Introduction
Methodology
Data acquisition
Creep data
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building orientation: 0\(^\circ \) , 45\(^\circ \) and 90\(^\circ \);
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scan strategy: stripe or meander;
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number of lasers: 1 or 4 lasers.
Test case | Orientation | Number of lasers | Scan strategy | Power (W) | Stripe point distance (\(\mu \)m) | Hatch time (\(\mu \)s) | Exposure distance (mm) | Hatch offset (mm) |
---|---|---|---|---|---|---|---|---|
VSM | \(90^\circ \) | 1 | Meander | 212.5 | 17 | 20 | 0.09 | 0 |
VS | \(90^\circ \) | 1 | Stripe | 212.5 | 17 | 20 | 0.09 | \(-\,0.2\) |
VM | \(90^\circ \) | 4 | Stripe | 212.5 | 17 | 20 | 0.09 | \(-\,0.2\) |
45S | \(45^\circ \) | 1 | Stripe | 212.5 | 17 | 20 | 0.09 | \(-\,0.2\) |
45M | \(45^\circ \) | 4 | Stripe | 212.5 | 17 | 20 | 0.09 | \(-\,0.2\) |
HS | \(0^\circ \) | 1 | Stripe | 212.5 | 17 | 20 | 0.09 | \(-\,0.4\) |
HM | \(0^\circ \) | 4 | Stripe | 212.5 | 17 | 20 | 0.09 | \(-\,0.4\) |
Porosity data
Extraction of material descriptors
Data preparation for machine learning models
Element | Weight % | Element | Weight % |
---|---|---|---|
Ni | 52.5 | Co | 0.04 |
Cr | 19.1 | C | 0.03 |
Fe | Bal. | Mn | 0.02 |
Nb + Ta | 4.89 | N | 0.01 |
Mo | 3.2 | Cu | 0.01 |
Ti | 0.86 | P | \(<0.01\) |
Al | 0.42 | S | 0.001 |
Si | 0.04 | B | \(<0.001\) |
Material descriptor | Description |
---|---|
Number of pores | The number of pores in the labelled image |
Area | Number of pixels in the connected labelled region |
Convex area | Convex hull image of labelled pixels, i.e. the smallest convex polygon that encloses the area |
Eccentricity | Eccentricity of the ellipse that has the same second-moments as the labelled area. The eccentricity is the ratio of the focal distance (distance between focal points) over the major axis length. The value is in the interval of 0 and 1. When it is 0, the ellipse becomes a circle |
Equivalent diameter | The diameter of a circle with the same area as the region |
Major axis length | Major axis length of the ellipse |
Minor axis length | Minor axis length of the ellipse |
Orientation | Angle between the row axis of image and the major axis of the ellipse that has the same second moments as the region, ranging from -pi/2 to pi/2 counter-clockwise |
Perimeter | Perimeter of object which approximates the contour as a line through the centers of border pixels using a 4-connectivity |
Density (solidity) | Ratio of pixels in the area to pixels of the convex hull image |
Inertia tensor | Inertia tensor of the area for the rotation around its mass |
Inertia tensor eigenvalues | The eigenvalues of the inertia tensor in decreasing order |
Machine learning models
Regularised linear regression
Random forest
Hyperparameter | Value |
---|---|
Number of trees | 500 |
Maximum material descriptors | \(\sqrt{p}\) |
Maximum depth of trees | 5 level |
Minimum samples before split | 2 |
Bootstrap | True |
Gradient boosted tree
Hyperparameter | Value |
---|---|
Number of trees | 500 |
Learning Rate | 0.1 |
Maximum depth of trees | 5 level |
