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Overview of Reliability Analysis and Design Capabilities in DAKOTA

Overview of Reliability Analysis and Design Capabilities in DAKOTA. Michael S. Eldred Sandia National Laboratories Barron J. Bichon Vanderbilt University Brian M. Adams Sandia National Laboratories http://endo.sandia.gov/DAKOTA.

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Overview of Reliability Analysis and Design Capabilities in DAKOTA

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  1. Overview of Reliability Analysis and Design Capabilities in DAKOTA Michael S. Eldred Sandia National Laboratories Barron J. Bichon Vanderbilt University Brian M. Adams Sandia National Laboratories http://endo.sandia.gov/DAKOTA Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,for the United States Department of Energy under contract DE-AC04-94AL85000.

  2. Harder problem Harder problem Introduction Uncertainties/variabilities must be properly modeled in order to quantify risk and design systems that are robust and reliable Uncertainty comes in different flavors: • Aleatory/irreducible: inherent uncertainty/variability with sufficient data probabilistic models • Epistemic/reducible: uncertainty from lack of knowledge nonprobabilistic models We use a UQ-based approach to optimization under uncertainty (OUU) • safety factors, multiple operating conditions, or local sensitivities are insufficient • Tailor OUU methods to strengths of different UQ approaches OUU methods encompass both: • design for robustness (moment statistics: mean, variance) • design for reliability (tail statistics: probability of failure)

  3. Introduction (cont.) Focus is simulation-based engineering applications • mostly PDE-based, often transient • response mappings are nonlinear and implicit • distinct from chance-constrained stochastic programming(often linear/explicit) Augment with general response statistics su(e.g.,m, s, z/b/p) using a linear mapping: Minimize f(d)+ Wsu(d) Subject to glg(d)gu h(d)=ht alAisu(d)au Aesu(d)=at dlddu [Nonlinear mappings (s2, s/m, etc.) via AMPL] Standard NLP: Minimize f(d) Subject to glg(d)gu h(d)=ht dlddu

  4. G(u) Outline Introduction Algorithms • DAKOTA software • Uncertainty quantification • Overview • Reliability methods • Sample benchmark results • Optimization under uncertainty • Overview • RBDO methods • Sample benchmark results Shape Optimization of Compliant MEMS • Bi-stable switch • RF switch Concluding Remarks

  5. Black Box: Sandia simulation codes Commercial simulation codes Semi-instrusive: SIERRA (multiphysics), SALINAS (structural dynamics), Xyce (circuits), Sage (CFD), MATLAB, Mathematica, ModelCenter, FIPER DAKOTA Optimization Uncertainty Quant. Parameter Est. Sensitivity Analysis Model Parameters Design Metrics DAKOTA Overview Goal: answer fundamental engineering questions • What is the best design? • How safe is it? • How much confidence in my answer? Technical themes • Large-scale optimization, UQ, and V&V • Algorithms for complex engineering applications: SBO, OUU • Balance algorithm research with production demands Impact • Internal: Large-scale ASC applics., multiphysics framework deployment • External: Tri-lab, WFO, GPL open source (~3000 download registrations) Optimized Nominal

  6. Parameters Responses DAKOTA Framework Model: Iterator Parameters Interface LeastSq DoE Design continuous discrete Functions Application system fork directgrid NLSSOL GN DDACE CCD/BB objectives constraints NL2SOL QMC/CVT Uncertain normal/logn uniform/logu triangular beta/gamma EV I, II, III histogram interval least sq. terms ParamStudy generic UQ Optimizer Gradients Approximationglobal polynomial 1/2/3, NN, kriging, MARS, RBFmultipoint – TANA3local – Taylor series hierarchical DSTE LHS/MC Vector List numericalanalytic SFEM Reliability Center MultiD Hessians numericalanalyticquasi State continuous discrete NLPQL DOT CONMIN NPSOL OPT++ COLINY JEGA Strategy: control of multiple iterators and models Coordination: Strategy Iterator Nested Layered Cascaded Concurrent Adaptive/Interactive Model Optimization Uncertainty Iterator Hybrid Parallelism: OptUnderUnc Asynchronous local Message passing Hybrid 4 nested levels with Master-slave/dynamic Peer/static Model SurrBased UncOfOptima Iterator Pareto/MStart 2ndOrderProb Branch&Bound/PICO Model

