2324 proceedings 2004 sdm45-nessus-paper

8/11/2019 2324 Proceedings 2004 SDM45-NESSUS-paper

1/17

NESSUS Capabilities for Ill-Behaved Performance

Functions

David S. Riha1and Ben H. Thacker2

Reliability & Materials Integrity Section , Southwest Research Institute, San Antonio, Texas, 78238

Simeon H. K. Fitch3

Mustard Seed Sotware, San Antonio, Texas, 782!"

Numerical simulation is now routinely used to predict the behavior and response of

comple systems! especially when consideration of nonlinear effects! multiple physics! or

comple "eometry is re#uired$ Better predictions of system performance re#uire the use of

probabilistic methods to account for uncertainty in model parameters$ %ost efficient

probabilistic analysis methods utili&e an optimi&ation al"orithm to locate the most probable

point$ 'he performance model defined by a computational model may be ill-behaved in the

sense of discontinuous! noisy or hi"hly non-linear responses$ 'his type of behavior can cause

difficulty in locatin" the %PP based on an optimi&ation approach$ In addition! difficultiesarise durin" "radient computations$ In many cases analytical derivatives are not possible

and finite difference approimations must be used$ (n %PP search failure al"orithm has

been developed and is demonstrated for a sample problem$ (pproaches for derivative

computations are outlined and demonstrated throu"h an application problem$ 'his paper

addresses these two issues throu"h a description of enhancements to the NESSUS

probabilistic analysis software and eample problems$

Nomenclature

b # safety indexCDF c!m!"ative distri#!tion f!nction

g "imit state f!nction

$%% most &ro#a#"y &oint$ &ro#a#i"ity of fai"!re! standard norma" variateu vector of standard norma" variates

x random varia#"eX vector of random varia#"es% &erformance meas!re

I$ Introduction

'!merica" sim!"ation is no( ro!tine"y !sed to &redict the #ehavior and res&onse of com&"ex systems) es&ecia""y(hen consideration of non"inear effects) m!"ti&"e &hysics) or com&"ex *eometry is re+!ired. The !se of

com&!tationa" sim!"ation is re"ied !&on increasin*"y more as &erformance re+!irements for en*ineered str!ct!res

increase and as a means of red!cin* testin*. Since str!ct!ra" &erformance is direct"y affected #y !ncertainties

associated (ith mode"s or in &hysica" &arameters and "oadin*s) the deve"o&ment and a&&"ication of &ro#a#i"isticana"ysis methods s!ita#"e for !se (ith com&"ex n!merica" mode"s is needed.

So!th(est Research ,nstit!te -S(R, has #een addressin* the need for efficient &ro#a#i"istic ana"ysis methods

and interfacin* to commercia" finite e"ement ana"ysis &acka*es for over ei*hteen years) #e*innin* (ith the

1 %rinci&a" /n*ineer) Re"ia#i"ity 0 $ateria"s ,nte*rity) %.. Dra(er 214) So!th(est Research ,nstit!te) San

5ntonio) T6 722) $em#er 5,55.2$ana*er R0D) Re"ia#i"ity 0 $ateria"s ,nte*rity) %.. Dra(er 214) So!th(est Research ,nstit!te) San 5ntonio)

T6 722) Senior $em#er 5,55.3(ner) $!stard Seed Soft(are) 1871 /ncino 9ay) San 5ntonio) Texas 728.

5merican ,nstit!te of 5erona!tics and 5strona!tics1


2/17

deve"o&ment of the '/SS:S1 &ro#a#i"istic ana"ysis com&!ter &ro*ram. '/SS:S can #e !sed to sim!"ate

!ncertainties in "oads) *eometry) materia" #ehavior) and other !ser;defined !ncertainty in&!ts to com&!te &ro#a#i"ity

of fai"!re and &ro#a#i"istic sensitivity meas!res. '/SS:S has a #!i"t;in finite e"ement str!ct!ra" mode"in* ca&a#i"ity

as (e"" as interfaces to many commercia""y avai"a#"e finite e"ement &ro*rams. '/SS:S (as initia""y deve"o&ed #y ateam "ed #y S(R, for the 'ationa" 5erona!tics and S&ace 5dministration -'5S5 to assess !ncertainties in critica"

s&ace sh!tt"e main en*ine com&onents.2C!rrent ca&a#i"ities of '/SS:S are sho(n in Fi*. 1. Severa" efficient fast

