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Multidisciplinary analysis and optimization under uncertainty

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Chen Liang

The document summarizes Chen Liang's doctoral dissertation research on multidisciplinary analysis and optimization under uncertainty. The research objectives are to develop efficient uncertainty quantification techniques for feedback-coupled multidisciplinary analysis and multidisciplinary design optimization that can account for both aleatory and epistemic sources of uncertainty. Specific areas of focus include representation of epistemic uncertainty, propagation of uncertainty through coupled analysis, and inclusion of uncertainty in high-dimensional multidisciplinary design optimization problems.

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Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 1 Multidisciplinary Analysis and Optimization under Uncertainty Chen Liang Dissertation Defense Adviser: Sankaran Mahadevan Department of Civil and Environmental Engineering Vanderbilt University, Nashville, TN Aug. 21st , 2015
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 2 MDA Overview Three-objective two-stage-to-orbit launch vehicle Heatsink Aircraft wing analysis Nodal Pressures Nodal Displacements Wing Backsweep Angle, Speed and Angle of Attack Lift, drag, stress FEA structure CFD fluid Compatibility  Fixed-point-iteration (FPI)
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 3 MDO under uncertainty  Presence of uncertainty sources  UQ  Sampling outside FPI SOFPI Repeated MDA  New design input values at each iteration Computationally unaffordable  Need efficient methods for MDA and MDO under uncertainty UQ / Reliability Analysis FEA CFD MDA Optimization
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 4 Three types of sources • Physical variability • Data uncertainty (e.g., sparse/interval data) • Model Uncertainty Forward problem • For a given input Uncertainty of output needs to be evaluated • Propagation of aleatory uncertainty is well-studied • Inclusion of epistemic uncertainty becomes more important • Little work regarding the propagation of epistemic uncertainty in feedback coupled MDA Uncertainty and errors in optimization Sources of Uncertainty and Errors Aleatory (Irreducible) Natural Variability Epistemic (Reducible) Data Uncertainty Sparse Data Interval Data Model Uncertainty Numerical error Discretization error Round off error Truncation error Surrogate model error UQ error Model Form Error
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 5 Overall Research Goal  Efficient UQ techniques for feedback coupled MDA and MDO  Combine information with both aleatory and epistemic sources of uncertainty  Particular emphasis on • Representation of epistemic sources of uncertainty • Propagation through feedback coupled analysis • Inclusion in the design optimization of multidisciplinary analysis with feedback coupling (high-dimensional)
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 6 Research objectives  MDA with epistemic uncertainty - Inclusion of data uncertainty and model error  MDA with high-dimensional coupling - Large number of coupling variables - Dependence among all variables - Efficient uncertainty propagation  Multi-objective optimization under uncertainty - Reliability-based design optimization - Solution enumeration (Pareto front construction)  Multidisciplinary design optimization - Concurrent interdisciplinary compatibility enforcement and objective/constraint functions evaluation A. Multidisciplinary analysis under uncertainty B. Multidisciplinary optimization under uncertainty Bayesian Framework
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 7 𝑔2 𝑓 𝑢12 𝑔1 𝑢21 Analysis 1 𝑨 𝟏(𝒙, 𝑢21) Analysis 2 𝑨 𝟐(𝒙, 𝑢12) Analysis 3 𝑨 𝟑(𝑔1, 𝑔2) 𝑥1 𝑥 𝑠 𝑥2  Uncertainty propagation under the compatibility condition  No need for full convergence analysis Multi-disciplinary multi-level system Review of MDA under uncertainty methods Sankararaman & Mahadevan, J. Mechanical Design, 2012 Approximation Method • First-order Second Moment (FOSM) approximations • Linear approximations of disciplinary analyses • PDF based on mean & variance • Du & Chen, Mahadevan & Smith • Fully Decoupled Approach • Calculate PDFs of u12 & u21 • Cut-off feedback  both directions • Ignores dependence between u12 & u21 • Lack of one-to—one correspondence between 𝑔1 and 𝑔2 in calculating f Likelihood-based approach for MDA (LAMDA) 𝑢21 𝑢12
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 8 𝑢12 𝑔2 𝑓 𝑢12 𝑔1 𝑢21 Analysis 1 𝑨 𝟏(𝒙, 𝑢21) Analysis 2 𝑨 𝟐(𝒙, 𝑢12) Analysis 3 𝑨 𝟑(𝑔1, 𝑔2) 𝑥1 𝑥 𝑠 𝑥2 Multi-disciplinary multi-level system Likelihood-based approach for MDA (LAMDA) Objective 1: MDA with epistemic uncertainty 𝑔2 𝑓 𝑔1 Analysis 1 𝑨 𝟏(𝒙, 𝑢21) Analysis 2 𝑨 𝟐(𝒙, 𝑢12) Analysis 3 𝑨 𝟑(𝑔1, 𝑔2) 𝑥1 𝑥 𝑠𝑥2 𝑢21
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 9 𝑈12 𝐹 𝑈12 𝐺Interdisciplinary compatibility: LAMDA 𝑢12 Analysis 1 𝑨 𝟏(𝒙, 𝑢21) Analysis 2 𝑨 𝟐(𝒙, 𝑢12) 𝑢21 𝑈12 Given a value of 𝑢12 what is 𝑃(𝑈12 = 𝑢12|𝑢12)  𝐿(𝑢12) 𝑓 𝑢12 = 𝐿(𝑢12) 𝐿(𝑢12)𝑑𝑢12 FORM is used to calculate the CDFs of the upper and lower bounds: 𝑃𝑢 and 𝑃𝑙 𝐿 𝑢12 ∝ 𝑢12− 𝜀 2 𝑢12+ 𝜀 2 𝑓𝑈12 𝑈12 𝑢12 𝑑𝑈12 𝐿 𝑢12 ∝ (𝑃𝑢 − 𝑃𝑙)  finite difference 𝑷 𝒖 𝑷𝒍 𝒖 𝟏𝟐 + 𝜺 𝟐 𝒖 𝟏𝟐 − 𝜺 𝟐
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 10 Sources of Uncertainty and Errors Aleatory (Irreducible) Natural Variability Epistemic (Reducible) Data Uncertainty Sparse Data Interval Data Model Uncertainty Numerical error Discretization error Round off error Truncation error Surrogate model error UQ error Model Form Error Uncertainty and errors in LAMDA Considering epistemic uncertainty sources
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 11 Data uncertainty (sparse and interval data) pi pQ X fX(x) θQθ3θ2θ1 θi p1 p2                  n i b a X m i iX dxPxfPxfPL i i 11 )|()|()( Parametric approach Non-parametric approach Likelihood Sparse data Interval data Convert sparse and interval data into a useable distribution (for propagation) Sankararaman & Mahadevan RESS 2011 Zaman, et. al, RESS 2011
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 12 Sources of Uncertainty and Errors Aleatory (Irreducible) Natural Variability Epistemic (Reducible) Data Uncertainty Sparse Data Interval Data Model Uncertainty Numerical error Discretization error Round off error Truncation error Surrogate model error UQ error Model Form Error Uncertainty and errors in LAMDA
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 13 Model uncertainty estimation Training points (e.g. FEA) Prediction Uncertainty Discretization error estimation • GP prediction ~ 𝑁(𝜇, 𝜎)  𝜇 and 𝜎 are input dependent Rangavajhala, et. al, AIAA Journal 2010 Richardson extrapolation At each input 𝒙
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 14  Auxiliary variable 𝑃ℎ~𝑈[0,1]  CDF of GP output  Stochastic model output  input random variable  FORM can be used for likelihood evaluation 𝑈12 CDF of GP output 𝐿 𝑢12 ∝ 𝑃(𝑈12 = 𝑢12|𝑢12) • Equation only calculable when 𝑈12 is deterministic given an 𝒙 and 𝑢12 Inclusion of model uncertainty in LAMDA 𝑢12 𝑢21 𝐴2 𝒙, 𝑢12 GP model (𝒙, 𝑢21) 𝒙 𝑃ℎ Deterministic 𝑃ℎ Extra loop of uncertainty propagation 
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 15 Electrical Parameters Component Heat Total Power Dissipation Heatsink Temperature Electrical Analysis Thermal Analysis Power Density: Total Power Dissipation Volume of the Heatsink Heatsink Size Parameters Numerical example: electronic packaging Model error in thermal Analysis • 2D steady state heat transfer equation (PDE) 𝛻𝑇𝑠𝑖𝑛𝑘 𝑥, 𝑦 + 𝑞(𝑥, 𝑦) 𝑘 = 0 • Solved by Finite Difference method • Limited computational resources  discretization error MDO test suite: Heatsink Data uncertainty Temperature Coefficient of resistance (𝜶) Data points Data intervals 0.0055 0.0057 [0.004,0.009] [0.0043,0.0085] [0.0045, 0.0088] 4 5 6 7 8 x 10 -3 0 200 400 600 800 1000 1200 Non-parametric PDF of 𝜶 PDF 𝜶 • Uncertainty estimated auxiliary variable
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 16 Temperature Thermal Analysis Electrical Analysis Power density 𝒙 𝟐𝒙 𝟏 Heat
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 17 Results Uncertainty of the coupling variables Comparison between FPI and LAMDA  FPI with stochastic model errors is difficult to converge  Only a few FPI realizations is affordable  LAMDA agrees well with the available data Liang & Mahadevan ASME JMD, 2015 Temerature(℃ ) PDF Component heat(Joule) PDF
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 18 Likelihood Approach for Multidisciplinary Analysis • Data uncertainty and model error in feedback coupled analysis. • Auxiliary variable  stochastic model error. Features of methodology • Likelihood-based approach for MDA (LAMDA) • FORM • GP estimation of model error • Auxiliary variable Objective-1 summary
