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Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense1
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 Defense2
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 Defense3
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 Defense4
Three types of sources
β’ Physical variability
β’ Data uncertainty
(e.g., sparse/interval data)
β’ Model Uncertainty
Forward problem
β’ For a given input Uncertainty of output needs to
be evaluated
β’ Propagation of aleatory uncertainty is well-studied
β’ Inclusion of epistemic uncertainty becomes
more important
β’ 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 Defense5
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
β’ Representation of epistemic sources of uncertainty
β’ Propagation through feedback coupled analysis
β’ Inclusion in the design optimization of multidisciplinary
analysis with feedback coupling (high-dimensional)
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense6
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 Defense7
π2
π
π’12
π1
π’21
Analysis 1
π¨π(π, π’21)
Analysis 2
π¨π(π, π’12)
Analysis 3
π¨π(π1, π2)
π₯1π₯π π₯2
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
β’ First-order Second Moment
(FOSM) approximationsβ’ Linear approximations of disciplinary
analyses
β’ PDF based on mean & variance
β’ Du & Chen, Mahadevan & Smith
β’ Fully Decoupled Approachβ’ Calculate PDFs of u12 & u21
β’ Cut-off feedback both directions
β’ Ignores dependence between u12 &
u21
β’ Lack of one-toβone correspondence
between π1 and π2 in calculating f
Likelihood-based approach for MDA
(LAMDA) π’21 π’12
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense8
π’12
π2
π
π’12
π1
π’21
Analysis 1
π¨π(π, π’21)
Analysis 2
π¨π(π, π’12)
Analysis 3
π¨π(π1, π2)
π₯1π₯π π₯2
Multi-disciplinary multi-level system
Likelihood-based approach for MDA
(LAMDA)
Objective 1: MDA with epistemic uncertainty
π2
π
π1
Analysis 1
π¨π(π, π’21)
Analysis 2
π¨π(π, π’12)
Analysis 3
π¨π(π1, π2)
π₯1π₯π π₯2
π’21
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense9
π12
πΉπ12
πΊInterdisciplinary compatibility:
LAMDA
π’12 Analysis 1
π¨π(π, π’21)
Analysis 2
π¨π(π, π’12)
π’21 π12
Given a value of π’12 what is π(π12 = π’12|π’12) πΏ(π’12)
π π’12 =πΏ(π’12)
πΏ(π’12)ππ’12
FORM is used to calculate the CDFs of
the upper and lower bounds: ππ’ and ππ
πΏ π’12 β π’12β
π2
π’12+π2ππ12
π12 π’12 ππ12
πΏ π’12 β (ππ’ β ππ) finite difference
π·π
π·π
πππ +πΊ
ππππ β
πΊ
π
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense10
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 Defense11
Data uncertainty (sparse and interval data)
pi pQ
X
fX(x)
ΞΈQΞΈ3ΞΈ2ΞΈ1ΞΈi
p1
p2
n
i
b
a
X
m
i
iX dxPxfPxfPLi
i11
)|()|()(
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 Defense12
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 Defense13
Model uncertainty estimation
Training
points
(e.g. FEA)
Prediction
Uncertainty
Discretization error estimation
β’ GP prediction ~ π(π, π) π and π are input dependent
Rangavajhala, et. al,
AIAA Journal 2010
Richardson
extrapolation
At each input π
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense14
Auxiliary variable πβ~π[0,1] CDF of GP output
Stochastic model output input random variable
FORM can be used for likelihood evaluation
π12
CDF of GP output
πΏ π’12 β π(π12 = π’12|π’12)
β’ Equation only calculable when π12 is deterministic given an π and π’12
Inclusion of model uncertainty in LAMDA
π’12 π’21π΄2 π, π’12
GP model
(π, π’21)
π πβ
Deterministic
πβ
Extra loop of uncertainty propagation
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense15
Electrical
Parameters
Component Heat
Total Power
Dissipation
Heatsink
Temperature
Electrical
Analysis
Thermal
Analysis
Power Density: Total Power DissipationVolume of the Heatsink
Heatsink Size
Parameters
Numerical example: electronic packaging
Model error in thermal Analysis
β’ 2D steady state heat transfer equation (PDE)
π»ππ πππ π₯, π¦ +π(π₯, π¦)
π= 0
β’ Solved by Finite Difference method
β’ Limited computational resources
discretization error
MDO test suite: Heatsink
Data uncertainty
Temperature Coefficient of resistance (πΆ)
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 πΆ
πΆ
β’ Uncertainty estimated auxiliary
variable
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense16
Temperature
Thermal
Analysis
Electrical
Analysis
Power density
ππππ
Heat
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense17
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(β )
PD
F
Component heat(Joule)
PD
F
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense18
Likelihood Approach for Multidisciplinary Analysis
β’ Data uncertainty and model error in feedback coupled
analysis.