&ro#a#i"ity inte*ration -F%, methods have #een im&"emented in '/SS:S inc"!din* the advanced mean va"!e

method and have #een sho(n to #e many times more efficient than $onte Car"o sim!"ation.3$any efficient &ro#a#i"istic ana"ysis methods have #een devised to a""eviate the need for $onte Car"o

sim!"ation) (hich is im&ractica" for "ar*e;sca"e hi*h;fide"ity &ro#"ems. These methods inc"!de) for exam&"e) the first

and second;order re"ia#i"ity method


3/17

$any standard o&timi?ation a"*orithms are avai"a#"e to &erform this task s!ch as the modified method of feasi#"e

directions -$$FD) se+!entia" "inear &ro*rammin* -S@%) and se+!entia" +!adratic &ro*rammin* -SA%. 7 Some

have a"so #een deve"o&ed s&ecifica""y for &ro#a#i"istic ana"ysis s!ch as the Rack(it?;Feiss"er method -RF


4/17

&ro#a#i"ity corres&ondin* to a s&ecific &erformance va"!e) or m!"ti&"e fai"!re &ro#a#i"ities s!ch that the com&"ete

c!m!"ative distri#!tion f!nction -CDF can #e constr!cted. 5"ternative"y) '/SS:S can com&!te a sin*"e

&erformance va"!e corres&ondin* to a s&ecific fai"!re &ro#a#i"ity. The choice of ana"ysis ty&e de&ends on the

&ro#"em #ein* so"ved.Traditiona" re"ia#i"ity ana"ysis invo"ves com&!tin* the &ro#a#i"ity of stress) S,exceedin* stren*th)R,%rRGS or

%rgG4) (hereg#RSis referred to as the "imit state f!nction. ,n *enera") g(i"" #e more com&"ex thang#RSand

(i"" #e *iven #yg#g'X() (here Xare the in&!t random varia#"es. ,n addition to the fai"!re &ro#a#i"ity) '/SS:S

com&!tes &ro#a#i"istic im&ortance factors) Iu) (here is inverse"y re"ated to $ and uare the in&!t random

varia#"es transformed into standard norma" s&ace) and &ro#a#i"istic sensitivity factors) I) (here are the

&arameters of the in&!t random varia#"es) e.*.) mean va"!e and standard deviation.

System eliability (nalysis

$ost en*ineerin* str!ct!res can fai" in more than one (ay. System re"ia#i"ity considers the &ossi#"e fai"!re ofm!"ti&"e com&onents of a system) or m!"ti&"e fai"!re modes of a com&onent. ,n '/SS:S) system re"ia#i"ity

&ro#"ems are form!"ated and so"ved !sin* a &ro#a#i"istic fa!"t tree ana"ysis -%FT5 method.11

5 fa!"t tree is constr!cted in '/SS:S #y connectin* #ottom events (ith 5'D and R *ates. /ach

#ottom event mode"s a se&arate fai"!re event) (hich can #e a com&"ex m!"ti;&hysics sim!"ation. The to&o"o*y of thefa!"t tree is defined #y the fai"!re modes #ein* sim!"ated. nce defined) severa" o&tions are avai"a#"e for so"vin* the

system re"ia#i"ity &ro#"em. First) direct $onte Car"o sim!"ation is avai"a#"e #!t may #e cost &rohi#itive if the "imit

state f!nctions of the #ottom events are com&!tationa""y ex&ensive. 5"ternative"y) '/SS:S can com&!te the&ro#a#i"ity of system fai"!re !sin* the 5dvanced $ean =a"!e -5$=J method or 5da&tive ,m&ortance Sam&"in*-5,S. Beca!se the '/SS:S %FT5 !ses a "imit state f!nction to re&resent each #ottom event) corre"ations d!e to

common random varia#"es #et(een the #ottom events is f!""y acco!nted for re*ard"ess of the &ro#a#i"istic method

!sed.