Loss function | Least square |
Maximum material descriptors | \(\sqrt{p}\) |
Splitting criterion | Friedman MSE |
Support vector regression
Hyperparameter | value |
---|---|
Kernel function | Radial Basis Function |
Margin of error | 0.1 |
Regularisation | 1.0 |
Maximum Iteration | No limit |
Deep neural network
Hyperparameter | Value |
---|---|
Number of hidden layers | 8 |
Number of nodes for each layer | 256, 128, 64, 32, 16, 8, 6, 4 |
Activation function | Rectified Linear Unit |
Loss function | MSE |
Optimiser | Rectified Adam with LookAhead (Tong et al. 2019) |
Learning rate | 0.001 |
L2 regulariser | 0.01 |
Evaluation metrics
Interpreting machine learning model outputs
Results
Creep data
Porosity data
Machine learning results
Samples | Density (%) | Number of pores |
---|---|---|
VSM | 99.943 | 3337 |
VS | 99.947 | 1841 |
VM | 99.984 | 1055 |
45S | 99.977 | 1265 |
45M | 99.980 | 1002 |
HS | 99.973 | 1371 |
HM | 99.960 | 1758 |
Evaluation metrics | MAD |
\(R^{2}\)
| % Error | MAE | RMSE |
---|---|---|---|---|---|
Random forest | 0.00 ± 0.00 | 1.00 ± 0.00 | 0.00 ± 0.00 | 0.00 ± 0.00 | 0.00 ± 0.00 |
Gradient boosted trees | 0.00 ± 0.00 | 1.00 ± 0.00 | 0.00 ± 0.00 | 0.00 ± 0.00 | 0.00 ± 0.00 |
Deep neural network | 0.44 ± 0.58 | 0.90 ± 0.11 | 30.45 ± 45.53 | 2.24 ± 1.90 | 4.98 ± 5.23 |
SVR | 0.56 ± 0.13 | 0.86 ± 0.09 | 5.07 ± 0.31 | 2.37 ± 0.12 | 4.39 ± 0.14 |
Ridge regressor | 7.48 ± 0.15 | 0.42 ± 0.04 | 20.65 ± 0.78 | 7.68 ± 0.23 | 8.99 ± 0.28 |
LASSO regressor | 8.09 ± 0.23 | 0.39 ± 0.02 | 20.22 ± 0.46 | 7.73 ± 0.18 | 9.18 ± 0.15 |
Discussion
Material descriptors affecting the creep rate
Effects of density
Models | Evaluation metrics | 45M | 45S | HM | HS | VM | VSM | VS |
---|---|---|---|---|---|---|---|---|
Random forest | MAD | 4.31 ± 0.01 | 4.24 ± 0.02 | 26.09 ± 0.00 | 1.55 ± 0.41 | 13.42 ± 0.08 | 100.10 ± 0.00 | 7.50 ± 0.76 |
MAE | 4.32 ± 0.01 | 4.19 ± 0.03 | 26.07 ± 0.02 | 3.33 ± 0.27 | 13.46 ± 0.07 | 100.10 ± 0.00 | 7.59 ± 0.49 | |
% Error | 14.17 ± 0.04 | 12.05 ± 0.11 | 46.06 ± 0.04 | 6.78 ± 0.55 | 49.32 ± 0.28 | 418.82 ± 0.00 | 18.80 ± 1.21 | |
RMSE | 4.32 ± 0.01 | 4.19 ± 0.03 | 26.07 ± 0.02 | 4.12 ± 0.27 | 13.46 ± 0.07 | 100.10 ± 0.00 | 8.75 ± 0.31 | |
Gradient boosted trees | MAD | 4.31 ± 0.00 | 4.28 ± 0.00 | 26.08 ± 0.00 | 7.41 ± 0.00 | 13.17 ± 0.00 | 100.09 ± 0.00 | 13.17 ± 0.07 |
MAE | 4.31 ± 0.00 | 4.35 ± 0.01 | 26.05 ± 0.01 | 10.47 ± 0.02 | 13.14 ± 0.00 | 100.09 ± 0.00 | 12.39 ± 0.13 | |
% Error | 14.19 ± 0.02 | 12.50 ± 0.02 | 46.02 ± 0.01 | 21.29 ± 0.04 | 48.14 ± 0.01 | 418.82 ± 0.00 | 30.67 ± 0.32 | |
RMSE | 4.32 ± 0.01 | 4.35 ± 0.01 | 26.05 ± 0.01 | 10.99 ± 0.02 | 13.14 ± 0.00 | 100.09 ± 0.00 | 12.52 ± 0.11 | |
Deep neural network | MAD | 20.96 ± 18.92 | 4.69 ± 3.16 | 25.39 ± 2.46 | 6.73 ± 5.10 | 20.07 ± 10.68 | 99.86 ± 0.18 | 14.62 ± 5.73 |
MAE | 21.02 ± 18.92 | 5.10 ± 3.16 | 24.88 ± 2.46 | 7.10 ± 5.10 | 20.76 ± 10.68 | 98.55 ± 0.18 | 21.30 ± 5.73 | |
% Error | 68.94 ± 61.69 | 14.65 ± 8.40 | 43.96 ± 4.80 | 14.43 ± 9.16 | 76.06 ± 35.55 | 412.38 ± 2.66 | 52.73 ± 8.30 | |
RMSE | 21.40 ± 18.5 | 5.69 ± 2.70 | 35.10 ± 2.79 | 7.97 ± 4.31 | 21.97 ± 10.37 | 98.62 ± 0.59 | 27.70 ± 4.91 | |
