Download Data Mining and Statistical Inference in Selective Laser Melting

Data Mining and Statistical Inference
in
Selective Laser Melting
Chandrika Kamath
Lawrence Livermore National Laboratory
Livermore, CA, USA
Symposium: Big Data and Predictive Computational Modeling
TU Munich Institute for Advanced Study
May 21, 2015
LLNL-PRES-670245: This work was performed under the auspices of the U.S. Department of Energy by
Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. The work was funded by the
Laboratory Directed Research and Development Program at LLNL under project tracking code 13-SI-002.
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
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Additive manufacturing allows us to build complex parts
Turbine blade
Medical implants
Lattices
Customized heat exchanger
Selective laser melting: building a 3-D part, layer by layer, using a high-energy laser beam
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
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However, the design freedom is accompanied by complexity
More than 130 parameters could affect the final quality of the part.
Modeling of the process is complicated: physical phenomena occur
over a broad range of length and time scales.
Laser Parameters
 Power
 Beam profile
 Scan speed
 Scan overlap
 Scan strategy
Material Properties
 Density
 Thermal conductivity
 Heat capacity
 Latent heat
 Electrical conductivity
Chandrika Kamath (LLNL)
Powder Bed Conditions
 Porosity
 Particle sizes
 Layer thickness
Data Mining and Statistical Inference in SLM
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Our goal is to understand and control the process better
Need:
Identify process parameters to create parts with desired properties
Understand how uncertainty in inputs affects the outputs
Control the process better
But, experiments and multi-scale simulations are both expensive
Approach:
Run complex sims and expts
Understand the design space using simulations
Identify important inputs
Build predictive models for code surrogates
Design experiments to validate simulations
Estimate uncertainty using data-driven models
q
Refine
sampling
v
q
Identify
important
inputs; build
surrogates for
prediction and
UQ
We use data mining and statistical inference
to intelligently exploit simulations and
experiments.
v
Sample
parameter
space
Run simple
sims/expts
Analyze
Extract
characteristics
Explore
parallel plots
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
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Our first task: identify process parameters for high-density
316L stainless steel parts
Challenge:
Prior work: machines with powers ≈100W and beam sizes D4σ ≈ 120µm
Our Concept Laser M2 machines have powers up to 400W and D4σ ≈ 54µm
We started with the simple, computationally-inexpensive Eagar-Tsai model† :
Thermal-conduction model - laser melting by a Gaussian beam on a flat plate
Span the design space using stratified sampling (⇒ 462 simulations)
Extract the melt-pool depth and width
†
Parameter
Minimum
Maximum
Levels
Power (W)
50
400
7
Speed (mm/s)
50
2250
10
Beam size D4σ (µm)
50
68
3
Absorptivity η
0.3
0.5
2
T. W. Eagar and N.-S. Tsai, “Temperature fields produced by traveling distributed heat sources,” Welding Research
Supplement, December 1983.
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
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Normalized values
Parallel coordinate plots indicate that speed and power are
important variables
Depth > 120 µm
Depth > 60 µm Depth < 120 µm (x1, x2, x3, x4, x5)
Depth < 60 µm Speed Power Beam size Absorptivity We consider a powder layer thickness of 30µm.
Actual powder thickness higher due to
porosity in powder bed
The energy density should melt the powder
into the substrate, but not too deep
Chandrika Kamath (LLNL)
Depth
Increase power or reduce speed
Data Mining and Statistical Inference in SLM
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We confirmed our choice of process parameters using
single-track experiments
A tilted plate with layer thickness 0 at
left end and 200µm at the right.
Track
Power
Speed
Depth
number
(W)
(mm/s)
(µm)
1
400
1800
105
2
400
1500
119
3
400
1200
182
4
300
1800
65
5
300
1500
94
6
300
1200
114
7
300
800
175
8
200
1500
57
9
200
1200
68
10
200
800
116
11
200
500
195
12
150
1200
30
13
150
800
67
14
150
500
120
Melt-pool depth at layer thickness 30µm
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
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We then used the parameters to build small pillars and
evaluated their density using the Archimedes’ method
Pillar: 10mm × 10mm × 7mm
Density for 48 316L pillars using CL powder; power 150-400W
100.0
Speed
150W
200W
250W
300W
350W
400W
Density (in percentage)
99.0
98.0
97.0
96.0
95.0
94.0
93.0
92.0
91.0
90.0
Power
500
1000
1500
2000
2500
3000
Speed (in mm/s)
Remaining parameters set to default values
C. Kamath, B. El-dasher, G. F. Gallegos, W. E. King, and A. Sisto, ”Density of additively-manufactured, 316L SS parts using
laser powder-bed fusion at powers up to 400 W,”. Int J Adv Manuf Technol. Volume 74, Issue 1 (2014), Page 65-78.
