Multidisciplinary analysis and optimization under uncertainty

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


under uncertainty

Chen Liang


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)


under uncertainty

Chen Liang


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


under uncertainty

Chen Liang


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


under uncertainty

Chen Liang


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)


under uncertainty

Chen Liang


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


under uncertainty

Chen Liang


𝑔2

𝑓

𝑢12

𝑔1

𝑢21

Analysis 1

𝑨𝟏(𝒙, 𝑢21)

Analysis 2

𝑨𝟐(𝒙, 𝑢12)

Analysis 3

𝑨𝟑(𝑔1, 𝑔2)

𝑥1𝑥𝑠 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


under uncertainty

Chen Liang


𝑢12

𝑔2

𝑓

𝑢12

𝑔1

𝑢21

Analysis 1

𝑨𝟏(𝒙, 𝑢21)

Analysis 2

𝑨𝟐(𝒙, 𝑢12)

Analysis 3

𝑨𝟑(𝑔1, 𝑔2)

𝑥1𝑥𝑠 𝑥2

Multi-disciplinary multi-level system

Likelihood-based approach for MDA

(LAMDA)

Objective 1: MDA with epistemic uncertainty

𝑔2

𝑓

𝑔1

Analysis 1

𝑨𝟏(𝒙, 𝑢21)

Analysis 2

𝑨𝟐(𝒙, 𝑢12)

Analysis 3

𝑨𝟑(𝑔1, 𝑔2)

𝑥1𝑥𝑠𝑥2

𝑢21


under uncertainty

Chen Liang


𝑈12

𝐹𝑈12

𝐺Interdisciplinary compatibility:

LAMDA

𝑢12 Analysis 1

𝑨𝟏(𝒙, 𝑢21)

Analysis 2

𝑨𝟐(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−

𝜀2

𝑢12+𝜀2𝑓𝑈12

𝑈12 𝑢12 𝑑𝑈12

𝐿 𝑢12 ∝ (𝑃𝑢 − 𝑃𝑙) finite difference

𝑷𝒖

𝑷𝒍

𝒖𝟏𝟐 +𝜺

𝟐𝒖𝟏𝟐 −

𝜺

𝟐


under uncertainty

Chen Liang




Natural Variability


Data Uncertainty

Sparse Data

Interval Data

Model Uncertainty

Numerical error


Round off error

Truncation error


UQ error

Model Form Error

Uncertainty and errors in LAMDA

Considering epistemic

uncertainty sources


under uncertainty

Chen Liang


Data uncertainty (sparse and interval data)

pi pQ

X

fX(x)

θQθ3θ2θ1θi

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


under uncertainty

Chen Liang




Natural Variability


Data Uncertainty

Sparse Data

Interval Data

Model Uncertainty

Numerical error


Round off error

Truncation error


UQ error

Model Form Error

Uncertainty and errors in LAMDA


under uncertainty

Chen Liang


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


under uncertainty

Chen Liang


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 𝑢21𝐴2 𝒙, 𝑢12

GP model

(𝒙, 𝑢21)

𝒙 𝑃ℎ

Deterministic

𝑃ℎ

Extra loop of uncertainty propagation


under uncertainty

Chen Liang


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

• 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


under uncertainty

Chen Liang


Temperature

Thermal

Analysis

Electrical

Analysis

Power density

𝒙𝟐𝒙𝟏

Heat


under uncertainty

Chen Liang


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(℃ )

PD

F

Component heat(Joule)

PD

F


under uncertainty

Chen Liang


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


under uncertainty

Chen Liang


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 ↑


under uncertainty

Chen Liang


Research objectives
















under uncertainty

Chen Liang


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)

𝑌𝑍

𝑋

𝑉

Uncertainty propagation (forward)

𝑣𝑜𝑏𝑠

𝑌𝑍

𝑋

Bayesian updating (inverse)


under uncertainty

Chen Liang


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)

𝐶𝛴𝐺𝑎𝑢𝑠𝑠 𝒖 = 𝚽𝚺 Φ−1 𝑢1 … Φ−1(𝑢𝑛)

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 = 𝑣𝑂𝑏𝑠

𝑌𝑍

𝑋

Bayesian updating (inverse)

If 𝑉 = 𝑉𝑜𝑏𝑠, the conditional samples of

𝑋, 𝑌 and 𝑍 can be obtained as following:

𝑥 = 𝐹𝑋−1 ΦΣ′ 𝑈𝑋|𝑉 = 𝑣𝑜𝑏𝑠

𝑦 = 𝐹𝑌−1 ΦΣ′ 𝑈𝑌|𝑉 = 𝑣𝑜𝑏𝑠

𝑧 = 𝐹𝑍−1 ΦΣ′ 𝑈𝑍|𝑉 = 𝑣𝑜𝑏𝑠


under uncertainty

Chen Liang


BNC-MDA:

𝒖𝟐𝟏 𝑼𝟐𝟏

𝜺𝒖𝟐𝟏

BN with 20 coupling variables

• Joint distribution of 𝑼𝟐𝟏 are evaluated by the conditional samples

• Given compatibility condition (𝜀21 = 0), generate samples from

the BN by copula-based sampling

𝑼𝟐𝟏𝒖𝟐𝟏 𝒖𝟏𝟐 Analysis 2

𝑨𝟐(𝒖𝟏𝟐, x)

Analysis 1

𝑨𝟏(𝒖𝟐𝟏, x)

𝒙

𝜺𝒖𝟐𝟏= 𝒖𝟐𝟏 − 𝑼𝟐𝟏

Interdisciplinary

compatibility= 𝟎

• Bayesian network is built using

samples of 𝑢21, 𝑈21 and 𝜀𝑢21

𝒖𝟐𝟏 𝑼𝟐𝟏

𝜺𝒖𝟐𝟏

One coupling variable


under uncertainty

Chen Liang


Challenge for BNC-MDA

High dimensional coupling

• 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 wing𝑵𝑷𝒊−𝟏 𝑵𝑷𝒊

𝜺

• BN with 774 nodes : enormous effort

• Variables are highly correlated Redundant

information, singularity of correlation coefficient

matrix of the copula


under uncertainty

Chen Liang


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 − 𝟏𝒕𝒉

iteration𝒊𝒕𝒉 PC at 𝒏𝒕𝒉

iteration

Difference

Updated by Interdisciplinary

compatibility (Difference = 0)


under uncertainty

Chen Liang


𝑵𝑷𝟐 𝑵𝑷𝟑

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

Iteration3rd

Iteration BNC

K-L divergence 0.21 0.18 0.16

Estimation without full convergence analysis

= 0

PCA Reduced

𝑵𝑷𝟐

Difference

PCA Reduced

𝑵𝑷𝟑

• 𝑁𝑃2: nodal pressure after 2nd iteration

• 𝑁𝑃3: nodal pressure after 3rd iteration

• 𝑁𝑃2 & 𝑁𝑃3 BNC-MDA (660 function calls)


under uncertainty

Chen Liang


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


• Bayesian Network

• Gaussian copula-based sampling

• Principal component analysis


under uncertainty

Chen Liang


Research objectives
















under uncertainty

Chen Liang


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

𝝈𝒐𝒃𝒋𝑰 𝝈𝒐𝒃𝒋

𝑰𝑰

Objective

Function

PDF

Constraint

Function

PDF


under uncertainty

Chen Liang


𝑚𝑖𝑛𝜇𝑋,𝑑

[𝜇𝑓 𝑋, 𝑃, 𝑑, 𝑝𝑑 ]

s.t.

Prob(𝑔𝑖(𝑋, 𝑑, 𝑃, 𝑝𝑑) ≥ 0)) ≥ 𝑝𝑡𝑖, i= 1,2 … , 𝑛𝑞

Prob(𝑋 ≥ 𝑙𝑏𝑋)) ≥ 𝑝𝑙𝑏𝑡

Prob(𝑋 ≤ 𝑢𝑏𝑋)) ≥ 𝑝𝑢𝑏𝑡

𝑙𝑏𝑑 ≤ 𝑑 ≤ 𝑢𝑏𝑑

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


under uncertainty

Chen Liang


Research objectives
















under uncertainty

Chen Liang


• Conflicting objectives One objective cannot be improved

without worsening others

Objective 3: Multi-objective optimization

• Pareto front tradeoff relationship

between different objectives

𝑚𝑖𝑛𝒙

[𝑓1 𝑿, 𝑃 , 𝑓2 𝑿, 𝑃 ]

s.t.

𝑙𝑏𝑋 ≤ 𝑿 ≤ 𝑢𝑏𝑋

• 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


under uncertainty

Chen Liang


𝑚𝑖𝑛𝜇𝑋,𝑑

[𝜇𝑓1 𝑋, 𝑑, 𝑃, 𝑝𝑑 , 𝜇𝑓2𝑋, 𝑑, 𝑃, 𝑝𝑑 , … , 𝜇𝑓𝑛

(𝑋, 𝑑, 𝑃, 𝑝𝑑)]

s.t.

