Tolerancing: Managing uncertainty from conceptual design to final product

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Edward MORSE, Jean-Yves DANTAN, Nabil ANWER, Rikard SÖDERBERG, Giovanni MORONI,

Ahmed Jawad QURESHI, Xiangqian JIANG, Luc MATHIEU - Tolerancing: Managing uncertainty

from conceptual design to final product - CIRP Annals - Manufacturing Technology - Vol. 67, n°2,

p.695-717 - 2018

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Tolerancing:

Managing

uncertainty

from

conceptual

design

to

final

product

Edward

Morse

(3)

*,

Jean-Yves

Dantan

(2)

,

Nabil

Anwer

(2)

,

Rikard

Söderberg

(2)

,

Giovanni

Moroni

(2)

_,

_Ahmed

_Qureshi

_,

_Xiangqian

_Jiang

₍₁₎

_,

_Luc

_Mathieu

₍₁₎

a

UniversityofNorthCarolinaatCharlotte,9201UniversityCityBlvd,Charlotte,NC28223,USA

b

ÉcoleNationaleSupérieured’ArtsetMétiers,4,rueAugustinFresnel,MetzTechnopole,57078MetzCedex3,France

c_{LURPA,ENSParis-Saclay,UniversitéParisSud,UniversitéParis-Saclay,94230Cachan,France} d_{ChalmersUniversityofTechnology,41296Göteborg,Sweden}

e

PolitecnicoMilano,viaLaMasa1,20156Milano,Italy

f

UniversityofAlberta116St.and85Ave.,Edmonton,ABT6G2R3,Canada

g

UniversityofHuddersfield,Queensgate,Huddersfield,EnglandHD13DH,UnitedKingdom

1. Introduction

Uncertainty is ubiquitous in any engineering system, at all

stagesof productdevelopmentand throughouttheproductlife

cycle.Thispresenceofuncertaintyincursrisks–totheproduct

performance,toprocessscheduling,tomarketacceptance,orto

thebusinessitself.Tomitigatetheserisks,strategiesthatbound

designvariables andtheirassociateduncertainty areemployed.

Theserelatedconcepts-uncertainty,risk,andtolerances-createthe

landscapewithinwhich manyengineering design activitiesare

performed.Intheclassicgeometricaldomain,uncertaintyappears

asdimensionalvariability,risk relatestonon-conformance,and

tolerancesareusedtolimittheallowablevariability.

Therisingdemandforhighreliability,robustnessandsafetyof

complex engineering systems, such asautomobiles and aircraft,

requiresengineerstounderstandandmanagevariousuncertainties duringthedesignprocess.Suchuncertaintiesincludeanticipated

manufacturing variation, imperfect numerical approximations,

impreciseestimatesofloading,andlimitedprototypesonwhich

to perform testing. These uncertainties,if incorrectlymanaged,could leadtosignificantdesignbias,costlymaintenance,evencatastrophic

consequences,especiallyformultidisciplinarysystems.Therefore,it

hasbecomeimperativetoidentifythesourcesofuncertaintyand

quantifytheimpactofmultipletypesofuncertaintiesin multidisci-plinarysystemsdesign[12,225,248,293,294].

Examples of uncertainty include manufacturingimprecision,

variationsinproductusage,andgeometricvariability;allofthese

are subject to imperfections and incomplete information. Such

uncertaintyhasasignificantimpactonproductperformance.The

ability to evaluate and improve product performance where

severaltypesofuncertaintyarepresentisveryimportanttoavoid warrantyreturnsandscraps[60].

V.Srinivasanidentifiedtwoaxiomsunderlyinghisdiscussionof

computationalmetrology[280,281].Theseare:

1) The axiom of manufacturing imprecision: “All manufacturing

processesareinherentlyimpreciseandproducepartsthatvary.”

2) Theaxiomofmeasurementuncertainty:“Nomeasurementcanbe

absolutelyaccurateandwitheverymeasurementthereissome

finite uncertaintyaboutthe measuredattribute ormeasured

value.”

Duetotheimprecisionassociatedwithmanufacturingprocess;

itisnotpossibletorepeatablyproducetheproduct'stheoretical

dimensions.Thisresultsina degradationof theproduct

perfor-mance. In order to ensure the desired behavior and the

ABSTRACT

Variabilityisunavoidableintherealizationofproducts.Whiledesignmustspecifyidealgeometry,itshall alsodescribelimitsofvariability(tolerances)thatmustbemetinordertomaintainproperproduct function.Althoughtolerancingisamaturefield,newmanufacturingprocessesanddesignmethodologies arecreating newavenuesof research, andmodelling standardsmust also evolve to supportthese processes.Inaddition,thestudyofuncertaintyhasproducedwidely-acceptedmethodsofquantifying variability,andmoderntolerancingtoolsshouldsupportthesemethods.Thechallengesintroducedby newprocessesanddesignmethodologiescontinuetomaketolerancingresearchafertileandproductive area.

performanceoftheengineeringsysteminspiteofuncertainty,the componentfeaturesareassignedtolerancelimitswithinwhichthe

characteristic of the feature – i.e. situation and intrinsic

characteristic–lies. Thisactivityis referredtoas“tolerancing”. Further,theinabilitytodeterminethetruevalueof actualpart

characteristics influences the ability to properly characterize

manufacturing processes. To manage the rate of out-tolerance

productsandtoevaluatetheimpactofcomponenttoleranceson

productperformance,designersneedtosimulatetheinfluencesof uncertaintywithrespecttothefunctionalrequirements. 1.1. Historyoftolerancing

Thedevelopmentoftolerancingcanbetracedbacktotheend

ofthe19thcenturyorthebeginningofthe20thcenturythrough

the need for more precisely engineered components to be

assembled interchangeably [99,47,131,240]. Since 1905, the

“Taylor Principle” or “envelope requirement” which is based

on the hard gauging practice,allowed the development of a

function-oriented approach for assembly, thus enabling the

foundationsforascientificapproachtotolerancing[287].

Sub-sequently,themilitaryandmanufacturing sectorsencouraged

the development of standards addressing limits and fits,

technicaldrawings, subcontractingdocuments, and also gave

moreconsiderationtomanufacturingoperationsandthecontrol

of workpiece variability in the practices of the design and

engineeringoffices[130].

AgeometricmodelfortolerancingwasdevelopedbyS.Parker

in1938through thedevelopment oftolerancesoflocationand

tolerancezones[230].Parker_’sworkisseenasthefoundationof

geometrictolerancingandhaspavedthewayfornewconcepts

such as the principle of the maximum material condition

developedbyChevroletin1940 [64].Atthesame time,efforts

to standardize the graphical symbolism of tolerancing for

technical drawing led to the GD&T (Geometric Dimensioning

andTolerancing)systemthrough thedevelopmentofAmerican

standards MIL-STD-8 (1949), ASA-Y14.5 (1957), USASI Y14.5

(1966),ANSI Y14.5 (1973), ANSIY14.5M (1982),ASME Y14.5M

(1994)[27],andASMEY14.5(2009)[24].Similarly,international tolerancingstandards(ISO)havealsoevolvedfromtheISOsystem

oflimits and fitsISO/R286 (1962)and standardsfor technical

drawingandgeometricaltolerancingISO/R1101(1969)toanew

systemofstandardsforGeometricalProductSpeci_fications(GPS)

whicharenowbeingdevelopedinthedifferentworkinggroups

oflSO/TC213[222].

The work in TC 213is based on the idea thatthe _field of

geometricalproductspecificationscanbedescribedasamatrix:

therowsarethevariousrequirementsandthecolumnsarethe

variouspiecesthathavetobeinplacetocreateanunambiguous

specification.Inthisnewapproach,specificationsaredefinedby

an ordered set of operations, each of which is applied to a

feature[167]according toMathieu andBallu [208], basedon

theseorderedsetofoperations(oroperators)theuncertainties

linkstotolerancingactivitiesaredevelopedin[169].Theideaof

theGPSsystemistoguaranteeandensuremechanicalproduct

propertiesintermsoffunctionality,reliabilityand interchange-ability.

Overthelast40years,theconfluenceofindustrialneed,therise

oftheCAxsoftware,andthedevelopmentofcoordinatemetrology

hasjustified both significant research and an evolution of the

tolerancing standards. The CIRP Seminar on Computer Aided

Tolerancing(CAT)wasconceivedduringthe1980sfollowingthe

growingdesireoftheCIRPcommunitytoundertakecooperative

projects on the topics of tolerancing and dimensioning of

mechanical parts, the functional meaning of tolerances,

uncer-taintyand standardization[132,232,314].Twomainneedswere

identifiedtobeemerging at thattime [313]: theintegration of

tolerancing procedures in the CAD/CAM environment, and the

assessment of geometrical errors of Coordinate Measuring

Machines(CMMs) andalgorithmsforanalyzingworkpiecedata.