,n addition to +!antifyin* the system re"ia#i"ity) '/SS:S a"so com&!tes &ro#a#i"istic sensitivities of the system&ro#a#i"ity of fai"!re (ith res&ect to the each random varia#"es mean va"!e and standard deviation.>These res!"ts

&rovide a rankin* #ased on the re"ative contri#!tion of each varia#"e to the tota" &ro#a#i"ity of fai"!re. The

sensitivities are a"so !sef!" in desi*n o&timi?ation) test &"annin* and reso!rce a""ocation.

%ro#a#i"istic fa!"t trees for system &ro#"ems are defined in '/SS:S !sin* a *ra&hica" editor. nce the system isdefined in the :,) the corres&ondin* Boo"ean a"*e#raic statement is transferred to the &ro#"em statement (indo()

(here the !ser then defines each event.

eliability %odelin" Process

'/SS:S !ses an o!t"ine str!ct!re to define the &ro#"em) as sho(n in the "eft hand side of Fi* 3. The !ser

navi*ates thro!*h the nodes of the o!t"ine from to& to #ottom to define the &ro#"em and &erform the ana"ysis. /ach

of these nodes is descri#ed in more detai" in the fo""o(in* sections. The ste&s needed to so"ve a re"ia#i"ity &ro#"emin '/SS:S inc"!deL

1. Deve"o& the f!nctiona" re"ationshi&s that define the mode")

2. Define the random varia#"e in&!ts)

3. Define the n!merica" mode"s needed in the f!nctiona" re"ationshi&)


5/17

statement &arser in '/SS:S identifies a"" of the inde&endent varia#"es in the &ro#"em statement (indo( and

transfers these varia#"es to the random varia#"e in&!t (indo( for f!rther definition.

ii) Rando+ -ariable In$ut and *robabilistic atabase

The random varia#"e in&!ts are defined in the random varia#"e definition (indo(. 5 *ra&hica" in&!t editor is&rovided for distri#!tions re+!irin* &arameters other than the mean and standard deviation) s!ch as !&&er and "o(er

#o!nds for tr!ncated distri#!tions. The &ro#a#i"ity density f!nction -%DF and c!m!"ative distri#!tion f!nction

-CDF &"ottin* ca&a#i"ity in '/SS:S &rovides a vis!a" ins&ection of the random varia#"es. Random varia#"es

a""o(ed in '/SS:S inc"!de norma") "o*norma") 9ei#!"") extreme va"!e ty&e ,) chi;s+!are) maxim!m entro&y)c!rve;fit) Frechet) tr!ncated norma" and tr!ncated 9ei#!"". The maxim!m entro&y and c!rve;fit distri#!tions can #e

!sed to mode" distri#!tions not direct"y s!&&orted.

'/SS:S maintains a "i#rary of re"evant %DFs in a &ro#a#i"istic data#ase. Random varia#"es can #e defined andstored !sin* a distri#!tion ty&e and associated &arameters. Distri#!tion fittin* f!nctions are &rovided to determine

the #est fit from ra( data. The entries can #e *ro!&ed and m!"ti&"e data#ases are s!&&orted. This a""o(s !sers to

deve"o& their o(n) &ossi#"y &ro&rietary) data#ases for !se in '/SS:S. Random varia#"e definitions from the

data#ase contents can #e inserted direct"y in the random definition ta#"e in '/SS:S !sin* a ri*ht mo!se c"ick.

iii) Res$onse Model einition

F!nctions defined in the &ro#"em statement (indo( are assi*ned in the res&onse mode" definition. The avai"a#"e

f!nction ty&es are se"ected from the mode" ty&e dro& do(n men! and inc"!de ana"ytica") re*ression) n!merica") and&redefined as sho(n in Fi*. 3. The ana"ytica" f!nction ty&e a""o(s mode"s to #e defined (ith standard mathematica"

o&erators) !sin* a format identica" to definitions in the &ro#"em statement (indo(. The n!merica" mode" ty&e a""o(s

the !se of interfaced codes or a !ser;defined code. Codes c!rrent"y interfaced to '/SS:S inc"!de 5B5A:S)

5merican ,nstit!te of 5erona!tics and 5strona!tics

Fi"ure .* NESSUS outline structure "uides the probabilistic problem setup and analysis$


6/17

5'SMS -5'SMS) ,nc.) DM'53D -@a(rence @ivermore 'ationa" @a#oratory) @S;DM'5 -@STC) ,nc.)

