Robust Design

Miks Robust Design?“Lockheed Martin used to spend an average

of 200 work-hours trying to get a part thatcovers the landing gear to fit. For yearsemployees had brainstorming sessions,which resulted in seemingly logicalsolutions. None worked. The statisticaldiscipline of Six Sigma discovered a partthat deviated by one-thousandth of aninch. Now corrected, the company saves$14,000 a jet.”1

1: Firms aim for Six Sigma efficiency; [FIRST Edition] Del Jones. USA TODAY. McLean, Va.: Jul 21, 1998. pg. 01.B

Miks Robust Design?“It will keep the company (Allied Signal) from

having to build an $85 million plant to fillincreasing demand for caprolactam usedto make nylon, a total savings of $30 -$40 million a year.”1

Raytheon figures it spends 25% of eachsales dollar fixing problems when itoperates at four sigma, a lower level ofefficiency. But if it raises its quality andefficiency to Six Sigma, it would reducespending on fixes to 1%.”1

1: Firms aim for Six Sigma efficiency; [FIRST Edition] Del Jones. USA TODAY. McLean, Va.: Jul 21, 1998. pg. 01.B

Miks Robust Design?“The reason to do DFSS is ultimately

financial. It generates shareholder valuebased on delivering customer value in themarketplace. Products developed underthe discipline and rigor of a DFSS-enabled product development processwill generate measurable value againstquantitative business goals and customerrequirements. DFSS helps fulfill thevoice of the business by fulfilling thevoice of the customer.”2

2: Design for Six Sigma in Technology and Product Development, C.M. Creveling, J. L. Slutsky, and D. Antis, Jr.

–Robust Design’i taust• Mis on Robust Design, DFSS, …?• Design for Quality• Robust Design inseneriarvutustes• Näited• Ebatäpsuse allikad• Ebatäpsuse efektid• Deterministliku ja tõenäosusliku lähenemise võrdlus• Tehnoloogiad

Robust DesignÜlevaade

Küsimused– “Is your company interested in lower its warranty costs and increasing

customer satisfaction?”– “Are you having infield failures that you do not understand given your

current analysis and you wish to understand and prevent these in thefuture?”

– “Is your company interested in the lowest total cost of producing aproduct?”

– “Do feel that you are currently over designing, but don’t have an ideaby how much?”

– “Are you currently holding all dimensions to the same tolerance justbecause this is what you have always done?”

– “Are you constantly testing new lots of material and having to rejectsome suppliers or shipments?”

– “Are you paying a premium to get material that meets or exceeds yourexacting specifications?”

– “Are you rejecting too many parts during the final Product Inspectionchecks?”

– “Are you spending too much money on Product Inspection?”– When you come to a financial commitment gate in your development

process, would you feel more secure making a decision if you had datafrom many design alternatives?"

Robust Design’i taustMis on Robust Design, DFSS, jne?

• Uncertainty AnalysisMääratakse muutuvate suuruste mõjutoote toimimisele (keskväärtused,standard hälve, jne)

• Reliability AnalysisMääratakse arvuliselt usaldusväärsus(tõrke tõenäosus, defekte miljoni kohta)

• Robust Design or Design For Six Sigma(DFSS)Optimeeritakse toodet selliselt, et ta eioleks tundlik muutuvate parameetritesuhtes (N. materjal, koormused, …)

• Reliability-based OptimizationToodet optimeeritakse selliselt, etusaldusväärsus oleks maksimaalne võisiis tõrke tõenäosus oleks minimaalne(defektide arv miljoni toote kohta)

Robust Design on sagelisünonüümks terminile“Design for Six Sigma”või “Reliability-basedOptimization”

Robust Design’i taustDesign for Quality

Six Sigma Quality = ainult 3.4 detaili 1’000’000 ei vasta nõuetele

Six Sigma Quality on oma olemuselt tõenäosuslik meetod

Gaussian Distribution

-6 -4 -2 0 2 4 6

Sigma-Value

Area=

FailureProbability

Product is ...

Bad Good

LSL USL

Product is ...

Good BadLSL = Lower

SpecificationLimit

USL = UpperSpecificationLimit

P.S.: Gaussi jaotus ei ole realistlik, kuid annab ideed edasi korrektselt

Six Sigma = Optimeerib tootmis protsessi selliselt, ettoodetakse automaatselt tooteid mis täidavad sixsigma qualityquality nõudeid

Design For Six Sigma = Optimeeritakse toodet selliselt, et toode täidakssix sigma qualityquality nõudeid, s.t. kvaliteetkvaliteet and usal-dusväärsus on kaasatud optimeerimis protsessi

Design for Six Sigma:• Achieve “Designed-In”

quality as opposed toletting customers findout about qualityproblems

• Make informed decisionthat are critical to qualityearly in the developmentprocess

