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Faculty of Engineering, University inKragujevac to

TEACHING AND SCIENTIFIC COUNCIL

Subject: Report of the Commission for the

assessment of the written work and the publicdefense of the doctoral dissertation by thecandidate Dipl.-Ing. Bojana Rosi

By decision of the teaching and scientificcouncil of the Faculty of Engineering, Uni-versity Kragujevac No. 01-1/1523/3, 14. 06.2012 we have been named as a members ofCommission for the assessment of written thework and public defense of the doctoraldissertation entitled "Variational Formulations

and Functional Approximation Algorithms inStochastic Plasticity of Mate-rials" written byDipl.-Ing. Bojana Rosi.

On the basis of the submitted doctoraldissertation, and the Regulations of the Unive-rsity in Kragujevac for the application, writtenform and public defense of the doctoraldissertation, the Commission submits thefollowing

REPORT

1. The significance and contribution of the

doctoral dissertation from the standpoint of

the current situation in a particular scien-

tific field

The doctoral dissertation entitled "VariationalFormulations and Functional ApproximationAlgorithms in Stochastic Plasticity of Mate-

rials" by candidate Dipl.-Ing. Bojana Rosi isthe result of the scientific research inmodelling elastoplastic materials described byrandom parameters. This dissertationrepresents a unique scientific work in thisregion of Europe, and opens a new venues forfuture scientific research.

A literature review has shown that existingmodels that describe a stochastic inelastic

behavior of materials are limited to a relatively

small number of practical situations. Theirmain disadvantages are mostly ineffectivealgorithms, a small number of random param-

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meters, and the application on the systemswith small input variance. Also, the existingliterature does not offer a mathematical formu-lation of the problem, proof of existence anduniqueness of the solution, etc. In view of this,the current dissertation starts from the theory

of deterministic variational inequalities toprovide a complete mathematical formulationof stochastic small deformation plasticity, withan extension to large deformations. At thesame time the candidate has formulated a morerealistic material model by including manyuncertain parameters into the description, and

proposed effective numerical algorithms thatcan find practical application in industry.

In the literature, so far, are only considered

simple stochastic elastoplastic models, mostlyin numerical manner. In fact, most papers dealwith the description of small deformationsdescribed by a single random parameter. Toovercome problems that arise in approxima-tion of random fields, authors used anapproach based on perturbation theory or

probability distributions. Therefore the modelsin the literature are not sufficiently accurate ortoo complicated mathematically to be appliedon multi-dimensional problems or systemsdescribed by random variables with largevariance. In order to find more efficientalgorithms the candidate promotes the originalidea of a functional approximation of randomvariables of a much more realistic models.Also, the candidate has developed a stochasticradial return algorithm and numerical methodsthat can be used in broadly defined areas. Thestochastic Galerkin method as presented in thisdissertation can be applied not only for solving

elastoplastic problems, but also for generallinear and nonlinear equations, as well as forother forms of variational inequalities andconvex optimization problems.

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2. Assessment that the doctoral thesis is in

original scientific research in the respective

field

The Commission considers that the doctoralthesis entitled "Variational Formulations and

Functional Approximation Algorithms in theStochastic Plasticity of Materials," bycandidate Dipl.-Ing. Bojana Rosi is the resultof original scientific work. The dissertation is avaluable contribution to the development ofelastoplastic models for the description ofheterogeneous materials. The candidate hasaddressed the topic with the help of the richtheoretical basis, and has demonstrated andconfirmed the required objectives andhypotheses. In addition, the scientific literature

on this has been thoroughly analysed andevaluated.

The originality of the scientific work, research,and the results achieved in this dissertation arereflected, among other things, in the followingelements:

- Starting from the classical theory of plasticitywith the help of convex variational analysis thecandidate has formulated the abstract

variational inequality describing elastoplasticbehavior with random parameters, and shownthe abstract similarity between deterministicand stochastic models. Specializing the generalclass of random variational inequalities (RVI)defined by a random monotone operator on arandom subset of a Hilbert space the mixedstochastic variational inequality for the elasto-

plastic problem is posed and the uniquenessand existence of the solution provided. Theconsidered problem is then reduced to the

minimisation of a convex functional whoseunique minimiser is found with the help of anovel approach - a stochastic closest point

projection algorithm based on "white noiseanalysis".

