Overview of Reliability Analysis and Design Capabilities in DAKOTAMichael S. EldredSandia National LaboratoriesBarron J. BichonVanderbilt UniversityBrian M. AdamsSandia National Laboratories

http://endo.sandia.gov/DAKOTASandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000.

IntroductionUncertainties/variabilities must be properly modeled in order to quantify risk and design systems that are robust and reliableUncertainty comes in different flavors:Aleatory/irreducible: inherent uncertainty/variability with sufficient data probabilistic modelsEpistemic/reducible: uncertainty from lack of knowledge nonprobabilistic modelsWe use a UQ-based approach to optimization under uncertainty (OUU)safety factors, multiple operating conditions, or local sensitivities are insufficientTailor OUU methods to strengths of different UQ approachesOUU methods encompass both:design for robustness (moment statistics: mean, variance)design for reliability (tail statistics: probability of failure)

Introduction (cont.)Focus is simulation-based engineering applicationsmostly PDE-based, often transient response mappings are nonlinear and implicitdistinct from chance-constrained stochastic programming (often linear/explicit)Standard NLP:

Minimizef(d)Subject to gl g(d) guh(d) = htdl d duAugment with general response statistics su (e.g., m, s, z/b/p) using a linear mapping:

Minimizef(d) + Wsu(d)Subject to gl g(d) guh(d) = htal Aisu(d) auAesu(d) = atdl d du

[Nonlinear mappings (s2, s/m, etc.) via AMPL]

OutlineIntroductionAlgorithmsDAKOTA softwareUncertainty quantificationOverviewReliability methodsSample benchmark resultsOptimization under uncertaintyOverviewRBDO methodsSample benchmark resultsShape Optimization of Compliant MEMSBi-stable switchRF switchConcluding Remarks

DAKOTA OverviewGoal: answer fundamental engineering questionsWhat is the best design?How safe is it?How much confidence in my answer?Technical themesLarge-scale optimization, UQ, and V&VAlgorithms for complex engineering applications: SBO, OUUBalance algorithm research with production demandsImpactInternal: Large-scale ASC applics., multiphysics framework deploymentExternal: Tri-lab, WFO, GPL open source (~3000 download registrations)NominalOptimized

Iterator Model Strategy: control of multiple iterators and modelsIterator Model Iterator Model Coordination:NestedLayeredCascadedConcurrentAdaptive/InteractiveParallelism:Asynchronous localMessage passingHybrid4 nested levels with Master-slave/dynamic Peer/staticDAKOTA FrameworkParametersModel:Design continuous discreteUncertain normal/logn uniform/logu triangular beta/gamma EV I, II, III histogram intervalState continuous discreteApplication system fork direct gridApproximation global polynomial 1/2/3, NN, kriging, MARS, RBF multipoint TANA3 local Taylor series hierarchicalFunctionsobjectivesconstraintsleast sq. termsgenericInterfaceHybridSurrBasedOptUnderUncBranch&Bound/PICOStrategyOptimizationUncertainty2ndOrderProbUncOfOptimaLHS/MCIterator OptimizerParamStudyCOLINYNPSOLDOTOPT++LeastSqDoEGNVectorMultiDListDDACECCD/BBUQReliabilityDSTEJEGAPareto/MStartCONMINNLSSOLNL2SOLQMC/CVTGradientsnumerical analyticHessiansnumerical analytic quasiNLPQLCenterSFEM

Uncertainty QuantificationActive UQ development (new, developing, planned).Sampling:LHS/MC, QMC/CVT, Bootstrap/Importance/Jackknife. Gunzburger collaboration.Reliability:MVFOSM, x/u AMV, x/u AMV+, FORM (RIA/PMA mappings), MVSOSM, x/u AMV2, x/u AMV2+, TANA, SORM (RIA/PMA) Renaud/Mahadevan collaborations.SFE: Polynomial chaos expansions (quadrature/cubiture extensions). Ghanem (Walters) collaborations.Metrics:Importance factors, partial correlations, main effects, and variance-based decomposition.Epistemic:2nd-order probability, Dempster-Schafer, Bayesian.Uncertainty applications: penetration, joint mechanics, abnormal environments, shock physics, LHS/MCUQSFEMReliabilityDSTE

Epistemic UQSecond-order probabilityTwo levels: distributions/intervals on distribution parametersOuter level can be epistemic (e.g., interval)Inner level can be aleatory (probability distrs)Strong regulatory history (NRC, WIPP).