Thesetwoareaswerebeingresearchedextensivelywithmostof

those contributions being published at the CIRP Annals

[241].Meanwhile,inthefieldofComputerAidedProcessPlanning

(CAPP), tolerance transfer and tolerance charting were being

computerizedin order tobe integratedinto CAD/CAM systems

[115,117].BearinginmindtherelevancethatCATwasacquiring, thenecessitytomeet,shareanddiscussthedevelopmentsofthis

fieldwas manifest.InDecember1989,inresponsetothis need,

Prof.R.D.Weillorganizedthefirsttwo-dayWorkingSeminaron

CATin Jerusalem,Israel.Since then,theseminar hasbeenheld

everytwoyears,takingplace15timesworldwideandwithover

600paperspublished.

1.2. Newchallengesintolerancing

The introduction of new manufacturing technologies has

broadened the scopeof both geometry and material attributes

that a designer may specify. With this specification naturally

comestheneedforcontrolofvariabilityinthesenewattributes.As

anexample,newadditivemanufacturingprocessescanproduce

assembliesinas-builtform,createcomplexlatticestructuresfor

support,andproducegradientsinthedensityandcompositionof

material throughout the workpiece. These potential workpiece

attributesintroducechallengesinthemodellingoftheworkpiece’s

nominal design, and until the nominal properties are defined,

variabilityintheseattributesisdifficulttocontrol.Forexample, considerthecomplexsupportstructureinFig.1:boththeexplicit

modelling of this geometry, and appropriate controls to the

support shapes represent challenges to conventional tools and

practice.However,thereisagreatopportunitytosimultaneously

considercontrol methodsas themodelling methodsare

devel-oped. If a particular representation is chosen to describe how

materialdensity changesthroughouta part,this representation

should accommodate the allowable variation in this density

attribute.

Inadditiontobroadeningthedomainofworkpiecespeci

fica-tion, the consistency and traceability of data throughout the

productlifecycleis ofincreasingimportanceasenterprises rely moreheavilyonadigitalrepresentationofnotonlytheworkpiece, buttheprocessesthatproduce,inspect,andmaintaintheproduct

through-out its lifecycle. Current standards describe how the

tolerancesassociatedwithfeaturesmaybepresentedtoahuman

user,asshowninFig.2,butdonotrequireaspecificunderlying modelorrepresentation.Theconceptofa“digitalthread”isthatall productinformationiscapturedinaformatthatisusablebythe design,manufacturing,andinspectionactivitiesoftheproduct's lifecycle,andthattheinformationisuniquelyidentifiable,sothat

thetraceabilityofinformationmaybemaintained.

Thesenewchallenges(andothers)arerevisitedinmoredetail inChapter7,whereaframeworkforfutureresearchisproposedin

the context of the information provided in the intervening

Chapters.

1.3. Extensionstotolerancing

Theconceptof toleranceissimilartotheconceptofmargin,

withthemain differencebeingthattheseconceptsareusedin

different fields. The term “engineeringtolerance” refers to the

permissiblelimitorlimitsofvariationinaphysicaldimensionorin

a physicalproperty ofa material,while a margin mayrefer to

modelparametersinavarietyofsituations.

Quantification of Margins and Uncertainty (QMU)[225,226]

focuses onthe identification and analysis of performances and

their margins that are evaluated under uncertainties using

computational modelling and simulation. QMU focuses on

rigorously quantifying model uncertainty in order to support

comparisontodesignmargins,andevaluatetheirimpactsonthe

response output variables. There is currently no standardized

methodology across the simulation community for conducting

QMU.

1.4. Subsequentchapters

Therearemanydifferentapproachestomanageuncertainties

whicharecloselyinterrelatedwithtolerancing.Theremainderof thispaperisorganizedasfollows:Chapter2containsareviewof

uncertaintytaxonomies,andthislanguageisusedthroughoutthe

followingchapters;Chapter3describeshowtheseuncertainties

arerelevanttothedifferentstagesoftheproductdesignprocess.

Theunambiguousspecificationoftolerancesrequiresa

standard-ized language for communication (Chapter 4). Techniques to

propagateandanalyzethetolerancesaredescribedinChapter5,

and methods toallocate thetolerances are coveredin Chapter

6. Newchallengesforeseeninthefieldoftolerancingarediscussed inChapter7,withconclusionsandastructuretoguidefuturework inChapter8.

2. Uncertaintytaxonomy

Inthis chapter,a general viewof theuncertainty conceptis

provided.Relevantpapersinthedomainofuncertaintyclassi

fica-tion,design underuncertainty,uncertainty, andtolerancingare

reviewed. Definition of uncertainty is widely different and is

greatlyinfluencedbycontextanddiscipline.

One of the most controversial discussions in uncertainty

analysisrelates tothe classification of uncertainty into several typesandofpossiblesourcesofuncertainty.Aclassicalclassi fica-tionistheseparationofuncertaintyintothetwotypes:aleatory andepistemic.Aleatoryuncertainty,alsoreferredtoasirreducible, objectiveor stochastic uncertainty, describestheintrinsic

vari-ability associated with a physical system or environment

[35].Accordingtotheprobabilitytheory,aleatoryuncertaintyis

modelledbyrandomvariablesorstochasticprocesses.Epistemic

uncertainty,ontheotherhand,isduetoanincompleteknowledge

about a physical system or environment. The definitions of

uncertaintyarebrieflyexplainedthroughSection2.1.Afterward, uncertaintytaxonomiesareprovidedinSection2.2.

2.1. Conceptofuncertainty

Uncertaintyisubiquitousinengineeringdesign.Asweaimfor

designingmoreandmorecomplicatedsystems,currenttoolsare

not capable of accurately predicting the behavior and design

parametersofthedesignedsystems.Thisinability,whichcanbe

duetovariousreasons,isdiscussedas“uncertainty”indesign.Lack

of knowledge about a system and its environment, imperfect

manufacturing,couplingofasystem’selements,errorsandmany

other issues cause the design to be uncertain. The concept of

uncertainty is discussed and classified in different engineering

domainssuchassystemsengineering[183],civilengineering[28],

structural engineering [211], aerospace [88] and mechanical

engineering[228].

Theterm‘uncertainty'hascometoencompassamultiplicityof

concepts. Basic definitions of uncertainty include “liability to

chanceoraccident”,“doubtfulnessorvagueness”,“wantofassurance orconfidence;hesitation,irresolution”,and“somethingnotdefinitely

knownorknowable”.

Uncertainties are things that are not known, known only

imprecisely,orincompletely.Thereisnovaluejudgmentinstating

that somethingis uncertain – it maybe worse or better than

expected.

In the field of production engineering, the concept of

uncertainty is associated with precision and metrology.

Uncer-tainties are factual and measurable; things are known, or not

known,orknowntoaquanti_fiabledegreeorwithinquanti_fiable

bounds. Measurement results are affected by measurement

uncertainty, which leads to technical and economic risks in

industrial companies.By assessing therisks and theconnected

consequences of the decisions (conformity verification), the

significance of the measurement result can be evaluated

[303,304].Thesimulationsofthefunctionalchainofconformity

assessments generate an estimation of the significance of

measurements independence of measurement uncertainty and

othertypesofuncertainties.Thisnotionofuncertaintyisbynow

wellentrenchedinmetrology.

Uncertaintyispresentinallareasofdesign,manufacturing

and metrology. ISO GPS standards established the duality

principleofspeci_ficationandveri_ficationandthatuncertainty

develops throughtheproductlifecycle.InISO/TS17450-2,the

concept of uncertainty is expanded to specification and

veri_fication. The uncertainties through the product life cycle

span from the design intent to the uncertainty in the

performance of the product as well as the environment in

whichitwillperform.TheclassificationisprovidedinFig.3.The

uncertaintyisdividedintocorrelationuncertaintyand

compli-ance uncertainty, which comprises specification uncertainty

andmeasurementuncertainty:

-Correlationuncertaintyisameasureofhowwellthefunctional requirements correlate toproduct specifications. If there is a

goodcorrelationbetweenthefunctionalrequirementsandthe

specification,thenthecorrelationuncertaintyislow.

-Complianceuncertaintyisthesumofspecificationuncertainty

andmeasurementuncertainty,withwhichitcanbeproventhata

workpiece complies with all possible interpretation of a

specification.

-Thespecificationuncertaintycharacterizestheambiguityinthe

specification expression. It is caused by poor definitions in

standardsandotherrequirementdocuments[200].

-Themeasurementuncertaintyisconsideredbythemetrologists

and well described in GUM. The measurement uncertainty

includesallthecausesofvariationofthequantityintendedtobe

measured,usuallythroughinspection.Thecomputationmethod

ofmeasurementuncertaintyisgiveninGUM[134]andsimpli_fied

inISO14253-2[166]. Fig.2.Exampleofmodelpresentationtotheuser.

-Thecombinationofthemeasurementuncertainty,thespeci fica-tionuncertaintyandthecorrelationuncertaintyiscalled“total uncertainty”.

Itcanbeseenthatevenwithalowuncertaintyinmeasurement,

thetotal uncertaintycouldbeverysignificantwhencorrelation

and/orspecificationuncertaintiesarelarge.