0.1

1

10

100

1000

Research Design DevelopmentPrototypeTests

Production

Product Development Phases

Rel

. Cos

t of D

esig

n C

hang

e

0%

20%

40%

60%

80%

100%

Design For Six Sigma Six Sigma

Deg

ree

of F

redo

m to

affe

ct

the

Prod

uct L

ifetim

e C

osts

Robust Design’i taustDesign for Quality

FROM:Reactive Quality

Management

• Extensive Design Rework• Assess Performance by

“build-test-build-test-…”• Fix performance/quality

problems aftermanufacturing

• Quality is “Tested-In”

TO:Predictive Quality

Management

• Controlled DesignParameters

• Estimate likelihood/rate ofperformance problems indesign & development

• Address quality problems indesign & development

• Designed for robustperformance and quality

• Quality is “Designed-In”

Robust Design’i taustDesign for Quality

Robust Design is a Paradigm Shift …

Purpose of Robust Design

InputInputInput ANALYSISANANALYSIALYSISS OutputOutputOutput

�Material Properties�Geometry�Boundary Conditions

�Deformation�Stresses / Strains�Fatigue, Creep,...

It’s a reality that inputparameters are subjected toscatter => automatically the

output parameters areuncertain as well!!

Robust Design’i taustRobust Design in Engineering Analysis

Questions answered with Robust Design:

• How large is the scatter of the output parameters?• What is the probability that output parameters do not fulfill design

criteria (failure probability – defects per million)?• How much does the scatter of the input parameters contribute to the

scatter of the output (sensitivities – critical-to-quality)?

Robust Design’i taustRobust Design in Engineering Analysis

ANALYSISANALYSISANALYSIS

Purpose of Robust Design

20Vibration loads40Acoustic loads10Deployment shock7.5Thermal loads50Transient loads5Launch vehicle , thrust8Honeycomb, face wrinkling10Honeycomb, shear, compression16Honeycomb, tension12Bond insert, axial load8Junction by screws, rivet, welding14Metallic shells, buckling strength17Carbon fiber composites, rupture15Metallic materiales, yieldSD/Mean %Property

Source: Klein, Schueller et.al. Probabilistic Approach to Structural Factors of Safety in Aerospace.

Proc. CNES Spacecraft Structures and Mechanical Testing Conf., Paris 1994

Robust Design’i taustSources of Uncertainty

CFDFEMCAD

FEMGeometry

Materials,Bound.-Cond.,

Loads, ...

Materials,Bound.-Cond., ... Materials,

Bound.-Cond.,

Loads, ...

LCF

Materials

± 0.1-10%

±5-50%

±5-100%

±30-60%

±??%

±5-100%

ThermalAnalysis

StructuralAnalysis

Robust Design’i taustEffects of Uncertainty

Elastsusmooduli ja termilise joonpaisumisteguri mõju termilistelepingetele:σthermal = E · α ·∆T

Deterministlik lähenemine:σMean = EMean · αMean · ∆T Mean = tavaliselt kasutatav lahend

Tõenäosuslik lähenemine:

Tõenäosus et (σ thermal >= 105% σ Mean) (σ thermal >= 110% σ Mean)‘E’ hajuvus ±5% 16% (~1 out of 6) 2.3% (~1 out of 40)

‘E’ ja ‘α‘ hajuvus ±5% 22% (~1 out of 4) 8% (~1 out of 12)

‘E’, ‘α‘ & ‘∆T’ hajuvus ±5% 28% (~1 out of 4) 13% (~1 out of 8)

Robust Design’i taustEffects of Uncertainty

Turbiini “What-If” analüüsi seeria

Robust Design’i taustCompare Deterministic/Probabilistic

Robust Design’i taustEnabling tehnoloogia: Parametriseerimine

• Robust Design for all parameters including:

– APDL Parameters

/syp,parabatch.exe,'testpb.rsx','testpb.cdb','location',%value%,'testpb_mod.cdb'

/inp,testpb_mod,cdb ! Input the modified geometry

Paramesh db Initial mesh

Parameter name

Parameter value

Output mesh

Import Output mesh

CAD Parameters (Workbench) APDL Parameters

ParaMesh Parameters

Robust Design’i taustVõimaldav tehnoloogia: DesignXplorer

DesignXplorermanages theparameters andthe uncertainties

CAD Geomeetria FEM Mesh FEMrajatingimused

Robust Design’i näide

Results for Maximum Principal StressPressure Side Suction Side

Peak Value σs

Tang.Leaning

AxialLeaning

Dove TailWidth

MaterialDensity

(Gaussian)

Fillet Radius(Lognormal)

Mass

Design Variables andUncertainties

Imbalance: (σp – σs)2 Avg.Stress: 0.5(σp + σs)

Peak Value σp

Robust Design’i näide

Teooria• Keskväärtus

• Standard hälve

• Asümeetria

• Kurtosis

Truncated Gaussian

Gaussian (Normal)

Kõige tavalisem statistiline jaotus

Kasutatakse paljude füüsikalisteparameetrite kirjeldamiseks.

Kõige Gaussi jaotuse puhul, kuid äärmisedpiirid lõigatakse, et elimineeridamõõtmisvigasid

Kasutatakse näiteks materjali omaduste võigeomeetria tolerantside kirjeldamiseks.