'5S5NRCNF/$ -'5S5 "enn Research Center) $5DM$ -T' 5!tomotive) $SC.'5STR5'

-$SC.Soft(are) %R'T -Sandia 'ationa" @a#oratories) and :S/RND/F,'/D. The !ser;defined n!merica"

mode" a""o(s the !ser to "ink the '/SS:S &ro#a#i"istic en*ine (ith any stand;a"one ana"ysis code. The re*ressionmode" ty&e a""o(s the !ser to in&!t f!nction coefficients or ra( &ert!r#ation data that can #e fit to "inear or +!adratic

f!nctions !sin* "inear re*ression. Fina""y) the &redefined mode" ty&e a""o(s "inkin* !ser;(ritten Fortran s!#ro!tines

(ith '/SS:S.

Fi*!re 3 sho(s an exam&"e !sin* the 5B5A:S finite e"ement &ro*ram. The exec!tion command (indo(&rovides the command or commands re+!ired to exec!te 5B5A:S. The in&!t and o!t&!t fi"es are a"so defined on

this in&!t screen. Defa!"t exec!tion o&tions for the s!&&orted codes are inserted a!tomatica""y #y '/SS:S from a

confi*!ra#"e tem&"ate fi"e) and can #e modified #y the !ser as needed. 5 #atch &rocessin* o&tion is &rovided to

a""o( &rocessin* on different com&!ters. This a""o(s '/SS:S to r!n on a "oca" (orkstation (hi"e the ana"ysiscodes r!n on a different (orkstation) c"!ster) s!&ercom&!ter) etc. Re"ated to the #atch &rocessin* feat!re) '/SS:S

a"so &rovides an a!tomatic restart o&tion. The restart ca&a#i"ity &rovides &ro#a#i"istic so"!tion refinement)

recoverin* from a#norma" so"ver termination) and eva"!atin* additiona" &erformance meas!res (itho!t rer!nnin*

&revio!s ste&s of the so"ver ana"yses.

$a&&in* Random =aria#"es to '!merica" $ode"s

5 rea"i?ation of a random varia#"e m!st #e ref"ected in the n!merica" mode"s in&!t (hen &erformin*

&ro#a#i"istic ana"ysis !sin* a n!merica" mode". The varia#"e may #e a random varia#"e or a com&!ted varia#"e fromanother code or ana"ytica" e+!ation. ,n *enera") the varia#"e can ma& to a sin*"e va"!e in the codes in&!t or to a

vector of va"!es s!ch as noda" coordinates in a finite e"ement mode". Ty&ica" exam&"es of sin*"e va"!e ma&&in*s

inc"!de Mo!n*Os mod!"!s or a concentrated &oint "oad. /xam&"es of vector ma&&in*s are a &ress!re fie"d actin* on a

set of e"ements or a *eometric &arameter that effects m!"ti&"e node "ocations.$a&&in* varia#"es to the n!merica" mode" in&!t in '/SS:S is achieved #y *ra&hica""y identifyin* the "ines and

co"!mns that are chan*ed (hen the varia#"e chan*es as sho(n in Fi*.

Fi"ure /* 'he numerical model definition screen in NESSUS defines the eecution command and re#uired

input0output files for eecutin" the numerical model$


7/17

=ector ma&&in*s re+!ire a f!nctiona" re"ationshi& #et(een the in&!t random varia#"e and the ana"ysis &ro*ram

in&!t. Beca!se different rea"i?ations of these varia#"es are re+!ired) a *enera" a&&roach is !sed in '/SS:S to re"atea chan*e in the in&!t random varia#"e va"!e to the codes in&!t. For exam&"e) if the random varia#"e is the radi!s of

a ho"e) chan*es to a set of noda" coordinate va"!es (i"" #e re+!ired each time the radi!s is chan*ed. 5 de"ta vector is

defined that re"ates ho( the coordinates chan*e (ith a chan*e in the varia#"e. Severa" other a&&roaches are avai"a#"efor definin* vector varia#"es. Some ana"ysis codes a""o( the finite e"ement mode" to #e &arametrica""y defined. ,nthis case) the varia#"es can #e ma&&ed direct"y (itho!t definin* the de"ta vector. 5nother o&tion is to inc"!de a finite

e"ement &re&rocessor !sin* the "inked mode" ca&a#i"ity. The varia#"es can #e ma&&ed to the &re&rocessor in&!t and

the res!"tin* mode" !sed for the ana"ysis.