Prob(𝑔𝑖(𝑋, 𝑑, 𝑃, 𝑝𝑑) ≥ 0)) ≥ 𝑝𝑡𝑖, i= 1,2 … , 𝑛𝑞

Prob(𝑋 ≥ 𝑙𝑏𝑋)) ≥ 𝑝𝑙𝑏𝑡

Prob(𝑋 ≤ 𝑢𝑏𝑋)) ≥ 𝑝𝑢𝑏𝑡

𝑙𝑏𝑑 ≤ 𝑑 ≤ 𝑢𝑏𝑑


• 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)

• Little work regarding the dependence relationships between

output variables (e.g. co-kriging for small number of outputs)


under uncertainty

Chen Liang


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


under uncertainty

Chen Liang


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

min𝝁𝒙

𝜇𝑊𝑒𝑖𝑔ℎ𝑡 & 𝜇𝑉𝑒𝑙𝑑𝑜𝑜𝑟

s.t. 𝑃 𝑖=19 (𝐶𝑜𝑛𝑖 < 𝐶𝑟𝑖𝑡𝑖) ≥ 0.99

0.5 ≤ 𝜇𝑥𝑖≤ 1.5, 𝑖 = 1 …7

0.192 ≤ 𝜇𝑥𝑗≤ 0.345, 𝑗 = 8 … 9


under uncertainty

Chen Liang


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


under uncertainty

Chen Liang


𝑪𝒐𝒔𝒕 𝑽𝒆𝒍𝒅𝒐𝒐𝒓 𝑳𝒐𝒂𝒅𝒂𝒃 𝑽𝒆𝒍𝑩𝑷𝑫𝑽𝟏 𝑫𝑽𝟐

𝑪𝒐𝒔𝒕 𝑽𝒆𝒍𝒅𝒐𝒐𝒓 𝑳𝒐𝒂𝒅𝒂𝒃 𝑽𝒆𝒍𝑩𝑷𝑫𝑽𝟏 𝑫𝑽𝟐

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 – IIOptimizer: NSGA-II

by VisualDOC


under uncertainty

Chen Liang


Pareto front - I

Weight

Do

or

Vel

oci

ty

Training values

Weight

Copula-generated samples


under uncertainty

Chen Liang


Training point selection (inverse)

𝑾𝒆𝒊𝒈𝒉𝒕 𝑽𝒆𝒍𝒅𝒐𝒐𝒓 𝑳𝒐𝒂𝒅𝒂𝒃 𝑽𝒆𝒍𝑩𝑷𝑫𝑽𝟏 𝑫𝑽𝟐

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


under uncertainty

Chen Liang


Weight

Do

or

Vel

oci

ty

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 front𝜇

𝑑𝑜𝑜𝑟𝑣𝑒𝑙

𝜇𝑤𝑒𝑖𝑔ℎ𝑡 𝜇𝑤𝑒𝑖𝑔ℎ𝑡

Approach II: selectively generated samplesApproach I: uniformly generated samples


under uncertainty

Chen Liang


Objective-3 summary


• Probabilistic graphical surrogate model

• Efficient joint probability estimation

• Concurrent training point selection for multiple outputs


• Bayesian network

• Gaussian copula sampling

• Sculpting


under uncertainty

Chen Liang


Research objectives
















under uncertainty

Chen Liang


Objective 4: MDO under uncertainty

𝑚𝑖𝑛𝒙

)𝑓(𝒙

s.t.

𝑔𝑖 𝒙, 𝒖 𝒙 , 𝒗 𝒙 ≤ 0, 𝑖 = 1, . . . , 𝑛𝑞

ℎ1 𝒙, 𝒖, 𝒗 = 0ℎ2 𝒙, 𝒖, 𝒗 = 0

𝐹𝐸𝐴(𝒙, 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒𝑠) − 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡𝑠 = 𝟎𝐶𝐹𝐷(𝒙, 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡𝑠) – 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒𝑠 = 𝟎

Interdisciplinary

compatibility

FEA CFD

UQ / Reliability Analysis

Optimization

Nodal

displacements

Nodal

pressures

Deterministic MDO


under uncertainty

Chen Liang


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?

𝑚𝑖𝑛𝒙 ,𝝃

𝜇(𝑓 𝒙, 𝝃 )

s.t.

