Robust Design Optimization Uncertainty Quantification and Design for
Robust Design Optimization, Uncertainty Quantification and Design for Reliability Mariapia Marchi – ESTECO Sp. A (Italy)
Design optimization and uncertainty • When facing “creative design” uncertainties in the performance prediction must be taken into account: – – Noise due to unpredictable environment Noise due to geometrical tolerances Numerical inaccuracies Uncertainty in material loadings and properties, etc • Uncertainties affect robustness and reliability of optimal solutions
Robustness & Reliability
Robustness measures • Typically output distribution first and second order moment, i. e. : Ø Expectation (mean) value (objective) Ø Variance/standard deviation (objective/constraint)
Reliability measures • Typically: Ø Failure probability Ø Reliability index Ø Percentile values
Computational techniques • Two major classes of methods Optimization Under Uncertainty (OUU): 1) Robust Design Optimization (RDO) 2) Reliability-Based Design Optimization (RBDO) • In this lecture, we consider the case of probabilistic uncertainties • Random input variables following probability distribution functions (PDFs) • System response is also stochastic, but its PDF is not known a priori • Uncertainty Quantification (UQ) methods estimate as precisely as possible unknown statistical properties of output responses. Used to estimate robustness and reliability measures
Comparison of Uncertainty Quantification Methods Monte Carlo/Latin Hypercube Sampling vs. Polynomial Chaos Expansion
Sampling methods Discretization of objective functions: n points are sampled in the range of fluctuation of stochastic variables Mean value Standard Deviation
Monte Carlo sampling •
Latin Hypercube sampling •
Monte Carlo vs. Latin Hypercube Monte Carlo Latin Hypercube
UQ method comparison Example: Uncertain Response Y=f(X)
UQ method comparison Deterministic Response Stochastic Response
UQ method comparison Former example: To reach 1% error in Mean PC needs 8 points Standard Deviation needs 12 points Convergence to exact statistical moments, with n=sample size, for Polynomial Chaos is reached as 1/en
Introduction to Reliability Analysis and Reliability-Based Design Optimization
Basic ideas (I) • Reliability analysis aims at finding failure probability Pf , i. e. probability that a structure, a mechanical component, etc will fail to meet pre-defined Limit State Function (LSF) criteria. Very popular in Structural Engineering • Reliability problem: X=(X 1, …, Xn) a vector of random input variables (uncertain quantities, such as loads, material properties, structure dimensions, environmental factors, etc. ) and g(X) a performance function that describes structure limit state in terms of X • By convention, g(X)≤ 0: failure domain. The boundary g(X)=0 is called limit state function
Basic ideas (II) • Reliability: R=P(g(X)>0) • Failure probability defined as: • f. X(X) joint probability density function of X • Integral can be solved with (smart) sampling techniques, but too demanding for small failure probabilities • Simplifications are necessary (especially in real-world problems)
Simplification I: input variable transforms • X→U, with U independent standard normal variables • In U–space, the integral becomes with product of standard (i. e. zero mean and unit standard deviation) Gaussian probability density functions • Complexity of variable transform depends on initial PDFs. • In some cases linear or straightforward change of variables are possible. In other cases Rosenblatt or Nataf transforms • If correlated input variables, then Cholesky decomposition or other methods to make them uncorrelated
Simplification II: integral domain • Usually, LSF approximated at design point or most probable point (MPP) (of failure) - minimum distance between origin in U-space and LSF. That distance is a reliability index • FORM (first-order approximation): LSF is linearized at the MPP • FORM failure probability becomes: • Second-order approximations (SORM) include curvature contributions
Direct and inverse reliability problem • Search of design point Direct reliability problem • Search of p–percentile value gp of performance function g given a probability of failure P(g(X)<gp)=p: Inverse reliability problem • They are constrained single objective optimization problems
FORM for direct reliability problems Three macro-steps are involved: 1. Variable transformation from starting random input variables X to U space variables 2. Search MPP and reliability index (solution of single objective optimization problem) 3. Compute FORM failure probability
SORM for direct reliability problems • SORM deals with non-linear LFSs • Some macro-steps are: 1. 2. 3. 4. • FORM analysis to find MPP Second order approximation of LSF at MPP Determination of LSF curvatures Computation of SORM failure probability Several approximations are possible. For instance Breitung’s formula: ki denote LSF main curvatures at design point, taken positive for convex failure sets
Challenges for Reliability Analysis • Highly-non linear LSFs • Multiple design points • High-dimensional problems • Very small failure probabilities, etc