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 19 High Dimensional Coupling CFD FEA UQ/Reliability Analysis Nodal Pressures Nodal Displacements  Multiple coupling variables in one direction  Joint distribution of the coupling variables in the same direction FORM-based LAMDA is inefficient because: • First-order approximation: dimension ↑, accuracy ↓ • Likelihood calculation (finite difference)  Number of function ↑
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 20 Research objectives  MDA with epistemic uncertainty - Inclusion of data uncertainty and model error  MDA with high-dimensional coupling - Large number of coupling variables - Dependence among all variables - Efficient uncertainty propagation  Multi-objective optimization under uncertainty - Reliability-based design optimization - Solution enumeration (Pareto front construction)  Multidisciplinary design optimization - Concurrent interdisciplinary compatibility enforcement and objective/constraint functions evaluation A. Multidisciplinary analysis under uncertainty B. Multidisciplinary optimization under uncertainty
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 21  Probabilistic graphical model: represents random variables and their conditional dependencies by nodes and edges.  Incorporate large number of variables with heterogeneous formats: distribution (continuous, discrete, empirical) & function.  Bayesian network is update by sampling approaches. An efficient Gaussian copula-based sampling method is adopted. Bayesian network and copula-based sampling (BNC) 𝑌 𝑍 𝑋 𝑉 Uncertainty propagation (forward) 𝑣 𝑜𝑏𝑠 𝑌 𝑍 𝑋 Bayesian updating (inverse)
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 22 Multi-variate Gaussian Copula • A copula is a function that joins the CDFs of multiple random variables as a joint CDF function. • Types: Gaussian, Clayton, Gumbel, … • The Multivariate Gaussian Copula (MGC) 𝐶𝛴 𝐺𝑎𝑢𝑠𝑠 𝒖 = 𝚽 𝚺 Φ−1 𝑢1 … Φ−1 (𝑢 𝑛) where 𝑢𝑖 is the CDF value of any arbitrary marginal 𝐹𝑋𝑖 (𝑥𝑖), • Gaussian copula assumption needs verification (done for all examples) • Other copulas are not as efficient in conditional sampling Hanea, et al., QREI, 2006 Efficient Conditional Sampling 𝑉 = 𝑣 𝑂𝑏𝑠 𝑌 𝑍 𝑋 Bayesian updating (inverse)  If 𝑉 = 𝑉𝑜𝑏𝑠, the conditional samples of 𝑋, 𝑌 and 𝑍 can be obtained as following: 𝑥 = 𝐹𝑋 −1 ΦΣ′ 𝑈 𝑋|𝑉 = 𝑣 𝑜𝑏𝑠 𝑦 = 𝐹𝑌 −1 ΦΣ′ 𝑈 𝑌|𝑉 = 𝑣 𝑜𝑏𝑠 𝑧 = 𝐹𝑍 −1 ΦΣ′ 𝑈 𝑍|𝑉 = 𝑣 𝑜𝑏𝑠
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 23 BNC-MDA: 𝒖 𝟐𝟏 𝑼 𝟐𝟏 𝜺 𝒖 𝟐𝟏 BN with 20 coupling variables • Joint distribution of 𝑼 𝟐𝟏 are evaluated by the conditional samples • Given compatibility condition (𝜀21 = 0), generate samples from the BN by copula-based sampling 𝑼 𝟐𝟏𝒖 𝟐𝟏 𝒖 𝟏𝟐 Analysis 2 𝑨 𝟐(𝒖 𝟏𝟐, x) Analysis 1 𝑨 𝟏(𝒖 𝟐𝟏, x) 𝒙 𝜺 𝒖 𝟐𝟏 = 𝒖 𝟐𝟏 − 𝑼 𝟐𝟏 Interdisciplinary compatibility = 𝟎 • Bayesian network is built using samples of 𝑢21, 𝑈21 and 𝜀 𝑢21 𝒖 𝟐𝟏 𝑼 𝟐𝟏 𝜺 𝒖 𝟐𝟏 One coupling variable
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 24 Challenge for BNC-MDA High dimensional coupling • Mesh resolution : 258 nodes  258 random variables 1 MAR 17 2015 09:30:53 ELEMENTS 1 MAR172015 12:54:37 ELEMENTS Nodal Pressures Nodal Displacements FEA CFD Aero-elastic analysis of an aircraft wing 𝑵𝑷𝒊−𝟏 𝑵𝑷𝒊 𝜺 • BN with 774 nodes : enormous effort • Variables are highly correlated  Redundant information, singularity of correlation coefficient matrix of the copula
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 25 Principal component analysis (PCA)  Correlated variables  linearly uncorrelated principal component space  First 15~20 PCs cover more than 99% of the original variances  Bayesian updating on the 15~20 individual uncorrelated principal components Reduces a giant Bayesian network into 15~20 small networks Bayesian update can be implemented in parallel 𝒊𝒕𝒉 PC at 𝒏 − 𝟏 𝒕𝒉 iteration 𝒊𝒕𝒉 PC at 𝒏 𝒕𝒉 iteration Difference Updated by Interdisciplinary compatibility (Difference = 0)
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 26 𝑵𝑷 𝟐 𝑵𝑷 𝟑 Numerical Example – MDA of an aircraft wing Input uncertainty • Backsweep angle 𝜃 ~ 𝑁 0.4,0.1 • 110 FPI analyses  Benchmark • 6 iterations till convergence  ~1300 function calls (FEA and CFD in total) FPI (secs) BNC-MDA (secs) Time saved (secs) 14,300 9,350 4,950 Total time consumed  Kullback-Leibler (K-L) divergence with benchmark solution : • Smaller value  Closer distributions Higher fidelity  more time saved Method 2nd Iteration 3rd Iteration BNC K-L divergence 0.21 0.18 0.16 Estimation without full convergence analysis = 0 PCA Reduced 𝑵𝑷 𝟐 Difference PCA Reduced 𝑵𝑷 𝟑 • 𝑁𝑃2: nodal pressure after 2nd iteration • 𝑁𝑃3: nodal pressure after 3rd iteration • 𝑁𝑃2 & 𝑁𝑃3  BNC-MDA (660 function calls)
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 27 Objective-2 summary Bayesian Approach for Multidisciplinary Analysis • Novel BNC-MDA for UQ in high-dimensional coupled MDA • Dimension and iteration reduction  Efficient while preserving the dependencies • Time for physics analysis ≫ time for stochastic analysis Features of methodology • Bayesian Network • Gaussian copula-based sampling • Principal component analysis
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 28 Research objectives  MDA with epistemic uncertainty - Inclusion of data uncertainty and model error  MDA with high-dimensional coupling - Large number of coupling variables - Dependence among all variables - Efficient uncertainty propagation  Multi-objective optimization under uncertainty - Reliability-based design optimization - Solution enumeration (Pareto front construction)  Multidisciplinary design optimization - Concurrent interdisciplinary compatibility enforcement and objective/constraint functions evaluation A. Multidisciplinary analysis under uncertainty B. Multidisciplinary optimization under uncertainty
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 29 Robustness-based Design Optimization (RDO) Reliability-based Design Optimization (RBDO) Attempts to minimize variability in the system performance due to variations in the inputs. Aims to maintain design feasibility at desired reliability levels. Background: Optimization under uncertainty Focuses on 𝜎𝑜𝑏𝑗 of the objective function Focuses on 𝑃𝑓 of the constraint function 𝝈 𝒐𝒃𝒋 𝑰 𝝈 𝒐𝒃𝒋 𝑰𝑰 Objective Function PDF Constraint Function PDF
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 30 𝑚𝑖𝑛 𝜇 𝑋,𝑑 [𝜇 𝑓 𝑋, 𝑃, 𝑑, 𝑝 𝑑 ] s.t. Prob(𝑔𝑖(𝑋, 𝑑, 𝑃, 𝑝 𝑑) ≥ 0)) ≥ 𝑝𝑡 𝑖 , i= 1,2 … , 𝑛 𝑞 Prob(𝑋 ≥ 𝑙𝑏 𝑋)) ≥ 𝑝𝑙𝑏 𝑡 Prob(𝑋 ≤ 𝑢𝑏 𝑋)) ≥ 𝑝 𝑢𝑏 𝑡 𝑙𝑏 𝑑 ≤ 𝑑 ≤ 𝑢𝑏 𝑑 Reliability-based design optimization • Natural variability • Data uncertainty • Sparse • Interval Model uncertainty • Numerical solution error • Model form error • The examples are formulated using the RBDO formulation • Proposed methodology is adaptable to solve RDO problems 𝑋: stochastic design variable 𝑃: stochastic non-design variable 𝑑: deterministic design variable 𝑃𝑑: deterministic non-design variable
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 31 Research objectives  MDA with epistemic uncertainty - Inclusion of data uncertainty and model error  MDA with high-dimensional coupling - Large number of coupling variables - Dependence among all variables - Efficient uncertainty propagation  Multi-objective optimization under uncertainty - Reliability-based design optimization - Solution enumeration (Pareto front construction)  Multidisciplinary design optimization - Concurrent interdisciplinary compatibility enforcement and objective/constraint functions evaluation A. Multidisciplinary analysis under uncertainty B. Multidisciplinary optimization under uncertainty
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 32 • Conflicting objectives  One objective cannot be improved without worsening others Objective 3: Multi-objective optimization • Pareto front  tradeoff relationship between different objectives 𝑚𝑖𝑛 𝒙 [𝑓1 𝑿, 𝑃 , 𝑓2 𝑿, 𝑃 ] s.t. 𝑙𝑏 𝑋 ≤ 𝑿 ≤ 𝑢𝑏 𝑋 • Existing methods  Weighted sum  𝜀-Constraint  Goal programming  Non-dominated sorting genetic algorithm (NSGA) Assign weights to each objective One as objective, others as constraints Global search and solution- ranking strategy Optimizes the weighted sum of the penalty
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 33 𝑚𝑖𝑛 𝜇 𝑋,𝑑 [𝜇 𝑓1 𝑋, 𝑑, 𝑃, 𝑝 𝑑 , 𝜇 𝑓2 𝑋, 𝑑, 𝑃, 𝑝 𝑑 , … , 𝜇 𝑓𝑛 (𝑋, 𝑑, 𝑃, 𝑝 𝑑)] s.t. Prob(𝑔𝑖(𝑋, 𝑑, 𝑃, 𝑝 𝑑) ≥ 0)) ≥ 𝑝𝑡 𝑖 , i= 1,2 … , 𝑛 𝑞 Prob(𝑋 ≥ 𝑙𝑏 𝑋)) ≥ 𝑝𝑙𝑏 𝑡 Prob(𝑋 ≤ 𝑢𝑏 𝑋)) ≥ 𝑝 𝑢𝑏 𝑡 𝑙𝑏 𝑑 ≤ 𝑑 ≤ 𝑢𝑏 𝑑 Multi-objective optimization under uncertainty • Challenges of existing MOO methods - Weights for objective aggregation is not intuitive - Goal programmingConstraint-based method may not produce Pareto solutions - NSGA is computationally expensive (surrogate models) • Little work regarding the dependence relationships between output variables (e.g. co-kriging for small number of outputs)
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 34 Dependence among Objectives/Constraints • Joint probability formulation for MOUU - Considered the dependence among objectives - Joint probability of the design threshold being satisfied  constraint - FORM Rangavajhala & Mahadevan JMD 2011 Proposed method  Graphical surrogate model that integrates design variables, uncertain variables, objectives and constraints in one Bayesian network  Gaussian copula-based sampling for efficient uncertainty propagation and reliability assessment (forward propagation)  Training samples selection to improve the Pareto front (inverse problem) Inefficient for high-dimensional problems
