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IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
Uncertainty Quantification in Performance Evaluation of Manufacturing Processes
Saideep Nannapaneni, Sankaran MahadevanVanderbilt University, Nashville, TN
IEEE International Conference on Big DataSpecial Session on Big Data and Analytics for Smart Manufacturing Systems
October 27, 2014
AcknowledgementFunding support: NIST (Cooperative Agreement NANB14H036, Monitor: Sudarsan Rachuri)Technical help: Seung-Jun Shin, Senthilkumaran Kumaraguru, Anantha Narayanan, Sudarsan Rachuri
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
2
Why Uncertainty Quantification ?
Information Fusion
Reliability data
Legacy system
data
Test data
Operational data
Simulation data
Expert Opinion
Mathematical models
Evaluation Uncertainty
Model uncertainty
Process variability
Material uncertainty
Sensor uncertainty
• Performance evaluation – affected by available data and uncertainty• Uncertainty sources – linear or non-linear combination, occur at different stages• Information sources – heterogeneous • A systematic approach is necessary for information fusion and uncertainty
quantification.
Uncertainty Sources Data Sources
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
3
Aleatory vs. epistemic uncertainty
Four types of quantities1. Deterministic known value (fixed constant)
2. Stochastic random variability– Distribution statistics are known– Aleatory, irreducible uncertainty
3. Quantity is deterministic, but unknown– Epistemic, reducible uncertainty
4. Quantity is stochastic, but distribution characteristics are unknown– Has both aleatory and epistemic uncertainty
• Unknown distribution type • Unknown distribution parameters
True value
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
4
Bayesian network
a
b
c
e
dg
f
a, b,….. component nodes (variables)
g – system-level output
U - set of all nodes { a, b,…, g }
π(U) = π(a)× π(b| a) × π(c| a) × π(d| c) × π(e| b, d) × π(f ) × π(g| e, f )
π(U, m) = π(U)× π(m| b)With new observed data m
π(g) = ∫ π(U) da db… dfPDF of node g
Joint PDF of all variables
a
b
c
e
d g
f
mData
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
5
Methodology for UQ (1/2)
Construction of Bayesian Network
• Physics-based Models– Use available models based on process physics (domain knowledge)
Eg: Differential equations
• Data-driven approach– Build surrogate models using data
e.g., Regression, Polynomial models, Gaussian Process– Structure Learning algorithms
Constraint-based and Score-based algorithms
• Hybrid approach– Combination of physics-based and data-driven approaches– Partial structure of the BN using domain knowledge– Learn other parts of network using data
Construction of Bayesian Network
Bayesian Model Calibration/Updating
Uncertainty Propagation
Bartram & Mahadevan, SCHM, 2014
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
6
Methodology for UQ (2/2)
Bayesian Model Calibration/Updating
• Estimate unknown (epistemic) parameters using data in a probabilistic manner
• Bayes’ theorem
Pr Pr PrPr ,
Sampling from Posterior: Markov Chain Monte Carlo (MCMC)
Construction of Bayesian Network
Bayesian Model Calibration, Updating
Uncertainty Propagation
Likelihood Prior
EvidencePosterior
Uncertainty Propagation• Obtain updated posterior distribution of the Quantity of
Interest (QoI)• Monte Carlo Sampling - samples from posterior distribution• Samples are propagated through the network
Posterior distribution of QoI (Performance Metric)
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
7
Scalability Dimension Reduction
Size of Production Network increases
Increase in Epistemic Parameters
Increase in Computational effort
Goal: Obtain a reduced set of variables such that there is no significant change in statistics of QoI
Approach: Global Sensitivity Analysis
Assess the variance contribution of each of the variables to the variance of the QoI.
, …
1
• Obtain prior sensitivity indices by performing sensitivity analysis using prior distributions
• Assume a threshold value for sensitivity index.
• If sensitivity index < threshold value, assume that variable to be deterministic at the nominal value.
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
8
Sensitivity Analysis under Epistemic Uncertainty
• Global Sensitivity Analysis– Applicable to Y = G(X) where G is deterministic– One value of X One value of Y– Define auxiliary variables– Can include both aleatory & epistemic sources
• Introduce auxiliary variable U represent variability U (0, 1)
F-1(U, P)U, P X
X
X dxPxfU )|(
Transfer Function
PDistribution
• Data uncertainty
Sankararaman & Mahadevan, RESS, 2013
G(X)X Y
• Model uncertainty
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
9
Handling Big Data
• HDFS : Hadoop Distributed File System– To store large amounts of data in databases
• MapReduce– A parallel processing framework for large-scale data processing.
