dakota/uq: a toolkit for uncertainty quantification in a ...structural dynamics research, dept. 9124...

S iaC d StatesD 000.

DAKO uantification ina M mputational

24

g Group

andia is a multiprogram laboratory operated by Sandorporation, a Lockheed Martin Company,for the Uniteepartment of Energy under contract DE-AC04-94AL85

TA/UQ: A Toolkit For Uncertainty Qultiphysics, Massively Parallel Co

Environment

Steven F. [email protected]

Structural Dynamics Research, Dept. 91Sandia-Albuquerque

LANL Uncertainty Quantification Workin

October 4, 2001

Uncertainty Quantification

Real

• terials,

•

•

•

• y

Unce

Engineering Sciences Center

Physical Systems:

Display random and systematic variation- geometry, ma

boundary conditions, initial conditions, excitations

Vary from one realization to the next

Display behavior that cannot be precisely measured

rtainty occurs in various forms:

Irreducible, variability, aleatoric

Reducible, epistemic, subjective, model form uncertaint

Uncertainty Quantification

Usefu

requirements.

I Program Plan,hich a computerm the perspective

• An

• Mo


l in:

alysis and Design

–To assess the reliability of physical systems.

–To establish designs that satisfy pre-established reliability

–To establish sensitivities to key uncertainties

del validation, certification, and accreditation

–As defined in the DOE Defense Programs (DOE/DP) ASCvalidation is the process of determining the degree to wmodel is an accurate representation of the real world froof the intended model applications.

–Convey confidence in predictions to decision makers

Uncertainty Quantification:Gen

Gene

s

and as

ually

f

Stati

• Mo(po

• Seob


eral Framework

ral Description:

tical Approach:

del components of as Random Variables or Fields, ssibly) Random External Input

ek quantities such as . However, what is acttained are conditional statistics .

M . X,( )f U

: vector of uncertain parameters

f

X

U : output(s) of system

M: a deterministic mapping

: input(s) to system

x

E g U( )[ ]E g U( ) M[ ]

Probabilistic/Statistical Approach: EssentialEle

Conc

o

PD

rization of the

acterization ofters

Esse

• Ra

• Pr

• Ch

Genethe re


ments of a Statistical Approach:

lusion: Need a Generalized Outlook.

ntial Elements of a Statistical Approach:

ndom External Inputs

pagation Techniques–Analytical Reliability Methods; Sampling; Response Surface

Approximations; Stochastic Finite Element Methods.

aracterization of Models–Verification and Validation.

E g U( )[ ] E E E g U( ) M X,[ ] M{ }⟨ ⟩=

ropagation Techniques (Note:ependency on M)

CharacteModel, M

Probabilistic charthe input parameral functions of

sponse, u

Random external loads

Anatomy of Global Uncertainty

E
⟨ ⟩

E M{ }

E g U( ) M X,[ ]

E g U( )[ ]=



E g U( ) M X,[ ]

- Random Vibration- Earthquake Engineering- Ocean Engineering- Weapons Applications: Launch Shocks/Re-entry Loads, Penetration Loads, Hostile Environments

Uncertainty due to External Loads: f( )


Un

- E


E M{ }

E g U( ) M X,[ ]

certainty Propagation:

ffects of parametric uncertainty:Intrinsic variabilities, Tolerances,Lack of repeatability

X( )


E
⟨ ⟩

E M{ }

E g U( ) M X,[ ]

E g U( )[ ]=

Uncertainty Quantification at Sandia-NM

• DAel problems,e approximation,

el Uncertainty

• Po

• Ep

• Se


KOTA (Design Analysis Kit for OpTimizAion)/UQ–Framework for multi-level, parallel computation: ASCI-lev

optimization, nondeterministic analysis, response surfacdesign of experiments, optimization under uncertainty

lynomial Chaos and Stochastic Finite Elements–Analysis of response of stochastic systems

istemic Uncertainty–Non-Probabilistic Approach, Probabilistic Approach, Mod

nsitivity Analysis

Objectives of Toolkit

Provi n a unifiedframe

s

e problems

n

t

non-gradient)

• Di

• AS

• Mi

• Fle

Why

• Ex

• Su

• Mu

• Ex

• Ex


de uncertainty quantification tools to the analyst community iwork to be used in the design and certification processes.