Reliability-based problems/measures • Failure probability may be taken as an objective to minimize • Probabilistic or chance constraints could be involved • Definition of reliability measures: 1. Sampling-based measures: computation of failure probability through (smart) sampling strategies – inefficient for high reliability requests 2. Optimization-based measures: determination of MPP
Approaches for finding MPP Based on inverse reliability problem. PMA (Performance Measure Approach) Based on direct reliability problem. RIA (Reliability Index Approach)
RBDO • A plethora of methods exists in the literature: 1. Double-loop approaches (reliability loops nested in the main optimization loop) 2. Single-loop approaches (combine both optimization levels by using approximations of the inner one – faster but less accurate than double-loop) 3. Decoupled methods (the two loops are performed sequentially) • Genetic algorithms and probabilistic constraints can be used too
Polynomial Chaos Expansion and adaptive approaches
Polynomial Chaos Expansion f function of an uncertain probabilistic d-dimensional X variable Expansion in a basis of orthogonal polynomials (truncated to k terms) Simple estimates for mean and standard deviation: ROBUSTNESS PCE as surrogate model provides empirical cumulative distribution function and percentiles RELIABILITY
Curse of dimensionality & sparse PCE To determine the unknown coefficients: N sampling points and solve least-square regression problem: Number of evaluations for Uncertainty Quantification: d=1 d=2 d=3 d=4 d=5 d=10 k=1 2 3 4 5 6 11 k=2 3 6 10 15 21 66 k=3 4 10 20 35 56 286 k=4 5 15 35 70 126 1001 Sparse polynomial chaos expansion* *Sudret et al. (ETH Zurich)
Optimization under uncertainty Optimization flow Uncertainty Quantification flow Robustness & reliability measures Mean value estimate Determine PCE coefficients Standard deviation estimate Step i Use PCE as surrogate model to compute empirical CDF Percentile values
Example 1 Robust Design Optimization of a racing car
Example with VI-Car_Real_Time Presented by Prof. Poloni at 2 nd VI-Grade Users' Conference (2008) • • Vehicle Dynamic p. o. v. Model: open wheel car Software: VI-Car_Real_Time: Targets: – Increase AVG speed – Minimize Steer Oscillations • Standard Parameters (10): – Car setup • • Front/rear ride_heights (2) Front/rear spring stiffnesses (2) Front/rear antirollbar stiffnesses (2) Front/rear dampers scaling factors (2) START – Limited Slip Differential setting: • • C 0 coefficient → preload (1) C 1 coefficient (1) – Stochastic Parameters (3) • • Driver behaviour: preview time (1) Tires: front/rear lateral grip factor (2) END
The Workflow in mode. FRONTIER
Optimization setup Scheduler: NSGA-II Initial population: 24 Designs Number of generations: 10 Number of concurrent designs evaluations: 28 Number of samples for UQ: 28. Full PCE of 2 nd order 28*28*10 = 7840 design evaluations About 4 minutes per analysis on a AMD Opteron 2. 4 Ghz Multiobjective RDO optimization time : 22 h
MORDO optimization results
Best design statistical properties
Best design performance • Optimize means improve performance: • . . . Even when uncertainties are dominant
Example 2 Reliability-based sizing optimization of an RC building with seismic isolation
Test case: RC structure with elastomeric bearings Presented at EURODYN 2017 in collaboration with Luca Rizzian and Numa Léger Location site: Reggio Calabria (Italy) Fixed-base or base-isolated structure: Response spectrum analysis with SAP 2000 SI elastomeric isolator from FIP Industriale catalog
Deterministic optimization problem • Sizing optimization of superstructure and isolators (discrete input variables) • Input and output constraints to comply with requirements by Italian law NTC 2008 • Three objectives: minimization of superstructure cost, top-floor acceleration and displacement
Uncertainties Uncertain Input Parameters Fixed-base Base-isolated Distribution Location parameter Other distr. parameters Permanent load Normal 3000 N/m 2 Co. V=10% Variable load Normal 2500 N/m 2 Co. V=10% Log-uniform 0. 1 Interval width=0. 05 Isolator damping Objective/constraints Beam and columns compliance checks Accelerations Displacements Fixed-base Base-isolated Reliability level 99. 9% - 99. 9% 99. 99%
Simulation details • Optimization with mode. FRONTIER software • Use of genetic algorithm MOGA-II (by Prof. Poloni et al. ) • Initial population from the optimal solutions of deterministic problem (100 elements) • 16 generations • Default crossover and mutation probability values • Fixed-base: full PCE of 3 rd order. Base-isolated: sparse PCE of 4 th order. In both cases: 20 samples for Uncertainty Quantification loop • Convergence checks
Deterministic vs. Stochastic outcomes = Deterministic result FIXED-BASE = Stochastic result BASE-ISOLATED
Reliability checks
Instead of conclusions…
Some challenges for Optimization Under Uncertainty • Reduce computational cost • Solve highly nonlinear or highly constrained problems • Solve high-dimensional problems • Solve problems with different kind of uncertainties (probabilistic, epistemic, stochastic black-boxes) • …
Thank you for your attention! Any questions?
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