressible

materialswith aprimaldomainde ompositionmethod. We extendedthe denitionofs alingma-

Figure5: Bu klingofonereinfor ement

10.5 10.6 10.7 10.8 10.9 11 11.1 11.2 11.3

0 0.5 1 1.5 2 2.5 3 3.5 4

u_y (mm)

Pressure (MPa) Buckling case Non buckling case

Figure6: Radialdispla ementofaninternalbound-

arypointofareinfor ement

Figure7: 17ade omposition Figure8: 3a-6rde omposition

Problem Aver. CPUtime(s) /sys It. nb Gain

De omposition Method Fa torization Total Aver/sys Seq./Par.

17a-1r(17pro .) Primal 14.9 139 162 62.3

17a-1r(17pro .) GIRKS 14.9 130 102 66.7

Table3: Numeri alperforman eParallel/Sequential(bu kling)

tri esto the ase wherepressure is ondensed, restoringthe s alability ofthe methodto this lass

ofproblems. Theresolutionofthe hallengingassessmentwassu essfullya hievedusingaNewton-

Krylov approa h. WeshowedthatthereuseofKrylovsubspa eswiththeGIRKSalgorithmalways

lead to better performan e, the speed-up ompared to the lassi al primal approa h an be

60%

^.

Comparedtothe dire tsequentialapproa h,theresolution is ondu tedinthebest ase

111

^times

fasterusingonly17pro essors. Inordertoavoidstagnationandrestartingasso iatedtoGIRKS,we

nowdevelopnewKrylovreusestrategiesbasedonanexa t oarsegridsolver. Duetothesigni ant

omputational ostofthisnewapproa hwe oupleitwithasele tivereuseofKrylovsubspa esbased

onaspe tralanalysisofthelinearsystems[4 ℄.

A knowledgements: The authors a knowledge support of omputational resour es by the Centre

InformatiqueNationalEnseignementSupérieurandthePledeCal ulParisSud.

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0 50 100 150 200

1 5 9 13 17 21 25 29 33 37 41 45

Average iteration number

Linear system Primal GIRKS

Figure9: GIRKS:nbof iterations

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1 5 9 13 17 21 25 29 33 37 41 45

Average CPU time (s)

Linear system Primal GIRKS

Figure10: GIRKS:CPUtime

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