β’ Auxiliary variable stochastic model error.
Features of methodology
β’ Likelihood-based approach for MDA (LAMDA)
β’ FORM
β’ GP estimation of model error
β’ Auxiliary variable
Objective-1 summary
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense19
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:
β’ First-order approximation: dimension β, accuracy β
β’ Likelihood calculation (finite difference) Number of function β
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense20
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 Defense21
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)
ππ
π
π
Uncertainty propagation (forward)
π£πππ
ππ
π
Bayesian updating (inverse)
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense22
Multi-variate Gaussian Copula
β’ A copula is a function that joins the CDFs of multiple random
variables as a joint CDF function.
β’ Types: Gaussian, Clayton, Gumbel, β¦
β’ The Multivariate Gaussian Copula (MGC)
πΆπ΄πΊππ’π π π = π½πΊ Ξ¦β1 π’1 β¦ Ξ¦β1(π’π)
where π’π is the CDF value of any arbitrary marginal πΉππ(π₯π),
β’ Gaussian copula assumption needs verification (done for all examples)
β’ Other copulas are not as efficient in conditional sampling
Hanea, et al.,
QREI, 2006
Efficient Conditional Sampling
π = π£πππ
ππ
π
Bayesian updating (inverse)
If π = ππππ , the conditional samples of
π, π and π can be obtained as following:
π₯ = πΉπβ1 ΦΣⲠππ|π = π£πππ
π¦ = πΉπβ1 ΦΣⲠππ|π = π£πππ
π§ = πΉπβ1 ΦΣⲠππ|π = π£πππ
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense23
BNC-MDA:
πππ πΌππ
πΊπππ
BN with 20 coupling variables
β’ Joint distribution of πΌππ are evaluated by the conditional samples
β’ Given compatibility condition (π21 = 0), generate samples from
the BN by copula-based sampling
πΌπππππ πππ Analysis 2
π¨π(πππ, x)
Analysis 1
π¨π(πππ, x)
π
πΊπππ= πππ β πΌππ
Interdisciplinary
compatibility= π
β’ Bayesian network is built using
samples of π’21, π21 and ππ’21
πππ πΌππ
πΊπππ
One coupling variable
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense24
Challenge for BNC-MDA
High dimensional coupling
β’ Mesh resolution : 258 nodes 258 random variables
1
MAR 17 201
5
09:30:53
ELEMENTS
1
MAR 17 2015
12:54:37
ELEMENTS
Nodal Pressures
Nodal Displacements
FEA CFD
Aero-elastic analysis of an aircraft wingπ΅π·πβπ π΅π·π
πΊ
β’ BN with 774 nodes : enormous effort
β’ Variables are highly correlated Redundant
information, singularity of correlation coefficient
matrix of the copula
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense25
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
πππ PC at π β πππ
iterationπππ PC at πππ
iteration
Difference
Updated by Interdisciplinary
compatibility (Difference = 0)
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense26
π΅π·π π΅π·π
Numerical Example β MDA of an aircraft wing
Input uncertainty
β’ Backsweep angle π ~ π 0.4,0.1
β’ 110 FPI analyses Benchmark
β’ 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 :
β’ Smaller value Closer distributions
Higher fidelity more time saved
Method 2nd
Iteration3rd
Iteration BNC
K-L divergence 0.21 0.18 0.16
Estimation without full convergence analysis
= 0
PCA Reduced
π΅π·π
Difference
PCA Reduced
π΅π·π
β’ ππ2: nodal pressure after 2nd iteration
β’ ππ3: nodal pressure after 3rd iteration
β’ ππ2 & ππ3 BNC-MDA (660 function calls)
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense27
Objective-2 summary
Bayesian Approach for Multidisciplinary Analysis
β’ Novel BNC-MDA for UQ in high-dimensional coupled MDA
β’ Dimension and iteration reduction Efficient while
preserving the dependencies
β’ Time for physics analysis β« time for stochastic analysis