2.2. Correlationuncertainty

Thecorrelationuncertaintyaimstoascertainthe

appropriate-nessof thegeometricalproduct specificationstoguaranteethe

functionalrequirements.Itisthedesigner’sresponsibilitytokeep

the correlation uncertainty as low as possible by the correct

expressionoftheintendedfunctionalrequirements.Intheactual ISOGPSstandards,the“correlationuncertainty”isreplacedbythe “ambiguityofthedescriptionofthefunction_”[169]whichrefersto

the“uncertaintyarisingfromthedifference betweentheactual

specificationoperatorandthefunctionaloperatorthatdefinesthe intendedfunctionoftheworkpiece”.

Functions such as the assembly of parts can becompletely

described by ISO GPS and ASME Y14.5 standards, and so the

correlationuncertaintyisthenconsideredtobelow[140].

Howev-er,somefunctionsareverycomplexanddependnotonlyonthe

geometrydescribedbytheshape,thesizeandthetexture,butalso

bythematerialproperties,themanufacturingmethods,andthe

operatingconditionsandmanysimplifyingassumptionsmustbe

made. This difficulty is described in Refs. [236] and [177] as

“perhapsthebiggestinverseprobleminmanufacturing”andalso

bySrinivasan[280],stating_{“Correlation}uncertainty,inparticular, isanunchartedterritory.Standardsdonottellushowtofindthis.”

Only fewresearchaddressedcorrelationuncertainty.Dantan

etal.[73]developedanapproachfortheexpressionandevaluation

of thecorrelationuncertainty for gearconformity. Weckenman

andHartman[311]proposedafunction-orientedmethodbasedon

mathematical-physicalmodelofthefunction,andintegratedthis

withtheentireprocesschainforthemicro-structuredsurfacesof aninkingroll.Toaccommodatethecorrelationuncertainty,Jiang

and Whitehouse [177,316] pointed out that the functional

performanceshouldbebasicallyclassifiedandnewtechnological

shiftsshouldbeaddressed.Thecharacterizationoffunctionand

thecorrelationwithgeometricparametershavebeenintensively

investigatedinthedomainofmicro-geometryandsurfacesusing

functionmapsthat permitsalsotoconsiderthemanufacturing

processwithonlyfewparameterstocorrelatethefunctionandto avoidcontributingto“theparameterrash”[315].

Theshifttowardsmicro-parts,freeformandstructuredsurfaces

for added-value manufacturing, and the optimization of the

performanceoftheproductsrepresentnewchallengesinmodern

designwhen tackling correlationuncertainty. Functionalities of

suchfreeformandstructuredsurfacescanbeclearlydefinedinthe firstplace,suchthatthefunctionalitiescanbedirectlyinterpreted intospecificationstodefinethesurfaces.Itisenvisaged thatin

suchcases,correlationuncertaintycanbecharacterized

mathe-matically and then be reduced significantly. Whilst design for

additivemanufacturing(AM)isgenuinelyfunction-oriented,the

challengesassociatedwithAMspecificationswithconsiderationof correlationuncertaintyarestillemerging(see Section7.1).New interdisciplinaryapproachestocorrelatethegeometryatmultiple

scales and the function and the efficient integration between

design,manufacturingandmetrologyarerecentlyexploredwith

promisingresults[289].

2.3. Uncertaintytaxonomy

A significant amount of research has been devoted to the

definitionandclassificationoftheuncertainty.Wewillconsider uncertaintiesfromthepointofviewofthedesigner.

The analysis of this paper is mainly based on the classical

taxonomy with considering the categorization of Thunnissen

[294]. With basis of aleatory and epistemic uncertainties,

Thunnissenadded twomorecategories: ambiguityand

Interac-tion.ThisstructureisshowninFig.4,andeachoftheuncertainty categoriesisdiscussedintheparagraphsthatfollow.

Ambiguityhasalsobeencalledimprecision,design

impreci-sion, linguistic imprecision,and vagueness. Althoughit can be

reducedbylinguisticconventionsandcarefuldefinitions,

ambi-guityremainsanunavoidableaspectofhumandiscourse.Inthe

contextoftolerancing,thespecificationuncertaintyissimilarto theambiguity.ThispointwillbedetailedintheChapter4,where

standardsplayanimportantrole.

Epistemic uncertainty is any lack of knowledge. The key

featureisthatthefundamentalcauseisincompleteinformationor

incompleteknowledgeofsomecharacteristicofthesystemorthe

environment. Thunnissen[294] classifiedepistemicuncertainty

intoModel,phenomenologicalandbehavioraluncertainty:

-Modeluncertaintyisintheaccuracyofthemodelofasystem

regardingtheactualsystem.Inotherwords,itisthedifference

betweenthemathematicalmodelandtheactualbehaviorofthe

system. Since the approach is to model the system; model

uncertaintyisanimportanttypeofepistemicuncertaintyhere.

Some of the most difficult problems arise from unintended

omissions from the model – these can be thoughtof as the

“unknownunknowns.” Model uncertainty can alsobe due to

approximationerrors,programmingerrorsornumericalerrors.

-Phenomenologicaluncertaintyisrelatedtothebehaviorofa

systemindifferentconditions.Itcanbeduetounknownimpacts

of the environment, unimaginable behavior of the system in

speci_ficcondition,orpossiblebehaviorofthesystemwhileusing

specific design technique. It may also be due to limited

understanding of the behavior of key parameters of the

phenomenon or their interactions. To model a system, it is

necessary to predict its behavior. Therefore, considering

phenomenologicaluncertaintyisessential.Thistypeof

uncer-tainty is mostly in qualitative form (related to the system’s

Fig.3.GPSuncertaintytaxonomy.

function) though it is important to consider this type of

uncertainty in modelling and designing the structure of the

system. So, for theelicitation of this type of uncertainty the

designerneedsamodelthatcreatesthelinkbetweenfunction

andphysicalstructure.

-Behavioraluncertaintyisrelatedtotheuncertainindividualor organizationalbehavior.Itcanbeduetothedesigndecisionsthat

aremadeduringthedesignprocess,customerorstakeholder

requirementuncertainty,uncertaintyinfutureactionoftheuser

or organization while interacting withthe system, or human

errorsduringdevelopmentofasystem.

Inthecontextoftolerancing,thevariabilityanalysisisaffected

by these uncertainties. In fact, variability analysis includes

variabilitymodelling,systembehaviormodellingandvariability

propagation.ThispointwillbedetailedintheChapter5.

Aleatory uncertainty is inherent variation associated with a

physical system or environment under consideration. Aleatory

uncertaintygoesbymanynames:variability,irreducible uncertain-ty,inherentuncertainty,orstochasticuncertainty.Inthecontextof tolerancing,thisuncertaintyisthemostwell-known[71].

Interactionuncertaintyarisesfromunanticipatedinteraction

of many factors and/or disciplines, each of which might, in

principle,beorshouldhavebeenforeseeable.Potentialtechniques todeal withthis typeofuncertaintyare simulation, multidisci-plinarydesignoptimization(MDO),andcomplexityscience.Inthe

context of tolerancing, this uncertainty affects the tolerance

allocation(Chapter6).

Althoughthis paperwillfocusonThunnissen’sclassification, thereareotherclassi_ficationsfor uncertainty[87,184]in design

andmodellingthat needtobementioned aswell.Walteretal.

[302]hasanothercategorizationwithfocusingonmodellingand

simulation.Theycategorizeduncertaintyinto_{“phenomenological}

uncertainty”, “uncertainty in human behavior”, “uncertainty in

data”and“uncertaintyinmodelandsimulation”(showninFig.5).

In comparison toThunnissen’s classification, phenomenological

uncertainty and uncertainty in human behavior are epistemic

uncertainties in the sub-category of phenomenological and

behavioraluncertainties respectively.Uncertaintyindatacanbe duetoitsvariation,whichinthiscaseisaleatoryuncertainty,or duetothevagueness,whichinthiscaseisepistemicuncertainty

[302].Modelandsimulationuncertaintyinthiscategorizationcan

beintheconcept(epistemic-phenomenological),the

mathemati-calmodel(epistemic-model),programming

(epistemic-behavior-al) or visualization of effect (aleatory). Structural engineering followsasomewhatanalogousclassi_fication[211].

Engelhardtetal.[104]categorizesuncertaintyintothreetypes, asshowninFig.6:

1) Stochasticuncertainties:thesearerelatedtotheprobabilityand propagationofanevent.Theyarealsorelatedtotheuncertain valuesforentitiesindesign.Thisisaleatoryuncertaintyinthe Thunnissen_’sclassi_fication.

2) Unknownuncertainties:Thisisregardingthelackofknowledge

about an event, effect or behavior of a system. This is an

epistemicuncertainty.

3) Estimateduncertainty:Thisuncertaintyis whentheeffectis

knownbuttheprobabilityoftheeventispartiallyquantified.

This represents a case where both aleatory and epistemic

uncertaintiesexist.