C. Kamath, “On the use of data mining techniques to build high-density, additively-manufactured parts”, in Information-Driven
Approaches to Materials Discovery and Design, T. Lookman, F. Alexander and K. Rajan (eds), Springer Materials series, 2015.
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
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Data analysis can help to reduce the number of simulations
and experiments required for high-density AM parts
For 316L stainless steel, were able to obtain relative density > 99%
over a range of power and speed values.
We also gained scientific insight.
We have used this approach successfully for other materials and
powder sizes.
Can we improve this approach to reduce costs and gain insight?
Feature selection to identify important variables
Surrogate models for simulations to reduce computational costs
Improved sampling to reduce number of sample points
More complex simulations for improved predictions
Uncertainty analysis for statistical inference
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
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Feature selection can be used for dimension reduction
Select important features using:
Correlation-based method: figure of merit of subset of k features
krcf
Merit = p
k + k(k − 1)rff
Mean-squared error method: also used in regression trees
MSE (A) = pL · s(tL ) + pR · s(tR )
where
rcf : average feature-output correlation
rff : average feature-feature correlation
tL and tR : subset of samples that go to the left and right
pL and pR : proportion of samples that go to the left and right
v
u
N(t)
u 1 X
2
s(t) = t
(ci − c(t) )
N(t) i=1
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
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Both methods select power and speed as important
Correlation-based method:
Melt-pool width
Speed
Power
Beam size
Absorptivity
Noise
5
4
2
3
1
Melt-pool length
3
5
2
4
1
Melt-pool depth
5
4
2
3
1
Power
Beam size
Absorptivity
Noise
Mean-squared error method:
Speed
Melt-pool width
5
4
2
3
1
Melt-pool length
3
5
2
4
1
Melt-pool depth
5
4
1
3
2
This confirms the results from the parallel coordinate plots.
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
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We can use data-driven models as surrogates for prediction
fi  ai
Y
N
Y
f j  aj
Y
f k  ak
N
Y
N
fi  bi
Y
f n  an
N
Y
N
X
Regression Trees:
Split decision based on minimizing the mean squared error
Ensembles can improve the accuracy of the models
use randomization to create multiple models and average the results
introduce randomization by sampling the instances at each node
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
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The leave-one-out metric illustrates benefit of ensembles
Random stratified; 1 tree
300
250
Predicted depth (in micron)
Predicted depth (in micron)
250
200
150
100
50
0
Random stratified; 10 trees
300
200
150
100
50
0
50
100
150
200
250
300
Actual depth (in micron)
0
0
50
100
150
200
250
300
Actual depth (in micron)
462 instances generated using the Eagar-Tsai model
5 runs of 5-fold cross-validation give a relative mean-squared error of
single tree: 8%
ensemble of 10 trees: 3.6%
Can we do just as well with fewer samples?
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
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The placement of initial sample points is important
Random
1
Random stratified
1
Poisson disk
1
0.9
0.9
0.9
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.6
0.6
0.6
0.6
0.5
0.5
0.5
0.5
0.4
0.4
0.4
0.4
0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Best candidate
1
0.9
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Random sampling: has over- and under-sampled regions
Stratified random sampling: better, but number of samples depends
on number of levels
Low discrepancy sampling: random, but samples are spread out
|P ∩ r | µ(r ) Discrepancy(P, R) = sup −
k
µ(X ) r ∈R
r
Poisson disk sampling
Mitchell’s best candidate algorithm
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
X : 0,12
point set P has k points
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Samples generated by the best candidate method have
more uniform distance to nearest neighbor
Random stratified
0.4
0.3
0.2
0.1
0
10
20
30
40
50
60
Best candidate
0.5
Distance to nearest neighbor
Distance to nearest neighbor
0.5
70
80
90
100
0.4
0.3
0.2
0.1
0
10
20
30
Sample point
40
50
60
70
80
90
100
Sample point
100 sample points for Eagar-Tsai model; 4 input dimensions
Both random stratified and best candidate scale to higher dimensions.
Best candidate can also support any number of samples as well as
incremental and subset sampling.
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
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As expected, fewer samples increase prediction error
100
100
90
80
70
60
50
90
80
70
60
50
100
90
90
80
70
60
50
40
40
30
30
30
20
20
20
30
40
50
60
70
80
90
100 110 120
Actual depth (in micron)
20
30
40
50
60
70
80
90
100 110 120
Best candidate; 10 trees
110
100
40
20
Best candidate; 1 tree
110
Predicted depth (in micron)
110
Predicted depth (in micron)
Predicted depth (in micron)
Random stratified; 10 trees
120
110
Predicted depth (in micron)
Random stratified; 1 tree
120
80
70
60
50
40
30
20
Actual depth (in micron)
30
40
50
60
70
80
90
100
110
20
Actual depth (in micron)
20
30
40
50
60
70
80
90
100
110
Actual depth (in micron)
Relative mean-squared error rate using 5 runs of 5-fold cross validation
Method
1 tree
10 trees
Random stratified (462 samples)
8.0%
3.6%
Random stratified (100 samples)
24.0%
12.4%
Best candidate (100 samples)
27.1%
12.3%
Since the Eagar-Tsai model is used mainly to identify a viable region, and
not for prediction, this error is tolerable.