Lognormal (option 1)

Samuti laialt levinud ja kasutatav jaotus.

Kasutatakse füüsikaliste suuruste kirjeldamiseks, milliste teatud andmetelogaritmid taanduvad normaaljaotusele.

Kasutatav näiteks väsimuse kirjeldamisel.

You provide values for the mean value µ and the standarddeviation σ of the random variable x. The PDS calculates thelogarithmic mean ξ and the logarithmic deviation δ:

Lognormal (option 2)

You provide values for the logarithmic mean value ξ and thelogarithmic deviation δ. The parameters ξ and δ are the mean valueand standard deviation of ln(x)

Triangular

Uniform

Ühtlane jaotus on fundamentaalne jaotusfunktsioonsellistes olukkordades kus muud info ei ole kättesaadavkui alumine ja ülemine piir. Väga kasulik geomeetriatolerantside kirjeldamiseks. Kasutatakse ka sellisteljuhtudel kui ei ole ühtegi tõendust juhuslike väärtustejaotuse kohta kindlas intervalis. Võidakse kasutada kajuhtudel kui "lack of engineering knowledge“ mängib rolli.

Kasutatakse juhuslike suuruste kirjeldamiseks kuiandmeid ei ole olemas. Väga tihti kasutatakseeksperthinnangute kirjeldamisel matemaatilises mudelisja ka koormuste kirjeldamisel. Olenemata probleemist onekspertidelt võimalik küsida näiteks "What is the one-in-a-thousand minimum and maximum case for this randomvariable? Või sarnaseid küsimusi.

Exponential The distribution is mostly used to describe time-related effects; for example, it describes the timebetween independent events occurring at a constantrate. It is therefore very popular in the area ofsystems reliability and lifetime-related systemsreliability, and it can be used for the life distribution ofnon-redundant systems. Typically, it is used if thelifetime is not subjected to wear-out and the failurerate is constant with time. Wear-out is usually adominant life-limiting factor for mechanicalcomponents, which would preclude the use of theexponential distribution for mechanical parts.However in cases where preventive maintenanceexchanges parts before wear-out can occur, then theexponential distribution is still useful to describe thedistribution of the time until exchanging the part isnecessary

Beta The Beta distribution is very useful for randomvariables that are bounded at both sides. If linearoperations are performed on random variables that areall subjected to a uniform distribution then the resultscan usually be described by a Beta distribution. Anexample is if you are dealing with tolerances andassemblies, where the components are assembledand the individual tolerances of the components followa uniform distribution. In this case the overalltolerances of the assembly are a function of adding orsubtracting the geometrical extension of the individualcomponents (a linear operation). Hence, the overalltolerances of the assembly can be described by a Betadistribution. Also, as previously mentioned, the Betadistribution can be useful for describing the scatter ofindividual geometrical extensions of components aswell. The uniform distribution is a special case of theBeta distribution

Gamma

Weibull

The Gamma distribution is again a more time-relateddistribution function. For example it describes thedistribution of the time required for exactly k events tooccur under the assumption that the events take place at aconstant rate. It is also used to describe the time to failurefor a system with standby components.

In engineering, the Weibull distribution is most oftenused for strength or strength-related lifetimeparameters, and it is the standard distribution formaterial strength and lifetime parameters for verybrittle materials (for these very brittle material the"weakest-link-theory" is applicable).

Monte Carlo simulatsioon

Direct Sampling Latin Hypercube Sampling

Matemaatiline mudel sisaldab juhuslikke sündmusi, juhuslikkesuurusi või nende arvkarakteristikuid (näiteks keskväärtust,dispersiooni). Sellise mudeli koostamist nimetataksestatistiliseks modelleerimisek laias tähenduses või MonteCarlo meetodi rakendamiseks laias tähenduses

Monte Carlo simulatsioon

Direct Sampling Latin Hypercube Sampling

The method is always applicable regardless of the physical effect modeled in a finiteelement analysis. It not based on assumptions related to the random outputparameters that if satisfied would speed things up and if violated would invalidate theresults of the probabilistic analysis. Assuming the deterministic model is correct anda very large number of simulation loops are performed, then Monte Carlo techniquesalways provide correct probabilistic results. Of course, it is not feasible to run aninfinite number of simulation loops; therefore, the only assumption here is that thelimited number of simulation loops is statistically representative and sufficient for theprobabilistic results that are evaluated. This assumption can be verified using theconfidence limits, which the PDS also provides.

Because of the reason mentioned above, Monte Carlo Simulations are the onlyprobabilistic methods suitable for benchmarking and validation purposes.

The individual simulation loops are inherently independent; the individual simulationloops do not depend on the results of any other simulation loops. This makes MonteCarlo Simulation techniques ideal candidates for parallel processing.

Monte Carlo simulatsioonDirect Sampling Latin Hypercube Sampling

Meetodil ei ole mälu 20% kuni 40% vähem katseid