The problem considered in the thesis is quasi-static elastoplasticity described by randomforce, constitutive tensor, convex domain andrandom hardening. As some random

parameters can not take negative values a

nonlinear function of Gaussian randomvariables is assumed for their models. In theintroduction of a new material model first it is

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assumed that the tensor is isotropic at the meanlevel, although its realisations are basicallyanisotropic. Such a model is then improved tothe fully anisotropic case.

- The candidate has developed her own

method of Galerkin projection of nonlinearsystem onto the polynomial basis as a part ofthe stochastic radial return algorithm. Themethod is implemented in two ways: directalgebraic and the indirect method based ondirect integration. Besides, the stochasticcollocation algorithm is also introduced.

- Through the conducted research thecandidate has examined the impact of therandomness of various parameters on the

response of the system, as well as theeffectiveness of the proposed numericalmethods.

Some of the results and conclusions drawn inthis work are of particular importance in thearea of uncertainty quantification and deve-lopment of stochastic methods, such as:

- direct integration methods are characterisedby very slow convergence and thus are

impractical

- the thesis confirms the hypothesis that thepresence of randomness in the materialcharacteristics has a significant effect on thesystem response. At the time the largest impactis coming from the yield strength, while the

bulk and shear moduli have much lessinfluence

- compared with conventional methods of

direct integration the stochastic Galerkinmethod gives approximate results with signi-ficantly better response time. In addition, toachieve sufficient accuracy the intrusivevariant of the method requires a much higher

polynomial order compared to the non-intrusive variant. If the main requirement isthe efficiency then the intrusive Galerkinmethod would be the proper choice. Whenefficiency is not the primary condition, butaccuracy, then intrusive methods with small

polynomial order and Monte Carlo methodsare considered a poor choice. On other side,the stochastic collocation method and non-

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intrusive Galerkin may be well suited when thesystem is described by large variance

- modelling of a random anisotropic materialtensor gives a more realistic prediction of thesystem in relation to material tensors which arein the mean isotropic and in realisations

basically anisotropic

- the correlation lengths describing thecovariance tensor of the material can bedetermined by identification procedures. If thecorrelation length of parameters are small theirrandom fluctuations are larger and thus thesystem becomes more sensitive to the presenceof uncertanties in inputs.

3. Achieved Results in the area of AppliedMechanics

Bojana Rosi graduated 29.09.2006. on the Fa-culty of Mechanical Engineering in Kragu-

jevac, the subject was Applied Mechanics andAutomatic Control. From 2007-2010 she didresearch at the Institute of Scientific Compu-ting, TU Braunschweig, as a stipend, and fromJanuary 2010 until June 2010 as a DAAD

stipend. Since June 2010 Miss Rosic isemployed as a research asistant at the Instituteof Scientific Computing, TU Braunschweig. In2010 she defended the access thesis " Stateof the Art Report: Stochastic Elastoplasticity"at the Faculty of Mechanical Engineering, Uni-versity Kragujevac.

The candidate has published five papers ininternational and two in national journals, two

in monographs, nine technical reports, and 12papers on international and national confe-rences.

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4. Evaluation of the fulfillment of the

volume and quality in relation to the

reported issue

The doctoral dissertation "Variational Formu-lations and Functional Approximation Algo-

rithms in the Stochastic Plasticity of Mate-rials", by Dipl.-Ing. Bojana Rosi fulfills therequired conditions and the quality in relationto the accepted report by the teaching councilof the Faculty of Mechanical Engineering andthe University in Kragujevac. The thesis meetsquality and quantity requirements as well asscientific and legal regulatives regarding thewriting of a doctoral dissertation.