• Model Building using all data computationally unaffordable
• Data Reduction and Feature Selection techniques required– Clustering– Selection of Data using Design of Experiments (DoE)– Apache Mahout : Machine learning library for Hadoop
• The reduced dataset can be used for Bayesian Network construction
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
10
Summary – Methodology
Construction of Bayesian Network
Bayesian Model Calibration, Updating
Uncertainty Propagation
Data Reduction, Dimension Reduction
Bayesian Model Calibration, Updating
Uncertainty Propagation
Construction of Bayesian Network
Without Dimension/Data Reduction
With Data/ Dimension Reduction
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
11
Demonstration Problem
ProductModule
Injection Molding
GMAW
Threaded Fastening
OR OR
GTAW
MetalPowder
PlasticGrains
e.g. Power Train
Die Casting
Machine3
Plastic Part
Metallic Module
Machine2
Machine1
Metal Part
Metal Part
Turning
Arc Welding
Process
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
12
Injection Molding
Three stages
• Melting of polymer
• Injection of polymer into the mold
• Cooling of polymer to form product
Source: http://www.maximintegrated.com/en/app-notes/index.mvp/id/4717
Goal:
UQ in Energy Consumption of an Injection Molding Process
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
13
Melting Process
• Power used for melting the dye
• Volume of shot
1 100Δ100
• Energy consumed for melting
• Flow rate and Average flow rate0.5
Legend
= specific gravity of polymer= heat capacity of polymer= Injection temperature= polymer temperature
= polymer heat of fusion
= Volume of mold= shrinkage
Δ = buffer= total volume injected
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
14
Injection Process
• Injection time2 1 Δ
1000
• Energy consumed in injection process
Legend
= number of cavities
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
15
Cooling Process
• Cooling Time
ln4
• Energy consumed in cooling process
Legend= maximum wall thickness of mold
= thermal diffusivity= Injection temperature= mold temperature= ejection temperature
COP = coefficient of performance of cooling equipment
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
16
Total Energy Consumption
• Energy consumed in making a part1 0.75 0.25 1 Δ
• Total Electricity Cost
3600 1000
• Reset Energy0.25
• Total Cycle Time
• Reset time t 1 1.75
Legend= maximum wall thickness of mold= Energy consumed for resetting
= Power required for basic energy consuming units, when machine is in stand‐by mode
= dry cycle time= maximum clamp stroke in the mold
= unit cost in USA
= Total cycle time, , , , = Efficiencies of
injection, reset, cooling, heating, machine power= depth of the part
= output per day= number of days
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
17
Injection Molding – Bayesian Network
Constants DeterministicAleatoryCalibration
Parameters (Epistemic)
Observations
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
18
Injection Molding Materials
Several types of materials used in injection molding• Metals• Glass• Elastomers• Thermoplastic polymers• Thermosetting polymers
Polyethylene properties (assumed)• Specific gravity = 960 kg/m3
• Heat capacity = 2250 J/kg-K• Thermal diffusivity = 2.27 x 10-7 m2/s• Shrinkage = 0.019• Heat of fusion = 240 kJ/kg
NylonPolyethylenePolypropylenePolyvinyl chloridePolystyreneAcrylicTeflon
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
19
Injection Molding Process Parameters – Synthetic Dataset
Parameter Value
Injection temperature 210 (oC)
Mold temperature 35 (oC)
Ejection temperature 50 (oC)
Injection pressure 90 MPa
Flow rate 1.67e-5 /
Coefficient of performance 0.7
All efficiency coefficients 0.7
Number of cavities 1
Fraction Δ 0.015
Thickness 0.0125 m
Volume of part 0.002048 m3
Unit Cost 10 cents/kWh
Parts per day 10,000
Number of days 28
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
20
Injection Molding Process - Synthetic dataset
• Error in temperature measurements (oC)~ 0,3
• Error in cycle time measurements (s)~ 0,2
• Error in injection pressure measurements (MPa)~ 0,4
• Initial temperature of polymer (oC)30,2
• Density of polymer (kg/m3)960,10
• Heat Capacity (J/kg-K)2250,20