2.4. Observations

Thischaptersummarizesuncertaintytaxonomiesand de

fini-tionsinthe_fieldsofprecisionengineeringandproductdesign.The classification theis used separates uncertainty intofourtypes: ambiguity,epistemic,aleatory,andinteraction.Epistemic

uncer-taintyis furtherbroken outintomodel,phenomenological, and

behavioraluncertainty.

The next chapter provides the position of the uncertainty

managementinthecontextofProductdesign.

3. Uncertaintymanagementindesign

Engineering Design problems are ‘ill-defined’ problems

[100].Innewproductdevelopment(NPD)processes,thenumbers

ofunknownsfaroutweighthenumberofknowns.Thisresultsina

processthathasahighdegreeofuncertaintyatthestart,whichis

progressivelyreducedthroughco-evolutionofproductspeci

fica-tionandproductdesignastheprocessprogresses.Inpractice,NPD

processes generallyfollow a divergentconvergent patternwith

stage-gatedproject processflowallowingforprogressivedesign

and development maturity [238]. This characteristic divergent

convergentpatternenablesthedesignertodeduceandreducethe

uncertainty throughout the process by narrowing the choices

towards feasible solutions throughout the process. The design

state,whichisde_finedastheincorporationofalltheinformation abouta designas itevolves, alsoprogressivelytransitionsfrom

being abstract (text descriptions, customer requirements), to

concrete(analyticalmodels,controlledgeometricrepresentation, firmspecifications)[227,324,325].

Using the uncertainty taxonomy described in the previous

section,andconsideringthe6phasesofagenericdesignprocess

[105]showninFig.7,thetypesofuncertaintyandtheirimpactin eachphasecanbeidentified.Eachdesignphaseisdescribedinthe paragraphsthatfollow,andsummarizedinFig.8.

0. Planning:Inthisphase,thedesignstateisinitsmostabstract

form. The design space is also at the point of being least

explored.Consequently, thisphase ischaracterized byahigh

degreeofambiguityandasignificantdegreeofepistemicand

interactionuncertainty.Duetotheabstractnatureofthedesign state,thealeatoryuncertainty cannotbewellde_fined inthis phase.

Fig.5.Walter’suncertaintytaxonomy.

Fig.6.Engelhardt’suncertaintytaxonomy[104].

Fig.7.GenericproductdevelopmentprocessaccordingtoUlrichandEppinger [105].

1. Conceptdevelopment:This is a critical divergent-convergent phaseoftheNPDprocess.Inthisphase,possibleconceptstothe designproblemaredetermined.Thisphaseischaracterizedbya

reducingambiguityasthedesignstateevolvesalongwiththe

co-evolutionofdesignproblemandsolution.Thisleadstoahigh levelinclusion ofinteractionuncertaintyas different

interac-tions between the elements of the system as well as the

interaction of the system with the different stakeholders

(environment,users, ...)isconsidered.Epistemicuncertainty

in designer’s decisions toward identifying a solution is

unavoidable.

2. System-level design: The concept obtained in the previous

phaseisfixedandsystemleveldecisionsaremadeinthisphase.

Thisphaseis characterizedbyreducingambiguityasthekey

productdecisionsareagreedupon,enabling thedesignersto

finalizekeyspecificationsaskey performanceindicators.The

availability of early stage analytical models as well as basic geometryallowsthedesignerstostartincorporatingestimates ofaleatoryuncertainty.

3. Detail design: The detail design entails a detailed focus on

discretecomponents.Inthisphasethedetailsincludingform, surfaces,tolerances,dimensions,andmaterialaredecidedupon

based on the design of the previous phase. This entails a

significantfocusonaleatoryuncertaintyasthedesignaswellas

manufacturing intents are finalized and key performance

indicatorsaresimulatedviamodelling.Theinteraction uncer-taintyisintegrated,controlledandmanagedinthisphasevia

developing more detailed models of the system and

under-standingthephysicalprocessesandtheirinteractionwitheach

other. In addition to the epistemic uncertainty due to the

decisions of this phase or previous phases, the existence of

aleatory uncertainty in the value of physical entities and

environmentalimpactsareunavoidable[229].

4. Testingandrefinement:Thisis anotablephaseinthedesign

processasoneofthemajorobjectivesofthephaseistoidentify, control,andreduceuncertainty.Thekeyactivitiesinthisphase aretodevelopandtestfunctionalprototypestoascertainthe

productperformance as pertherequirements inpresence of

differentuncertaintiesmanagedintheearlierphases. Addition-ally,thisphasealsoallowstoidentifytheuncertaintiesthatmay nothavebeenaddressedorconsidered.Assuch,theinteraction and aleatory uncertainties are significantlyaddressed in this phaseandanyoutstandingambiguityorepistemicuncertainties areaccountedfor.

5. Production ramp-up: This phase is characterized by a high

degreeoffocusonaleatoryuncertaintyasthekeyproduction

plansareputintoactionandanyunexpectedprocessvariations

and uncertainties are managed and resolved before the

productionrampupprocessisconvertedintofullproduction.

Fig. 8 shows the relative influence of the differenttypes of

uncertaintythroughthephasesofthedesignprocess.Inaddition totheclassicalcomponentdesignprocess,itisimportanttoalso

consider how uncertainty plays a role in higher-level project

design.Thedesignofcomplexsystemsismoreheavilyweighted

towardtheearlystages,andtheinteractionuncertainty–drivenby

complexity – plays a large role in the risk of the process. A

discussionoftheevolutionofuncertaintiesincostoveraproduct lifecycleisdescribedbySchwabeetal.[261].

3.1. Uncertaintyandrisk

Toclarifythescopeofuncertaintymanagementindesign,itis importanttoidentifytheconsequencesoftheseuncertainties:The uncertaintiesleadtorisks.ConciseOxfordDictionarydefines‘risk’ means:“hazard,chanceofbadconsequences,loss,exposuretochance of injury orloss”. In the context of system engineering, risk is classifiedintotechnical(feasibility,operability,manufacturability,

and systems effectiveness), cost (estimates, goals), schedule

(technology/material availability, technical achievements,

mile-stones),andprogrammatic(resources,contractual)[150].

Brown-ing [43] proposes an overview on the link between risks and

uncertainties. The sources of risk in product development are

dividedintosixcategories,described belowandsummarizedin

Fig.9:

1. Performancerisk—Uncertaintyintheabilityofadesigntomeet

desiredqualitycriteria(alonganyoneormoredimensionsof

merit, including price and timing) and the consequences

thereof.

2. Schedulerisk—Uncertaintyintheabilityofaprojecttodevelop anacceptabledesign (i.e.,to sufficientlyreduceperformance

risk)withinaspanoftimeandtheconsequencesthereof.

3. Developmentcostrisk—Uncertaintyintheabilityofaprojectto

develop an acceptable design (i.e., to sufficiently reduce

performancerisk)withinagivenbudgetandtheconsequences

thereof.

4. Technologyrisk—Asubsetofperformancerisk:uncertaintyin

capability of technology to provide performance benefits

(within cost and/or schedule expectations) and the

conse-quencesthereof.

5. Marketrisk—Uncertaintyintheanticipatedutilityorvalueto themarketofthechosen“designto”specifications(including

priceandtiming)andtheconsequencesthereof.

6. Businessrisk—Uncertaintyinpolitical,economic,labor,

socie-tal, or other factors in the business environment and the

consequencesthereof.

Alltypesofuncertaintycauserisksinthattheyaffectthedesign process[133].Manytechniquesexisttoreduceandmitigatethese risks[56].Toclarifythescopeofthesetechniques,McManusand Hasting[209]proposeaframework:“UncertaintiesleadtoRisksor

Opportunities, which are handled technically by Mitigations or

Exploitations,whichhopefullyleadtodesiredOutcomes_”.Basedon

this framework, the taxonomy of uncertainty management is

provided in Fig. 10. In the next section of this chapter, this

taxonomyisusedtodefinetheaim,thescopeandthepositionof

variousapproaches:Robustdesign,Reliabilitybaseddesign,and

tolerancing. Fig. 8. Effectofuncertaintyindifferentphasesofproductdevelopmentprocess.

3.2. Robustdesign

Robustness refers to performance stability in engineering.

Robustdesignisasetofengineeringmethodswidelysuccessfulin

reducing sensitivity to such noise factors as design parameter

variation,customeruseconditions,manufacturingvariability,and

degradationofasystemovertime.

Robust Design Optimization (RDO) aims at minimizing the

sensitivity of the performance under variability. RDO methods

intend to achieve systems with slight performance variations

aroundtheirnominalvalues.TheprimaryobjectiveofRDOisto

reducetheuncertaintythroughselectionofthevaluesofdesign

variablesXwhileconsideringtheperformancesYsothatthemean valueoftheperformance

m

s

arewithinacceptablerangeofthedesignoutcomes.

