Overview of Reliability Analysis and Design Capabilities in DAKOTA

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
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/variabi
  lity 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
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
DAKOTA Overview
Optimized
Nominal

• 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 VV
• 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)

6
DAKOTA Framework
Model
Iterator
Parameters
Interface
LeastSq
DoE
Design continuous discrete
Application system fork direct grid
Functions
GN
NLSSOL
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
Approximation global polynomial 1/2/3,
NN, kriging, MARS, RBF multipoint
TANA3 local Taylor series hierarchical
Gradients
DSTE
LHS/MC
Vector
List
numericalanalytic
SFEM
MultiD
Reliability
Center
Hessians
numericalanalyticquasi
State continuous discrete
COLINY
NPSOL
DOT
OPT
JEGA
CONMIN
NLPQL
Strategy control of multiple iterators and models
Coordination
Strategy
Iterator
Nested Layered Cascaded Concurrent Adaptive/Intera
ctive
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
BranchBound/PICO
Model
7
Uncertainty Quantification
UQ
DSTE
LHS/MC
SFEM
Reliability

• Active UQ development (new, developing, planned).
• Sampling LHS/MC, QMC/CVT, Bootstrap/Importance/Ja
  ckknife. 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
UQ with Reliability Methods
Mean Value Method
Surprisingly popular with analysts
10
Reliability Algorithm VariationsFirst-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
Reliability Algorithm VariationsSecond-Order
Methods
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
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 VariationsSample of
Results to Date
Analytic benchmark test problems lognormal
ratio, short column, cantilever
43 z levels
43 p 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
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
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

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
RBDO Algorithms
16
RBDO Algorithm VariationsSample 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 bh s.t. b gt 2.5
Kuschel Rackwitz, 1997
17
OUU Progress To Date

• 2003 Surrogate-based OUU with sampling methods
• 2004 Bi-level RBDO with numerical reliability
  gradients
• 2005 Fully analytic bi-level RBDO Sequential/s
  urrogate-based RBDO (1st-order, 2nd-order)

OUU/SBOUU
RBDO
BL
S1
FA-BL
S2
18
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 DeploymentShape
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
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
MVFOSM-based RBDO d
Problematic 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/mu
ltimodal 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
Trust-Region Surrogate-Based Optimization
27
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
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 5-7x less 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 to gi ?
0, for i 1,2,3 mr2 3sr2 ? 1.6e5
Conference papers at AIAA MAO, SIAM CSE,
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 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
31
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 gt 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