Features of methodology
β’ Bayesian Network
β’ Gaussian copula-based sampling
β’ Principal component analysis
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense28
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 Defense29
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 ππππ of the
objective function
Focuses on ππ of the
constraint function
πππππ° ππππ
π°π°
Objective
Function
Constraint
Function
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense30
πππππ,π
[ππ π, π, π, ππ ]
s.t.
Prob(ππ(π, π, π, ππ) β₯ 0)) β₯ ππ‘π, i= 1,2 β¦ , ππ
Prob(π β₯ πππ)) β₯ ππππ‘
Prob(π β€ π’ππ)) β₯ ππ’ππ‘
πππ β€ π β€ π’ππ
Reliability-based design optimization
β’ Natural variability
β’ Data uncertainty
β’ Sparse
β’ Interval
Model uncertainty
β’ Numerical solution error
β’ Model form error
β’ The examples are formulated using the RBDO formulation
β’ Proposed methodology is adaptable to solve RDO problems
π: stochastic design variable
π: stochastic non-design variable
π: deterministic design variable
ππ: deterministic non-design variable
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense31
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 Defense32
β’ Conflicting objectives One objective cannot be improved
without worsening others
Objective 3: Multi-objective optimization
β’ Pareto front tradeoff relationship
between different objectives
ππππ
[π1 πΏ, π , π2 πΏ, π ]
s.t.
πππ β€ πΏ β€ π’ππ
β’ Existing methods
Weighted sum
π-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 Defense33
πππππ,π
[ππ1 π, π, π, ππ , ππ2π, π, π, ππ , β¦ , πππ
(π, π, π, ππ)]
s.t.
Prob(ππ(π, π, π, ππ) β₯ 0)) β₯ ππ‘π, i= 1,2 β¦ , ππ
Prob(π β₯ πππ)) β₯ ππππ‘
Prob(π β€ π’ππ)) β₯ ππ’ππ‘
πππ β€ π β€ π’ππ
Multi-objective optimization under uncertainty
β’ Challenges of existing MOO methods
- Weights for objective aggregation is not intuitive
- Goal programming\Constraint-based method may not produce
Pareto solutions
- NSGA is computationally expensive (surrogate models)
β’ 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 Defense34
Dependence among Objectives/Constraints
β’ 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 Defense35
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
minππ
πππππβπ‘ & ππππππππ
s.t. π π=19 (πΆπππ < πΆπππ‘π) β₯ 0.99
0.5 β€ ππ₯πβ€ 1.5, π = 1 β¦7
0.192 β€ ππ₯πβ€ 0.345, π = 8 β¦ 9
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense36
Design variables
Uncertain
Sources
Constraints
Objectives
Optimization with Bayesian network
Optimizer
π₯1_π π‘ π₯2_π π‘ π₯3_π π‘ π₯4_π π‘ π₯5_π π‘ π₯6_π π‘ π₯7_π π‘ π₯8_π π‘ π₯9_π π‘ π₯10 π₯11
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense37
πͺπππ π½πππ πππ π³πππ ππ π½πππ©π·π«π½π π«π½π
πͺπππ π½πππ πππ π³πππ ππ π½πππ©π·π«π½π π«π½π
Conditionalization (forward)
β’ Conditional samples are used to
estimate objectives and joint
probability (constraint)
β’ Optimizer generates a set of design
values to the BN
β’ Bayesian network is conditionally
sampled using Gaussian copula
Posterior distributions of
objectives and constraints
In each BN evaluation:
Optimizer: NSGA β IIOptimizer: NSGA-II
by VisualDOC
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense38
Pareto front - I
Weight
Do
or
Vel
oci
ty
Training values
Weight
Copula-generated samples
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense39
Training point selection (inverse)
πΎπππππ π½πππ πππ π³πππ ππ π½πππ©π·π«π½π π«π½π
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 Defense40
Weight
Do
or
Vel
oci
ty
Pareto front β IIβ’ For comparison, another 20 samples are chosen by Latin Hypercube
sampling calculate outputs.