  7. d UQ Uncertainty Quantification DSTE LHS/MC SFEM Reliability Active 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, …

  8. Epistemic UQ Second-order probability • Two levels: distributions/intervals on distribution parameters • Outer level can be epistemic (e.g., interval) • Inner level can be aleatory (probability distrs) • Strong regulatory history (NRC, WIPP). Dempster-Schafer theory of evidence • Basic 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] 2005 2006

  9. MPP search methods G(u) Reliability Index Approach (RIA) Performance MeasureApproach (PMA) Find min dist to G level curve Used for fwd map z p/b Find min G at b radius Better for inv map p/b z Variations: MPP search alg, Approximations, Integrations, Warm starting … UQ with Reliability Methods Mean Value Method Surprisingly popular with analysts

  10. AMV: u-space AMV: AMV+: u-space AMV+: FORM: no linearization Reliability Algorithm Variations:First-Order Methods Limit state linearizations Integrations 1st-order: MPP search algorithm [HL-RF], Sequential Quadratic Prog. (SQP), Nonlinear Interior Point (NIP) Warm starting When: AMV+ iteration increment, z/p/b level increment, or design variable change What: linearization point & assoc. responses (AMV+) and MPP search initial guess PMA Projection: RIA Projection: MPP initial guess benefits from projection since KKT conditions w.r.t. u still satisfied for new level at previous optimum 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.

  11. 2nd-order local limit state approximations • e.g., x-space AMV2+: • Hessians may be full/FD/Quasi • Quasi-Newton Hessians may be BFGS or SR1 Failureregion • Multipoint limit state approximations • e.g., TPEA, TANA: G(u) Reliability Algorithm Variations:Second-Order Methods Synergistic features: Hessian data needed for SORM integration can enablemore rapid MPP convergence [QN] Hessian data accumulated during MPP search can enable more accurate probability estimates 2nd-order integrations Also, AIS, … curvature correction Eldred, 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.

  12. Reliability Algorithm Variations:Sample of Results to Date Analytic benchmark test problems: lognormal ratio, short column, cantilever 43 p levels 43 z levels • Limit 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) Note: 2nd-order PMA with prescribed p level is harder problem  requires b(p) update/inversion

  13. G(u) Outline Introduction Algorithms • DAKOTA software • Uncertainty quantification • Overview • Reliability methods • Sample benchmark results • Optimization under uncertainty • Overview • RBDO methods • Sample benchmark results Shape Optimization of Compliant MEMS • Bi-stable switch • RF switch Concluding Remarks

  14. Opt {d} {S } u Data Fit {d} {S } u UQ {u} {R } u Data Fit/Hier {d} {R } u {u} Sim Optimization Under Uncertainty OUU techniques categorized based on UQ approach: • Sampling-based (noise-tolerant opt.; design for robustness) • TR-SBOUU: trust region surrogate-based • Nongradient-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 structure • Leverage SBO with global surrogates • Epistemic uncertainty • Evidence theory-based (Agarwal) • Bayesian inference: model calibration under uncertainty • 2nd-order probability: 3-level OUU approaches • Intrusive OUU (all at once approaches) • SFE + SAND: intrusive PCE variant amenable to SAND • Unilevel RBDO + SAND 2003 20042005 Augment NLP with response statistics su(m, s, p/b/z) using a linear mapping: Minimize f(d)+ Wsu(d) Subject to glg(d)gu h(d)=ht alAisu(d)au Aesu(d)=at dlddu 2006

  15. Bi-level RBDO • Constrain RIA z p/b result • Constrain PMA p/b z result PMARBDO RIARBDO • Fully analytic Bi-level RBDO • Analytic reliability sensitivitiesavoid numerical differencing at design level If d = distr param, then expand (1st order) • Sequential/Surrogate-based RBDO: • Break nesting: iterate between opt & UQ until target is met.Trust-region surrogate-based approach is non-heuristic. 1st-order (also 2nd-order, …) • Unilevel RBDO: • All at once: apply KKT conditions of MPP search as equality constraints • Opt. increases in scale (d,u) • Requires 2nd-order info for derivatives of 1st-order KKT KKT of MPP RBDO Algorithms