Se"ectin* Res&onses for '!merica" $ode"s

The fina" ste& in definin* the n!merica" mode" is to identify the res&onse +!antity or +!antities that are to #e

ret!rned to '/SS:S. The a&&roach !sed in '/SS:S is to read the ana"ysis res!"ts for a *iven set of node) e"ement

and time ste&s direct"y from the ana"ysis codes res!"ts fi"e. '/SS:S s!&&orts a!tomated extraction for mosten*ineerin* +!antities of interest inc"!din* dis&"acements) ve"ocities) acce"erations) stresses) strains) etc. 9hen

m!"ti&"e +!antities are re+!ested) '/SS:S &rovides f!rther o&tions to red!ce the res!"ts do(n to a sin*"e va"!e

!sin* f!nctions s!ch as maxim!m) minim!m) avera*e) etc. For dynamic codes) se"ection of the res&onse from a

res!"t time series is &rovided across m!"ti&"e times s!ch as maxim!m) "ast) and !ser s&ecified. ,n some cases) theres&onse time series can #e fi"tered to smooth the res&onse #efore !se in the &ro#a#i"istic ana"ysis.

i.) *ara+eter -ariation Analysis

%arameter variation ana"ysis is another !sef!" too" to !nderstand ho( the &erformance varies (ith chan*es in therandom varia#"es. '/SS:S &rovides severa" methods for definin* varia#"e &ert!r#ations inc"!din* #ack(ard)

centra") and for(ard differences as (e"" as varia#"e s(ee&s. ,n addition) s&ecific &ert!r#ation va"!es can #e in&!t

direct"y to define ex&erimenta" desi*ns. =is!a"i?ation of the res&onse variation is &rovided in &redefined 6M scatter

&"ots.


Fi"ure 1* NESSUS provides a "raphical mappin" tool to identify the portions of the code2s input that chan"e

when the random variable chan"es$ 'he mappin" can include multiple lines and columns in the code2s input$


8/17

.) *robabilistic Analysis einitions

$any efficient &ro#a#i"istic ana"ysis methods have #een devised to a""eviate the need for $onte Car"o

sim!"ation) (hich is im&ractica" for "ar*e;sca"e hi*h;fide"ity &ro#"ems. The traditiona" methods inc"!de) for

exam&"e) the first and second;order re"ia#i"ity methods


9/17

The most &ro#a#"y &oint search is ty&ica""y &erformed !sin* an o&timi?ation a"*orithm that "ocates the minim!m

distance) ) from the fai"!re s!rface to the ori*in in a transformed &ro#a#i"ity -! s&ace -Fi*. >. $any standard

o&timi?ation a"*orithms are avai"a#"e to &erform this task s!ch as the modified method of feasi#"e directions-$$FD) se+!entia" "inear &ro*rammin* -S@%) and se+!entia" +!adratic &ro*rammin* -SA%. For "ocatin* the

$%%) '/SS:S !ses a modified version of the standard Rack(it?;Feiss"er -RF method. The modified RF method

&rovides conver*ence checks in addition to the standard check for conver*ence of the safety index) ) that is)

1

1

4.41i i

i

-1

'/SS:S a"so checks for conver*ence on the res&onse) P) and that the an*"e #et(een s!ccessive $%%s is (ithin

a !ser;s&ecified to"erance) i.e.)

1

1

4.41i i

i

% %

%

-2

and

1

1cos - 34

o

i i

= =

-3

The RF method is a 'e(ton #ased method and is not *!aranteed to conver*e. Ho(ever) (hen the RF methodconver*es it *enera""y conver*es in far fe(er f!nction eva"!ations than other o&timi?ation a"*orithms s!ch as SA%.

Therefore an a&&roach is needed to identify (hen the RF method is fai"in* and idea""y as ear"y in the search

&rocess as &ossi#"e. 5 second *oa" is to !se a"" information from the search as a startin* &oint for a more ro#!st yetcom&!tationa""y intensive search a"*orithm. The a!thors ex&erience has fo!nd that most fai"!res of the RF method

&rovide a characteristic cyc"ic $%% search &attern. This ty&e of search #ehavior is sho(n in Fi*. 7. 5n a"*orithm has

#een deve"o&ed #ased on a!tocorre"ation of the minim!m distance) ) for ste&s in the search.