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 35 Numerical example: vehicle side impact model  FEA data is unavailable  Step-wise regression model  generate training data & validate the proposed method  115 training points by Latin Hypercube sampling Vehicle side impact model Gu, et al. , IJVD, 2001 Problem description * Design variables have variability, 2 additional uncertain variables min 𝝁 𝒙 𝜇 𝑊𝑒𝑖𝑔ℎ𝑡 & 𝜇 𝑉𝑒𝑙 𝑑𝑜𝑜𝑟 s.t. 𝑃 𝑖=1 9 (𝐶𝑜𝑛𝑖 < 𝐶𝑟𝑖𝑡𝑖) ≥ 0.99 0.5 ≤ 𝜇 𝑥 𝑖 ≤ 1.5, 𝑖 = 1 … 7 0.192 ≤ 𝜇 𝑥 𝑗 ≤ 0.345, 𝑗 = 8 … 9
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 36 Design variables Uncertain Sources Constraints Objectives Optimization with Bayesian network Optimizer 𝑥1_𝑠𝑡 𝑥2_𝑠𝑡 𝑥3_𝑠𝑡 𝑥4_𝑠𝑡 𝑥5_𝑠𝑡 𝑥6_𝑠𝑡 𝑥7_𝑠𝑡 𝑥8_𝑠𝑡 𝑥9_𝑠𝑡 𝑥10 𝑥11
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 37 𝑪𝒐𝒔𝒕 𝑽𝒆𝒍 𝒅𝒐𝒐𝒓 𝑳𝒐𝒂𝒅 𝒂𝒃 𝑽𝒆𝒍 𝑩𝑷𝑫𝑽 𝟏 𝑫𝑽 𝟐 𝑪𝒐𝒔𝒕 𝑽𝒆𝒍 𝒅𝒐𝒐𝒓 𝑳𝒐𝒂𝒅 𝒂𝒃 𝑽𝒆𝒍 𝑩𝑷𝑫𝑽 𝟏 𝑫𝑽 𝟐 Conditionalization (forward) • Conditional samples are used to estimate objectives and joint probability (constraint) • Optimizer generates a set of design values to the BN • Bayesian network is conditionally sampled using Gaussian copula Posterior distributions of objectives and constraints In each BN evaluation: Optimizer: NSGA – II Optimizer: NSGA-II by VisualDOC
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 38 Pareto front - I Weight DoorVelocity Training values Weight Copula-generated samples
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 39 Training point selection (inverse) 𝑾𝒆𝒊𝒈𝒉𝒕 𝑽𝒆𝒍 𝒅𝒐𝒐𝒓 𝑳𝒐𝒂𝒅 𝒂𝒃 𝑽𝒆𝒍 𝑩𝑷𝑫𝑽 𝟏 𝑫𝑽 𝟐 Identify the input samples that relate to the desired outputs Weight Copula-generated samples Velocity Sculpting Select 20 input samples of the  calculated outputs Cooke, Zang, Mavis, Tai MAO Conf., 2015 Sample-based conditioning Rebuild a BN with the additional training samples
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 40 Weight DoorVelocity Pareto front – II • For comparison, another 20 samples are chosen by Latin Hypercube sampling  calculate outputs. • Rebuild a BN with the additional samples • Recalculate the Pareto front𝜇𝑑𝑜𝑜𝑟𝑣𝑒𝑙 𝜇 𝑤𝑒𝑖𝑔ℎ𝑡 𝜇 𝑤𝑒𝑖𝑔ℎ𝑡 Approach II: selectively generated samplesApproach I: uniformly generated samples
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 41 Objective-3 summary Multi-objective optimization under uncertainty • Probabilistic graphical surrogate model • Efficient joint probability estimation • Concurrent training point selection for multiple outputs Features of methodology • Bayesian network • Gaussian copula sampling • Sculpting
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 42 Research objectives  MDA with epistemic uncertainty - Inclusion of data uncertainty and model error  MDA with high-dimensional coupling - Large number of coupling variables - Dependence among all variables - Efficient uncertainty propagation  Multi-objective optimization under uncertainty - Reliability-based design optimization - Solution enumeration (Pareto front construction)  Multidisciplinary design optimization - Concurrent interdisciplinary compatibility enforcement and objective/constraint functions evaluation A. Multidisciplinary analysis under uncertainty B. Multidisciplinary optimization under uncertainty
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 43 Objective 4: MDO under uncertainty 𝑚𝑖𝑛 𝒙 )𝑓(𝒙 s.t. 𝑔𝑖 𝒙, 𝒖 𝒙 , 𝒗 𝒙 ≤ 0, 𝑖 = 1, . . . , 𝑛 𝑞 ℎ1 𝒙, 𝒖, 𝒗 = 0 ℎ2 𝒙, 𝒖, 𝒗 = 0 𝐹𝐸𝐴(𝒙, 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒𝑠) − 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡𝑠 = 𝟎 𝐶𝐹𝐷(𝒙, 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡𝑠) – 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒𝑠 = 𝟎 Interdisciplinary compatibility FEA CFD UQ / Reliability Analysis Optimization Nodal displacements Nodal pressures Deterministic MDO
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 44 MDO under Uncertainty Surrogate models are commonly used: • Training data can be expensive to get • Curse of dimensionality • Enforce compatibility • Evaluate (mean) objectives and (probability) constraints Simultaneously achieved without fully converged physics analysis? 𝑚𝑖𝑛 𝒙,𝝃 𝜇(𝑓 𝒙, 𝝃 ) s.t. 𝑃(𝑔𝑖 𝒙, 𝝃, 𝒖 𝒙, 𝝃 , 𝒗 𝒙, 𝝃 ≥ 0) ≤ 𝛼𝑖 𝑖 = 1, . . . , 𝑛 𝑞 ℎ1 𝒙, 𝜉, 𝒖, 𝒗 = 0 ℎ2(𝒙, 𝜉, 𝒖, 𝒗) = 0 𝝃 : a vector of random variables 𝝃 𝟏, … , 𝝃 𝒎 𝜉 : one realization of the random variable 𝜉 𝛼𝑖 : desired reliability for 𝑔𝑖 Problem formulation Inclusion of epistemic uncertainty
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 45  BNC-MDA + Optimization 𝑼 𝟐𝟏 𝒖 𝟐𝟏 Analysis2Analysis 1 𝒐𝒃𝒋 𝑪𝒐𝒏𝒔𝒕𝒓 𝑫𝑽, 𝑼𝑽 BNC-MDO  BN is built with samples of the one-iteration analysis  Optimization framework on the top 𝐷𝑉 𝑈𝑉 𝑢21 𝑈21 𝐶𝑜𝑛𝑠𝑡𝑟 𝑂𝑏𝑗 𝐷𝑖𝑓𝑓 OptimizationOne-iteration analysis MDA 𝑫𝒊𝒇𝒇 = 𝑼 − 𝒖
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 46 𝑫𝒊𝒇𝒇 𝒖 𝟐𝟏 𝑼 𝟐𝟏 𝑶𝒃𝒋𝑫𝑽 𝑼𝑽 𝑪𝒐𝒏 Concurrently enforce compatibility and estimate outputs  In each call of BN: • 𝐷𝑉 = 𝑑𝑒𝑠𝑖𝑔𝑛 𝑣𝑎𝑙𝑢𝑒, 𝑑𝑖𝑓𝑓 = 0  compatibility Conditional samples of objective and constraint are generated for further analysis Interdisciplinary compatibility and objective/constraint evaluation simultaneously achieved using BNC
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 47 MDO with High-dimensional Coupling 𝐷𝑉 𝑈𝑉 𝐶𝑜𝑛𝑠𝑡𝑟 𝑂𝑏𝑗 𝑃𝐶 𝑈21 𝑖𝑃𝐶 𝑢21 𝑖 𝜀 𝑃𝐶 𝑖 𝑃𝐶 𝑈21 2𝑃𝐶 𝑢21 2 𝜀 𝑃𝐶 𝑖 𝑃𝐶 𝑈21 𝑙 𝑃𝐶 𝑢21 𝑙 𝜀 𝑃𝐶 𝑖 • The size of the Bayesian network becomes very large • Including all coupling variables in one BN is unwieldy for training and sampling. BN reduction with PCA 1 MAR 17 2015 09:30:53 ELEMENTS 1 MAR172 12:54 ELEMENTS Nodal Pressures Nodal Displacements FEA CFD Aeroelastic wing analysis MDA 𝒊 𝒕𝒉 PC at 𝒏 − 𝟏 𝒕𝒉 iteration 𝒊 𝒕𝒉 PC at 𝒏𝒕𝒉 iteration Difference Small uncorrelated
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 48 Electrical Parameters Component Heat Total Power Dissipation Heatsink Temperature Electrical Analysis Thermal Analysis Watt Density: Total Power Dissipation Volume of the Heatsink Heatsink Size Parameters Example-1 : Electronic packaging MDO test suite: Heatsink Design variables: 𝑥1: heat sink width 𝑥2: heat sink length 𝑥3: fin length 𝑥4: fin width Uncertain variables: Variability of 𝑥1~𝑥4 𝑥5: nominal resistance at temperature 𝑇 𝑜 𝑥6: temperature coefficient of electrical resistance Likelihood-based non-parametric distribution
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 49 Case I: MDO with data uncertainty (coarsest mesh size for thermal analysis) max 𝜇 𝒙 𝜇 𝑃𝐷 𝑿 s.t. 𝑃 𝑇𝑒𝑚𝑝 < 56 𝑜 𝐶 ∩ 𝑉𝑜𝑙𝑢𝑚𝑒 < 6𝐸 − 4 𝑚3 ≥ 0.95 𝑙𝑏𝑖 ≤ 𝜇 𝑋 𝑖 ≤ 𝑢𝑏𝑖 𝑖 = 1, … , 4 𝑇ℎ𝑒𝑟𝑚𝑎𝑙 ℎ𝑒𝑎𝑡, 𝒙 − 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 = 0 𝐸𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 − ℎ𝑒𝑎𝑡 = 0 Joint probability RBDO: BN for optimization Design variables Uncertainty sources Coupling variables System output Compatibility condition • Solved using the proposed BNC-MDO • Component temperature is used to enforce the compatibility Optimizer: DIviding RECTangle (DIRECT) 𝑥1_𝑠𝑡 𝑥2_𝑠𝑡 𝑥3_𝑠𝑡 𝑥4_𝑠𝑡 𝑥5 𝑥6
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 50 Results Number of training samples for BN = 800 (5 seconds) Time for optimization with BNC = ~4 min RBDO using SOFPI (feedback): 10,000 / obj(con) evaluation No. of function evaluations till convergence ~= 5 Total number of function evaluations = 6,950,000 (~ 4 hours) • BN gives larger (hence better) objective values • SOFPI produces suboptimal solution (insufficient samples, unreliable) SOFPI (original model) BNC-MDO (surrogate) 𝒙 𝟏 0.056 0.052 𝒙 𝟐 0.056 0.052 𝒙 𝟑 0.021 0.021 𝒙 𝟒 0.039 0.024 objective 73172 73781 re-evaluate with original model objective N/A 84997 constraint 0.962
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 51 Case II: MDO with model uncertainty Model error in thermal Analysis • 2D steady state heat transfer equation (PDE) 𝛻𝑇𝑠𝑖𝑛𝑘 𝑥, 𝑦 + 𝑞(𝑥, 𝑦) 𝑘 = 0 • Solved by Finite Difference method • Limited computational resources  discretization error At each realization of input 𝒙: Training points GP Prediction  Auxiliary variable 𝑃ℎ representation of the stochastic model output  Sample 𝑃ℎ from 𝑈(0,1)  inverse CDF  deterministic output • GP prediction ~ 𝑁(𝜇, 𝜎)  𝜇 and 𝜎 are input dependent
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 52 Representation of Model Error in BN Design variables Uncertainty sources Coupling variables System output Compatibility condition (including 𝑷 𝒉) 𝑥1_𝑠𝑡 𝑥2_𝑠𝑡 𝑥3_𝑠𝑡 𝑥4_𝑠𝑡 𝑥5 𝑥6 𝑃ℎ
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 53 Results Design Variables RBDO using FPI Design Variables 𝑥1 0.077 𝑥2 0.149 𝑥3 0.021 𝑥4 0.010 Objective 𝝁 𝑾𝑫 11170 Constraint Joint Probability 0.99 • Cannot implement SOFPI since the FPI is hard to converge with stochastic model output. Iteration 𝝁 𝑾𝑫 • Optimizer: Genetic algorithm • 100 populations, 15 iterations. • Build BN with 120 samples (240 function evaluations).
