Stochastic models of uncertainties in computational mechanics

134 Pages·2012·1.714 MB·English

by  Christian, Ph.d Soize

LectureNotesinMechanics2
Stochastic Models
of Uncertainties in
Computational
Mechanics
Christian Soize, Ph.D.
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Lecture Notes in Mechanics
AristotleandArchytasdefinedmechanicsasthe“organizationofthought
towardssolvingperplexingproblemsthatareusefultohumanity.”Inthespirit
Aristotle and Archytas, Lecture Notes in Mechanics (LNMech) tran-
scends the traditional division of mechanics and provides a forum for
presentingstate-of-the-artresearchthattacklestherangeofcomplexis-
suesfacingsocietytoday.
LNMech provides for the rapid dissemination of comprehensive
treatmentsofcurrentdevelopmentsinmechanics,servingasarepository
andreferenceforinnovationinmechanics,acrossallrelevantapplication
domains.
LNMech publishes original contributions, including monographs,
extended surveys, and collected papers from workshops and confer-
ences. All LNMech volumes are peer reviewed, available in print and
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abstractingservices.
SeriesEditor
RogerGhanem
EditorialBoard
YounaneAbousleiman,Ph.D.,UniversityofOklahoma
RobertoBallarini,Ph.D.,P.E.,UniversityofMinnesota
RonaldoI.Borja,Ph.D.,StanfordUniversity
ShiyiChen,Ph.D.,PekingUniversity
TerryFriesz,Ph.D.,PennsylvaniaStateUniversity
BojanB.Guzina,Ph.D.,UniversityofMinnesota
IoannisKevrekidis,Ph.D.,PrincetonUniversity
MohammadA.Khaleel,P.E.,Ph.D.,PacificNorthwestNational
Laboratory
PeterKuhn,Ph.D.,UniversityofCaliforniaatSanDiego
ArifMasud,Ph.D.,UniversityofIllinois,Urbana-Champaign
IgorMezic,Ph.D.,UniversityofCaliforniaatSantaBarbara
KaramSab,Ph.D.,E´coleNormaledesPontsetChausse´es
AndrewW.Smyth,Ph.D.,ColumbiaUniversity
ChristianSoize,Ph.D.,Universite´ Paris-Est
PolD.SpanosPh.D.,P.E.,RiceUniversity
LizhiSun,Ph.D.,UniversityofCaliforniaatIrvine
ZhigangSuo,Ph.D.,HarvardUniversity
PaulTorrens,Ph.D.,UniversityofMaryand
Franz-JosefUlm,Ph.D.,P.E.,MassachusettsInstituteofTechnology
LNMech1 PracticalApproximateAnalysisofBeamsandFrames
NabilFares,Ph.D.
LNMech2 StochasticModelsofUncertaintiesinComputational
Mechanics
ChristianSoize,Ph.D.
Contents
1 Introduction 1
2 Shortoverviewof probabilistic modeling of
uncertaintiesand related topics 3
2.1 Uncertaintyandvariability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Typesofapproachforprobabilisticmodelingofuncertainties. . . . . . . 4
2.3 Typesofrepresentationfortheprobabilisticmodeling
ofuncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Constructionofpriorprobabilitymodelsusingthe maximumentropy
principleundertheconstraintsdefinedbytheavailableinformation . . . 9
2.5 RandomMatrixTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.6 Propagationofuncertaintiesandmethodstosolvethestochastic
dynamicalequations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.7 Identificationofthepriorandposteriorprobabilitymodels
ofuncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8 Robustupdatingofcomputationalmodelsandrobustdesignwithun-
certaincomputationalmodels . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Parametric probabilistic approach to uncertainties in
computationalstructuraldynamics 21
3.1 Introductionofthemeancomputationalmodelincomputationalstruc-
turaldynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Introductionofthereducedmeancomputationalmodel . . . . . . . . . . 22
3.3 Methodology for the parametric probabilistic approach of model-
parameteruncertainties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Constructionofthepriorprobabilitymodelofmodel-parameteruncer-
tainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Estimationoftheparametersofthepriorprobabilitymodeloftheun-
certainmodelparameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.6 Posterior probability model of uncertainties using output-prediction-
errormethodandtheBayesianmethod . . . . . . . . . . . . . . . . . . . . 27
v
vi StochasticModelsofUncertaintiesinComputationalMechanics
4 Nonparametricprobabilistic approachto
uncertaintiesin computationalstructural
dynamics 29
4.1 Methodology to take into account both the model-parameter uncer-
taintiesandthemodeluncertainties(modelingerrors) . . . . . . . . . . . 29
4.2 Constructionofthepriorprobabilitymodeloftherandommatrices . . . 30
4.3 Estimationoftheparametersofthepriorprobabilitymodel
ofuncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.4 Commentsabouttheapplicationsandthevalidationofthenonpara-
metricprobabilisticapproachofuncertainties . . . . . . . . . . . . . . . . 32