Find: X

Minimizing: w

m

s

Choi [65] categorizes robust design into four types. Type-I

robustdesign isrelated toidentifyingdesignvariablevalues to

satisfy the required performance requirements despite the

variations in noise factors [2]. Type-I RDO manages Aleatory

uncertainty from noise variables. Type-II RDO deals with

uncertainty in the value of design variables while considering

noise,thereby,addressingtwosourcesofaleatoryuncertainty.As showninfigureXbytheyellowregion,bothTypeI,andIIRDOaim

to stabilize the system response while considering aleatory

uncertainty. Type-III RDO deals with managing the effects of

uncertaintyintroducedbyidentifyingadjustablerangesfordesign variables,thatsatisfysetsofperformancerequirementrangesand areinsensitivetothevariabilitywithinthemodel.Thisisshownby

the orange region of decreased system response variability in

Fig.11.InadditiontoaddressingAleatoryuncertainty,Type-IIIRDO

also deals with the epistemic uncertainty related to the

phenomenon and behaviorof system. Type-IVis related tothe

modeluncertaintyincludingerrorsindecisionsandaccumulated

errorsbyseriesofuncertainsubsystemmodels[65].

The search for robust solutions has led to analyses and

modellingofuncertainties due tomanufacturingimperfections,

externaluncertaintyanderrormodelling.Theseuncertaintiescan bealeatoryorepistemicbynature[226].Inthepastfewyears,

manyapproacheshavebeendevelopedtodealwithuncertainty

suchas robustdesign methodology(RDM)[212].Mostof these

approaches,however,focusondownstreamdesignphases3–5and

areinformofanalyticalornumericalmodels,intendedmainlyto

study the impact of uncertainty on key design parameters

[213]. This is due to presence of ambiguity and epistemic

uncertainty in earlier phases,which hinders the application of

concretenumericalmodelsinearlierphases.Afewmodelsaimto

resolve early stage uncertainty through integration of abstract

modelling techniques such as function modelling with

meta-modelsofsystemswithuncertaintyintegration[83,103,204].Some

approaches haveaddressed the integration of robust design in

early phases with downstream phases while considering the

impactofthedesignparametersonthesensitivitytovariationas wellascost[101].

It was mentioned at the start of this chapter that design

problems are ‘ill-defined’. Robust design strategies help built

robustnessintheprocesswhichisoneofthecriticaloutcomesof thedesignprocess.Robustnessofadesignensuresthatdespitethe

types of uncertainties that havebeen consideredin thedesign

processtheproduct’sperformancewillnotbeaffected.Incurrent industrial practice,robustness isintegratedinactivitiessuchas design verification, tolerances,and design choices. Thisin turn

primarily reducesthe performance risk, developmentcost risk,

andhasasubsequenteffectonthemarketandbusinessrisk[44]. 3.3. Reliabilitybaseddesign

Reliabilitycanbedefinedasthelikelihoodthatacomponent

(orasystem)willperformitsintendedfunctionwithoutfailurefor

a specified period of time under stated operating conditions.

Reliabilityischaracterizedbytheprobabilitythatthesystemwill performforacertaintime.ReliabilityBasedDesignOptimization

(RBDO) methods are based on the probability distributions to

describe variability of design variables and model parameters.

They intend to achieve systems with an acceptable level of

reliability (failureprobabilities)anda satisfyinglevel of perfor-mance(Fig.12).

Asolutionissaidtobereliableiftheprobabilityofsatisfying eachconstraintisgreaterthanaspecifiedreliabilitylevel.RBDO methodsconsistofdesignoptimizationwithareliability assess-ment, Find: X Maximizing: ProbðZðXÞ2½ZLL; ZULÞ subjectto: HðXÞ>0 X2½LL;UL ð2Þ with, X Designvariables

Z PerformanceswhichdependonX

HðXÞ>0 Designconstraints

ProbðZðXÞ2½ZLL;ZULÞ Conformityprobability

LL_;UL Lowerandupperlimits

Reliabilityanalysisisanessentialactivityemployedtomanage

andreduceperformancerisk.Itdoessobymitigatingepistemic

andaleatoryuncertaintytoaprobabilitythatisacceptablegiven Fig.10.Uncertaintymanagementtaxonomy.

thetechnology,market,andbusinessrisks.RBDOtechniquesare

extensivelyusedinphase3ofthedesignprocess,mostlyinthe

formofstatisticalandstochasticmodelstoestimatetheproduct functionintermsofitsmeanlife.Severalqualitativetechniquesare alsousedintheearlierdesignphases(phases1and2)tomitigate theinteractionuncertaintybyuncouplingthedesignvariablesand theireffectonreliablesystemperformance.

3.4. Tolerancedesign

All manufacturingprocessesexhibitvariation.This variationis

described in terms of aleatory uncertainty in terms of process

probability density function of a bounded process accuracy or

precisionwithameanandvariability.Duetothisuncertainty,itis impossible to attain theoretically nominal dimensions. Tolerancing is a setofactivitiesinvolvingvarioustoolsandmethodswhichallowthe

designers to manage the manufacturing uncertainty during the

productdevelopmentprocess.Tolerancingdiffersfromrobustdesign

andreliabilitybased design byfocusing and emphasizingon the

aleatoryuncertaintyintroducedduetomanufacturingprocesses.

Thealeatoryuncertainty inthemanufacturingprocess

trans-lates into two fundamental aleatory uncertainties in the final

geometryoftheproduct:uncertaintyinsize,i.e.,attainingnominal dimension,anduncertaintyinform,i.e.,attaininganominalshape. Increasesineitherorbothoftheseuncertaintiesdirectlyresultin anincreaseinrisk.Thismaybe:performanceriskinformoflossin

productperformanceduedeparturefromintendedfunction,with

animpactonrobustnessandreliability;scheduleriskduetolossof interchangeabilityofdifferentcomponentsofthepart,resulting

into lower flexibility, evolvability, and interoperability, and

subsequentlymissed deadlines and adherence to schedules;or

developmentcostrisk,byunforeseenbatchrejectsandreworks.

These risks increase the product’s market and business risk,

therebyriskingtheprofitabilityofthecompany.Tolerancingallows

the mitigation of these risks, contributing significantly to

robustness,reliability,flexibility,evolvability,andinteroperability ofthe_finalproduct.

Tolerancedesigncomprisesthreeiterativeactivities:tolerance specification,allocation,andanalysis.Theseactivitiesaredrivenby

requirements of the product's performance, customer

require-ments, manufacturability, and interchangeability (Fig.13). Cur-rentlytheyareextensivelyusedinformofanalytical,numerical, stochastic,andstatisticalmethodsappliedinPhase3:detaildesign

tospecify tolerancewhile considering theaboveconstraints to

minimizeandmanageuncertainty.Therearelimitedapplications

oftolerancinginupstreamdesignprocess.Dantanetal.[83,237]

haveprovidedanintegratedmethodtoconsiderrobustdesignand

tolerancedesign activities.Ebro and Howard[101] haverelated

tolerancesensitivityanalysistoaproductscorefunctionaswellas

manufacturabilityandsuccessintheearlydesignphases.Zhang

et al. [328]also propose a concurrent method to develop the structureoftoleranceassignmentindesign.

Thetolerancingactivityisahighly coupledprocessthataddresses

andguides allowable manufacturinguncertainty in theprocess.

However,thisalsodirectlyaffectsthefunctionoftheproductdueto theacceptablechangeintheallowedvariation.Tolerancingitselfis linkedtoanumberofdesigndecisionssuchascost,manufacturing process selection, and required design speci_fication [110]. It is presentsignificantlyinthedownstreamproductdesignprocessand hasacriticaleffectontheproductdevelopmentandmanufacturing, eventuallyeffectingproducts_’success.

3.5. Observations

Itisimportanttopointoutthedifferenceintheprimaryfociof differenttechniques discussed in this section. Firstly,all of the

above techniques are primarily sets of tools and methods to

identify, model, simulate, and optimize uncertainties in the

productdesignanddevelopmentprocesses;however,theprimary

focus of each method is different. Robust design techniques

primarilymanageandmitigateuncertaintyrelatedtotheselection ofvaluesofdesignvariablesandaimtoreducetheuncertaintyin

theproduct performance.The reliability basedmethods aimto

reducetheuncertaintyoftheproductfailureintermsofitsmean life.Tolerancedesignactivitiesprimarilyfocusonmanagingand

reducingtheriskinproductdesignbymanagingmanufacturing

uncertaintywhileassuringtheproductperformanceandcustomer

requirements. It must be noted that all of these activities are

mutuallycoupledandareiterativelyandconcurrentlycarriedout

tominimizeuncertaintyinthedesignprocess.Thesetechniques

areanintermediateactivityinthedesignprocessthattranslates

the customer requirements into a successful design with the

requiredoutcomesasshownin(Fig.10atendofSection3.1).

4. Specificationandverificationofallowableuncertainty

A clear picture of the importance of standardizing the

specification, verification and exchange of product geometryis

given by Srinivasan [282]. The picture is completed by the

discussionoftheevolutionofthesestandardsdrivenbytheneed ofreducingtheiruncertaintiesandambiguities.