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 54 𝑵𝑷 𝟑 𝑵𝑷 𝟒 Example-2 Aeroelastic wing design Design variables: • Backsweep angle 𝜃: [0 , 0.5] • Input variability: 𝜎 𝜃 ~ 𝑁(0, 0.03) • FPI takes 10 iterations to converge Coupling variables: nodal pressure • 𝑁𝑃3: nodal pressure after 3rd iteration • 𝑁𝑃4: nodal pressure after 4th iteration 𝑵𝑷 𝟑 after PCA 𝑵𝑷 𝟒 after PCA Difference = 0 I/O BN with 30 principal components
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 55 Optimization problem and solution Optimal Value Design variable 𝝁 𝒃𝒘 0.405 Objective 𝝁𝒍𝒊𝒇𝒕 1707.5 Constraint 𝑃(𝑠𝑡𝑟𝑒𝑠𝑠) 0.998  Optimizer: DIRECT  67 calls of BN  547 seconds max 𝝁 𝒃𝒘 𝐸 𝐿 s.t 𝑃 𝑆 ≥ 3 ∗ 105 𝑃𝑎 ≤ 10−3 0.05 ≤ 𝜇 𝑏𝑤 ≤ 0.45 Optimization formulation Optimal solution Optimization history 0 5 10 1695 1700 1705 1710 No. of iterations Lift BN is trained with samples without full convergence analysis
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 56 Objective-4 summary BNC-MDO under uncertainty • Efficiently integrates of MDA and optimization under uncertainty • Simultaneously enforces the interdisciplinary compatibility and evaluates objectives and constraints Features of methodology • BNC • PCA • Optimization algorithm (DIRECT/GA)
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 57 Future Work (1)Scalability of the proposed BNC-MDA approach needs to be investigated by solving larger problems. (2)Extension in multi-level analyses, and multi-disciplinary feedback coupled analyses (for more than two disciplines). (3)Extension to robustness-based design optimization under both aleatory and epistemic uncertainty. (4)Analytical multi-normal integration of the Gaussian copula instead of the sampling-based strategy for reliability assessment. (5)Improve the efficiency for non-Gaussian copulas. (6)Extension to time-dependent problems.
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 58 List of journal manuscripts 1. Liang, C. and Mahadevan, S., Stochastic Multi-Disciplinary Analysis under Epistemic Uncertainty, Journal of Mechanical Design, Vol. 137, Issue 2, 2015. 2. Liang, C. and Mahadevan, S., Bayesian Sensitivity Analysis and Uncertainty Integration for Robust Optimization, Journal of Aerospace Information Systems, Vol 12, Issue 1, 2015. 3. Rangavajhala, S., Liang, C., Mahadevan, S. and Hombal, V., Concurrent optimization of mesh refinement and design parameters in multidisciplinary design, Journal of Aircraft, Vol. 49, No. 6, 2012. 4. Liang, C. and Mahadevan, S., Stochastic Multidisciplinary Analysis with High- Dimensional Coupling, AIAA Journal, under review. 5. Liang, C. and Mahadevan, S., Pareto Surface Construction for Multi-objective Optimization under Uncertainty, ready for submission. 6. Liang, C. and Mahadevan, S., Probabilistic Graphical Modeling for Multidisciplinary Optimization, ready for submission.
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 59 List of conference proceedings 1. Liang, C. and Mahadevan, S., Reliability-based Multi-objective Optimization under Uncertainty, 16th AIAA/ISSMO Multidisciplinary Analysis and Optmization Conference, Dallas, Texas, 2015. 2. Liang, C. and Mahadevan, S., Bayesian Framework for Multidisciplinary Uncertainty Quantification and Optimization, 16th AIAA Non-Deterministic Approaches Conference, National Harbor, Maryland, 2014. 3. Liang, C. and Mahadevan, S., Multidisciplinary Analysis and Optimization under Uncertainty, 11th International Conference on Structural Safety and Reliability, New York, New York, 2013. 4. Liang, C. and Mahadevan, S., Multidisciplinary Analysis under Uncertainty, 10th World Congress of Structural and Multidisciplinary Optimization, Orlando, Florida, 2013. 5. Liang, C. and Mahadevan, S., Design Optimization under Aleatory and Epistemic Uncertainty, 10th World Congress of Structural and Multidisciplinary Optimization, Orlando, Florida, 2013. 6. Liang, C. and Mahadevan, S., Inclusion of Data Uncertainty and Model Error in Multi- disciplinary Analysis and Optimization, 54th Structures, Structural Dynamics, and Materials Conference, Boston, Massachusetts, 2013. 7. Liang, C. and Mahadevan, S., Design Optimization under Aleatory and Epistemic Uncertainties, 14th Multidisciplinary Analysis and Optimization Conference, Indianapolis, Indiana, 2012.
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 60 Acknowledgement Committee members: Dr. Prodyot Basu (CEE) Dr. Mark N. Ellingham (Math) Dr. Mark P. McDonald (Lipscomb) Dr. Dimitri Mavris (GT) Dr. Roger M. Cooke (TU Delft) University of Melbourne: Dr. Anca Hanea Dan Ababei Vanderbilt University: Dr. Sirisha Rangvajhala Dr. Shankar Sankararaman Dr. You Ling Dr. Vadiraj Hombal Adviser: Dr. Sankaran Mahadevan Ghina Nakad Absi, Dr. Bethany Burkhart, Beverly Piatt Defense preparation: Great friends at Vanderbilt University !
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 61 Acknowledgement Funding support: (1) NASA Langley National Laboratory (2) Sandia National Laboratory (3) Vanderbilt University, Department of Civil and Environmental Engineering Software Licenses: (1) UNINET by LightTwist Inc. (Dan Ababei) (2) VisualDOC by Vanderplaats R&D Inc. (Garret Vanderplaats, Juan-Pablo Leiva)
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 62
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 63 Fit a Gaussian process model to 𝒇 with 𝑥 𝑇 and 𝑦 𝑇 as inputs, and predict the function value at desired points 𝑥 𝑃: GP  Model the underlying covariance in the data instead of the functional form: GP Surrogate Modeling 𝑝 𝑓𝑝 𝒚 𝑻, 𝒙 𝑻, 𝒙 𝑷, 𝚯 ~𝑁(𝑚, 𝑆) Function value Training data Prediction Point GP Parameters Gaussian distribution 𝒎 = 𝑲 𝑷𝑻 𝑲 𝑻𝑻 + 𝝈 𝒏 𝟐 𝑰 −𝟏 𝒚 𝑻 𝑺 = 𝑲 𝑷𝑷 − 𝑲 𝑷𝑻 𝑲 𝑻𝑻 + 𝝈 𝒏 𝟐 𝑰 −𝟏 𝑲 𝑻𝑷 • Models that evaluate 𝒈 are substituted by GP
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 64 Vine copula based sampling 𝑋1 𝑌1 𝑌2 𝑋2 M 𝑋1 𝑋2 𝑌1 𝑌2 Goal: estimate 𝑓 𝑋1, 𝑋2, 𝑌1, 𝑌2 𝜌𝑖𝑗 = 2sin( 𝑟𝑖𝑗 𝜋 6 ) 𝜌12;3…𝑛 = 𝜌12;3…,𝑛−1 − 𝜌1𝑛;3,…,𝑛−1 ∗ 𝜌2𝑛;3,…,𝑛−1 1 − 𝜌1𝑛;3,…,𝑛−1 2 1 − 𝜌2𝑛;3,…,𝑛−1 2 𝑐 𝑅 𝑷 = 1 det 𝑅 exp − 1 2 Φ−1 𝑃𝑥1 Φ−1 𝑃𝑥2 Φ−1 𝑃𝑦1 Φ−1 𝑃𝑦2 ∙ 𝑅−1 − 𝐼 ∙ Φ−1 𝑃𝑥1 Φ−1 𝑃𝑥2 Φ−1 𝑃𝑦1 Φ−1 𝑃𝑦2 where 𝜌𝑖𝑗 are the elements of 𝑅 𝑟𝑖𝑗
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 65 Kullback-Leibler Divergence For continuous distributions 𝑝 and 𝑞 𝐷 𝐾𝐿(𝑝||𝑞) = −∞ +∞ 𝑝(𝑥)l n( 𝑝 𝑥 )𝑞(𝑥 )𝑑𝑥 Numerical implementation 𝐷 𝐾𝐿(𝑝||𝑞) = 𝑖=1 𝑛 ln 𝑝 𝑥𝑖 𝑞 𝑥𝑖 𝑝 𝑥𝑖 ∗ (𝑥𝑖 − 𝑥𝑖−1)
Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 66 Vine copula based sampling 𝑋1 𝑌1 𝑌2 2 5 4 6 𝑋2 1 M 𝑋1 𝑋2 𝑌1 𝑌2 Goal: estimate 𝑓 𝑋1, 𝑋2, 𝑌1, 𝑌2 𝑟𝑋2 𝑌2 𝑟𝑋1 𝑌1 𝑟𝑋1 𝑋2 𝑟𝑋2 𝑌1|𝑋1 𝑟𝑌1 𝑌2|𝑋1 𝑋2 𝑋1 𝑋2𝑌1 𝑌2 𝑟𝑋1 𝑌2|𝑋2 𝜌𝑖𝑗 = 2sin( 𝑟𝑖𝑗 𝜋 6 ) 𝜌12;3…𝑛 = 𝜌12;3…,𝑛−1 − 𝜌1𝑛;3,…,𝑛−1 ∗ 𝜌2𝑛;3,…,𝑛−1 1 − 𝜌1𝑛;3,…,𝑛−1 2 1 − 𝜌2𝑛;3,…,𝑛−1 2 𝑐 𝑅 𝑷 = 1 det 𝑅 exp − 1 2 Φ−1 𝑃𝑥1 Φ−1 𝑃𝑥2 Φ−1 𝑃𝑦1 Φ−1 𝑃𝑦2 ∙ 𝑅−1 − 𝐼 ∙ Φ−1 𝑃𝑥1 Φ−1 𝑃𝑥2 Φ−1 𝑃𝑦1 Φ−1 𝑃𝑦2 where 𝜌𝑖𝑗 are the elements of 𝑅

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Multidisciplinary analysis and optimization under uncertainty

  • 1. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 1 Multidisciplinary Analysis and Optimization under Uncertainty Chen Liang Dissertation Defense Adviser: Sankaran Mahadevan Department of Civil and Environmental Engineering Vanderbilt University, Nashville, TN Aug. 21st , 2015
  • 2. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 2 MDA Overview Three-objective two-stage-to-orbit launch vehicle Heatsink Aircraft wing analysis Nodal Pressures Nodal Displacements Wing Backsweep Angle, Speed and Angle of Attack Lift, drag, stress FEA structure CFD fluid Compatibility  Fixed-point-iteration (FPI)
  • 3. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 3 MDO under uncertainty  Presence of uncertainty sources  UQ  Sampling outside FPI SOFPI Repeated MDA  New design input values at each iteration Computationally unaffordable  Need efficient methods for MDA and MDO under uncertainty UQ / Reliability Analysis FEA CFD MDA Optimization
  • 4. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 4 Three types of sources • Physical variability • Data uncertainty (e.g., sparse/interval data) • Model Uncertainty Forward problem • For a given input Uncertainty of output needs to be evaluated • Propagation of aleatory uncertainty is well-studied • Inclusion of epistemic uncertainty becomes more important • Little work regarding the propagation of epistemic uncertainty in feedback coupled MDA Uncertainty and errors in optimization Sources of Uncertainty and Errors Aleatory (Irreducible) Natural Variability Epistemic (Reducible) Data Uncertainty Sparse Data Interval Data Model Uncertainty Numerical error Discretization error Round off error Truncation error Surrogate model error UQ error Model Form Error