The work and orginal results in the framework

of this dissertation are reported on 273 pagesin 9 chapters:

1. Introduction2. Determinstic theory of plasticity3. Plasticity described by uncertain parameters4. Discretisation5. Numerical Approaches6. Polynomial Chaos Algebra7. Numerical implementation- PLASTON8. Numerical Results9. Conclusion

The thesis contains 53 figures and 58 tables.The candidate's bibliography lists 241 refere-nces.

The thesis is organised as follows:

In the first introductory chapter the applicanthas indicated the need for the introduction ofrandom variables in the modelling of

elastoplastic materials, and engineeringproblems in general. Materials that do not varymuch can be described by a proper calibrationof constitutive models, while rock-type mate-rials, concrete, and bone are of inherentlyvariable nature and as such can not bedescribed within already existing mathematicalmodels. This section provides the historicaldevelopment of stochastic finite elementmethods and an overview of research in thefield of non-elastic material.

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The second chapter gives a mathematical for-mulation of the problem of mixed hardeningelastoplasticity. Starting with the classicaltheory an abstract formulation of the infini-tesimal problem is introduced, basically therandom variational inequality of a second

order. With the help of convex and duallitytheory the formulation is transformed into amore appropriate description of the mixed

problem. With certain assumptions, the dualformulation is reduced to a minimization

problem of convex functionals, a quadraticfunction with stress as an argument. Thisformulation is then extended to the field oflarge displacements as a natural extension ofthe theory of small deformations.

The third chapterrelates to the formulation ofstochastic elastoplastic materials. The chapterbegins with a description of uncertanties thatcharacterize the irreversible processes and

principles of their modeling via full or reducedparametric methods. Such modelled param-eters are then introduced to the general formu-lation of standard materials. In a similar wayas in the second chapter the abstract stochasticvariational inequality is formulated and thentransformed into the so-called mixed de-

scription.

The fourth chapter gives an overview of theprocess of discretisation. As the inputquantities are dependent on time, space andstohastic input this chapter describes thenumerical approximation strategy of timestepping, of the geometric domain and ofrandom fields. Time discretization is

performed using the implicit Euler method,and the spatial by the finite element method.

Similarly, the stochastic discretization isachieved via the Karhunen-Love expansion,followed by polynomial chaos.

The main focus of Chapter 5 is the directstochastic Galerkin method and its non-intrusive alternative as novel procedures, whilethe numerical methods already existing in theliterature are only shortly reviewed. As thesolution of the stochastic variational inequality

is found by projection onto the polynomialchaos basis, this chapter introduce the sto-chastic variant of already very well known clo-

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(PLAsticity - STOchastic aNalysis)

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sest point projection algorithm.

In order to enrich the structure of Chapter 5and make it more understandable, Chapter 6

provides more details about functionalapproximation and polynomial chaos algebra

widely used throughout this dissertation. Itcontributes the basic polynomial chaosoperations of linear and nonlinear types for

both scalar and tensor valued random variablestogether with the various numerical appro-aches for their computation.

Furthermore, Chapter 7outlines the structureof the library PLASTON (PLAsticity -STOchastic aNalysis), providing a concise de-scription of main routines and library modules

by offering the most important user interface.

Chapter 8 verifies the method through a seriesof test examples which are commonly used totest new material models, such as a plate withhole, Cooke's membrane and etc. Examples are

by increasing complexity of input classified onrandom variable and random field examples.They are intended to demonstrate the basicadvantages and drawbacks of the proposedalgorithms and to extract the best variant for

practical applications. As the most complex, aproblem described by anisotropic tensor is alsotested.

Finally, the conclusions give a conciseevaluation and review of the proposed theoryand numerical applications presented in the

previous chapters.

