Variational Formulations and Functional Approximation Algorithms in Stochastic Plasticity of ...

308 Pages·2013·16.6 MB·English

Variational Formulations and Functional
Approximation Algorithms in
Stochastic Plasticity of
Materials
Dissertation
Bojana Rosi´c
Universityof
Kragujevac

Variational Formulations and Functional
Approximation Algorithms in Stochastic
Plasticity of Materials
Vonder
Carl-Friedrich-Gauß-Fakulta¨t
TechnischeUniversita¨tCarolo-WilhelminazuBraunschweig
und
derFakulta¨tfu¨rIngenieurwissenschaften
Universita¨tKragujevac
zurErlangungdesGrades
Doktor-Ingenieurin(Dr.-Ing.)
genehmigteDissertation
von
BojanaRosic´
geborenam16.August1982
inKragujevac,Serbien
Eingereichtam: 20.Juni2012
Mu¨ndlichePru¨fungam: 16.November2012
1. Referent: Prof.HermannG.Matthies,Ph.D.
2. Referent: Prof.Dr.Miroslav Zˇivkovic´
3. Referent: Prof.Dr.AdnanIbrahimbegovic´
2013
Copyright c byB.V.Rosic´
(cid:13)
Allrightsreserved.
AlleRechtevorbehalten.
Svapravazadrzˇana.
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Tomyfamily.“Youaretheonly
reasonIam.Youareallmy
reasons.Thankyou”
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Abstract
A class of abstract stochastic variational inequalities of the second kind described
by uncertain parameters is considered within the framework of infinitesimal and
large displacement elastoplasticity theory. Particularly the focus is set on the rate-
independentevolutionaryproblemwithgeneralhardeningwhosematerialcharacter-
isticsareassumedtohavepositively-definitedistributions. Byexhibitingthestruc-
tureoftheevolutionaryequationsinaconvexsettingthemathematicalformulation
iscarriedovertothecomputationallymoresuitablemixedvariationaldescriptionfor
which the existence and uniqueness of the solution is studied. Time discretised as
usualwithbackwardEuler,theinequalityisreducedtoaminimisationproblemfor
a convex functional on discrete tensor product subspaces whose unique minimiser
isobtainedviaastochasticclosestpointprojectionalgorithmbasedon“whitenoise
analysis”.Tothisendadescriptioninthelanguageofnon-dissipativeanddissipative
operatorsisused,bothemployingthestochasticGalerkinmethodinitsfullyintrusive
or non-intrusive variant. The former method represents the direct, purely algebraic
way of computing the response in each iteration of Newton-like methods. As the
solutionisgiveninaformofpolynomialchaosexpansion,i.e.anexplicitfunctional
relationship between the independent random variables, the subsequent evaluations
of its functionals (the mean, variance, or probabilities of exceedence) are shown to
be very cheap, but with limited accuracy. Due to this reason, the intrusive method
iscontrastedtothelessefficientbutmoreaccuratenon-intrusivevariantwhicheval-
uatestheresiduumineachiterationviahigh-dimensionalintegrationrulesbasedon
random or deterministic sampling - Monte Carlo and related techniques. In addi-
tion to these, the problem is also solved with the help of the stochastic collocation
method via sparse grid techniques. Finally, the methods are validated on a series
oftestexamplesinplainstrainconditionswhosereferencesolutioniscomputedvia
directintegrationmethods.
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Rezime
U okviru teorije malih i velikih plasticˇnih deformacija razmatrana je klasa apstrak-
tnih stohasticˇkih varijacionih nejednakosti opisanih slucˇajnim promenljivim. Pose-
ban fokus je stavljen na asocijativni evolucioni problem sa generalnim ojacˇanjem
cˇijematerijalnekarakteristikeimajudistribucijuodredenuzakonommaksimalneen-
tropije. Proucˇavajuc´i strukturu evolucionih jednacˇina uz pomoc´ konveksne teorije
analizirani su uslovi za postojanje i jedinstvenost resˇenja uz dodatnu matematicˇku
reformulaciju problema u numericˇki prikladan mesˇoviti varijacioni opis. Dobijena
nejednakost se nakon implicitne diskretizacije svodi na minimizaciju konveksnog
funkcionala definisanog u tenzorskom prostoru. Resˇenje tako postavljenog prob-
lemasemozˇedobitinovouvedenomstohasticˇkommetodomprojekcijenajblizˇetacˇke
uz pomoc´ teorije analize belog sˇuma. Pomenuta metoda se sastoji od dva koraka:
elasticˇnogiplasticˇnog,cˇijisualgoritmibaziraninastohasticˇkojGalerkinovojmetodi.
Uzpomoc´ funkcionalneanalizeGalerkinovmetodjeformulisannadvanacˇina: di-
rektan(intruzivan)iposredan(neintruzivan). Intruzivnimetodpredstavljadirektan,
algebarskinacˇindobijanjaresˇenjausvakojiteracijiNjutnovemetode. Zahvaljujuc´i
polinomnoj formi resˇenja sve predstojec´e evaluacije njegovih funkcionala, kao sˇto
su srednja vrednost, varijansa itd. postaju racˇunski jako efikasne, ali ogranicˇene
tacˇnosti. U cilju unapredenja tacˇnosti Galekinova methoda je implementirana i u
svojoj manje efikasnoj, neintruzivnoj varijanti, koja racˇuna rezidual u svakoj Njut-
novoj iteraciji numericˇkom (deterministicˇkom ili stohasticˇkom) integracijom. Obe
varijanteGalerkinovihmetodasuuporedenesametodomstohasticˇkekolokacijeza-
snovanenapravilu“sparsegrid”-a. Konacˇno,svepredstavljenemetodesuverifiko-
vane na seriji test primera u ravanskom stanju deformacije i za referentno resˇenje
dobijenouzpomoc´ direktneintegracije.
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