In thefollowing, thecurrent situation in termsof tolerance

specificationandverificationwillbesummarized. 4.1. Tolerancespecification

TheInternationalOrganizationofStandards(ISO)hasproduced

many standards related to dimensioning and tolerancing

[153,154,158–165,167–169,174]. Each of these standards cover

oneormorecellsintheGeometricalProductSpecification(GPS)

matrix[172].TheGPS matrixde_finesnine geometricalproperty

categories(sixrelated tothegeometry,i.e. size,distance,form,

orientation, location, run-out, and three related tothe surface

finish, that is profile surface texture, arealsurface texture, and surfaceimperfections).Foreachproperty,a‘chainofstandards’is Fig.12.Reliabilityillustration.

defined,groupingallthestandardsrelatedtosuchpropertyand

definingsymbols(syntax),meaning(semantics,i.e.mathematical

foundation), and measurement procedures for it. The chain is

formedby‘chainlinks’,whichgroupthestandardsrelatedtosome specificstepinthepropertydefinition.Therecentlyredesigned

[223] GPS matrix reserves the first three chain links to the tolerancespecificationissue(A–C),thelastthreechainlinkstothe toleranceverificationissue(E–G),whilechainlinkDconsistsofISO

GPSstandardsdefiningtherequirementsforcomparisonbetween

specificationrequirementsandverificationresults.

The American Society of Mechanical Engineers (ASME) has

collected the majority of GD&T requirements in ASME Y14.5

standards[24].AlthoughASMEGD&T[24]doesnotexplicitlycover themathematicaldefinitionandtheverificationissueandthelink

between tolerance specification and verification, other ASME

standardsdealwiththeseinstrument-specifictopics[19–22].

ISO GPS _‘chain of standards_’ clearly testi_fies one important

issue: the need to harmonizeverification with specification of

producttolerances,thusreducinguncertaintiesintheinformation

flowfromdesigntomanufacturing.Arewesurethatthedesign

intentisclearlytransferredtomanufacturing?

Itisworthnotingthatthisissuehasbeenafundamentalreason

for developing manufacturing oriented standards [24–

26,173].Thesestandardsrefertowell-establishedmanufacturing

fields:casting, forging,andtheproductionofmoldedpartsand

composites.Nevertheless,problemsmayariseifnew

manufactur-ingtechnologiesenabletheproductionofnewproducts,suchas

thosethataremicro-ornano-scaled,oradditively-manufactured. Consideringthelatter(AM)case, itis possibletoaddressthese

problemsdistinguishingbetweenprocess-drivenissuesandissues

highlightedbythecapabilitiesofadditivemanufacturing,despite theallegedbenefitof“complexityforfree”[4,321].

4.2. ASME—ISOmaindifferencesintolerancespecification

The best-known difference between the ASME and ISO

tolerance specification standards is in the governing principle

thatsize controlsform(called Rule #1)in ASME,where sizeis

independentof form in ISO. However,ASME allows touse the

independency symbol to override Rule #1, and ISO uses the

envelopedprincipletoinvokeRule#1ifrequired.

Despitethisfundamentaldifference,ASMEandISOstandards

havemore similarities than differences,and a convergentpath

seemsinprogress.HeysiattalabandMorsehaverecentlyreviewed

the main differences [141] that the two standards have with

respecttoterminology(Table1)andsymbols(Table2).Themost relevantarethedifferentinterpretationsthatASMEandISOhave forsomeidenticaltolerancesymbols,actualvalues,andmaterial conditions.Table3showsthosedifferences.

4.3. 3DTolerancespecification

GD&Trequirementsneedtobetransferredtomanufacturing

and inspection. However, until recently, GD&T information

consistsintwo dimensionalannotationsondrawings.Since3D

models have almost completely replaced 2D drawings as the

master for product technical data in manufacturing industry

[282,299],theneedaroseforstandardizedindicationsof dimen-sionsandtoleranceson3Dmodels[119]or,better,inside

model-basedengineering(MBE)packages.

Thisneedhasledtothedevelopment–forover30years–ofa familyofstandards,theISO10303series,knownasSTEP(STandard

fortheExchangeofProductmodeldata)thatdescribes

standard-ized data models in several Application Protocols (AP). In

particular,AP242entitled‘ManagedModelBased3DEngineering’

isthemostcomprehensiveproductmodel-baseddefinition(MBD)

ofSTEP,asitcontainsseveraltypesof3Dmodeldata,including dimensionalandgeometrictolerances[171].Itisdevelopedusinga

modulararchitecture[118].ItsmodulesusetheEXPRESSschema

language,asappropriatefortheintendedapplications,todefine thedatamodels[231].

Thetransitiontodigitalmanufacturingisrisingtheimportance

ofincorporatingProductandManufacturingInformation(PMI)in

the (MBE) packages [322]. It also enables Computer Aided

Manufacturing(CAM) softwaretodefineand validate

machine-readable instructions for manufacturing and Computer Aided

Engineering(CAE)softwaretovalidateandoptimizetheproduct

definition.ManufacturersarerecognizingthebenefitsofMBDand movingawayfromrelianceon2DdrawingstorepresentPMI[194].

Asastandardlanguage forPMIdoesnotexistyet,CAD/CAM

softwarevendorsdevelopandimplementtheirownPMIintheir

software.Thesoftwarevendors’implementationsaretestedinan

‘Implementer Forum’ [61,62,63] to ensure that PMI has been

correctlyimplementedand canbeexchangedsmoothly, evenif

partially,usingtheSTEPstandards.TheCAxImplementerForum

(CAx-IF) defines recommendedpractices for interoperable data

exchangeusingSTEPfiles[194].

An example of PMI related to GD&T is the ANSI “Quality

Information Framework” (QIF) [11]. It adopts the modern

Table1

ASMEandISOproprietarysymbols.

Standards Descriptions Symbol

ASME Modifyingsymbols

Dimensioningsymbols

ISO Additionalsymbols

Table2

ComparisonofANSIY14.5andISOterminology.

ASME ISO

Basicdimension Theoreticalexactdimension(TED)

Innerboundary –

Outerboundary –

Featurecontrolframe Toleranceframe

Trueposition(TP) TheoreticalExactPosition(TEP)

Circularity Roundness

Referencedimension Auxiliarydimension

Table3

Differentinterpretationsofstandards.

Tolerance ASME ISO

Flatness Appliedonlytoonesurface Appliedtooneortwo surfaces Orientation appliedtoaxis ormedian plane Appliedtoaperfect-form featureaxisorplane (matingenvelope)

Appliedtotheextracted axis,lineormediansurface

MMC*

orLMC**

Notappliedtoconcentricity andsymmetry

Appliedtoconcentricity andsymmetry Symmetryand

concentricity

Appliedtothemedian points

Appliedtotheextracted medianlineorextracted mediansurface Position AppliedonlytoaFOS***

AppliedtoaFOSortoa plane

Run-out Tolerancezonealways normaltothenominal profile

Tolerancezonenormalor non-normaltothenominal surfaceofthepart Profile Tolerancezonedefinedby

twoequallyorunequally disposedsurfacesorlines aboutthetrueprofilethat extendtointersection points

Tolerancezonedefinedby twoequallyorunequally disposedsurfacesorlines formedbysweepinga sphereoracirclearound thenominalprofile Composite

tolerancing

Usedforpositionaland profiletolerances

Meanstwoindependent tolerances

ExtensibleMarkupLanguageSchemaDefinition(XSD)as informa-tionmodelinglanguage,anditcovers(quality)metrologysystems

[141].It is a feature- and characteristic-based dataformat that definesfouraspectsofacharacteristic:Definition,Nominal,Item

andActual.ThisallowsQIFtobothdefinePMIrequirementsand

reportmeasurementsresultsinacommondatamodel,allowing

thelinkingofresultstotheoriginaldesign.

Anotherlanguagetoexpressthespecificationfromfunctionto verificationisGeoSpelling[32].TheobjectiveoftheGeoSpelling

language is to enable the semantics of specifications to be

expressedand todefine theirmeaningclearly. The conceptsof

GeoSpelling have been integratedinto the ISO 17450-1 and -2

standards[167,169].

Thecontinuousreviewandrevisionofthedifferentstandards

andthegrowingindustrialinterestindigitalmanufacturing,itis expectedtoreducetheinformationlossindatatranslationamong differentMBEpackagesand,withthehelpofPMI,tomitigatethe

uncertaintiesintheinformationcontent.However,boththeASME

and ISO systems continue to struggle with the increased

complexityofproductsandtheirrequirements.

4.4. Toleranceverification

AlthoughASMEGD&T[25]doesnotcoverthisaspect,theneed

for tolerance verification is clearly stated in the ISO GPS

[148,167].Therecentlyredesigned[223]GPSmatrixreservesthe lastthreechainlinkstotheproblemoftoleranceverification:

– ChainlinkE,‘measurement’,referstotheoptimalprocedures

andrequirementfortheperformanceofmeasurementstoverify

conformancetotolerances;

– Chain link F, ‘measurement equipment’, defines the

require-mentsoftheequipmentusedtoverifytolerances;

– Chain link G, ‘calibration’, states how the measurement

equipment must be managed to guarantee the accuracy of

toleranceverification.