The cyc"ic #ehavior is identified #y ca"c!"atin* the a!tocorre"ation #et(een each ' iterations. The ' is the "a*#et(een search &oints and indicates if the search a"*orithm is contin!a""y "ocatin* the same &oints. ,f the

a!tocorre"ation is s!fficient"y "ar*e -e.*.) *reater than 4. then fai"!re is detected and an a"ternate so"!tion a&&roach

can #e !sed. 5n a&&"ication &ro#"em sho(in* the detai"s of this a&&roach is &rovided #e"o(.,n '/SS:S) the modified RF method is em&"oyed first since it is the most efficient. ,f conver*ence diffic!"ties

are enco!ntered) the method is s(itched to SA%.


u=

u;

f *u.u

ost !robable!oint *!!.

1pproimate 4imit-/tate

Eact 4imit-/tate

g(x) = 0

Fi"ure 7* 8oint probability density function 59pdf6! eact and approimate limit-state! and most probable

point 5%PP6 for two random variables in transformed 5u6 space$


10/17

(pproaches for Noisy esponse Functions

5na"ytica" derivatives are *enera""y not avai"a#"e for non"inear n!merica" ana"yses or third &arty soft(are (hen

so!rce code is not avai"a#"e. Therefore finite difference a&&roximations to the derivatives are ty&ica""y !sed. Thefinite difference can take the form of for(ard) #ack(ard and centra" differences (ith an a&&ro&riate se"ection of the

ste& si?e. Derivatives can a"so #e a&&roximated from m!"ti&"e &oints !sin* "inear re*ression. For noisy res&onse

f!nctions) the ste& si?e se"ection is critica" to o#tainin* acce&ta#"e derivative a&&roximations. ,n a &ro#a#i"istic

ana"ysis context) ste& si?es for derivative a&&roximation are sometimes s&ecified in terms of standard deviation. ,naddition for &ro#a#i"istic ana"ysis) varia#"e ran*es (i"" ty&ica""y fa"" (ithin Q> standard deviations from the mean at

most. Therefore for this disc!ssion) the varia#"e x is mode"ed as a norma" random varia#"e (ith a mean of and

standard deviation of 4..

5 sim!"ated noisy res&onse f!nction is sho(n in Fi*. . The mode"ed f!nction is ?x2

(ith a noise term ofS,'-144x. This ty&e of #ehavior is common"y seen in transient dynamic ana"yses s!ch as im&act and #"ast

ana"ysis. The overa"" res&onse) ?) a&&ears fair"y (e"" #ehaved #!t "ar*e errors in derivative a&&roximations can

occ!r. The exact derivative for the f!nction exc"!din* the noise is 2x. The derivative a&&roximations for the ran*e of

interest are sho(n in Fi*. 8 for t(o ste& si?es !sin* for(ard difference. 5 ste& si?e of 4.1*enerates "ar*e errors in

the derivative) even res!"tin* in an o&&osite *radient direction. By examinin* ho( the res&onse chan*es (ith x) the

noise can #e some(hat e"iminated #y se"ectin* a "ar*er ste& si?e. By examination) a ste& si?e 4. sho!"d &rovide

reasona#"y acc!rate derivatives (hi"e sti"" ca&t!rin* the "oca" sensitivity.'/SS:S &rovides a &arameter variation ca&a#i"ity to *ain a #etter !nderstandin* ho( the res&onse chan*es (ith

chan*es in the random varia#"es. This vis!a"i?ation too" &rovides #etter ste& si?e se"ection for derivative

a&&roximations (hen ana"ytica" derivatives are not &ossi#"e. 5n a&&"ication &ro#"em is &resented #e"o( to

demonstrate this a&&roach.


atin hypercube simulation$


17/17

>9!) M.;T.) Com&!tationa" $ethod for /fficient Str!ct!ra" Re"ia#i"ity and Re"ia#i"ity Sensitivity 5na"ysis) 5,55

o!rna") =o". 32) 188

2324 proceedings 2004 sdm45-nessus-paper

Documents