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This presentation slides will help to make bridge with knowledge and reality in traffic flow modelling based on real understanding of mathematical terms in modelling equations. I hope it will make good contribution to improve our knowledge level for performing simulation of any model based on numerical method e.g., finite difference scheme.
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Recently, the machine learning community has expressed strong interest in applying latent variable modeling strategies to causal inference problems with unobserved confounding. Here, I discuss one of the big debates that occurred over the past year, and how we can move forward. I will focus specifically on the failure of point identification in this setting, and discuss how this can be used to design flexible sensitivity analyses that cleanly separate identified and unidentified components of the causal model.
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2018 IMSM: Computational Techniques to Enhance Laser-based Characterization of Thin Film Thermomechanical Properties - Sandia National Labs Working Group, July 25, 2018
1. Computational Techniques to Enhance
Laser-based Characterization of Thin
Film Thermomechanical Properties
Esther Amfo, Jiahui Chen, Rasika Rajapakshage, Klajdi
Sinani, John Wakefield, Meng Zhang
Problem Presenter: Dr. Jordan E. Massad
Faculty Mentor: Drs. Ralph Smith and Paul Miles
Industrial Mathematical & Statistical Modeling Workshop for Graduate Students
Raleigh, NC
July 25, 2018
2. What’s our Problem?
• How can we remove distortion from existing
measurements?
• How can we better understand the deflectometer
through mathematical modeling?
• What can we do to obtain more accurate results in
the future?
𝑅 𝑄𝑁2
𝑅 𝑁𝑄?
How does the thermal enclosure affect laser
deflectometry of thin film warpage?
Team Laserwarp 2
9. Data Analysis
• 35 wafer samples measured.
• Multiple tests done on each sample.
• Data can vary with duration between tests.
• Four test configurations identified:
1. Q - uses the quartz window
2. NQ - does not use the quartz window
3. QN2 - uses the quartz window and a nitrogen flow
4. NQN2 - does not use the quartz window, uses nitrogen flow
Team Laserwarp 9
10. Effect of Quartz Window?
The quartz makes the wafer happier!
No Quartz Quartz
Team Laserwarp
R < 0
R > 0
10
12. Another Factor: Nitrogen Flow
Team Laserwarp 12
Thermal Enclosure
• Pure nitrogen flows through the enclosure to maintain
tight thermal and moisture control.
• Observation: Nitrogen flow affects measurements.
13. Quantifying N2 Flow Effect
Team Laserwarp 13
𝑅 𝑄 = 𝑐𝑅 𝑄𝑁2
+ 𝑏
c = 1.000443
𝑏 = 6.7 × 10−5
m
A linear response!
14. Mapping 𝑅 𝑄𝑁2
to 𝑅 𝑁𝑄
Team Laserwarp 14
N2 Flow Effect
Quartz Plate
Effect
𝑹 𝑵𝑸 =
𝒂 𝟏 𝑹 𝑸𝑵 𝟐
+ 𝒂 𝟐
𝒂 𝟑 − 𝒂 𝟒 𝑹 𝑸𝑵 𝟐
𝑎1 = 1.0004
𝑎2 = 6.7 × 10−5
m
𝑎3 = 1.004
𝑎4 = 4.465 × 10−3 m-1
• 𝑅 𝑄𝑁2
is the measured radius of curvature (warpage)
with quartz plate and N2 flow.
• 𝑅 𝑁𝑄 is the actual warpage of the wafer.
15. Statistical Model Validation
Known
Flat Wafer
Known
Curved Wafer
Measured Radius with
Quartz/N2 Error (m)
[254.50, 265.03] [40.57, 40.77]
Computed Actual Radius (m) [-1925.24, -1479.14] [49.32, 49.62]
Measured Actual Radius (m) [-1925.34, -1479.09] [49.31, 49.63]
Team Laserwarp 15
Model predictions are within 1 standard deviation or measurements.
The model is reasonably predictive.
16. Recommendations
• Use the statistical model to estimate the actual
warpage from thermal test measurements.
• To improve the model fit, acquire measurements of
wafers with 16-500 m radius of curvature.
• To continually calibrate the model fit, run a series of
measurements with/without the quartz plate and N2
flow at the beginning and/or end of each thermal test
series.
• The mirror may significantly impact quartz error:
manage uncertainty in its position and orientation.
• Continue developing mathematical models to predict
trends in data
Team Laserwarp 16