– Furthermore,chainlinkD‘conformanceandnon-conformance’

actsasaliaisonbetweenthelinksrelativetothedefinitionof tolerancesandthelinksrelatedtotheverificationofthesame.

The useof lowaccuracy verification systemscanlead toan

apparentreduction of theprocess capability [186], which is of

courseofgreatimpactonthetolerancingprocess.

Uncertaintyhasbeendeeplystudiedinthefieldofmetrology,

andquiteoftentheterm“uncertainty”directlyreferstotheGPS

measurement uncertainty [318], which is aleatory uncertainty.

Whentryingtoprovetheconformancetogeometrictolerance,the

uncertaintyreducesthereliabilityofanystatement.Thisissueis treatedinGPSchainlinkD,andthereferencestandardistheISO 14253-1[170],whichstatesrulesforprovingtheconformanceof partstoatolerance.Themainruleisthattheuncertaintyalways

‘playsagainst’who isperforming thetest. Thismeansthatthe

conformancezoneisreducedinwhenasupplierperformsthetest

tryingtoproveconformity,andisenlargedwhenacustomeraims

atprovingnonconformity.

ChainlinkGisingeneralcoveredbytheISO14253-2standard

[166].Anyway,therulesstatedinthisstandardareverygeneric.

Morespecificproceduresfortheevaluationofthemeasurement

uncertaintyarecoveredintheISO15530series[155].However,

these standards do not cover the GPS method uncertainty

(ambiguity) arising from the risk of misunderstanding the

tolerances, in particular, when high precision part or complex

surfacesareinvolved.The definition ofa commonlanguagefor

tolerancing and tolerance verification is yet to be realized. A

significantstepinthisdirection,inthecaseoffeaturesofsize,has

beenachievedwiththeintroductionoftheISO14405-1standard

[174].Thenewsyntaxinthisstandardallowsforthedefinitionof thetypeofsize(twopoints,least-squares,etc.)togetherwiththe

size value. An effort in this direction (the reduction of the

ambiguitywhenmovingfromspeci_ficationtoveri_fication)hasalso beenundertakenwiththedefinitionoftheskinmodel[13,258,257]

andoftheGeoSpellinglanguage[32,201,202]proposestosolvethis

problemthrough a revision of the STEPstandard. Examples of

integrateddefinitionofthetoleranceandverificationhavebeen proposedinthefieldofgears[46,75,78,80,298].

4.5. Observations

Fig. 14 summarizes this chapter. The main objective of

specification model is to provide a language to limit the

manufacturing imperfections (aleatory uncertainty). These

lan-guages or models are affected by ambiguity. To reduce this

ambiguity,newstandardsandmodelshavebeendeveloped,but

their complexity increases the epistemic uncertainty (lack of

knowledge).Ingeneral,designersarenotfamiliarwithallofthe newconceptsthatappearinthestandards.Oneofthechallengesis

tofindacompromisebetweentheambiguityandtheepistemic

uncertaintyduetothespecificationmodelsandlanguages.

5. Toleranceanalysis—uncertainties

Inthischapter,toleranceanalysisandvariationsimulationis

describedfromthreemainperspectives:(1)tolerancemodelsfor

representing the geometrical deviations on individual parts

(Section5.1),(2) Systembehaviormodels,forrepresentinghow

variationpropagatesinaproductoranassembly(Section5.2).(3) toleranceandvariationanalysistechniques(Section5.3).Section

5.4discussandsummarizesthedifferentuncertaintiesinvolvedin toleranceanalysisusingtheclassificationgiveninChapter2. 5.1. Tolerancemodels

Tolerancemodelsarethefirststeptowardstranslationofthe

functional requirements and geometric relations in form of

quantifiable mathematical expression. A significant amount of

researcheffortshasbeencarriedoutinthelastdecadetoexplorethe

mathematical models for geometric deviation representation:

variationalgeometryapproach,skinmodelshape,modal

represen-tation,andothers.Amongthemostcommonlyusedarevariational

geometryapproaches. Inthese approaches, theformdefects are

neglected.Theseapproachesarebasedontheparameterizationof

deviationsfromtheoreticgeometry.Therealgeometryofpartsis consideredbyavariationofnominaldimensionoritisboundedbya

variation (position and orientation) of the nominal geometry

[135].Theorientationandpositiondeviationsofeachsurfacecould berepresentedbyTTRS[6,36,39,66,67,91],kinematicformulation

[101,122,197,245,250], small displacement torsor (SDT)

[42,92,127,128,288], matrix representation[120,142], or vectorial

tolerancing [123,139,319]. In the TTRS model, any part can be

representedasasuccessionofbinarysurfacesassociationsforminga tree.Additionally,eachsurfaceassociation,termedasaTTRSobject,

is representedbya setof minimumgeometricdatum elements

(MGDE). Once established, each TTRS can be givenappropriate

Fig.14. Mappingbetweenthe main conceptsof specification model andthe uncertainties.

geometricdimensioningandtolerancing(GD&T)symbolsthrougha general procedure making useof GD&T tables andcombinationrules.

Thirteen different constraints for dimensioning and tolerancing

wereproposedbyCleméntetal.[67].

Thegeometricordimensioningtolerancesarerepresentedby

deviationdomain[120,127,129,180],Tolerance-Map1[85,86,220]

orspecificationhull[76,79,81,246,247,336].Thesethreeconcepts

area hypotheticalEuclidean volumein parameterspacewhich

representspossibledeviationsinthesize,orientationandposition

of features. The T-Map point-space corresponds to possible

locations and variations of a feature which can arise from

toleranceson size,form, and orientation ona part.The model

hasbeenappliedtotheASMEand ISOStandardsfor geometric

tolerances[3,38,85,86,220,268].

TheTTRSmodel,theDeviationDomainmodelandtheT-map

model were analyzed and compared by Ameta et al. [6]. The

methodsarequitesimilarintheiraim,usingdifferent

mathemati-cal approaches. No model is fully complete when it comes to

representingthetolerancingstandardbutprovideagoodsupport

fortoleranceanalysis.Specifically,modellingofformerrorsisnot possible.

Often,inearlyconcept phases,beforeany physicalpartsare

manufactured,the exact variation behavior of the parts is not

known. However, typical manufacturing behavior or expected

formerrorsneedstobeincludedinthetoleranceanalysis.Methods

tomodelpartvariationbysuperposingdifferentvariationmodes

hadbeenproposedbyChaseetal.[15,40,49,57,58,84,143,147,109, 205,242,258].Manufacturingsignaturefortoleranceanalysiswas

addressedinRef.[234].Theskinmodel,proposedbyBalluand

Mathieu [30], is a comprehensive framework that includes

position,orientationandformdefects.Themodalrepresentation

methodof geometricaldeviationdecompositionhasextensively

beenstudied.HuangandCeglarek[147]proposed

discrete-cosine-transformation(DCT)basedondecompositionmethodfor form

defects modeling. Samper et al. [251] developed the Discrete

Modal Decomposition (DMD) considering modal shapes of a

discretizedfeature.Usually,thetechnicalinterpretationofthese modalrepresentationsisnoteasilyachieved.

To captureand model partvariation, Designof Experiments

(DOE)incombinationwithmanufacturingsimulation(stamping,

molding, forging) and principal component analysis (PCA) has

been proposed in Ref. [198,192]. Methods based on morphing

technologiesandinspectiondatafromsimilarprojectshavebeen

proposedinRef.[300].Thedescriptionof partvariationisused

togetherwithMonteCarlosimulationin theassemblyvariation

simulation.Allofthesemethodsuseasimpli_fiedtriangularformat (VRML,STL,orFEAmeshes)torepresentpartgeometries,similarto theskinmodelapproach.Allofthesemodelsareaffectedbymodel

uncertainty as it is not possible to model and identify all

manufacturingimprecisions.

5.2. Systembehaviormodels

Fig. 15 shows a typical tolerance analysis situation where

tolerancesonindividualparts accumulateintotoleranceonthe

assembly.Orconversely,asdescribedinchapter6,theallowable limitsontheassemblymustbeassuredbythelimitsontheparts.

Analyzingthe effect of variation in a product, model, oran

assembly requires establishing relations between the allocated

tolerancesandthecriticalproductdimensionsinformofModels.

Thesemodelsfortoleranceanalysiscanroughlybedividedinto

analyticalmodelsandnumericalsimulationmodels.

5.2.1. Analyticalmodels

Intermsofdegreesoffreedom,mechanismscanbedividedinto

two main categories: iso-constrained mechanisms, and

over-constrainedmechanisms.Giventheirimpactonthemathematical

formulation for the problem of tolerance analysis, a brief

discussionofthesetwotypesisgivenbyBalluetal.[33]: – “Isoconstrainedmechanismsarequiteeasytograsp.Geometrical

deviationswithinsuchproductsdonotleadtoassemblyproblems; thedeviationsareindependentandthedegreesoffreedomcatchthe deviations.When consideringsmall deviations, functional devia-tionsmaybeexpressedbylinearfunctionsofthedeviations.”