Dempster-Schafer theory of evidenceBasic probability assignment (interval-based)Solve opt. problems (currently sampling-based) to compute belief/plausibility for output intervals

[Also, could do basic black-box response interval estimation with min/max global optimization]20052006

UQ with Reliability MethodsMean Value MethodSurprisingly popular with analysts

Reliability Algorithm Variations:First-Order MethodsLimit state linearizationsIntegrations1st-order:Warm startingWhen: AMV+ iteration increment, z/p/b level increment, or design variable changeWhat: linearization point & assoc. responses (AMV+) and MPP search initial guessPMA Projection:RIA Projection:MPP initial guess benefits from projection since KKT conditions w.r.t. u still satisfied for new level at previous optimumMPP search algorithm[HL-RF], Sequential Quadratic Prog. (SQP), Nonlinear Interior Point (NIP)Eldred, M.S., Agarwal, H., Perez, V.M., Wojtkiewicz, S.F., Jr., and Renaud, J.E., Investigation of Reliability Method Formulations in DAKOTA/UQ, (to appear) Structure and Infrastructure Engineering, Taylor & Francis.

Reliability Algorithm Variations:Second-Order Methods2nd-order integrationscurvature correctionAlso, AIS, Synergistic features:Hessian data needed for SORM integration can enable more rapid MPP convergence[QN] Hessian data accumulated during MPP search can enable more accurate probability estimatesEldred, M.S. and Bichon, B.J., Second-Order Reliability Formulations in DAKOTA/UQ, (in review) Structure and Infrastructure Engineering, special issue on uncertainty in aerospace systems, Taylor & Francis.

Reliability Algorithm Variations:Sample of Results to DateAnalytic benchmark test problems: lognormal ratio, short column, cantileverNote: 2nd-order PMA with prescribed p level is harder problem requires b(p) update/inversionLimit state surrogate approaches are significantly more effective than general purpose optimizers (which may use internal linear/quadratic approxs.)Best (practical) performer to date:AMV2+ (with SR1 Hessian updates)More efficient (RIA, PMA) and more robust (PMA with 2nd-order p levels)43 z levels43 p levels

OutlineIntroductionAlgorithmsDAKOTA softwareUncertainty quantificationOverviewReliability methodsSample benchmark resultsOptimization under uncertaintyOverviewRBDO methodsSample benchmark resultsShape Optimization of Compliant MEMSBi-stable switchRF switchConcluding Remarks

Optimization Under Uncertainty

OUU techniques categorized based on UQ approach:Sampling-based (noise-tolerant opt.; design for robustness)TR-SBOUU: trust region surrogate-basedNongradient-based (Trosset)Robust design of experiments (Taguchi)Reliability-based (exploit structure; design for reliability)Bi-level RBDO (nested)Sequential RBDO (iterative)Unilevel RBDO (all at once)Stochastic finite element-based (multiphysics)Exploit PCE coeffs & random process structureLeverage SBO with global surrogatesEpistemic uncertaintyEvidence theory-based (Agarwal)Bayesian inference: model calibration under uncertainty2nd-order probability: 3-level OUU approachesIntrusive OUU (all at once approaches)SFE + SAND: intrusive PCE variant amenable to SANDUnilevel RBDO + SANDAugment NLP with response statistics su (m, s, p/b/z) using a linear mapping:

Minimizef(d) + Wsu(d)Subject to gl g(d) guh(d) = htal Aisu(d) auAesu(d) = atdl d du20032004 20052006

RBDO Algorithms

RBDO Algorithm Variations:Sample of Results to DateShort Column min bh s.t. b > 2.5P = N(500, 100)rP,M = 0.5M = N(2000, 400)bnom = 5Y = LogN(5, 0.5)hnom = 15Kuschel & Rackwitz, 1997Analytic benchmark test problems: short column, cantilever, steel column

OUU Progress To Date2003:Surrogate-based OUU with sampling methods2004:Bi-level RBDO with numerical reliability gradients2005: Fully analytic bi-level RBDO Sequential/surrogate-based RBDO (1st-order, 2nd-order)BLFA-BLS1S2OUU/SBOUURBDO

Chart5

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Bi-level OUU

TR-SBOUU2

TR-SBOUU4

Bi-level RBDO

Sequential RBDO

Total Resp Fn Evals

OUU Performance vs. Time - Cantilever Problem

Sheet1

DateTotal Fn EvalsTotal Grad EvalsTotal Fn EvalsTotal Grad EvalsTotal Fn EvalsTotal Grad EvalsTotal Fn EvalsTotal Grad EvalsTotal Fn EvalsTotal Grad EvalsAlg