𝑃(𝑔𝑖 𝒙, 𝝃, 𝒖 𝒙, 𝝃 , 𝒗 𝒙, 𝝃 ≥ 0) ≤ 𝛼𝑖

𝑖 = 1, . . . , 𝑛𝑞

ℎ1 𝒙, 𝜉, 𝒖, 𝒗 = 0

ℎ2(𝒙, 𝜉, 𝒖, 𝒗) = 0

𝝃 : a vector of random variables 𝝃𝟏, … , 𝝃𝒎

𝜉 : one realization of the random variable 𝜉𝛼𝑖 : desired reliability for 𝑔𝑖

Problem formulation

Inclusion of epistemic uncertainty


under uncertainty

Chen Liang


BNC-MDA + Optimization

𝑼𝟐𝟏𝒖𝟐𝟏

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


under uncertainty

Chen Liang


𝑫𝒊𝒇𝒇 𝒖𝟐𝟏 𝑼𝟐𝟏 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


under uncertainty

Chen Liang


MDO with High-dimensional Coupling

𝐷𝑉 𝑈𝑉

𝐶𝑜𝑛𝑠𝑡𝑟 𝑂𝑏𝑗

𝑃𝐶𝑈21

𝑖𝑃𝐶𝑢21𝑖

𝜀𝑃𝐶𝑖

𝑃𝐶𝑈21

2𝑃𝐶𝑢212

𝜀𝑃𝐶𝑖

𝑃𝐶𝑈21

𝑙𝑃𝐶𝑢21𝑙

𝜀𝑃𝐶𝑖

• 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

MAR 17 2015

12:54:37

ELEMENTS

Nodal Pressures

Nodal

Displacements

FEA CFD

Aeroelastic wing analysis

MDA

𝒊𝒕𝒉 PC at 𝒏 − 𝟏𝒕𝒉

iteration𝒊𝒕𝒉 PC at 𝒏𝒕𝒉

iteration

Difference

Small uncorrelated


under uncertainty

Chen Liang


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:

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 𝑇𝑜

𝑥6: temperature coefficient of electrical resistance

Likelihood-based non-parametric distribution


under uncertainty

Chen Liang


Case I: MDO with data uncertainty (coarsest mesh size for thermal analysis)

max𝜇𝒙

𝜇𝑃𝐷 𝑿

s.t.

𝑃 𝑇𝑒𝑚𝑝 < 56𝑜𝐶 ∩ 𝑉𝑜𝑙𝑢𝑚𝑒 < 6𝐸 − 4 𝑚3 ≥ 0.95

𝑙𝑏𝑖 ≤ 𝜇𝑋𝑖≤ 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


under uncertainty

Chen Liang


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)

𝒙𝟏 0.056 0.052

𝒙𝟐 0.056 0.052

𝒙𝟑 0.021 0.021

𝒙𝟒 0.039 0.024

objective 73172 73781

re-evaluate with

original model

objectiveN/A

84997

constraint 0.962


under uncertainty

Chen Liang


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

pointsGP

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


under uncertainty

Chen Liang


Representation of Model Error in BN

Design

variables

Uncertainty

sources

Coupling

variables

System

output

Compatibility

condition

(including 𝑷𝒉)

𝑥1_𝑠𝑡 𝑥2_𝑠𝑡 𝑥3_𝑠𝑡 𝑥4_𝑠𝑡 𝑥5 𝑥6 𝑃ℎ


under uncertainty

Chen Liang


Results

Design

Variables

RBDO using

FPI

Design

Variables

𝑥1 0.077

𝑥2 0.149

𝑥3 0.021

𝑥4 0.010

Objective 𝝁𝑾𝑫 11170

Constraint Joint Probability 0.99

• Cannot implement SOFPI since the FPI is hard to converge with stochastic

model output.

Iteration

𝝁𝑾𝑫

• Optimizer: Genetic algorithm

• 100 populations, 15 iterations.