SVR | MAD | 19.33 ± 0.00 | 0.75 ± 0.00 | 20.12 ± 0.00 | 0.51 ± 0.00 | 23.68 ± 0.00 | 14.64 ± 0.00 | 10.26 ± 0.00 |
MAE | 19.11 ± 0.00 | 0.81 ± 0.00 | 20.26 ± 0.00 | 0.69 ± 0.00 | 23.68 ± 0.00 | 14.50 ± 0.00 | 10.37 ± 0.00 | |
% Error | 62.64 ± 0.00 | 2.34 ± 0.00 | 35.80 ± 0.00 | 1.40 ± 0.00 | 86.74 ± 0.00 | 60.68 ± 0.00 | 25.69 ± 0.00 | |
RMSE | 19.12 ± 0.00 | 0.93 ± 0.00 | 20.26 ± 0.00 | 0.91 ± 0.00 | 23.68 ± 0.00 | 14.51 ± 0.00 | 10.41 ± 0.00 | |
Ridge regressor | MAD | 22.29 ± 0.00 | 19.23 ± 0.00 | 28.20 ± 0.00 | 12.06 ± 0.00 | 28.44 ± 0.00 | 22.67 ± 0.00 | 4.22 ± 0.00 |
MAE | 22.25 ± 0.00 | 18.87 ± 0.00 | 28.39 ± 0.00 | 11.75 ± 0.00 | 28.49 ± 0.00 | 24.11 ± 0.00 | 8.15 ± 0.00 | |
% Error | 72.96 ± 0.00 | 54.22 ± 0.00 | 50.17 ± 0.00 | 23.89 ± 0.00 | 104.36 ± 0.00 | 100.89 ± 0.00 | 20.17 ± 0.00 | |
RMSE | 22.26 ± 0.00 | 18.95 ± 0.00 | 28.40 ± 0.00 | 11.89 ± 0.00 | 28.49 ± 0.00 | 24.90 ± 0.00 | 11.51 ± 0.00 | |
LASSO regressor | MAD | 19.27± 0.00 | 12.71 ± 0.00 | 24.80 ± 0.00 | 10.57 ± 0.00 | 17.88 ± 0.00 | 16.92 ± 0.00 | 2.99 ± 0.00 |
MAE | 19.27 ± 0.00 | 12.71 ± 0.00 | 24.80 ± 0.00 | 10.57 ± 0.00 | 17.88 ± 0.00 | 16.92 ± 0.00 | 2.99 ± 0.00 | |
% Error | 63.20 ± 0.00 | 36.52 ± 0.00 | 43.82 ± 0.00 | 21.48 ± 0.00 | 65.51 ± 0.00 | 70.82 ± 0.00 | 7.41 ± 0.00 | |
RMSE | 19.27 ± 0.00 | 12.71 ± 0.00 | 24.80 ± 0.00 | 10.57 ± 0.00 | 17.88 ± 0.00 | 16.92 ± 0.00 | 2.99 ± 0.00 |
Effects of LPBF build parameters
Interpretation of machine learning models’ predictions
Conclusions and future work
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Various ML models were able to predict the creep rate using porosity data and LPBF build parameters despite a limited amount of data.
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Random Forest and Gradient Boosted Trees were able to accurately predict the creep rate with 100% accuracy when all test cases were used to train the model. Support Vector Regressor achieved the 98.6% accuracy when one case was left out at its best.
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The top material descriptors affecting the creep rate were identified to be density, number of pores and build orientation, in descending order of importance. The number of lasers was found to have only an insignificant effect on the creep rate.
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Density and the number of pores serve as crack initiation points and negatively affect the creep rate; the scan strategy was found to affect component density and the build orientation affected the failure of samples.
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The main disadvantage of using ML models is the need for large sample size in order to increase accuracy. However, accurate ML models were trained by using data augmentation to increase sample size and image analysis to extract material descriptors. But this could be solved by more collaboration between researchers.
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Finally, despite using a small data set, some ML models were able to offer insight into the effect of process parameters on creep properties as well as to predict creep rates.