β’ Rebuild a BN with the additional samples
β’ Recalculate the Pareto frontπ
πππππ£ππ
ππ€πππβπ‘ ππ€πππβπ‘
Approach II: selectively generated samplesApproach I: uniformly generated samples
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense41
Objective-3 summary
Multi-objective optimization under uncertainty
β’ Probabilistic graphical surrogate model
β’ Efficient joint probability estimation
β’ Concurrent training point selection for multiple outputs
Features of methodology
β’ Bayesian network
β’ Gaussian copula sampling
β’ Sculpting
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense42
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 Defense43
Objective 4: MDO under uncertainty
ππππ
)π(π
s.t.
ππ π, π π , π π β€ 0, π = 1, . . . , ππ
β1 π, π, π = 0β2 π, π, π = 0
πΉπΈπ΄(π, ππππ π π’πππ ) β πππ πππππππππ‘π = ππΆπΉπ·(π, πππ πππππππππ‘π ) β ππππ π π’πππ = π
Interdisciplinary
compatibility
FEA CFD
UQ / Reliability Analysis
Optimization
Nodal
displacements
Nodal
pressures
Deterministic MDO
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense44
MDO under Uncertainty
Surrogate models are commonly used:
β’ Training data can be expensive to get
β’ Curse of dimensionality
β’ Enforce compatibility
β’ Evaluate (mean) objectives and
(probability) constraints
Simultaneously achieved without
fully converged physics analysis?
ππππ ,π
π(π π, π )
s.t.
π(ππ π, π, π π, π , π π, π β₯ 0) β€ πΌπ
π = 1, . . . , ππ
β1 π, π, π, π = 0
β2(π, π, π, π) = 0
π : a vector of random variables ππ, β¦ , ππ
π : one realization of the random variable ππΌπ : desired reliability for ππ
Problem formulation
Inclusion of epistemic uncertainty
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense45
BNC-MDA + Optimization
πΌπππππ
Analysis2Analysis 1
πππ πͺπππππ
π«π½, πΌπ½
BNC-MDO
BN is built with samples of the one-iteration analysis
Optimization framework on the top
π·π ππ
π’21 π21
πΆπππ π‘π πππ
π·πππ
OptimizationOne-iteration analysis
MDA
π«πππ = πΌ β π
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense46
π«πππ πππ πΌππ πΆπππ«π½ πΌπ½ πͺππ
Concurrently enforce compatibility and estimate outputs
In each call of BN:
β’ π·π = πππ πππ π£πππ’π, ππππ = 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 Defense47
MDO with High-dimensional Coupling
π·π ππ
πΆπππ π‘π πππ
ππΆπ21
πππΆπ’21π
πππΆπ
ππΆπ21
2ππΆπ’212
πππΆπ
ππΆπ21
πππΆπ’21π
πππΆπ
β’ The size of the Bayesian network
becomes very large
β’ 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
MAR 17 2015
12:54:37
ELEMENTS
Nodal Pressures
Nodal
Displacements
FEA CFD
Aeroelastic wing analysis
MDA
πππ PC at π β πππ
iterationπππ PC at πππ
iteration
Difference
Small uncorrelated
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense48
Electrical
Parameters
Component Heat
Total Power
Dissipation
Heatsink
Temperature
Electrical
Analysis
Thermal
Analysis
Watt Density: Total Power DissipationVolume of the Heatsink
Heatsink Size
Parameters
Example-1 : Electronic packaging
MDO test suite: Heatsink
Design variables:
π₯1: heat sink width
π₯2: heat sink length
π₯3: fin length
π₯4: fin width
Uncertain variables:
Variability of π₯1~π₯4
π₯5: nominal resistance at temperature ππ
π₯6: temperature coefficient of electrical resistance
Likelihood-based non-parametric distribution
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense49
Case I: MDO with data uncertainty (coarsest mesh size for thermal analysis)
maxππ
πππ· πΏ
s.t.