• Flow Rate (m3/s)~ 0,1.5 6
.100 data points are generated and used in calibration process
Parameter Prior Distribution
(oC) Uniform (205,220)
(oC) Uniform (45,60)
(oC) Uniform (30,45)
(MPa) Uniform (88,95)
(m3/s) Uniform (1.6e-5,1.75e-5)
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
21
Results – Prior Sensitivity Analysis
Variable Type of Uncertainty Prior – Individual effect Prior - Total effect Remarks
Epistemic 0.4857 0.5829 SignificantAleatory 0.0161 0.0363 Significant
Epistemic 0.2412 0.3617 SignificantAleatory 0.1226 0.1264 Significant
Epistemic 0.0028 0.0078 InsignificantAleatory 0.0015 0.0035 InsignificantAleatory 0.0134 0.0137 Significant
Sensitivity index < 0.01 (Threshold value) Insignificant
Variables retained for Calibration : Injection Temperature, Ejection Temperature
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
22
Results - Prior and Posterior Plots (Epistemic variables)
Injection Temperature
Ejection Temperature
Energy consumed per part
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
23
Results – Prior and Posterior Sensitivity Analysis
Variable Type of Uncertainty
Prior –Individual
effect
Prior -Total effect
RemarksPosterior –Individual
effect
Posterior – Total effect
Remarks
Epistemic 0.4857 0.5829 Significant 0.0083 0.0125 Insignificant
Aleatory 0.0161 0.0363 Significant 0.0686 0.0961 Significant
Epistemic 0.2412 0.3617 Significant 0.0063 0.0126 Insignificant
Aleatory 0.1226 0.1264 Significant 0.7997 0.818 Significant
Epistemic 0.0028 0.0078 Insignificant 0.0018 0.0019 Insignificant
Aleatory 0.0015 0.0035 Insignificant 0.0107 0.0418 Significant
Aleatory 0.0134 0.0137 Significant 0.0414 0.0798 Significant
Uncertainty in and greatly reduced after calibration process (Effects Significant to Insignificant)
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
24
Illustrative Example – Welding ProcessEnergy Consumption (1/2)
• Volume of the Weld 34 2
• Theoretical Minimum Energy (filler and metal are the same)
• Input Energy (Laser)
• Efficiency of welding process
LegendL = Length of weld= density of material = Heat capacity of material
= Final Temperature= Initial Temperature= Latent Heat
= Voltage= Current= Welding Speed
Goal: UQ in Energy Consumption of a Welding Process
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
25
Energy Consumption in Welding Process (2/2)
• Total Power
• Total Electricity Cost
Legend= Total weld time= Number of days in service= Unit cost for electricity in USA
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
26
Welding Process – Bayesian Network
Constants DeterministicAleatoryCalibration
Parameters(Epistemic)
Observations
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
27
Synthetic Dataset (1/2)
Parameter ValueInitial Temperature ( ) (K) 303,0.3Final Temperature ( ) (K) 1628,10
Heat Capacity ( ) (J/kgK) 500,5
Density ( ) (kg/m3) 8238,10
Latent Heat ( ) (kJ/kg) 270,3
Weld Zone parameters (mm)8.5,0.52.6,0.52,0.115,0.511,1
Length of weld ( ) (mm) 500,10
Voltage ( ) (V) 20
Current ( ) (A) 250
Weld Speed ( ) (mm/min) 700
26
10 cents/kWh
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
28
Synthetic Dataset(2/2)
• Error in length measurement (mm), ~ 0,0.01
• Error in current measurement (A), ~ 0,2
Observation Data
• 8.5,0.5 0,0.01
• 2.6,0.5 0,0.01
• 11,1 0,0.01
• 250 0,2
Parameter Prior Distributions
Uniform (8.3,8.6)Uniform (0.2,0.7)
Uniform (2.5,2.8)Uniform (0.3,0.6)
Uniform (10,13)Uniform (0.8,1.3)
Uniform (235,260)
Uniform (1.7,2.5)
Prior Distributions
- Standard deviation in measurement error of current
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
29
Results – Prior Sensitivity Analysis
Parameter Type of uncertainty
Prior –Individual
effect
Prior – Total effect Remarks
Aleatory 0.0745 0.0872 SignificantEpistemic 0.0026 0.0027 InsignificantEpistemic 6.44 x 10-7 7.75 x 10-3 InsignificantAleatory 0.2147 0.2238 Significant
Epistemic 0.0073 0.0074 InsignificantEpistemic 7.056 x 10-8 0.00826 InsignificantAleatory 0.3153 0.3227 Significant
Epistemic 0.2019 0.2032 SignificantEpistemic 1.225 x 10-6 0.00569 InsignificantAleatory 0.044 0.0442 SignificantAleatory 0.1651 0.1655 SignificantAleatory 0.000124 0.000125 InsignificantAleatory 0.00435 0.0044 InsignificantAleatory 0.00091 0.00093 InsignificantAleatory 2.393 x 10-6 2.421 x 10-6 InsignificantAleatory 0.00225 0.00228 InsignificantAleatory 0.0347 0.0351 Significant