cipline independent

CI (Accelerated Strategic Computing Initiative)-scal

imize number of function evaluations

xibility in uncertainty model

ie UQ tools to the DAKOTA framework?

isting, proven software framework

ccessfully linked with over 20 application codes

ltilevel parallelism

tensive optimization algorithm library (gradient and

tensive selection of approximation strategies

Answ• W

oo

eeo sSe

DAKOTA toolkitDes

Contours of GaMe3on Reacting Surface

Streamlines

Inlet:

GaMe3

AsH3

H2

Outlet:GaMe3

AsH3

H2

CH4

• H• H

Addi• R• L• N• A• B


er fundamental engineering questions:hat is the best design?w safe is it?w much confidence in my answer

tional motivations:use tools and interfacesverage optimization, UQ, et al.nconvex, nonsmooth design spaces → state-of-the-art methodologieCI-scale applications and architectures → scalable parallelism a pathfinder in enabling M&S-based culture change at Sandia

DAKOTAOptimizationUncertainty Quant.Parameter Est.Sensitivity Analysis

DesignModelParameters Metrics

MERCURY

SALINASALEGRA

PRONTO

GOMAEAGLE

NG ion opticsNG power supplyAF&F subsystemlaydown, gas transfer

Coatings consortiumGoodyear CRADA

CTH explosives performance

ign optimization of engineering applications

Ove

Responses

Functionsobjectivesconstraints

Gradientsnumericalanalyticmixed

Hessians

least sq. termsgeneric

analytic

I

e

ainty

odelFormExtrap

DO

Strat

DDADopt

N

LHS/MAMFOR


rview of DAKOTA framework

Iterator

Optimizer ParamStudy

SGOPTNPSOLDOT OPT++

Least Sq.E

InterfaceParameters

Model:

Designcontinuousdiscrete

Uncertainnormal/lognuniform/logu

histogramState

continuousdiscrete

Application

system call, fork

direct

Approximationpolynomialneural networksplines

synch & asynch

synch & asynch

terator

Model

gy: control of multiple iterators and models

Iterator

Model

Iterator

Model

Coordination:

Cascaded

Nested

ConcurrentAdaptive/Interactive

Hybrid

SeqApprox

OptUnderUnc

Branch&Bound/PICO

Strategy

Optimization Uncert

M

UncOfOptima

message passing

kriging

weibull

BCGN

Vector

Centered MultiD

List

CECCD/FF

onDeterm

C

SFEM

Taylor series

Parallelism:

Peer (static, distr. sched.)Master-slave (self-sched.)

4 nested levels

V+M

Boot/Imp

Layered (surrogates, hierarchical)

hierarchicalrSQP++

Optimization/UQ Projects

DAKO-7471

ary):

5482bina

P-7471

s):-747196

itus Leung

rch libraries):v

houcq,illiams

DAKO

SGO

PICO

OPT+


TA project (optimization with engineering simulations):Sandia manager - David Womble, 9211, [email protected], 845PI - Mike Eldred, 9211, [email protected], 844-6479Team members - Tony Giunta, Bill Hart, Bart van Bloemen Waandershttp://endo.sandia.gov/DAKOTA/

TA/UQ project (analytic reliability, sampling, and SFE UQ librSandia manager - Martin Pilch, 9133, [email protected], 845-3047PI - Steve Wojtkiewicz, 9124, [email protected], 284-Team members - Mike Eldred, Rich Field, John Red-Horse, Angel Ur

T project (stochastic global optimization):Sandia manager - David Womble, 9211, [email protected], 845PI - Bill Hart, 9211, [email protected], 844-2217http://www.cs.sandia.gov/~wehart/main.html

project (mixed integer programming, scheduling and logisticSandia manager - David Womble, 9211, [email protected], 845PI - Cindy Phillips, 9211, [email protected], 845-72Team members - Bob Carr, Jonathan Eckstein (Rutgers), Bill Hart, Vhttp://www.cs.sandia.gov/~caphill/proj/pico.html

+/DDACE/APPS/IDEA projects (NLP, sampling, & pattern seaSandia manager - Chuck Hartwig(acting), 8950, [email protected] - Juan Meza, 8950, [email protected], 294-2425Team members - Paul Boggs, Patty Hough, Tamara Kolda, Leslea Le