  • 5. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 5 Overall Research Goal  Efficient UQ techniques for feedback coupled MDA and MDO  Combine information with both aleatory and epistemic sources of uncertainty  Particular emphasis on • Representation of epistemic sources of uncertainty • Propagation through feedback coupled analysis • Inclusion in the design optimization of multidisciplinary analysis with feedback coupling (high-dimensional)
  • 6. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 6 Research objectives  MDA with epistemic uncertainty - Inclusion of data uncertainty and model error  MDA with high-dimensional coupling - Large number of coupling variables - Dependence among all variables - Efficient uncertainty propagation  Multi-objective optimization under uncertainty - Reliability-based design optimization - Solution enumeration (Pareto front construction)  Multidisciplinary design optimization - Concurrent interdisciplinary compatibility enforcement and objective/constraint functions evaluation A. Multidisciplinary analysis under uncertainty B. Multidisciplinary optimization under uncertainty Bayesian Framework
  • 7. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 7 𝑔2 𝑓 𝑢12 𝑔1 𝑢21 Analysis 1 𝑨 𝟏(𝒙, 𝑢21) Analysis 2 𝑨 𝟐(𝒙, 𝑢12) Analysis 3 𝑨 𝟑(𝑔1, 𝑔2) 𝑥1 𝑥 𝑠 𝑥2  Uncertainty propagation under the compatibility condition  No need for full convergence analysis Multi-disciplinary multi-level system Review of MDA under uncertainty methods Sankararaman & Mahadevan, J. Mechanical Design, 2012 Approximation Method • First-order Second Moment (FOSM) approximations • Linear approximations of disciplinary analyses • PDF based on mean & variance • Du & Chen, Mahadevan & Smith • Fully Decoupled Approach • Calculate PDFs of u12 & u21 • Cut-off feedback  both directions • Ignores dependence between u12 & u21 • Lack of one-to—one correspondence between 𝑔1 and 𝑔2 in calculating f Likelihood-based approach for MDA (LAMDA) 𝑢21 𝑢12
  • 8. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 8 𝑢12 𝑔2 𝑓 𝑢12 𝑔1 𝑢21 Analysis 1 𝑨 𝟏(𝒙, 𝑢21) Analysis 2 𝑨 𝟐(𝒙, 𝑢12) Analysis 3 𝑨 𝟑(𝑔1, 𝑔2) 𝑥1 𝑥 𝑠 𝑥2 Multi-disciplinary multi-level system Likelihood-based approach for MDA (LAMDA) Objective 1: MDA with epistemic uncertainty 𝑔2 𝑓 𝑔1 Analysis 1 𝑨 𝟏(𝒙, 𝑢21) Analysis 2 𝑨 𝟐(𝒙, 𝑢12) Analysis 3 𝑨 𝟑(𝑔1, 𝑔2) 𝑥1 𝑥 𝑠𝑥2 𝑢21
  • 9. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 9 𝑈12 𝐹 𝑈12 𝐺Interdisciplinary compatibility: LAMDA 𝑢12 Analysis 1 𝑨 𝟏(𝒙, 𝑢21) Analysis 2 𝑨 𝟐(𝒙, 𝑢12) 𝑢21 𝑈12 Given a value of 𝑢12 what is 𝑃(𝑈12 = 𝑢12|𝑢12)  𝐿(𝑢12) 𝑓 𝑢12 = 𝐿(𝑢12) 𝐿(𝑢12)𝑑𝑢12 FORM is used to calculate the CDFs of the upper and lower bounds: 𝑃𝑢 and 𝑃𝑙 𝐿 𝑢12 ∝ 𝑢12− 𝜀 2 𝑢12+ 𝜀 2 𝑓𝑈12 𝑈12 𝑢12 𝑑𝑈12 𝐿 𝑢12 ∝ (𝑃𝑢 − 𝑃𝑙)  finite difference 𝑷 𝒖 𝑷𝒍 𝒖 𝟏𝟐 + 𝜺 𝟐 𝒖 𝟏𝟐 − 𝜺 𝟐
  • 10. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 10 Sources of Uncertainty and Errors Aleatory (Irreducible) Natural Variability Epistemic (Reducible) Data Uncertainty Sparse Data Interval Data Model Uncertainty Numerical error Discretization error Round off error Truncation error Surrogate model error UQ error Model Form Error Uncertainty and errors in LAMDA Considering epistemic uncertainty sources
  • 11. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 11 Data uncertainty (sparse and interval data) pi pQ X fX(x) θQθ3θ2θ1 θi p1 p2                  n i b a X m i iX dxPxfPxfPL i i 11 )|()|()( Parametric approach Non-parametric approach Likelihood Sparse data Interval data Convert sparse and interval data into a useable distribution (for propagation) Sankararaman & Mahadevan RESS 2011 Zaman, et. al, RESS 2011
  • 12. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 12 Sources of Uncertainty and Errors Aleatory (Irreducible) Natural Variability Epistemic (Reducible) Data Uncertainty Sparse Data Interval Data Model Uncertainty Numerical error Discretization error Round off error Truncation error Surrogate model error UQ error Model Form Error Uncertainty and errors in LAMDA
  • 13. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 13 Model uncertainty estimation Training points (e.g. FEA) Prediction Uncertainty Discretization error estimation • GP prediction ~ 𝑁(𝜇, 𝜎)  𝜇 and 𝜎 are input dependent Rangavajhala, et. al, AIAA Journal 2010 Richardson extrapolation At each input 𝒙
  • 14. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 14  Auxiliary variable 𝑃ℎ~𝑈[0,1]  CDF of GP output  Stochastic model output  input random variable  FORM can be used for likelihood evaluation 𝑈12 CDF of GP output 𝐿 𝑢12 ∝ 𝑃(𝑈12 = 𝑢12|𝑢12) • Equation only calculable when 𝑈12 is deterministic given an 𝒙 and 𝑢12 Inclusion of model uncertainty in LAMDA 𝑢12 𝑢21 𝐴2 𝒙, 𝑢12 GP model (𝒙, 𝑢21) 𝒙 𝑃ℎ Deterministic 𝑃ℎ Extra loop of uncertainty propagation 
  • 15. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 15 Electrical Parameters Component Heat Total Power Dissipation Heatsink Temperature Electrical Analysis Thermal Analysis Power Density: Total Power Dissipation Volume of the Heatsink Heatsink Size Parameters Numerical example: electronic packaging Model error in thermal Analysis • 2D steady state heat transfer equation (PDE) 𝛻𝑇𝑠𝑖𝑛𝑘 𝑥, 𝑦 + 𝑞(𝑥, 𝑦) 𝑘 = 0 • Solved by Finite Difference method • Limited computational resources  discretization error MDO test suite: Heatsink Data uncertainty Temperature Coefficient of resistance (𝜶) Data points Data intervals 0.0055 0.0057 [0.004,0.009] [0.0043,0.0085] [0.0045, 0.0088] 4 5 6 7 8 x 10 -3 0 200 400 600 800 1000 1200 Non-parametric PDF of 𝜶 PDF 𝜶 • Uncertainty estimated auxiliary variable
  • 16. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 16 Temperature Thermal Analysis Electrical Analysis Power density 𝒙 𝟐𝒙 𝟏 Heat
  • 17. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 17 Results Uncertainty of the coupling variables Comparison between FPI and LAMDA  FPI with stochastic model errors is difficult to converge  Only a few FPI realizations is affordable  LAMDA agrees well with the available data Liang & Mahadevan ASME JMD, 2015 Temerature(℃ ) PDF Component heat(Joule) PDF
  • 18. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 18 Likelihood Approach for Multidisciplinary Analysis • Data uncertainty and model error in feedback coupled analysis. • Auxiliary variable  stochastic model error. Features of methodology • Likelihood-based approach for MDA (LAMDA) • FORM • GP estimation of model error • Auxiliary variable Objective-1 summary
  • 19. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 19 High Dimensional Coupling CFD FEA UQ/Reliability Analysis Nodal Pressures Nodal Displacements  Multiple coupling variables in one direction  Joint distribution of the coupling variables in the same direction FORM-based LAMDA is inefficient because: • First-order approximation: dimension ↑, accuracy ↓ • Likelihood calculation (finite difference)  Number of function ↑
  • 20. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 20 Research objectives  MDA with epistemic uncertainty - Inclusion of data uncertainty and model error  MDA with high-dimensional coupling - Large number of coupling variables - Dependence among all variables - Efficient uncertainty propagation  Multi-objective optimization under uncertainty - Reliability-based design optimization - Solution enumeration (Pareto front construction)  Multidisciplinary design optimization - Concurrent interdisciplinary compatibility enforcement and objective/constraint functions evaluation A. Multidisciplinary analysis under uncertainty B. Multidisciplinary optimization under uncertainty
  • 21. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 21  Probabilistic graphical model: represents random variables and their conditional dependencies by nodes and edges.  Incorporate large number of variables with heterogeneous formats: distribution (continuous, discrete, empirical) & function.  Bayesian network is update by sampling approaches. An efficient Gaussian copula-based sampling method is adopted. Bayesian network and copula-based sampling (BNC) 𝑌 𝑍 𝑋 𝑉 Uncertainty propagation (forward) 𝑣 𝑜𝑏𝑠 𝑌 𝑍 𝑋 Bayesian updating (inverse)
