Variational formulations and functional approximation algorithms in stochastic plasticity of materials
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Within the framework of innitesimal and large displacement elastoplasticity
theory we consider a class of abstract stochastic variational inequalities
of the second kind described by uncertain parameters. Particularly we focus
on the rate-independent evolutionary problem with general hardening whose
material characteristics are assumed to be distributed according to the maximum
entropy law. By exhibiting the structure of the evolutionary equations
in a convex setting we study the existence and uniqueness of the solution
and carry the mathematical formulation over to the computationally more
suitable mixed variational description. Within one time-backward Euler step
the inequality reduces to a minimisation problem for smooth convex energy
functional on discrete tensor product subspaces whose unique minimizer is
obtained via a stochastic closest point projection algorithm based on white
noise analysis". To this end we use a description in the language of nondissipative and dissipative op...erators and introduce the stochastic Galerkin
method into the computational process in fully intrusive and 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
solution is given in a form of polynomial chaos expansion, i.e. an explicit
functional 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. Furthermore,
the method is contrasted to the less-ecient but more accurate non-intrusive variant which evaluates the residuum in each iteration via highdimensional
integration rules based on random or deterministic sampling - Monte Carlo and related techniques. In addition to these, we also present the stochastic collocation method via sparse grid techniques. Finally the methods are validated on a series of test examples in plain strain conditions whose reference solution is computed via direct integration methods.
U okviru teorije malih i velikih plasticnih deformacija razmatrana je
klasa apstraktnih stohastičkih varijacionih nejednakosti opisanih slučajnim
promenljivama. Poseban fokus je stavljen na asocijativni evolucioni problem
sa generalnim ojačanjem čije materijalne karakteristike imaju distribuciju
odredenu zakonom maksimalne entropije. Proučavajući strukturu evolucionih
jednačina uz pomoć konveksne teorije uslovi za postojanje i jedinstvenost
rešenja su analizirani uz dodatnu matematičku reformulaciju problema
u numerički prikladan mešoviti varijacioni opis. Dobijena nejednakost se
nakon implicitne diskretizacije svodi na minimizaciju konveksnog funkcionala
denisanog u tenzorskom prostoru dobijenom kao proizvod determinističkog
i stohastičkog podprostora. Rešenje tako postavljenog problema se može
dobiti novouvedenom stohastiškom metodom projekcije najbliže tačke uz
pomoć teorije analize belog šuma. Pomenuta metoda se sastoji od dva
koraka: elastičnog i plastičnog, koji zajedno čine sto...hastičku Galerkinovu
metodu, ovde formulisanu na dva načina: direktan (intruzivan) i posredan
(neintruzivan). Prva varijanta predstavlja direktan, algebarski način dobijanja
rešenja u svakoj iteraciji Njutnove metode. Zahvaljujući polinomnoj
formi rešenja sve predstojeće evaluacije njegovih funkcionala kao što su srednja
vrednost, varijansa itd. postaju računski jako efikasne, ali ograničene
tačnosti. U cilju unapredenja tačnosti Galekinova methoda je implementirana
i u svojoj manje efikasnoj neintruzivnoj varijanti, koja računa rezidual
u svakoj Njutnovoj iteraciji numeričkom (determinističkom ili stohastičkom)
integracijom. Obe varijante Galerkinovih metoda su uporedene sa metodom
stohastičke kolokacije zasnovane na sparse grid pravilu. Konačno sve predstavljene
metode su verikovane na seriji test primera u ravanskom stanju deformacije i za referentno rešenje dobijeno uz pomoć direktne integracije.