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5. Scientific results of the doctoral disse-

rtation

The work has scientific significance in severalfields, some of which are extracted here:

- this dissertation gives a complete mathe-matical formulation of the problem finfinitesimal plasticity described by random

parameters,

- shows the difference in output if thematerial parameters are approximated byrandom variables or random fields. Alsoisotropic and anisotropic material tensors arediscussed and their modelling procedures,

- the paper presents the process of functionalapproximations of random variables by

polynomial chaos algebra,

- the presented mathematical model isdiscretized in the stochastic sense in the mosteffective way by using the Karhunen-Loveexpansion

- several numerical methods are proposed,of which two are the product of thisdissertation. Namely, the intrusive and theindirect Galerkin methods used to solve theequilibrium equa-tions in each iteration of

Newton's method. In doing so, the candidateproposed a stochastic radial return algorithmas the most appropriate method for solving theoptimization problem.

- quantification of randomness in the vonMises type of model is fully described, starting

from the influence of random input on theoutput, the impact of the correlation lengthsand accuracy of the input approximation on thefinal result.

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6. The siginificance and contribution of

results in theory and practice

As it is becoming increasingly difficult toignore the presence of uncertainty in thedescription of materials, the main significance

of this paper is to improve the existingmathematical models of elastoplastic pheno-mena by including uncertanties in the problem.In recent years there has been an increasinginterest in stochastic linear problems taking theform of linear elastic equations, while onlyfew studies are consi-dering more complexnonlinear phenomena. In this regard, thisdissertation offers a study of an elastoplasticsystem, or mathematically speaking a varia-tional inequality described by a random

constitutive tensor. Such a model can be usedas the surrogate for the description of rocks orsoils, concrete, and many biological materialssuch as bone tissue. In these situations thestochastic models are more informative thanthe deterministic ones. They produce the fulldistribution of possible outcomes, give theconfidence levels that a certain outcome willhappen, provide correlations between theevents, and so on. In this manner thedescription of real situations such as thedevelopment of the tumor in bone tissue, the

breaking risk of a concrete dam, the confi-dence of materials in civil structures under theinfluence of known or uncertain excitations(seismic phenomena, wind, and snow) can be

provided.

Furthermore, the stochastic elastoplastic fo-rmulation may also become very important inthe process of identification of material pro-

perties such as yield stress, hardening etc.Once the set of noisy and incompletemeasurement data is provided one may alter(update) the apriori assumed probability modelto a more realistic one with the help ofBayesian probabilistic models.

It can be concluded that the work in thisdoctoral dissertation has provided number of

practical and specific engineering solutionsthat can be integrated into practical processes.

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7.

. , :

[1] H. G. Matthies and B. Rosi. InelasticMedia under Uncertainty: Stochastic Models

and Computational Approaches. In DayaReddy, IUTAM Bookseries, 11:185-194, ISBN-13:978-1-4020-9089-9, e-ISBN-13:978-1-4020-9090-5, 2008, url:http://www.springerlink.com/content/q1w8642101632425/ (M33)

[2] B. Rosi and H. G. Matthies. ComputationalApproaches to Inelastic Media with Uncertain

Parameters. Journal of Serbian Society forComputational Mechanics, 2(1): 28-43, ISSN1820-6530, 2008, url: http://www.sscm.kg.ac.rs/jsscm/downloads/ (M53)

[3] B. Rosi and H. G. Matthies and M.ivkovi. Uncertainty Quantification of Infi-nitesimal Elastoplasticity. Scientific TechnicalReview, 61(2): 3-9, 2011, ISSN: YU ISSN 18200206, url:http://www.vti.mod.gov.rs/ntp/rad2011/2-11/1/1.pdf (M52)

[4] B. Rosi and H. G. Matthies. StochasticGalerkin method for the elastoplasticity

problem with uncertain parameters. In DanaMueller-Hoeppe, Stefan Loehnert, StefanieReese (Eds.) Recent Developments and

Innovative Applications in Computational

Mechanics, pp. 303-310, ISBN 978-3-642-17483-4, e-ISBN 978-3-642-17484-1, Springer,Berlin, Heidelberg, 2011, doi: 10.1007/978-3-642-17484-1, (M14)

