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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 1 / 24 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 2 / 24 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 3 / 24 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 4 / 24 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 5 / 24 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 6 / 24 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 7 / 24 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 8 / 24 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 9 / 24 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 10 / 24 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 11 / 24 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 12 / 24 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 13 / 24 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 : 0,12 point set P has k points 14 / 24 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 15 / 24 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 16 / 24 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 17 / 24 ... 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 18 / 24 ... 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 20 / 24 ... 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 21 / 24 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 22 / 24 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 23 / 24 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 24 / 24