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

  17. With tuning of initial TR size,3 RBDO benchmarks solved in ~40 fn evals per limit state: • 35 for 1 limit state in short column • 75 for 2 limit states in cantilever • 45 for 1 limit state in steel column Two orders of magnitude improvement (so far) over “brute force” OUU OUU Progress To Date • 2003: Surrogate-based OUU with sampling methods • 2004: Bi-level RBDO with numerical reliability gradients • 2005: Fully analytic bi-level RBDOSequential/surrogate-based RBDO (1st-order, 2nd-order) OUU/SBOUU RBDO BL S1 FA-BL S2

  18. G(u) Outline Introduction Algorithms • DAKOTA software • Uncertainty quantification • Overview • Reliability methods • Sample benchmark results • Optimization under uncertainty • Overview • RBDO methods • Sample benchmark results Shape Optimization of Compliant MEMS • Bi-stable switch • RF switch Concluding Remarks

  19. Engineering Application Deployment:Shape Optimization of Compliant MEMS • MEMS designs are subject to substantial variabilities and lack historical knowledge base • Sources of uncertainty • Material properties, manufactured geometries, residual stresses • Data can be obtained  aleatoric uncertainty, probabilistic approaches • Resulting part yields can be low or have poor cycle durability • Goals: • Achieve prescribed reliability • Minimize 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 reliable Bi-stable MEMS Switch RF MEMS Switch

  20. m) m) m) m) Bi-Stable Switch: Problem Formulation 13 design vars: Wi, Li, qi 2 random vars:  reliable + robust

  21. Bi-Stable Switch: Results MVFOSM-based RBDO AMV+/FORM-based RBDO Reliability:target achieved for AMV+/FORM; target approximated for MV Robustness: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/RBDO

  22. Bi-Stable Switch: Observed Challenges Problematic d MVFOSM-based RBDO d* • Nonlinear limit state • Higher-order integration needed • Nonsmooth due to sim failures in left tail of edge bias (flimsy structure) • Smooth, linear limit state (in range of interest) • First-order integration OK In general, need support of nonlinear/nonsmooth/multimodal limit states and careful attention to problem formulation (e.g., supporting simulation requirements)

  23. Conclusions & Future Directions General UQ/OUU Aspects: • Nonlinear, large-scale, expensive simulations w/ implicit, noisy response metrics • Aleatory and epistemic uncertainties • Design for robustness and reliability Reliability Analysis: • Explored limit state approxs, integrations, MPP search algs, Hessian approxs, warm starting • New algorithms for 2nd-order PMA, AMV2/AMV2+, and QN integrations • Recommendation 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 sensitivities • New 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. characterization • model calibration/inversion under uncertainty (matching output CDFs) • explore TR linkages between MPP and design levels

  24. Extra Slides

  25. Optimization with Surrogate Models • Purpose: • Reduce the number of expensive, high-fidelity simulations by using a succession of approximate (surrogate) models • Approximations generally have a limited range of validity • Trust regions adaptively manage this range based on efficacy during opt • With trust region globalization and local 1st-order consistency,SBO algorithms are provably-convergent • Surrogate models of interest: • Data fits • Multifidelity (special case: multigrid optimization) • Reduced-order models • Future connections to multi-scale for managing approximated scales

  26.  Multifidelity Data Fit  ROM New area Data fit surrogates: • Global: polynomial regress., splines, neural net, kriging, radial basis fn • Local: 1st/2nd-order Taylor • Multipoint: TANA, … Data fits in SBO • Smoothing: extract global trend • DACE: number of des. vars. limited • Local consistency must be balanced with global accuracy Multifidelity surrogates: • Coarser discretizations, looser conv. tols., reduced element order • Omitted physics: e.g., Euler CFD, panel methods Multifidelity SBO • HF evals scale better w/ des. vars. • Requires smooth LF model • Design vector maps may be reqd. • Correction quality is crucial ROM surrogates: • Spectral decomposition (str. dynamics) • POD/PCA w/ SVD (CFD, image analysis) • KL/PCE (random fields, stoch. proc.) • RBGen/Anasazi ROMs in SBO • Key issue: capture parameter changes • Extended ROM, Spanning ROM • Shares features of data fit and multifidelity cases Trust-Region Surrogate-Based Optimization