5 Generalizedprobabilisticapproachtouncertaintiesin
computationalstructuraldynamics 35
5.1 Methodologyofthegeneralizedprobabilisticapproach . . . . . . . . . . 35
5.2 Constructionofthepriorprobabilitymodeloftherandommatrices . . . 37
5.3 Estimationoftheparametersofthepriorprobabilitymodel
ofuncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.4 PosteriorprobabilitymodelofuncertaintiesusingtheBayesian
method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6 Nonparametricprobabilistic approachto
uncertaintiesin structural-acoustic
models for the low- and medium-frequency
ranges 41
6.1 Reducedmeanstructural-acousticmodel. . . . . . . . . . . . . . . . . . . 42
6.2 Stochastic reduced-order model of the computational structural-
acousticmodelusingthenonparametricprobabilisticapproachofun-
certainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.3 Constructionofthepriorprobabilitymodelofuncertainties . . . . . . . . 47
6.4 Modelparameters,stochasticsolverandconvergenceanalysis . . . . . 49
6.5 Estimationoftheparametersofthepriorprobabilitymodel
ofuncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.6 Commentsabouttheapplicationsandtheexperimentalvalidationof
thenonparametricprobabilisticapproachofuncertaintiesinstructural
acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7 Nonparametricprobabilistic approachto
uncertaintiesin computationalnonlinear
structuraldynamics 51
7.1 Nonlinearequationfor3Dgeometricallynonlinearelasticity . . . . . . . 52
7.2 Nonlinearreducedmeanmodel . . . . . . . . . . . . . . . . . . . . . . . . . 52
7.3 Algebraicpropertiesofthenonlinearstiffnesses . . . . . . . . . . . . . . 54
StochasticModelsofUncertaintiesinComputationalMechanics vii
7.4 Stochastic reduced-order model of the nonlinear dynamical system
usingthenonparametricprobabilisticapproachofuncertainties . . . . . 54
7.5 Comments aboutthe applicationsofthe nonparametric probabilistic
approachofuncertaintiesincomputationalnonlinearstructural
dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
8 Identificationofhigh-dimensionpolynomialchaosex-
pansions with random coefficients for non-Gaussian
tensor-valued random fields using partial and limited
experimentaldata 57
8.1 Definitionoftheproblemtobesolved . . . . . . . . . . . . . . . . . . . . . 58
8.2 Constructionofafamilyofprioralgebraicprobabilitymodels(PAPM)
forthetensor-valuedrandomfieldinelasticitytheory. . . . . . . . . . . . 60
8.3 Methodology for the identification of a high-dimension polynomial
chaosexpansionusingpartialandlimitedexperimentaldata. . . . . . . 70
8.4 Computational aspects for constructing realizations of polynomial
chaosinhighdimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
8.5 PriorprobabilitymodeloftherandomVVC . . . . . . . . . . . . . . . . . . 78
8.6 Posterior probability model of the random VVC using the classical
Bayesianapproach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8.7 PosteriorprobabilitymodeloftherandomVVCusinganewapproach
derivedfromtheBayesianapproach . . . . . . . . . . . . . . . . . . . . . . 86
8.8 Commentsabouttheapplicationsconcerningtheidentificationofpoly-
nomialchaosexpansionsofrandomfieldsusingexperimentaldata . . 88
9 Conclusion 89
References 91
Index 109
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Chapter 1
Introduction
This volume presents an overview of the main concepts, formulations,
and recentadvancesin the areaof stochastic modeling of uncertainties
incomputationalmechanics.Amathematical-mechanicalmodelingpro-
cess is useful for predicting the responses of a real structural system
in its environment. Computational models are subject to two types of
uncertainties—variabilities in the real system and uncertainties in the
modelitself—so,tobeeffective,thesemodelsmustsupportrobustopti-
mization,design,andupdating.
A probabilistic approach to uncertainties is the most powerful, ef-
ficient, and effective tool for computational modeling, as described in
Chapter 2. The next three chapters focus on parametric (chapter 3),
nonparametric(chapter4),andgeneralized(chapter5)probabilisticap-
proaches to linear and nonlinear structural dynamics. Chapter 6 exam-
ines a nonparametric probabilistic approachto structuralacoustics and
vibration. Chapter 7 presents an extension of the nonparametric prob-
abilistic approachto geometrically nonlinear elasticity in structural dy-
namics. Chapter 8 describes recent results in constructing prior proba-
bilitymodelsofnon-Gaussiantensor-valuedrandomfieldsbasedonthe
use of polynomial chaos expansion with applications to the mesoscale
modelingofheterogeneouselasticmicrostructures.
1