Sep-029750138001260OUU1

Feb-039750106501008TR-SBOUU2

Jul-0382506450882TR-SBOUU4

Jun-04544544Bi-level numerical RBDO

Jan-05235259Bi-level analytic RBDO

Mar-05200227TR-SB-RBDO (1st-order)

Oct-056885TR-SB-RBDO (2nd-order)

Sheet1

Bi-level OUU

TR-SBOUU2

TR-SBOUU4

Bi-level RBDO

Sequential RBDO

Total Resp Fn Evals

OUU Performance vs. Time - Cantilever Problem

Sheet2

Sheet3

OutlineIntroductionAlgorithmsDAKOTA softwareUncertainty quantificationOverviewReliability methodsSample benchmark resultsOptimization under uncertaintyOverviewRBDO methodsSample benchmark resultsShape Optimization of Compliant MEMSBi-stable switchRF switchConcluding Remarks

Engineering Application Deployment:Shape Optimization of Compliant MEMS MEMS designs are subject to substantial variabilities and lack historical knowledge baseSources of uncertaintyMaterial properties, manufactured geometries, residual stressesData can be obtained aleatoric uncertainty, probabilistic approachesResulting part yields can be low or have poor cycle durabilityGoals:Achieve prescribed reliabilityMinimize sensitivity to uncertainties (robustness)Nonlinear FE simulations~20 min. desktop simulation expense (SIERRA codes: Adagio, Aria, Andante)Remeshing with FASTQ/CUBIT or smooth mesh movement with DDRIV(semi-analytic) p/b/z gradients appear to be reliableRF MEMS SwitchBi-stable MEMS Switch

Bi-Stable Switch: Problem Formulation13 design vars: Wi, Li, qi2 random vars: reliable + robust

Bi-Stable Switch: ResultsReliability: target achieved for AMV+/FORM; target approximated for MVRobustness: variability in Fmin reduced from 5.7 to 4.6 mN per input s [mFmin/b]Continuing: quantity of interest error estimates error-corrected UQ/RBDOMVFOSM-based RBDOAMV+/FORM-based RBDO

Bi-Stable Switch: Observed ChallengesMVFOSM-based RBDO d*Problematic dIn general, need support of nonlinear/nonsmooth/multimodal limit states and careful attention to problem formulation (e.g., supporting simulation requirements)Smooth, linear limit state (in range of interest)First-order integration OKNonlinear limit stateHigher-order integration neededNonsmooth due to sim failures in left tail of edge bias (flimsy structure)

Conclusions & Future DirectionsGeneral UQ/OUU Aspects:Nonlinear, large-scale, expensive simulations w/ implicit, noisy response metricsAleatory and epistemic uncertaintiesDesign for robustness and reliabilityReliability Analysis:Explored limit state approxs, integrations, MPP search algs, Hessian approxs, warm startingNew algorithms for 2nd-order PMA, AMV2/AMV2+, and QN integrationsRecommendation for AMV2+ (with SR1 limit state Hessian updates)Future work: noisy limit states, multiple MPPs global surrogates, TR approaches, global opt.RBDO Algorithms:Explored bi-level and sequential methods exploiting probabilistic sensitivitiesNew sequential RBDO approaches employing TRs and 2nd-order local approxs.Recommendation for 2nd-order sequential RBDO (with SR1 metric Hessian updates)Future work:enhancements to input distr. parameterization and output distr. characterizationmodel calibration/inversion under uncertainty (matching output CDFs)explore TR linkages between MPP and design levels

Extra Slides

Optimization with Surrogate ModelsPurpose:Reduce the number of expensive, high-fidelity simulations by using a succession of approximate (surrogate) modelsApproximations generally have a limited range of validityTrust regions adaptively manage this range based on efficacy during optWith trust region globalization and local 1st-order consistency, SBO algorithms are provably-convergentSurrogate models of interest:Data fitsMultifidelity (special case: multigrid optimization)Reduced-order modelsFuture connections to multi-scale for managing approximated scales

Trust-Region Surrogate-Based Optimization

SBOUU FormulationsFor surrogate-based OUU, the surrogate can appearat the optimization level (fit S(d))at the UQ level (fit R(d, u))at both levels (fit S(d) and R(d, u))Surrogate can belocal/global/multipoint data fit (either level)model hierarchy approximation (UQ level only)