• Build BN with 120 samples (240 function evaluations).


under uncertainty

Chen Liang


𝑵𝑷𝟑 𝑵𝑷𝟒

Example-2 Aeroelastic wing design

Design variables:

• Backsweep angle 𝜃: [0 , 0.5]

• Input variability: 𝜎𝜃 ~ 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

𝑵𝑷𝟑 after PCA 𝑵𝑷𝟒 after PCA

Difference = 0

I/O

BN with 30 principal components


under uncertainty

Chen Liang


Optimization problem and solution

Optimal Value

Design

variable𝝁𝒃𝒘 0.405

Objective 𝝁𝒍𝒊𝒇𝒕 1707.5

Constraint 𝑃(𝑠𝑡𝑟𝑒𝑠𝑠) 0.998

Optimizer: DIRECT

67 calls of BN

547 seconds

max𝝁𝒃𝒘

𝐸 𝐿

s.t

𝑃 𝑆 ≥ 3 ∗ 105𝑃𝑎 ≤ 10−3

0.05 ≤ 𝜇𝑏𝑤 ≤ 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


under uncertainty

Chen Liang


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


• BNC

• PCA

• Optimization algorithm (DIRECT/GA)


under uncertainty

Chen Liang


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.


under uncertainty

Chen Liang


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.


under uncertainty

Chen Liang


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.


under uncertainty

Chen Liang


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 !


under uncertainty

Chen Liang


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)


under uncertainty

Chen Liang



under uncertainty

Chen Liang


Fit a Gaussian process model to 𝒇 with 𝑥𝑇 and 𝑦𝑇 as inputs, and

predict the function value at desired points 𝑥𝑃:

GP Model the underlying covariance in the data instead of the functional

form:

GP Surrogate Modeling

𝑝 𝑓𝑝 𝒚𝑻, 𝒙𝑻, 𝒙𝑷, 𝚯 ~𝑁(𝑚, 𝑆)

Function

valueTraining

dataPrediction

Point

GP

Parameters

Gaussian distribution

𝒎 = 𝑲𝑷𝑻 𝑲𝑻𝑻 + 𝝈𝒏𝟐𝑰

−𝟏𝒚𝑻

𝑺 = 𝑲𝑷𝑷 − 𝑲𝑷𝑻 𝑲𝑻𝑻 + 𝝈𝒏𝟐𝑰

−𝟏𝑲𝑻𝑷

• Models that evaluate 𝒈 are substituted by GP


under uncertainty

Chen Liang


Vine copula based sampling

𝑋1

𝑌1 𝑌2

𝑋2

M

𝑋1

𝑋2

𝑌1

𝑌2

Goal: estimate 𝑓 𝑋1, 𝑋2, 𝑌1, 𝑌2

𝜌𝑖𝑗 = 2sin(𝑟𝑖𝑗𝜋

6) 𝜌12;3…𝑛 =

𝜌12;3…,𝑛−1 − 𝜌1𝑛;3,…,𝑛−1 ∗ 𝜌2𝑛;3,…,𝑛−1

1 − 𝜌1𝑛;3,…,𝑛−12 1 − 𝜌2𝑛;3,…,𝑛−1

2

𝑐𝑅 𝑷 =1

det 𝑅exp −

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


under uncertainty

Chen Liang


Kullback-Leibler Divergence

For continuous distributions 𝑝 and 𝑞

𝐷𝐾𝐿(𝑝||𝑞) = −∞

+∞

𝑝(𝑥)l n(𝑝 𝑥

)𝑞(𝑥)𝑑𝑥

Numerical implementation

𝐷𝐾𝐿(𝑝||𝑞) =

𝑖=1

𝑛

ln𝑝 𝑥𝑖

𝑞 𝑥𝑖𝑝 𝑥𝑖 ∗ (𝑥𝑖 − 𝑥𝑖−1)


under uncertainty

Chen Liang


Vine copula based sampling

𝑋1

𝑌1 𝑌2

25

4

6

𝑋21

M

𝑋1

𝑋2

𝑌1

𝑌2

Goal: estimate 𝑓 𝑋1, 𝑋2, 𝑌1, 𝑌2

𝑟𝑋2𝑌2𝑟𝑋1𝑌1 𝑟𝑋1𝑋2

𝑟𝑋2𝑌1|𝑋1

𝑟𝑌1𝑌2|𝑋1𝑋2

𝑋1 𝑋2𝑌1 𝑌2

𝑟𝑋1𝑌2|𝑋2

𝜌𝑖𝑗 = 2sin(𝑟𝑖𝑗𝜋

6)

𝜌12;3…𝑛 =𝜌12;3…,𝑛−1 − 𝜌1𝑛;3,…,𝑛−1 ∗ 𝜌2𝑛;3,…,𝑛−1

1 − 𝜌1𝑛;3,…,𝑛−12 1 − 𝜌2𝑛;3,…,𝑛−1

2

𝑐𝑅 𝑷 =1

det 𝑅exp −

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

Multidisciplinary analysis and optimization under uncertainty

Documents