[5] O. Pajonk and B. Rosi and A. Litvinenkoand H. G. Matthies. A Deterministic Filter fornon-{G}aussian State Estimation. PAMM Proc.Appl. Math. Mech., 11: 703 -704, ISSN: 1617-7061, 2011, doi: 10.1002/pamm.201110341(M33)

[6] B. Rosi and A. Litvinenko and O. Pajonkand H. G. Matthies. Sampling-free Bayesian

7. Presentation of the results to the scientific

community

A number of results presented in this thesis arealready published and verified through paperssent to international and national journals, as

well as on public meetings and conferences.Here are mentioned only the most importantones in last five years:

[1] H. G. Matthies and B. Rosi. Inelastic Me-dia under Uncertainty: Stochastic Models and

Computational Approaches. In Daya Reddy,IUTAM Bookseries, 11:185-194, ISBN-13:978-1-4020-9089-9,e-ISBN-13:978-1-4020-9090-5, 2008, url:http://www.springerlink.com/content/q1w8642101632425/ (M33)

[2] B. Rosi and H. G. Matthies. Computatio-nal Approaches to Inelastic Media with Unce-

rtain Parameters. Journal of Serbian Societyfor Computational Mechanics, 2(1): 28-43,ISSN 1820-6530, 2008, url:http://www.sscm.kg.ac.rs/jsscm/downloads/ (M53)

[3] B. Rosi and H. G. Matthies and M. ivko-vi. Uncertainty Quantification of Infinitesi-mal Elastoplasticity. Scientific TechnicalReview, 61(2): 3-9, ISSN: YU ISSN 1820-0206, ,2011, url: http://www.vti.mod.gov.rs/ntp/rad2011/2-11/1/1.pdf (M52)

[4] B. Rosi and H. G. Matthies. StochasticGalerkin method for the elastoplasticity prob-lem with uncertain parameters. In Dana Mue-ller-Hoeppe, Stefan Loehnert, Stefanie Reese(Eds.) Recent Developments and Innovative

Applications in Computational Mechanics, pp.

303-310, ISBN 978-3-642-17483-4, e-ISBN978-3-642-17484-1, Springer, Berlin, Heide-lberg, 2011, (M14)

[5] O. Pajonk and B. Rosi and A. Litvinenkoand H. G. Matthies.A Deterministic Filter fornon-{G}aussian State Estimation. PAMMProc. Appl. Math. Mech., 11: 703 -704, ISSN:1617-7061, 2011, doi:10.1002/pamm.201110341(M33)

[6] B. Rosi and A. Litvinenko and O. Pajonkand H. G. Matthies. Sampling-free Bayesianupdate of polynomial chaos representations.

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update of polynomial chaos representations.Journal of Computational Physics, 231(17):5761-5787, ISSN: 0021-9991, 2012, doi: 10.1016/j.jcp.2012.04.044 (M21)

[7] O. Pajonk and B. Rosi and A. Litvinenko

and H. G. Matthies. A deterministic filter fornon-Gaussian Bayesian estimation Applica-tions to dynamical system estimation with noisy

measurements. Physica D: Nonlinear Pheno-mena, 241 (7): 775788, ISSN: 0167-2789,2012, doi:10.1016 /j.physd.2012.01.001, (M21)

[8] A. Kuerov and J. Skora and B. Rosi andH. G. Matthies. Acceleration of UncertaintyUpdating in the Description of Transport

Processes in Heterogeneous Materials. Journal

of Computational and Applied Mathematics,ISSN: 0377-0427, 2012, Available online, doi:10.1016/j.cam. 2012.02.003, (M21)

[9] O. Pajonk and B. Rosi and H. G. Matthies.Sampling-free linear Bayesian upda-ting of

model state and parameters using a square root

approach. Computers and Geo-sciencies, ISSN:0098-3004, 2012, Available online, doi:10.1016/ j.cageo.2012.05.017, (M22)