Optimization under Uncertainty with SurrogatesFormulations 2 & 4 amenable to trust-region approachesGoals: maintain quality of results, provable convergence (for a selected confidence level)Formulation 1: NestedFormulation 2: Surrogate containing NestedFormulation 3: Nested containing SurrogateFormulation 4: Surrogate containing Nested containing Surrogate

TR-SBOUU ResultsDirect nested OUU is expensive and requires seed reuseSBOUU expense much lower (up to 100x), but unreliable.TR-SBOUU maintains quality of results and reduces expense ~10xEx. 1: formulation 4 with TR 5-7x less expensive than direct nestingEx. 2: formulation 4 with TR 8-12x less expensive than direct nestingICF Ex.: formulations 2/4 with TR locate vicinity of a min in a single cycleAdditional benefits:Navigation of nonsmooth engineering problemsLess sensitive to seed reuse: variable patterns OK and often helpful, possibility of exploitation reducedLess sensitive to starting point: data fit SBO provides some global ident.Minimizef + pfail_r1 + pfail_r3Subject to gi 0, for i = 1,2,3mr2 + 3sr2 1.6e5Conference papers at AIAA MA&O, SIAM CS&E, USNCCM:Eldred, M.S., Giunta, A.A., Wojtkiewicz, S.F., Jr., and Trucano, T.G., "Formulations for Surrogate-Based Optimization Under Uncertainty."

Trust Region Surrogate-Based Optimization under Uncertainty (TR-SBOUU)From SBO to SBOUU:SBO is provably convergent with TR globalization(at least) 1st order consistency with correctionverification of approx. steps

Extensions to SBOUU1st order consistency, assuming a worthwhile stoch. gradientverification of stats. in relative sense. Three levels of verification rigor:Least: nominal statistics.Most: ordinal opt. (Chen/Romero) nonoverlap confidence bounds on every step (provable convergence for a selected confidence level).

Affordable compromise: stochastic approximation (Igusa) probability of erroneous TR steps is decreased in proportion to iteration count.Sequence of trust regions

Intrusive OUU: DAKOTA/MOOCHO w/ SIERRA/NEVADANext-generation multi-physics simulation architectures:SIERRA: mechanics framework (S. DAKOTA)NEVADA: physics framework (N. DAKOTA)Architecture extensions underway for:Opt.: SAND optimization (MOOCHO)UQ: (intrusive) stochastic finite elementsOther: Stability analysis (LOCA), Nonlinear equations (NOX), Fully-coupled MDA Impact:Performance enhancements for existing nested methodsModel I/O in core in parallelSPMD execution on compute nodes of ASCI MPPsNext-generation, tightly-coupled opt. & UQDirect/adjoint sensitivities & ADIntrusive OUU:SFE + SANDUnilevel RBDO + SAND

RBDO ResultsWu et al., 20012 design vars: w, t4 uncorr. normal uncertain vars: E, R, X, YCantilever min wt s.t. bD, bS > 3Limit state eqns (unnormalized):

Concluding RemarksDAKOTA/UQ provides a flexible object-oriented framework for investigating algorithmic variations in reliability analysis and RBDO

Novel aspects:Full CDF/CCDFLimit state linearizations in PMAWarm starting by projection (avoids premature convergence)SQP vs. NIPUQ Performance:Relative to FORM:MV 2 orders of magnitude reduction but only accurate near meansAMV 1 order of magnitude reduction but MPP not convergedAMV+ ~3x reduction with full accuracyWarm starts: significant win, as expectedNIP vs. SQP: mixed results, but NIP promising (can avoid u-space excursions)x-space vs. u-space: mixed results, application dependentRBDO Performance:MV/AMV RBDO is cheap and gets in the solution vicinityRIA zb RBDO: preferred to RIA zp RBDO, slight advantage over PMA RBDOAMV+ RBDO 3.5x more efficient than FORM RBDOGreat for rough stats

Quick transition: DAKOTA/UQ continues to be a highly active area and a big programmatic driver.Have already touched on new 2nd-order reliability methods, will highlight new epistemic methodsROM area has some attractive advantages, in that it is more physics-based than a regression-based data fit, and does not require multiple computational models or meshes which are not always available/practical. But a key issue that must be solved is capturing of parameter changes in the ROM.TO DO: add ROM case to this