  27. Opt {d} {S } u Data Fit {d} {S } u SBOUU with two surrogate levels: UQ Simple Nested OUU: {u} {R } u Data Fit/Hier {d} {R } u {u} Sim SBOUU Formulations For surrogate-based OUU, the surrogate can appear • at 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 be • local/global/multipoint data fit (either level) • model hierarchy approximation (UQ level only)

  28. Nested model: internal iterators/models execute a complete iterative study as part of every evaluation. Surrogate model: internal iterators/models used for periodic update and verification of data fit (global/local/multipoint) or hierarchical (variable fidelity) surrogates. Nested/Surrogate models can recurse DakotaModel Single Layered Nested Data Fit Hierarchical Opt {d} {S } u Data Fit Opt Opt {d} {S } {d} u {d} {S } {S } u u UQ Data Fit UQ {R } {u} {d} {S } u {u} {R } Data Fit/Hier u u Data Fit/Hier {d} UQ {R } {d} u {u} {R } {u} u {u} {R } u Sim Sim Sim Optimization under Uncertainty with Surrogates Formulation 1: Nested Formulation 2: Surrogate containing Nested Formulation 3: Nested containing Surrogate Formulation 4: Surrogate containing Nested containing Surrogate Formulations 2 & 4 amenable to trust-region approaches Goals: maintain quality of results, provable convergence(for a selected confidence level)

  29. TR-SBOUU Results • Direct nested OUU is expensive and requires seed reuse • SBOUU expense much lower (up to 100x), but unreliable. • TR-SBOUU maintains quality of results and reduces expense ~10x • Ex. 1: formulation 4 with TR 5-7xless expensive than direct nesting • Ex. 2: formulation 4 with TR 8-12x less expensive than direct nesting • ICF Ex.: formulations 2/4 with TR locate vicinity of a min in a single cycle • Additional benefits: • Navigation of nonsmooth engineering problems • Less sensitive to seed reuse: variable patterns OK and often helpful,possibility of exploitation reduced • Less sensitive to starting point: data fit SBO provides some global ident. Minimize f + pfail_r1 + pfail_r3 Subject togi 0, for i = 1,2,3 mr2 + 3sr2 1.6e5 Conference papers atAIAA 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."

  30. 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 correction • verification of approx. steps Extensions to SBOUU • 1st order consistency, assuming a worthwhile stoch. gradient • verification of stats. in relativesense. 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

  31. transientdynamics comp fluid Fire shock physics structural dynamics Sierra radiation Dakota Optimization Uncertainty Quantification Parameter Estimation Sensitivity Analysis MOOCHO Veltisto O3D SFE Common Code Ar c hitecture Opt./UQ DSMC Incomp Fluids Solid Mech Thermal Intrusive OUU: DAKOTA/MOOCHO w/ SIERRA/NEVADA Next-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 elements • Other: Stability analysis (LOCA), Nonlinear equations (NOX), Fully-coupled MDA Impact: • Performance enhancements for existing nested methods • Model I/O in core in parallel • SPMD execution on compute nodes of ASCI MPPs • Next-generation, tightly-coupled opt. & UQ • Direct/adjoint sensitivities & AD • Intrusive OUU: • SFE + SAND • Unilevel RBDO + SAND

  32. RBDO Results Cantilevermin wts.t. bD, bS> 3 Limit state eqns (unnormalized): • Wu et al., 2001 • 2 design vars: w, t • 4 uncorr. normal uncertain vars: E, R, X, Y

  33. Concluding Remarks • DAKOTA/UQ provides a flexible object-oriented framework for investigating algorithmic variations in reliability analysis and RBDO • Novel aspects: • Full CDF/CCDF • Limit state linearizations in PMA • Warm starting by projection (avoids premature convergence) • SQP vs. NIP • UQ Performance: • Relative to FORM: • MV 2 orders of magnitude reduction but only accurate near means • AMV 1 order of magnitude reduction but MPP not converged • AMV+ ~3x reduction with full accuracy • Warm starts: significant win, as expected • NIP vs. SQP: mixed results, but NIP promising (can avoid u-space excursions) • x-space vs. u-space: mixed results, application dependent • RBDO Performance: • MV/AMV RBDO is cheap and gets in the solution vicinity • RIA zb RBDO: preferred to RIA zp RBDO, slight advantage over PMA RBDO • AMV+ RBDO 3.5x more efficient than FORM RBDO Great for rough stats

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