π ππππ < 56ππΆ β© ππππ’ππ < 6πΈ β 4 π3 β₯ 0.95
πππ β€ πππβ€ π’ππ
π = 1, β¦ , 4
πβπππππ βπππ‘, π β π‘πππππππ‘π’ππ = 0
πΈππππ‘πππππ π‘πππππππ‘π’ππ β βπππ‘ = 0
Joint probability RBDO:
BN for optimization
Design
variables
Uncertainty
sources
Coupling
variables
System output
Compatibility
condition
β’ Solved using the proposed BNC-MDO
β’ Component temperature is used to enforce the compatibility
Optimizer:
DIviding RECTangle
(DIRECT)
π₯1_π π‘ π₯2_π π‘ π₯3_π π‘ π₯4_π π‘ π₯5 π₯6
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense50
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)
β’ BN gives larger (hence better) objective values
β’ SOFPI produces suboptimal solution (insufficient samples, unreliable)
SOFPI
(original model)
BNC-MDO(surrogate)
ππ 0.056 0.052
ππ 0.056 0.052
ππ 0.021 0.021
ππ 0.039 0.024
objective 73172 73781
re-evaluate with
original model
objectiveN/A
84997
constraint 0.962
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense51
Case II: MDO with model uncertainty
Model error in thermal Analysis
β’ 2D steady state heat transfer equation (PDE)
π»ππ πππ π₯, π¦ +π(π₯, π¦)
π= 0
β’ Solved by Finite Difference methodβ’ Limited computational resources
discretization error
At each realization of input π:
Training
pointsGP
Prediction
Auxiliary variable πβ representation of the stochastic model output
Sample πβ from π(0,1) inverse CDF deterministic output
β’ GP prediction ~ π(π, π)
π and π are input dependent
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense52
Representation of Model Error in BN
Design
variables
Uncertainty
sources
Coupling
variables
System
output
Compatibility
condition
(including π·π)
π₯1_π π‘ π₯2_π π‘ π₯3_π π‘ π₯4_π π‘ π₯5 π₯6 πβ
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense53
Results
Design
Variables
RBDO using
FPI
Design
Variables
π₯1 0.077
π₯2 0.149
π₯3 0.021
π₯4 0.010
Objective ππΎπ« 11170
Constraint Joint Probability 0.99
β’ Cannot implement SOFPI since the FPI is hard to converge with stochastic
model output.
Iteration
ππΎπ«
β’ Optimizer: Genetic algorithm
β’ 100 populations, 15 iterations.
β’ Build BN with 120 samples (240 function evaluations).