Threshold value – 0.01
Significant Epistemic
variables -
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
30
Results – Prior and Posterior Plots (Epistemic Variables)
Prior and Posterior no major change, most of the uncertain variables are aleatory
(uncertainty can not be reduced)
Weld parameter ‘e’ Theoretical energy consumption per weld
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
31
Results – Prior and Posterior Sensitivity Analysis
Parameter Type of uncertainty
Prior –Individual
effect
Prior – Total effect Remarks
Posterior –Individual
effect
Posterior –Total effect Remarks
Aleatory 0.0745 0.0872 Significant 0.1193 0.1226 SignificantEpistemic 0.0026 0.0027 Insignificant 0.00012 0.00013 Insignificant
Epistemic 6.44 x 10-7 7.75 x 10-3 Insignificant 1.062 x 10-7 6.005 x 10-5 Insignificant
Aleatory 0.2147 0.2238 Significant 0.2995 0.3001 SignificantEpistemic 0.0073 0.0074 Insignificant 0.00025 0.000247 InsignificantEpistemic 7.056 x 10-8 0.00826 Insignificant 7.208 x 10-10 0.000135 InsignificantAleatory 0.3153 0.3227 Significant 0.2967 0.2989 Significant
Epistemic 0.2019 0.2032 Significant 0.00034 0.00034 InsignificantEpistemic 1.225 x 10-6 0.00569 Insignificant 2.991 x 10-9 0.000147 InsignificantAleatory 0.044 0.0442 Significant 0.0493 0.0494 SignificantAleatory 0.1651 0.1655 Significant 0.2069 0.2075 InsignificantAleatory 0.000124 0.000125 Insignificant 0.000153 0.000154 InsignificantAleatory 0.00435 0.0044 Insignificant 0.00549 0.00554 InsignificantAleatory 0.00091 0.00093 Insignificant 0.00107 0.00108 Insignificant
Aleatory 2.393 x 10-6 2.421 x 10-6 Insignificant 2.054 x 10-6 2.564 x 10-6 Insignificant
Aleatory 0.00225 0.00228 Insignificant 0.00313 0.00316 InsignificantAleatory 0.0347 0.0351 Significant 0.04097 0.0413 Significant
Uncertainty in greatly reduced after calibration process (Significant to Insignificant)
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
32
Comprehensive framework for uncertainty integration and management
Bayesian network– Include all available models and data– Include calibration, verification and
validation results at multiple levels– Heterogeneous data of varying precision
and cost– Models of varying complexity, accuracy,
cost– Data on different but related systems Bayesian Network
• Forward problem: UQ in overall system-level prediction– Integrate all available sources of information and results of modeling/testing/monitoring
• Inverse problem: Resource allocation at various stages of system life cycle– Model development, data collection, system design, manufacturing, operations, health
monitoring, risk management
Facilitates
g
e d
c r
s
n
o
j
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
33
Summary of Analyses
• Information Retrieval– retrieve required information from database
• Dimension Reduction – Select a subset of features/parameters
• Data Reduction – Reduce amount of data through clustering
• Model Building – Build a Bayesian Network
• Uncertainty Propagation – Monte Carlo simulation
• Sensitivity Analysis (Aleatory vs. Epistemic) – Variance decomposition
• Performance Prediction – Monte Carlo simulation
• Temporal variation -- Dynamic Bayesian Network
• Diagnosis, Prognosis – Classification, Particle filtering
• Decision-Making, Resource allocation
IEEE International Conference on Big Data Special Session on Big Data and Analytics for Smart
Manufacturing Systems, Oct 27, 2014
UQ in Performance Evaluation of Manufacturing ProcessesPresenter: Sankaran Mahadevan
34
Future Work
• Derive Bayesian network from process and model libraries
• Include text and image data, along with numerical data in UQ
• Apply to production network with multiple processes
• Analysis over time with streaming data (Dynamic Bayesian Network)
• Explore various problems in manufacturing– Process Monitoring (Dynamic Tracking) – For diagnosis– Stochastic Process Optimization – Adjust parameters to meet the requirements– Risk Management – Reduce performance variability to be within desired limits– Resource Allocation – maximize information maximize reduction in uncertainty