  • 22. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 22 Multi-variate Gaussian Copula • A copula is a function that joins the CDFs of multiple random variables as a joint CDF function. • Types: Gaussian, Clayton, Gumbel, … • The Multivariate Gaussian Copula (MGC) 𝐶𝛴 𝐺𝑎𝑢𝑠𝑠 𝒖 = 𝚽 𝚺 Φ−1 𝑢1 … Φ−1 (𝑢 𝑛) where 𝑢𝑖 is the CDF value of any arbitrary marginal 𝐹𝑋𝑖 (𝑥𝑖), • Gaussian copula assumption needs verification (done for all examples) • Other copulas are not as efficient in conditional sampling Hanea, et al., QREI, 2006 Efficient Conditional Sampling 𝑉 = 𝑣 𝑂𝑏𝑠 𝑌 𝑍 𝑋 Bayesian updating (inverse)  If 𝑉 = 𝑉𝑜𝑏𝑠, the conditional samples of 𝑋, 𝑌 and 𝑍 can be obtained as following: 𝑥 = 𝐹𝑋 −1 ΦΣ′ 𝑈 𝑋|𝑉 = 𝑣 𝑜𝑏𝑠 𝑦 = 𝐹𝑌 −1 ΦΣ′ 𝑈 𝑌|𝑉 = 𝑣 𝑜𝑏𝑠 𝑧 = 𝐹𝑍 −1 ΦΣ′ 𝑈 𝑍|𝑉 = 𝑣 𝑜𝑏𝑠
  • 23. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 23 BNC-MDA: 𝒖 𝟐𝟏 𝑼 𝟐𝟏 𝜺 𝒖 𝟐𝟏 BN with 20 coupling variables • Joint distribution of 𝑼 𝟐𝟏 are evaluated by the conditional samples • Given compatibility condition (𝜀21 = 0), generate samples from the BN by copula-based sampling 𝑼 𝟐𝟏𝒖 𝟐𝟏 𝒖 𝟏𝟐 Analysis 2 𝑨 𝟐(𝒖 𝟏𝟐, x) Analysis 1 𝑨 𝟏(𝒖 𝟐𝟏, x) 𝒙 𝜺 𝒖 𝟐𝟏 = 𝒖 𝟐𝟏 − 𝑼 𝟐𝟏 Interdisciplinary compatibility = 𝟎 • Bayesian network is built using samples of 𝑢21, 𝑈21 and 𝜀 𝑢21 𝒖 𝟐𝟏 𝑼 𝟐𝟏 𝜺 𝒖 𝟐𝟏 One coupling variable
  • 24. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 24 Challenge for BNC-MDA High dimensional coupling • Mesh resolution : 258 nodes  258 random variables 1 MAR 17 2015 09:30:53 ELEMENTS 1 MAR172015 12:54:37 ELEMENTS Nodal Pressures Nodal Displacements FEA CFD Aero-elastic analysis of an aircraft wing 𝑵𝑷𝒊−𝟏 𝑵𝑷𝒊 𝜺 • BN with 774 nodes : enormous effort • Variables are highly correlated  Redundant information, singularity of correlation coefficient matrix of the copula
  • 25. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 25 Principal component analysis (PCA)  Correlated variables  linearly uncorrelated principal component space  First 15~20 PCs cover more than 99% of the original variances  Bayesian updating on the 15~20 individual uncorrelated principal components Reduces a giant Bayesian network into 15~20 small networks Bayesian update can be implemented in parallel 𝒊𝒕𝒉 PC at 𝒏 − 𝟏 𝒕𝒉 iteration 𝒊𝒕𝒉 PC at 𝒏 𝒕𝒉 iteration Difference Updated by Interdisciplinary compatibility (Difference = 0)
  • 26. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 26 𝑵𝑷 𝟐 𝑵𝑷 𝟑 Numerical Example – MDA of an aircraft wing Input uncertainty • Backsweep angle 𝜃 ~ 𝑁 0.4,0.1 • 110 FPI analyses  Benchmark • 6 iterations till convergence  ~1300 function calls (FEA and CFD in total) FPI (secs) BNC-MDA (secs) Time saved (secs) 14,300 9,350 4,950 Total time consumed  Kullback-Leibler (K-L) divergence with benchmark solution : • Smaller value  Closer distributions Higher fidelity  more time saved Method 2nd Iteration 3rd Iteration BNC K-L divergence 0.21 0.18 0.16 Estimation without full convergence analysis = 0 PCA Reduced 𝑵𝑷 𝟐 Difference PCA Reduced 𝑵𝑷 𝟑 • 𝑁𝑃2: nodal pressure after 2nd iteration • 𝑁𝑃3: nodal pressure after 3rd iteration • 𝑁𝑃2 & 𝑁𝑃3  BNC-MDA (660 function calls)
  • 27. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 27 Objective-2 summary Bayesian Approach for Multidisciplinary Analysis • Novel BNC-MDA for UQ in high-dimensional coupled MDA • Dimension and iteration reduction  Efficient while preserving the dependencies • Time for physics analysis ≫ time for stochastic analysis Features of methodology • Bayesian Network • Gaussian copula-based sampling • Principal component analysis
  • 28. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 28 Research objectives  MDA with epistemic uncertainty - Inclusion of data uncertainty and model error  MDA with high-dimensional coupling - Large number of coupling variables - Dependence among all variables - Efficient uncertainty propagation  Multi-objective optimization under uncertainty - Reliability-based design optimization - Solution enumeration (Pareto front construction)  Multidisciplinary design optimization - Concurrent interdisciplinary compatibility enforcement and objective/constraint functions evaluation A. Multidisciplinary analysis under uncertainty B. Multidisciplinary optimization under uncertainty
  • 29. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 29 Robustness-based Design Optimization (RDO) Reliability-based Design Optimization (RBDO) Attempts to minimize variability in the system performance due to variations in the inputs. Aims to maintain design feasibility at desired reliability levels. Background: Optimization under uncertainty Focuses on 𝜎𝑜𝑏𝑗 of the objective function Focuses on 𝑃𝑓 of the constraint function 𝝈 𝒐𝒃𝒋 𝑰 𝝈 𝒐𝒃𝒋 𝑰𝑰 Objective Function PDF Constraint Function PDF
  • 30. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 30 𝑚𝑖𝑛 𝜇 𝑋,𝑑 [𝜇 𝑓 𝑋, 𝑃, 𝑑, 𝑝 𝑑 ] s.t. Prob(𝑔𝑖(𝑋, 𝑑, 𝑃, 𝑝 𝑑) ≥ 0)) ≥ 𝑝𝑡 𝑖 , i= 1,2 … , 𝑛 𝑞 Prob(𝑋 ≥ 𝑙𝑏 𝑋)) ≥ 𝑝𝑙𝑏 𝑡 Prob(𝑋 ≤ 𝑢𝑏 𝑋)) ≥ 𝑝 𝑢𝑏 𝑡 𝑙𝑏 𝑑 ≤ 𝑑 ≤ 𝑢𝑏 𝑑 Reliability-based design optimization • Natural variability • Data uncertainty • Sparse • Interval Model uncertainty • Numerical solution error • Model form error • The examples are formulated using the RBDO formulation • Proposed methodology is adaptable to solve RDO problems 𝑋: stochastic design variable 𝑃: stochastic non-design variable 𝑑: deterministic design variable 𝑃𝑑: deterministic non-design variable
  • 31. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 31 Research objectives  MDA with epistemic uncertainty - Inclusion of data uncertainty and model error  MDA with high-dimensional coupling - Large number of coupling variables - Dependence among all variables - Efficient uncertainty propagation  Multi-objective optimization under uncertainty - Reliability-based design optimization - Solution enumeration (Pareto front construction)  Multidisciplinary design optimization - Concurrent interdisciplinary compatibility enforcement and objective/constraint functions evaluation A. Multidisciplinary analysis under uncertainty B. Multidisciplinary optimization under uncertainty
  • 32. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 32 • Conflicting objectives  One objective cannot be improved without worsening others Objective 3: Multi-objective optimization • Pareto front  tradeoff relationship between different objectives 𝑚𝑖𝑛 𝒙 [𝑓1 𝑿, 𝑃 , 𝑓2 𝑿, 𝑃 ] s.t. 𝑙𝑏 𝑋 ≤ 𝑿 ≤ 𝑢𝑏 𝑋 • Existing methods  Weighted sum  𝜀-Constraint  Goal programming  Non-dominated sorting genetic algorithm (NSGA) Assign weights to each objective One as objective, others as constraints Global search and solution- ranking strategy Optimizes the weighted sum of the penalty
  • 33. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 33 𝑚𝑖𝑛 𝜇 𝑋,𝑑 [𝜇 𝑓1 𝑋, 𝑑, 𝑃, 𝑝 𝑑 , 𝜇 𝑓2 𝑋, 𝑑, 𝑃, 𝑝 𝑑 , … , 𝜇 𝑓𝑛 (𝑋, 𝑑, 𝑃, 𝑝 𝑑)] s.t. Prob(𝑔𝑖(𝑋, 𝑑, 𝑃, 𝑝 𝑑) ≥ 0)) ≥ 𝑝𝑡 𝑖 , i= 1,2 … , 𝑛 𝑞 Prob(𝑋 ≥ 𝑙𝑏 𝑋)) ≥ 𝑝𝑙𝑏 𝑡 Prob(𝑋 ≤ 𝑢𝑏 𝑋)) ≥ 𝑝 𝑢𝑏 𝑡 𝑙𝑏 𝑑 ≤ 𝑑 ≤ 𝑢𝑏 𝑑 Multi-objective optimization under uncertainty • Challenges of existing MOO methods - Weights for objective aggregation is not intuitive - Goal programmingConstraint-based method may not produce Pareto solutions - NSGA is computationally expensive (surrogate models) • Little work regarding the dependence relationships between output variables (e.g. co-kriging for small number of outputs)
  • 34. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 34 Dependence among Objectives/Constraints • Joint probability formulation for MOUU - Considered the dependence among objectives - Joint probability of the design threshold being satisfied  constraint - FORM Rangavajhala & Mahadevan JMD 2011 Proposed method  Graphical surrogate model that integrates design variables, uncertain variables, objectives and constraints in one Bayesian network  Gaussian copula-based sampling for efficient uncertainty propagation and reliability assessment (forward propagation)  Training samples selection to improve the Pareto front (inverse problem) Inefficient for high-dimensional problems
  • 35. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 35 Numerical example: vehicle side impact model  FEA data is unavailable  Step-wise regression model  generate training data & validate the proposed method  115 training points by Latin Hypercube sampling Vehicle side impact model Gu, et al. , IJVD, 2001 Problem description * Design variables have variability, 2 additional uncertain variables min 𝝁 𝒙 𝜇 𝑊𝑒𝑖𝑔ℎ𝑡 & 𝜇 𝑉𝑒𝑙 𝑑𝑜𝑜𝑟 s.t. 𝑃 𝑖=1 9 (𝐶𝑜𝑛𝑖 < 𝐶𝑟𝑖𝑡𝑖) ≥ 0.99 0.5 ≤ 𝜇 𝑥 𝑖 ≤ 1.5, 𝑖 = 1 … 7 0.192 ≤ 𝜇 𝑥 𝑗 ≤ 0.345, 𝑗 = 8 … 9