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense54
π΅π·π π΅π·π
Example-2 Aeroelastic wing design
Design variables:
β’ Backsweep angle π: [0 , 0.5]
β’ Input variability: ππ ~ π(0, 0.03)
β’ FPI takes 10 iterations to converge
Coupling variables: nodal pressure
β’ ππ3: nodal pressure after 3rd iteration
β’ ππ4: nodal pressure after 4th iteration
π΅π·π after PCA π΅π·π after PCA
Difference = 0
I/O
BN with 30 principal components
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense55
Optimization problem and solution
Optimal Value
Design
variableπππ 0.405
Objective πππππ 1707.5
Constraint π(π π‘πππ π ) 0.998
Optimizer: DIRECT
67 calls of BN
547 seconds
maxπππ
πΈ πΏ
s.t
π π β₯ 3 β 105ππ β€ 10β3
0.05 β€ πππ€ β€ 0.45
Optimization formulation
Optimal solution Optimization history
0 5 101695
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 Defense56
Objective-4 summary
BNC-MDO under uncertainty
β’ Efficiently integrates of MDA and optimization under
uncertainty
β’ Simultaneously enforces the interdisciplinary
compatibility and evaluates objectives and constraints
Features of methodology
β’ BNC
β’ PCA
β’ Optimization algorithm (DIRECT/GA)
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense57
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 Defense58
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 Defense59
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 Defense60
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 Defense61
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 Defense62
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense63
Fit a Gaussian process model to π with π₯π and π¦π as inputs, and
predict the function value at desired points π₯π:
GP Model the underlying covariance in the data instead of the functional
form:
GP Surrogate Modeling
π ππ ππ», ππ», ππ·, π― ~π(π, π)
Function
valueTraining
dataPrediction
Point
GP
Parameters
Gaussian distribution
π = π²π·π» π²π»π» + ππππ°
βπππ»
πΊ = π²π·π· β π²π·π» π²π»π» + ππππ°
βππ²π»π·
β’ Models that evaluate π are substituted by GP
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense64
Vine copula based sampling
π1
π1 π2
π2
M
π1
π2
π1
π2
Goal: estimate π π1, π2, π1, π2
πππ = 2sin(ππππ
6) π12;3β¦π =
π12;3β¦,πβ1 β π1π;3,β¦,πβ1 β π2π;3,β¦,πβ1
1 β π1π;3,β¦,πβ12 1 β π2π;3,β¦,πβ1
2
ππ π· =1
det π exp β
1
2
Ξ¦β1 ππ₯1
Ξ¦β1 ππ₯2
Ξ¦β1 ππ¦1
Ξ¦β1 ππ¦2
β π β1 β πΌ β
Ξ¦β1 ππ₯1
Ξ¦β1 ππ₯2
Ξ¦β1 ππ¦1
Ξ¦β1 ππ¦2
where πππ are the elements of π
πππ
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense65
Kullback-Leibler Divergence
For continuous distributions π and π
π·πΎπΏ(π||π) = ββ
+β
π(π₯)l n(π π₯
)π(π₯)ππ₯
Numerical implementation
π·πΎπΏ(π||π) =
π=1
π
lnπ π₯π
π π₯ππ π₯π β (π₯π β π₯πβ1)
Multidisciplinary analysis and optimization
under uncertainty
Chen Liang
Doctoral Dissertation Defense66
Vine copula based sampling
π1
π1 π2
25
4
6
π21
M
π1
π2
π1
π2
Goal: estimate π π1, π2, π1, π2
ππ2π2ππ1π1 ππ1π2
ππ2π1|π1
ππ1π2|π1π2
π1 π2π1 π2
ππ1π2|π2
πππ = 2sin(ππππ
6)
π12;3β¦π =π12;3β¦,πβ1 β π1π;3,β¦,πβ1 β π2π;3,β¦,πβ1
1 β π1π;3,β¦,πβ12 1 β π2π;3,β¦,πβ1
2
ππ π· =1
det π exp β
1
2
Ξ¦β1 ππ₯1
Ξ¦β1 ππ₯2
Ξ¦β1 ππ¦1
Ξ¦β1 ππ¦2
β π β1 β πΌ β
Ξ¦β1 ππ₯1
Ξ¦β1 ππ₯2
Ξ¦β1 ππ¦1
Ξ¦β1 ππ¦2
where πππ are the elements of π