– “Considering overconstrained mechanisms is much more

complex.Assemblyproblemsoccurandtheexpressionofthe

functionalclearanceisnomorelinear.Dependingonthevalueof

themanufacturingdeviations:theassemblyisfeasibleornot;

theworst configuration of contactsis not uniquefor a given

functionaldeviation.Foreachoverconstrainedloop,eventson

thedeviationshavetobedetermined:eventsensuringassembly,

eventscorrespondingtothedifferentworst configurations of

contacts.Astherearedifferentconfigurations,theexpressionof thefunctionaldeviationcannotbelinear."

Therefore,isoconstrainedtolerancescanbemodeledinformof explicitanalyticfunctionsofgeneralformY=f(X),wherefisthe response(characteristicsuchasgaporfunctionalcharacteristics)

oftheassembly.

A commonly used method for 3D variation simulation and

tolerance analysis in industry is the so called “point-based

method”. The method is used in many of the commercial

computer-aidedtolerancing(CAT) tools.In thismethod,mating

conditionsbetweenpartsaredescribedbydefiningpoint-based

masterlocatingschemes.Thelocatingschemesdefinecoordinate

systemsthatarealignedduringassembly.Foramasterlocation

scheme,correspondingtoanABCdatumframe,typicallya

point-basedorthogonal3-2-1locatingschemeisused(seeFig.16left).A numberofdifferentlocatingschemesexistandareusedinvarious industrialsituations,seeSöderberg,Lindkvistetal.[277].

Subordi-nate (local) locating schemes can be defined to describe

dependenciesonasinglepart.Tolerancesareappliedasvariation

in the locating points, correspondingto theallowed tolerance,

definedbythetoleranceforthespecificfeature.Typically,holes,

slots, planes and surfaces are used as locating features and

tolerancesaretypicallyposition,flatness,surfaceprofileetc.For non-rigidparts,over-constrainedlocatingschemeswithadditional supportpoints,areused(seeFig.16right).Theconceptutilizes

transformationmatricestocalculatehowvariationpropagatesin

theassembly.Criticaldimensionsintheassembly(objectforthe analysis)suchasposition,clearance,parallelism,angles,etc.are alsoevaluatedfromthesepointlocations.

The point-based method is a straight forward method that captures rotations,translationsandnon-linearityandisoftencombinedwith

Monte Carlo Simulation. The method is not limited to normal

distributiontolerancesbutuseanydistributionorsamplesofdata, Fig.15.Toleranceanalysis.

such as inspection data, as input. Robustness optimization by

optimizingof locator positions has beenpresented in Wang and

Pelinescu[305].Duringmodelling,GD&Tspecificationsarebroken downtovariationinindividualpointsonpartfeatures.Whenusing triangularformatssuchasSTLorVRML(orFEAmeshesfornon-rigid analysis) all points/nodes of the part featurescan beassigned a tolerance.However,usuallyonlypointsonfeaturescontributingto variationpropagation,orfeaturestobestudiedintheanalysis,are assigned a tolerance. Tolerances assigned to points on the same feature areoftendefinedwithadependency.CATtoolssuchasRD&T,VSA,and 3DCSusethisapproachtosupporttheproductdevelopmentprocess

andbridgethegapbetweentolerancingandproductdevelopment

[97,102,185,271,275,276]. Fig.17shows anexample fromvariation simulationinRD&Twherethecolorcodingoftherearlampindicates therobust(blue)andsensitive(red)areasduetovariationinlocators (matingpoints).Thestatisticaldistributioninthefigureinsertshows theexpectedvariationinacriticalproductdimension_–inthiscase

flushbetweentherearwindowandtherearlamp.

In the generalcaseofanalytic formulation,the mathematical

formulationoftoleranceanalysistakesintoaccountthein_fluenceof

geometricaldeviationsonthegeometricalbehaviorofthe

mecha-nismandonthegeometrical productrequirements;allthesephysical

phenomenaaremodeledbyconstraintsontheparameters[74]:

– Compositionrelationsofdisplacementsinthevarious

topologi-calloopsexpressthegeometricalbehaviorofthemechanism.

Theydefinecompatibilityequationsbetweenthedeviationsand

thegaps. The setof compatibilityequations, obtainedbythe

applicationofcompositionrelationtothevariousloops,results

into a system of linear equations. Successful solutionof this

systemofequationsindicatessolution.

– Interface constraints model the assembly constraints. These

constraints characterize non-interference or association

be-tweensubstitutesurfaces,whicharenominallyincontact.These constraintsalsolimitthegapsbetweensubstitutesurfaces.In thecaseofclearancecontact,therelativepositionsofsubstitute

surfaces are constrained technologically by the

non-interfer-ence. The interface constraints result in a system of linear

inequalities.Inthecaseofslippingandfixedcontact,therelative positionsofsubstitutesurfacesareconstrainedtechnologically inagivenconfigurationbymechanicalaction.

– The functional requirements model the core functional

con-straints bylimiting theorientation and thelocation between

surfaces,whichareinfunctionalrelation.Thisrequirementisa conditionontherelativedisplacementsbetweenthesesurfaces. Thisconditionisalsomodelledbyasystemoflinearinequalities.

5.2.2. Numericalsimulationmodels

In somecases,thegeometricaldeviationsimpactsome

non-geometricalfunctionalrequirements.Tosimulatetheinfluencesof

geometrical deviations on these requirements, an analytic

formulation cannot possibly be employed [45]. Instead, it is

necessarytousenumericalsimulationforwhichitispossibleto

compute a value for Y given values of deviations and gaps:

Y=fnumericasimulation(X)orY=fnumericasimulation(X,G).

Formodellingvariationpropagationin assemblieswith

non-rigidparts,finiteelementanalysis(FEA)maybeused.Anassembly stiffnessmatrix,basedonthestiffnessmatrixesfortheindividual parts,describestheresponseintheoutputparameters.FEAisoften

combined with the point-based method and Monte Carlo

Simulation.Themethodallowsover-constrainedlocatingschemes

thatresultinbendingduringassemblyduetovariationinpartsand fixtures.Fig.16(right)showsa17-7-1non-rigidlocatingscheme forabodysideofacar.

To reduce computational time, the method of influence

coefficient(MIC)isusedinmostMCbasedvariationsimulation

approaches [195]. The main idea of MIC is to find a linear

relationship between part deviations and assembly deviations

after spring-back. A sensitivity matrix, calculated using FEA,

describesthatlinearrelationship.Thesensitivitymatrix isthen usedtocalculatetheresponseineachMCiteration.

In non-rigid analysis, mating conditions between parts are

definedbyconstraintsbetweenpartsand/or partsand fixtures.

ThisisnormallydonebyconstrainingnodesintheFEAmodel.Due

topartvariationandbendingduringassembly,newcontactsmay

occurduringthisprocess.Thesecontactsactasnewconstraints

andmustthereforebetakenintoconsideration.Contactmodeling

fornon-rigidanalysiswasaddressedbyDahlströmandLindkvist

[70],Wärmefjordetal.[308],andLindauetal.[193].

Variation simulation for non-rigid sheet metal parts and

assembliesisdescribedinRefs.[193,206,306,307,309].Atolerance

analysismethodologyforsheetmetalassemblybasedonphysical/

functionalmodellingofthefabricationerrorusingthebeam-based

modelwasdescribedinRef.[55].Themodellingmethodincludes

principlesof decouplingautomotive parts intobeam members,

modellingofbeam-to-beamjointgeometry,andidentificationof

part locating points. Modelling variation propagation of

multi-stationassemblysystemswithcompliantpartswasdescribedin

Ref.[48].Theproblemofmodelgrowthinvariationsimulationis discussedandtreatedinRef.[191].

Inmanytypesofassemblies,thejoiningprocessisacontributor tovariationandalinkinthechainfrompartvariationtoassembly

variation. The joining process then needs to be modelled and

includedintheassemblymodel.Fornon-rigidparts,thejoining

sequence is crucial for how variation in the individual parts,

fixtures and welding equipment will affect the final assembly.

Fig.18showsanexamplewherethesametwoparts,withthesame

fixture,arejoinedtogetherusingtwodifferentsequences.Ascan beseen,onesequenceresultsinquitelargedeviation(redarea) whiletheotherdoesnot.Inasense,thelattercanthereforebeseen

asthemorerobustone.Joiningsequenceoptimizationisa

non-linearproblem,andrequirescontactmodelling[308].Therefore,

geneticalgorithmsareoften usedtofindtheoptimalsequence

[310,263]. Furthermore, in Ref. [52] the cycle time is simulta-neouslyoptimizedandinRef.[53]theassemblyfeasibilityof

non-nominal parts is considered. An important aspect is also the

positionvariationoftheweldinggun[274]. Fig.17.VariationsimulationinRD&T[276].