  • 36. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 36 Design variables Uncertain Sources Constraints Objectives Optimization with Bayesian network Optimizer 𝑥1_𝑠𝑡 𝑥2_𝑠𝑡 𝑥3_𝑠𝑡 𝑥4_𝑠𝑡 𝑥5_𝑠𝑡 𝑥6_𝑠𝑡 𝑥7_𝑠𝑡 𝑥8_𝑠𝑡 𝑥9_𝑠𝑡 𝑥10 𝑥11
  • 37. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 37 𝑪𝒐𝒔𝒕 𝑽𝒆𝒍 𝒅𝒐𝒐𝒓 𝑳𝒐𝒂𝒅 𝒂𝒃 𝑽𝒆𝒍 𝑩𝑷𝑫𝑽 𝟏 𝑫𝑽 𝟐 𝑪𝒐𝒔𝒕 𝑽𝒆𝒍 𝒅𝒐𝒐𝒓 𝑳𝒐𝒂𝒅 𝒂𝒃 𝑽𝒆𝒍 𝑩𝑷𝑫𝑽 𝟏 𝑫𝑽 𝟐 Conditionalization (forward) • Conditional samples are used to estimate objectives and joint probability (constraint) • Optimizer generates a set of design values to the BN • Bayesian network is conditionally sampled using Gaussian copula Posterior distributions of objectives and constraints In each BN evaluation: Optimizer: NSGA – II Optimizer: NSGA-II by VisualDOC
  • 38. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 38 Pareto front - I Weight DoorVelocity Training values Weight Copula-generated samples
  • 39. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 39 Training point selection (inverse) 𝑾𝒆𝒊𝒈𝒉𝒕 𝑽𝒆𝒍 𝒅𝒐𝒐𝒓 𝑳𝒐𝒂𝒅 𝒂𝒃 𝑽𝒆𝒍 𝑩𝑷𝑫𝑽 𝟏 𝑫𝑽 𝟐 Identify the input samples that relate to the desired outputs Weight Copula-generated samples Velocity Sculpting Select 20 input samples of the  calculated outputs Cooke, Zang, Mavis, Tai MAO Conf., 2015 Sample-based conditioning Rebuild a BN with the additional training samples
  • 40. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 40 Weight DoorVelocity Pareto front – II • For comparison, another 20 samples are chosen by Latin Hypercube sampling  calculate outputs. • Rebuild a BN with the additional samples • Recalculate the Pareto front𝜇𝑑𝑜𝑜𝑟𝑣𝑒𝑙 𝜇 𝑤𝑒𝑖𝑔ℎ𝑡 𝜇 𝑤𝑒𝑖𝑔ℎ𝑡 Approach II: selectively generated samplesApproach I: uniformly generated samples
  • 41. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 41 Objective-3 summary Multi-objective optimization under uncertainty • Probabilistic graphical surrogate model • Efficient joint probability estimation • Concurrent training point selection for multiple outputs Features of methodology • Bayesian network • Gaussian copula sampling • Sculpting
  • 42. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 42 Research objectives  MDA with epistemic uncertainty - Inclusion of data uncertainty and model error  MDA with high-dimensional coupling - Large number of coupling variables - Dependence among all variables - Efficient uncertainty propagation  Multi-objective optimization under uncertainty - Reliability-based design optimization - Solution enumeration (Pareto front construction)  Multidisciplinary design optimization - Concurrent interdisciplinary compatibility enforcement and objective/constraint functions evaluation A. Multidisciplinary analysis under uncertainty B. Multidisciplinary optimization under uncertainty
  • 43. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 43 Objective 4: MDO under uncertainty 𝑚𝑖𝑛 𝒙 )𝑓(𝒙 s.t. 𝑔𝑖 𝒙, 𝒖 𝒙 , 𝒗 𝒙 ≤ 0, 𝑖 = 1, . . . , 𝑛 𝑞 ℎ1 𝒙, 𝒖, 𝒗 = 0 ℎ2 𝒙, 𝒖, 𝒗 = 0 𝐹𝐸𝐴(𝒙, 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒𝑠) − 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡𝑠 = 𝟎 𝐶𝐹𝐷(𝒙, 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡𝑠) – 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒𝑠 = 𝟎 Interdisciplinary compatibility FEA CFD UQ / Reliability Analysis Optimization Nodal displacements Nodal pressures Deterministic MDO
  • 44. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 44 MDO under Uncertainty Surrogate models are commonly used: • Training data can be expensive to get • Curse of dimensionality • Enforce compatibility • Evaluate (mean) objectives and (probability) constraints Simultaneously achieved without fully converged physics analysis? 𝑚𝑖𝑛 𝒙,𝝃 𝜇(𝑓 𝒙, 𝝃 ) s.t. 𝑃(𝑔𝑖 𝒙, 𝝃, 𝒖 𝒙, 𝝃 , 𝒗 𝒙, 𝝃 ≥ 0) ≤ 𝛼𝑖 𝑖 = 1, . . . , 𝑛 𝑞 ℎ1 𝒙, 𝜉, 𝒖, 𝒗 = 0 ℎ2(𝒙, 𝜉, 𝒖, 𝒗) = 0 𝝃 : a vector of random variables 𝝃 𝟏, … , 𝝃 𝒎 𝜉 : one realization of the random variable 𝜉 𝛼𝑖 : desired reliability for 𝑔𝑖 Problem formulation Inclusion of epistemic uncertainty
  • 45. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 45  BNC-MDA + Optimization 𝑼 𝟐𝟏 𝒖 𝟐𝟏 Analysis2Analysis 1 𝒐𝒃𝒋 𝑪𝒐𝒏𝒔𝒕𝒓 𝑫𝑽, 𝑼𝑽 BNC-MDO  BN is built with samples of the one-iteration analysis  Optimization framework on the top 𝐷𝑉 𝑈𝑉 𝑢21 𝑈21 𝐶𝑜𝑛𝑠𝑡𝑟 𝑂𝑏𝑗 𝐷𝑖𝑓𝑓 OptimizationOne-iteration analysis MDA 𝑫𝒊𝒇𝒇 = 𝑼 − 𝒖
  • 46. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 46 𝑫𝒊𝒇𝒇 𝒖 𝟐𝟏 𝑼 𝟐𝟏 𝑶𝒃𝒋𝑫𝑽 𝑼𝑽 𝑪𝒐𝒏 Concurrently enforce compatibility and estimate outputs  In each call of BN: • 𝐷𝑉 = 𝑑𝑒𝑠𝑖𝑔𝑛 𝑣𝑎𝑙𝑢𝑒, 𝑑𝑖𝑓𝑓 = 0  compatibility Conditional samples of objective and constraint are generated for further analysis Interdisciplinary compatibility and objective/constraint evaluation simultaneously achieved using BNC
  • 47. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 47 MDO with High-dimensional Coupling 𝐷𝑉 𝑈𝑉 𝐶𝑜𝑛𝑠𝑡𝑟 𝑂𝑏𝑗 𝑃𝐶 𝑈21 𝑖𝑃𝐶 𝑢21 𝑖 𝜀 𝑃𝐶 𝑖 𝑃𝐶 𝑈21 2𝑃𝐶 𝑢21 2 𝜀 𝑃𝐶 𝑖 𝑃𝐶 𝑈21 𝑙 𝑃𝐶 𝑢21 𝑙 𝜀 𝑃𝐶 𝑖 • The size of the Bayesian network becomes very large • Including all coupling variables in one BN is unwieldy for training and sampling. BN reduction with PCA 1 MAR 17 2015 09:30:53 ELEMENTS 1 MAR172 12:54 ELEMENTS Nodal Pressures Nodal Displacements FEA CFD Aeroelastic wing analysis MDA 𝒊 𝒕𝒉 PC at 𝒏 − 𝟏 𝒕𝒉 iteration 𝒊 𝒕𝒉 PC at 𝒏𝒕𝒉 iteration Difference Small uncorrelated
  • 48. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 48 Electrical Parameters Component Heat Total Power Dissipation Heatsink Temperature Electrical Analysis Thermal Analysis Watt Density: Total Power Dissipation Volume of the Heatsink Heatsink Size Parameters Example-1 : Electronic packaging MDO test suite: Heatsink Design variables: 𝑥1: heat sink width 𝑥2: heat sink length 𝑥3: fin length 𝑥4: fin width Uncertain variables: Variability of 𝑥1~𝑥4 𝑥5: nominal resistance at temperature 𝑇 𝑜 𝑥6: temperature coefficient of electrical resistance Likelihood-based non-parametric distribution
  • 49. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 49 Case I: MDO with data uncertainty (coarsest mesh size for thermal analysis) max 𝜇 𝒙 𝜇 𝑃𝐷 𝑿 s.t. 𝑃 𝑇𝑒𝑚𝑝 < 56 𝑜 𝐶 ∩ 𝑉𝑜𝑙𝑢𝑚𝑒 < 6𝐸 − 4 𝑚3 ≥ 0.95 𝑙𝑏𝑖 ≤ 𝜇 𝑋 𝑖 ≤ 𝑢𝑏𝑖 𝑖 = 1, … , 4 𝑇ℎ𝑒𝑟𝑚𝑎𝑙 ℎ𝑒𝑎𝑡, 𝒙 − 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 = 0 𝐸𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 − ℎ𝑒𝑎𝑡 = 0 Joint probability RBDO: BN for optimization Design variables Uncertainty sources Coupling variables System output Compatibility condition • Solved using the proposed BNC-MDO • Component temperature is used to enforce the compatibility Optimizer: DIviding RECTangle (DIRECT) 𝑥1_𝑠𝑡 𝑥2_𝑠𝑡 𝑥3_𝑠𝑡 𝑥4_𝑠𝑡 𝑥5 𝑥6
  • 50. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 50 Results Number of training samples for BN = 800 (5 seconds) Time for optimization with BNC = ~4 min RBDO using SOFPI (feedback): 10,000 / obj(con) evaluation No. of function evaluations till convergence ~= 5 Total number of function evaluations = 6,950,000 (~ 4 hours) • BN gives larger (hence better) objective values • SOFPI produces suboptimal solution (insufficient samples, unreliable) SOFPI (original model) BNC-MDO (surrogate) 𝒙 𝟏 0.056 0.052 𝒙 𝟐 0.056 0.052 𝒙 𝟑 0.021 0.021 𝒙 𝟒 0.039 0.024 objective 73172 73781 re-evaluate with original model objective N/A 84997 constraint 0.962
  • 51. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 51 Case II: MDO with model uncertainty Model error in thermal Analysis • 2D steady state heat transfer equation (PDE) 𝛻𝑇𝑠𝑖𝑛𝑘 𝑥, 𝑦 + 𝑞(𝑥, 𝑦) 𝑘 = 0 • Solved by Finite Difference method • Limited computational resources  discretization error At each realization of input 𝒙: Training points GP Prediction  Auxiliary variable 𝑃ℎ representation of the stochastic model output  Sample 𝑃ℎ from 𝑈(0,1)  inverse CDF  deterministic output • GP prediction ~ 𝑁(𝜇, 𝜎)  𝜇 and 𝜎 are input dependent
  • 52. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 52 Representation of Model Error in BN Design variables Uncertainty sources Coupling variables System output Compatibility condition (including 𝑷 𝒉) 𝑥1_𝑠𝑡 𝑥2_𝑠𝑡 𝑥3_𝑠𝑡 𝑥4_𝑠𝑡 𝑥5 𝑥6 𝑃ℎ
  • 53. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 53 Results Design Variables RBDO using FPI Design Variables 𝑥1 0.077 𝑥2 0.149 𝑥3 0.021 𝑥4 0.010 Objective 𝝁 𝑾𝑫 11170 Constraint Joint Probability 0.99 • Cannot implement SOFPI since the FPI is hard to converge with stochastic model output. Iteration 𝝁 𝑾𝑫 • Optimizer: Genetic algorithm • 100 populations, 15 iterations. • Build BN with 120 samples (240 function evaluations).