Kevin Long, Monica Martinez-Canales, and Pam Whttp://csmr.ca.sandia.gov/~meza/research.html

Current Dakota/UQ Capabilities

Samp

r

Order Reliabil-

s

mize

• Ra

• St

Analy

• Me

• FOity

Robu

Stoch

Respo

• Apco


ling Techniques:

ndom Sampling (Monte Carlo)

atified Sampling (LHS (Latin Hypercube Sampling)

tical Reliability Techniques:

an Value (MV), Advanced Mean Value (AMV/AMV+)

RM (First Order Reliability Method)/SORM (Second Method)

tness Analysis

astic Finite Element/ Polynomial Chaos Expansions

nse Surface Approximations:

plication of UQ tools to a surrogate function to minimputational expense.

Sampling Techniques

Mont

iscipline ofo

capabilities

)

• Gepr

• Ea

• Co

• Tw

• Un


e Carlo-Style (Sampling-based) Analysis:

neral, simple to implement and robust to size and dblem being investigated

sily wrapped around current deterministic analysis

mputationally expensive (many function evaluations

o current options:–Traditional Monte Carlo–Latin Hypercube Sampling

der investigation:–Bootstrap Sampling–Importance Sampling Techniques–Quasi-Monte Carlo SImulation–Markov Chain Monte Carlo

Overview of Analytically BasedRel

• Inv d normal ran-do

ization problem distance func-ethod.

ble space.

ues about anp

• Na

• MVwhtio

• MV

• FO

• Eq“o


iability Methods

olve a transformation to unit variance, uncorrelatem variable space.

taf Transformation used in DAKOTA/UQ.

, AMV/AMV+, FORM all solve a constrained optimere the objective function is always this minimumn with the constraint function depending on the m

and AMV/AMV+ work in the original random varia

RM/SORM work in the transformed space.

uivalent to Polynomial Response Surface Techniqtimally” selected expansion point

Probabilistic Robustness Analysis

• “G e of outputfun

and itsxU

• Po

such

wherlowe


iven the bounds on the input parameters, what rangction is possible?”

se two global optimization problems:

that

e is the size of uncertain input vector, denoter and upper bounds, respectively.

gupper max g M f x,( )( )=x

glower min g M f x,( )( )=x

xi( )L xi xi( )U≤ ≤ i 1…N=∀

N xL

Probabilistic Robustness Analysis

Answ

m variables of SDM 2002)

• Reun


er:

cently extended to mixed case of intervals and randoknown dependence (to appear in Wojtkiewicz, AIAA

g u( ) glower gupper,[ ]∈

SFEM/Polynomial Chaos Techniques

• Ap

p of randomar

e ransforma-io

s

o

• Ov

• Rt

• E

• C


proximation of full stochastic representation

timal approximation in inner product spaces, spaceiables.

presents a more general alternative to the Rosenblatt tn

–avoid assuming full distribution when faced with limited input data

timating coefficients is the key issue–requires realizations of the function it replaces

nvergence issues–are there sufficient samples to compute coefficients?

–possibility of non-physical realizations

–mean square convergence

L2

SFEM/Polynomial Chaos Techniques

•

•

r+ )

uni x( )

δi x( )-----------=


Consider PCE of general random process,

, where

– th order polynomial in , where

–function of underlying random variables

Solve for the Fourier coefficients,

can be solved in closed-form

u

u x Φ;( ) u x Φ;( )P( )

ui x( )Γ i ξ( )i 0=

P

∑≡≈ P1s!---- m(

r 0=

s 1–

∏

s 1=

q

∑=

q ξ ξ ξ1 ξ2 … ξm

T=

m

ui x( )

i x( )u x h ξ[ ],( ) Γ i ξ( )⟨ ⟩

Γ i2 ξ( )⟨ ⟩

-------------------------------------------

… u x h ξ[ ],( ) Γ i ξ( ) f ξ ξ( ) ξd

∞–

∞

∫∞–

∞

∫∞–

∞

∫

… Γ i2 ξ( ) f ξ ξ( ) ξd

∞–

∞

∫∞–

∞

∫∞–

∞

∫-----------------------------------------------------------------------------------= =

δi x( )

Epistemic Uncertainty

•

•

bases do

P

•

Non-


Epistemic Uncertainty results from a lack of information.