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
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We then considered a more complex simulation ...
The Verhaeghe model† includes a powder bed:
models the changing substrate structure: dense metal to powder bed
takes into account phase transitions, including evaporation
uses a simplified model for laser absorption
ignores Marangoni convection and surface-tension driven shape
evolution of the liquid
Approach - iteratively refine the samples:
Run Eagar-Tsai on the 100 samples from the best candidate sampling
Run the Verhaeghe model at 34 samples with E-T depth > 60µm
Range of parameters: power 150-400W; speed 500-1600 mm/s
BUT, too few samples to use regression trees
†
F. Verhaeghe, T. Craeghs, J. Heulens, and L. Pandelaers, “A pragmatic model for selective laser melting with evaporation,”
Acta Materialia, 57, PP 6006-6012, 2009.
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
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... coupled with a more sophisticated surrogate model...
A Gaussian process model:
Extends multi-variate Gaussian distributions to infinite dimensions
Consider N observations as a single sample from an N-variate
Gaussian distribution.
Relate one observation to another using the covariance function
−(x − x 0 )2
0
2
k(x, x ) = σf exp
2l 2
Assume that the mean of the GP is zero
y
K K∗T
= N 0,
y∗
K∗ K∗∗
Given y, what is the probability of y∗ ?
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
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... that predicts the mean and variance at a new sample
The best estimate for y∗ is
y ∗ = K∗ K −1 y
and the uncertainty in the estimate is given by the variance
var(y∗ ) = K∗∗ − K∗ K −1 K∗T
Interpolated function
Training data
Chandrika Kamath (LLNL)
New data point
Data Mining and Statistical Inference in SLM
Prediction with error bar
19 / 24
The Gaussian process model provides insight on depth as a
function of power and speed ...
200
150
100
50
0
100
400
200
150
100
50
0 600
400
200
150
100
50
0 600
150.0 W
400
200
150
100
50
0 600
250.0 W
200
150
100
50
0
100
Depth as a function of speed for set power values, 2σ=26µm, η=0.40
350.0 W
1200.00 m/s
200
150
100
50
0
100
1500.00 m/s
900.00 m/s
Depth as a function of power for set speed values, 2σ=26µm, η=0.40
150
150
150
200 250 300
Power in Watts
200 250 300
Power in Watts
200 250 300
Power in Watts
350
350
350
800 1000 1200 1400 1600 1800
Speed in mm/s
800 1000 1200 1400 1600 1800
Speed in mm/s
800 1000 1200 1400 1600 1800
Speed in mm/s
Results obtained using GPy: http://sheffieldml.github.io/GPy/
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
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... as well as depth as a function of 2σ and η
100
50
26
28
30
32
2σ in micron
50
025
34
150
150
100
100
2σ=30.00
η=0.40
2σ=25.00
150
100
0
50
0
η=0.45
Depth as a function of η for set 2σ values; 250W, 1200mm/s
150
26
28
30
32
2σ in micron
150
150
100
100
50
0
26
28
30
32
2σ in micron
Chandrika Kamath (LLNL)
34
30
45
35
40
η as a percentage
50
55
30
45
35
40
η as a percentage
50
55
30
45
35
40
η as a percentage
50
55
50
025
34
2σ=35.00
η=0.35
Depth as a function of 2σ for set η values; 250W, 1200mm/s
50
025
Data Mining and Statistical Inference in SLM
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A comparison of predicted values for the 14 track plate
indicates that the Verhaeghe model is promising
Gaussian process prediction vs. experiment
250
Experiment
GP prediction using Verhaeghe
Depth in micron
200
150
100
50
0
0
2
4
6
8
10
12
14
Track number
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
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We will be applying these data analysis ideas to more
complex simulations and experiments
Iteratively refine the GP surrogate by adding points suitably
Understand the effect of other variables: powder size distribution,
void fraction in powder bed, material parameters, ...
Use Gaussian processes for calibration and validation
Perform a quantitative comparison of models
Analyze the uncertainty in part properties for qualification
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
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Acknowledgements
Wayne King - implementation and running of the Eagar-Tsai model
John W. Gibbs - implementation of the Verhaeghe model
Paul Alexander - machinist
Cheryl Evans, Gil Gallegos, Mark Pearson - metallographic
preparation and measurements
Sheffield machine learning group - GPy software
ACAMM project - funding
For more details
Contact: Chandrika Kamath, [email protected]
Web page: ckamath.org
Project web page: acamm.llnl.gov
Chandrika Kamath (LLNL)
Data Mining and Statistical Inference in SLM
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