  • 54. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 54 𝑵𝑷 𝟑 𝑵𝑷 𝟒 Example-2 Aeroelastic wing design Design variables: • Backsweep angle 𝜃: [0 , 0.5] • Input variability: 𝜎 𝜃 ~ 𝑁(0, 0.03) • FPI takes 10 iterations to converge Coupling variables: nodal pressure • 𝑁𝑃3: nodal pressure after 3rd iteration • 𝑁𝑃4: nodal pressure after 4th iteration 𝑵𝑷 𝟑 after PCA 𝑵𝑷 𝟒 after PCA Difference = 0 I/O BN with 30 principal components
  • 55. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 55 Optimization problem and solution Optimal Value Design variable 𝝁 𝒃𝒘 0.405 Objective 𝝁𝒍𝒊𝒇𝒕 1707.5 Constraint 𝑃(𝑠𝑡𝑟𝑒𝑠𝑠) 0.998  Optimizer: DIRECT  67 calls of BN  547 seconds max 𝝁 𝒃𝒘 𝐸 𝐿 s.t 𝑃 𝑆 ≥ 3 ∗ 105 𝑃𝑎 ≤ 10−3 0.05 ≤ 𝜇 𝑏𝑤 ≤ 0.45 Optimization formulation Optimal solution Optimization history 0 5 10 1695 1700 1705 1710 No. of iterations Lift BN is trained with samples without full convergence analysis
  • 56. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 56 Objective-4 summary BNC-MDO under uncertainty • Efficiently integrates of MDA and optimization under uncertainty • Simultaneously enforces the interdisciplinary compatibility and evaluates objectives and constraints Features of methodology • BNC • PCA • Optimization algorithm (DIRECT/GA)
  • 57. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 57 Future Work (1)Scalability of the proposed BNC-MDA approach needs to be investigated by solving larger problems. (2)Extension in multi-level analyses, and multi-disciplinary feedback coupled analyses (for more than two disciplines). (3)Extension to robustness-based design optimization under both aleatory and epistemic uncertainty. (4)Analytical multi-normal integration of the Gaussian copula instead of the sampling-based strategy for reliability assessment. (5)Improve the efficiency for non-Gaussian copulas. (6)Extension to time-dependent problems.
  • 58. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 58 List of journal manuscripts 1. Liang, C. and Mahadevan, S., Stochastic Multi-Disciplinary Analysis under Epistemic Uncertainty, Journal of Mechanical Design, Vol. 137, Issue 2, 2015. 2. Liang, C. and Mahadevan, S., Bayesian Sensitivity Analysis and Uncertainty Integration for Robust Optimization, Journal of Aerospace Information Systems, Vol 12, Issue 1, 2015. 3. Rangavajhala, S., Liang, C., Mahadevan, S. and Hombal, V., Concurrent optimization of mesh refinement and design parameters in multidisciplinary design, Journal of Aircraft, Vol. 49, No. 6, 2012. 4. Liang, C. and Mahadevan, S., Stochastic Multidisciplinary Analysis with High- Dimensional Coupling, AIAA Journal, under review. 5. Liang, C. and Mahadevan, S., Pareto Surface Construction for Multi-objective Optimization under Uncertainty, ready for submission. 6. Liang, C. and Mahadevan, S., Probabilistic Graphical Modeling for Multidisciplinary Optimization, ready for submission.
  • 59. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 59 List of conference proceedings 1. Liang, C. and Mahadevan, S., Reliability-based Multi-objective Optimization under Uncertainty, 16th AIAA/ISSMO Multidisciplinary Analysis and Optmization Conference, Dallas, Texas, 2015. 2. Liang, C. and Mahadevan, S., Bayesian Framework for Multidisciplinary Uncertainty Quantification and Optimization, 16th AIAA Non-Deterministic Approaches Conference, National Harbor, Maryland, 2014. 3. Liang, C. and Mahadevan, S., Multidisciplinary Analysis and Optimization under Uncertainty, 11th International Conference on Structural Safety and Reliability, New York, New York, 2013. 4. Liang, C. and Mahadevan, S., Multidisciplinary Analysis under Uncertainty, 10th World Congress of Structural and Multidisciplinary Optimization, Orlando, Florida, 2013. 5. Liang, C. and Mahadevan, S., Design Optimization under Aleatory and Epistemic Uncertainty, 10th World Congress of Structural and Multidisciplinary Optimization, Orlando, Florida, 2013. 6. Liang, C. and Mahadevan, S., Inclusion of Data Uncertainty and Model Error in Multi- disciplinary Analysis and Optimization, 54th Structures, Structural Dynamics, and Materials Conference, Boston, Massachusetts, 2013. 7. Liang, C. and Mahadevan, S., Design Optimization under Aleatory and Epistemic Uncertainties, 14th Multidisciplinary Analysis and Optimization Conference, Indianapolis, Indiana, 2012.
  • 60. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 60 Acknowledgement Committee members: Dr. Prodyot Basu (CEE) Dr. Mark N. Ellingham (Math) Dr. Mark P. McDonald (Lipscomb) Dr. Dimitri Mavris (GT) Dr. Roger M. Cooke (TU Delft) University of Melbourne: Dr. Anca Hanea Dan Ababei Vanderbilt University: Dr. Sirisha Rangvajhala Dr. Shankar Sankararaman Dr. You Ling Dr. Vadiraj Hombal Adviser: Dr. Sankaran Mahadevan Ghina Nakad Absi, Dr. Bethany Burkhart, Beverly Piatt Defense preparation: Great friends at Vanderbilt University !
  • 61. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 61 Acknowledgement Funding support: (1) NASA Langley National Laboratory (2) Sandia National Laboratory (3) Vanderbilt University, Department of Civil and Environmental Engineering Software Licenses: (1) UNINET by LightTwist Inc. (Dan Ababei) (2) VisualDOC by Vanderplaats R&D Inc. (Garret Vanderplaats, Juan-Pablo Leiva)
  • 62. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 62
  • 63. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 63 Fit a Gaussian process model to 𝒇 with 𝑥 𝑇 and 𝑦 𝑇 as inputs, and predict the function value at desired points 𝑥 𝑃: GP  Model the underlying covariance in the data instead of the functional form: GP Surrogate Modeling 𝑝 𝑓𝑝 𝒚 𝑻, 𝒙 𝑻, 𝒙 𝑷, 𝚯 ~𝑁(𝑚, 𝑆) Function value Training data Prediction Point GP Parameters Gaussian distribution 𝒎 = 𝑲 𝑷𝑻 𝑲 𝑻𝑻 + 𝝈 𝒏 𝟐 𝑰 −𝟏 𝒚 𝑻 𝑺 = 𝑲 𝑷𝑷 − 𝑲 𝑷𝑻 𝑲 𝑻𝑻 + 𝝈 𝒏 𝟐 𝑰 −𝟏 𝑲 𝑻𝑷 • Models that evaluate 𝒈 are substituted by GP
  • 64. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 64 Vine copula based sampling 𝑋1 𝑌1 𝑌2 𝑋2 M 𝑋1 𝑋2 𝑌1 𝑌2 Goal: estimate 𝑓 𝑋1, 𝑋2, 𝑌1, 𝑌2 𝜌𝑖𝑗 = 2sin( 𝑟𝑖𝑗 𝜋 6 ) 𝜌12;3…𝑛 = 𝜌12;3…,𝑛−1 − 𝜌1𝑛;3,…,𝑛−1 ∗ 𝜌2𝑛;3,…,𝑛−1 1 − 𝜌1𝑛;3,…,𝑛−1 2 1 − 𝜌2𝑛;3,…,𝑛−1 2 𝑐 𝑅 𝑷 = 1 det 𝑅 exp − 1 2 Φ−1 𝑃𝑥1 Φ−1 𝑃𝑥2 Φ−1 𝑃𝑦1 Φ−1 𝑃𝑦2 ∙ 𝑅−1 − 𝐼 ∙ Φ−1 𝑃𝑥1 Φ−1 𝑃𝑥2 Φ−1 𝑃𝑦1 Φ−1 𝑃𝑦2 where 𝜌𝑖𝑗 are the elements of 𝑅 𝑟𝑖𝑗
  • 65. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 65 Kullback-Leibler Divergence For continuous distributions 𝑝 and 𝑞 𝐷 𝐾𝐿(𝑝||𝑞) = −∞ +∞ 𝑝(𝑥)l n( 𝑝 𝑥 )𝑞(𝑥 )𝑑𝑥 Numerical implementation 𝐷 𝐾𝐿(𝑝||𝑞) = 𝑖=1 𝑛 ln 𝑝 𝑥𝑖 𝑞 𝑥𝑖 𝑝 𝑥𝑖 ∗ (𝑥𝑖 − 𝑥𝑖−1)
  • 66. Multidisciplinary analysis and optimization under uncertainty Chen Liang Doctoral Dissertation Defense 66 Vine copula based sampling 𝑋1 𝑌1 𝑌2 2 5 4 6 𝑋2 1 M 𝑋1 𝑋2 𝑌1 𝑌2 Goal: estimate 𝑓 𝑋1, 𝑋2, 𝑌1, 𝑌2 𝑟𝑋2 𝑌2 𝑟𝑋1 𝑌1 𝑟𝑋1 𝑋2 𝑟𝑋2 𝑌1|𝑋1 𝑟𝑌1 𝑌2|𝑋1 𝑋2 𝑋1 𝑋2𝑌1 𝑌2 𝑟𝑋1 𝑌2|𝑋2 𝜌𝑖𝑗 = 2sin( 𝑟𝑖𝑗 𝜋 6 ) 𝜌12;3…𝑛 = 𝜌12;3…,𝑛−1 − 𝜌1𝑛;3,…,𝑛−1 ∗ 𝜌2𝑛;3,…,𝑛−1 1 − 𝜌1𝑛;3,…,𝑛−1 2 1 − 𝜌2𝑛;3,…,𝑛−1 2 𝑐 𝑅 𝑷 = 1 det 𝑅 exp − 1 2 Φ−1 𝑃𝑥1 Φ−1 𝑃𝑥2 Φ−1 𝑃𝑦1 Φ−1 𝑃𝑦2 ∙ 𝑅−1 − 𝐼 ∙ Φ−1 𝑃𝑥1 Φ−1 𝑃𝑥2 Φ−1 𝑃𝑦1 Φ−1 𝑃𝑦2 where 𝜌𝑖𝑗 are the elements of 𝑅