Epistemic Uncertainty manifests itself in several ways

–Uncertainty in parameters for which statistically significant datanot exist

–The form of the model is not known exactly

robabilistic Approach

Variety of approaches investigated:

–Interval analysis–Possibility Theory–Evidence Theory (Dempster-Shafer)–Imprecise Probability–Probability Bounds–Interval-valued Probability Distributions–Convex Sets of Probability Distributions

The Penetrator Problem

• Pro

ent failure,

n with simplified

P f

u SRS ω( )=} )

codeslinear transient dynamicslinear soil mechanicsr filter

tain parametersangle-of-attack soil parameter

wind


blem statement

–During the penetration event, predict the probability of compon

–Consider a nonlinear, full-body, 3D, coupled-physics simulatioprobabilistic properties.

α

-γ

D

CLv

f 0=M . ; α D{ ,(

M

∼ASCI FE

• structure - non• boundary - non• SRS - nonlinea

Uncer• α, • D,

The model, M: a complex, cascaded system

102

eq (Hz)

α (deg)h

f )

✓ Nodyn

✓ 50,

✓ Sp✓ Lo

sim

✓ 33 sim

Struc

Soil


P

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09−400

−300

−200

−100

0

100

200

300

400

101

300

400

500

600

700

800

900

1000

Filters

Time (s)A

ccel

(g)

Pea

kA

ccel

(g)

Natural Fr

D (in)

Pea

kA

ccel

(g)

nlinear transientamics FEA000 DOF

erical cavity expansionads couple with mechanicsulation

LP, SRS

CPU hours perulation

✓ 49 total runs performed

✓ Performed simultaneouslyon network of Sun Ultra IIs

Component Response

tural Mechanics

Mechanics

Approximate RS Models

u(

Overview of the Shock Response Spectrum (SRS)

Why u

• potential

•

•

•

103

104

105

Frequency (Hz)

SRS

0−8000

−6000

−4000

−2000

0

2000

4000

6000

8000

Acc

el (

g)


se SRS?

measure of shock severity; indicative of shock damage

frequency-domain representation of shock response

long history of use in weapon design; test-based spec

used for component qualification - compare to

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01Time (sec)

101

102

100

101

102

103

104

105

Natural

Pea

k A

ccel

(g)

Time history

SRSref

UQ analysis of Penetrator System

• Tw

e model
• Us
(res


o design variables, :– , angle of attack is a normal random variable with mean 1

and standard deviation of 1.

– , soil depth is a lognormal random variable with mean 25and standard deviation 16.

ing the results from simulations, build a approximat

ponse surface approximation) for .

Xα

D

u mini

SRSref f i( ) SRS f i( )–( )=

Z g U( ) I U( )= =

P f P Z 0≤( ) 1 E g U( )[ ]–= =

u

UQ analysis of Penetrator System

• Ap :

LHS

0/0.02400

0/0.06767

0/0.05581

0/0.05071

Z 0( )
ply MC/LHS to these surrogate models to evaluate

and

Response SurfaceApproximation Method

MC

Kriging 0.02000/0.02300 0.0200

Splines 0.06900/0.06781 0.0672

Neural Net 0.05024/0.05588 0.0550

Quadratic Polynomial 0.04960/0.05077 0.0507

F

Ns 14×10= Ns 5

6×10=

Sum

Curre

n s

r

pling

-

Anal

Sam

Prob

Poly

Futu

Enha

Non


mary

nt Capabilities

ytical Reliability Techniques:

•MV, AMV, AMV+, FORM/SORM

pling techniques:

•Pure Random Sampling (Monte Carlo)

•Stratified Sampling (LHS)

abilistic Robustness Analysis

omial Chaos Expansions/Stochastic Finite Element Technique

e Capabilities:

nced sampling methods:

•Importance Sampling, Bootstrap Sampling,

Quasi-Monte Carlo Sampling, Markov Chain Monte Carlo Sam

traditional uncertainty methodologies

dakota/uq: a toolkit for uncertainty quantification in a ...structural dynamics research, dept. 9124...

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