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Journal of Computational Physics, 231(1):5761-5787, ISSN: 0021-9991 ,2012, doi:10.1016/j.jcp.2012.04.044 (M21)

[7] O. Pajonk, B. Rosi, A. Litvinenko and H.G. Matthies. A deterministic filter for non-

Gaussian Bayesian estimation Applicationsto dynamical system estimation with noisy

measurements. Physica D: Nonlinear Pheno-mena, 241 (7): 775788, ISSN: 0167-2789,2012 , 2012, doi:10.1016/j.physd.2012.01.001, (M21)

[8] A. Kuerov, J. Skora, B. Rosi and H. G.Matthies. Acceleration of UncertaintyUpdating in the Description of Transport

Processes in Heterogeneous Materials. Journal

of Computational and Applied Mathematics,ISSN: 0377-0427, 2012, Available online, doi:10.1016/j.cam. 2012.02.003, (M21)

[9] O. Pajonk, B. Rosi and H. G. Matthies.Sampling-free linear Bayesian updating of

model state and parameters using a square

root approach. Computers and Geosciencies,ISSN: 0098-3004, 2012, Available online, doi:10.1016/j.cageo. 2012.05.017, (M22)

The comission agrees that the researchreported in this work results in large number ofscientific results, which can be further

presented at scientific meetings or in journalpapers. The work can be very inspiring for thefuture work of researchers and scientists inarea of probability theory, identification orelastoplastic theory.

Based on the above the Commission makes thefollowing

CONCLUSION

The submitted doctoral dissertation meets theproposal theme accepted by the Teaching andResearch Council of the Faculty of Enginee-ring and University in Kragujevac.

To present the research the candidate used the

common terminology. The structure of thedissertation and the method of its presentationis in accordance with university regulations.

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The candidate wrote the thesis respecting themethodology of scientific research, used a

proven systematic approach to the problem,and showed the skill of rational reasoning andthe ability to create visions for future work.

Fulfilling the research requests given in thedissertation application, Dipl.-Ing. BojanaRosi has done original scientific work whichrepresents a significant contribution to thefield of stochastic variational inequalitiesdescribing elastoplastic problems.

On the basis of the foregoing the Commissionfor the review and defense of the doctoraldissertation of Bojana Rosi agreed byconsensus that according to the quality and

quantity of research the thesis entitled"Variational Formulations and FunctionalApproximation Algorithms in the StochasticPlasticity of Materials" fully meets thescientific, technical and legal requirements fora doctoral dissertation. The Commissionagrees that the applicant's dissertation is asignificant contribution to the field ofstochastic variational inequalities appearing inthe elastoplastic problem. We are pleased to

propose to the Teaching and Scientific Councilof the Faculty of Engineering Sciences in Kra-gujevac to accept the dissertation of BojanaRosi as successfully done and to invite thecandidate to the public defense.

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COMMISSION MEMBERS

Dr. Hermann G. Matthies, ProfessorTechnische Universitiit Braunschweig,Cari-Friedrich-GauB-Fakultiit

Scientific field: Applied mathematics and informatics,computer sci ~ 7 Y r~ ~

Dr. Miroslav Zivkovic, ProfessorUniversity in Kragujevac,Faculty ofEngineering

Scientific field: applied mechanics,applied i n f o ~ M F engineeringDr. Adnan Ibrahimbegovic, Professor

E N s - G t ~ J . , . , . J LMT-Cachan, Paris, FranceScientific field: theoreti a! and applied mechanics

Dr. Radovan Slavkovic, ProfessorUniversity in Kragujevac,

Faculty of EngineeringScientific field: applied.inechanics,applied informatics and computer science in engineeringp ~

Dr. Dirk Lorenz, ProfessorTechnische Universitiit Braunschweig,Cari-Friedrich-GauB-Fakultiit

Scientific field: theoretical and applied mathematics"'""'- '~ ) ~ L

' .