2.1 Overview
The part deflection due to warpage considered in this work is based on the analysis from Armillotta et al. [21]. Here the authors provided new hypotheses based on multiple-layer shrinkage and plastic deformation to justify the observed influence of part height on warpage. The new analytic formulation provides a better warpage prediction than the basic model obtained by Armillotta et al. and consistent with the assumptions of analytic models present in the literature [22]. Nonetheless, the improved model gives a poor fit for long and thick parts built with small layer thickness, leading the authors to the conclusion that the model will have to be revised considering additional physical mechanisms.
Hence, in this work, the basic model formulation is considered since it allows to describe the proposed methodology in a much simpler fashion and to obtain data samples with limited computational effort. Once a better predictive tool is available (either analytic or finite element-based), the same approach can easily be adapted. A range of material commonly used in FDM was selected, namely ABS, PLA, PEEK, and Nylon (PA12).
ML is applied for both classification and regression purposes.
Classification is performed to identify the combination of parameters, such as part length, part height, and material glass-transition temperature, which ensure the part printability according to a user-defined warpage threshold. This, in turn, allows distinguishing between relevant and irrelevant areas of the design space. In other words, it indicates whether a selected material for a chosen part geometry can be manufactured without exceeding the selected part warpage.
Regression is used on these relevant design space areas to provide a capability profile based on the considered parameters. The capability profile can then be used to predict the part warpage and to quickly evaluate the effect of both material and process parameters. To this aim, the validated capability profile is used to ascertain the effect of the chamber temperature on the part warpage.
2.2 Data samples generation
The considered equation of the inter-layer warpage δ is [21]
![](data:image/png;base64,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)
in which α is the linear shrinkage coefficients, TG is the material glass transition temperature, TC is the chamber temperature, L is the length of the part, Δh is the thickness of every deposited layer, and h is the height of the part.
Based on equation (1), two parameters P1 and P2 are defined, as the linear shrinkage coefficients α and the difference (TG –TC), respectively. All the other parameters represent the user initial input and are hence fixed. Equation (1) can be rewritten as
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAqsAAABTCAIAAAA3J9n7AAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAABJkSURBVHhe7d3fbxRVGwdweO9poeuVoYTQkkjUSKSlBhaSmlhqMeY1Cl0pIU1s0LaJF4htU2iIkRS20Zp40WUJJL0A3RouuLC1a5OaWDS0XUk3YiSRbQwxXO12rfwBvF/3OZ13OrM7+6vtzux8P4njnNlfM2fLPs85c+bM5qdPn24iIiIil/mP+j8RERG5CTMAIiIiN2IGQERE5EbMAIiIiNyIGQAREZEbMQMgIiJyI2YAREREbsQMgIiIyI2YARAREbkR5wQsvUQiMTU1dfv27b///nvXrl0nTpzwer3qMSIiovXBDKD0qqqqsBwZGamoqLhw4UIkEonFYkgF5FEiIqL1wLMApZdMJhH+fT5fS0vLJ598gi1TU1PyUF6i0WjzisXFRbW1TJ07d06OdGhoSG0iIqJ8OCwDQJDDL7789CNkhkIh9YDt3blzR3ZbqK0pMzMzr732mqxXVFRgWV1dLUUDifHBYFCVV9u+fXt7e3s4HJ6bmyuDLgR8uThY1Jsqr3b8+HEscbCVlZWyhYiI8vPUOWpqarDD/f39CJnj4+NHjhxBsbOzUz1sb7FYrLW1NVXlm7DzaqsJDqqurk4VTHDseDnqQZXTwRPwJqrgZKgHHIvF9+v3+/EE/DGoMhER5cNJGQB+7hECVeHp03g8LjmBU2KARCyLkIaHEPZwXKpssi0Fb7KwsKA2rYaqwKP4IFV2LBwgDkSOV20ykRRQFYiIKE9OOguA8HbmzBlV2LTJ4/HU1tZi5bfffpMtjnD48GG1tlpXV9f8/Pzk5CSOS21aTU55XLp0Cctvvvkmtc1IquLAgQNSdK5gMNja2trU1JRMJicmJtTW1ebm5iQJICKiAjgpA/B6vYbomEgksHzllVekaHPT09NY7t+/X4p6CP8I8GNjY5nCP4yOjvp8PhkxgGfKRoNoNIrlnj17pJgvGWegCqWDrxW18d8UFL/99lvZrre4uIjkYN++faq8Gt4BByJ/HkRElJ7qC3Ag6Sh20Dlv7G3aPu2vv/4a27WO/ZmZGWyRdU0sFsPL5TlygjztiYCamhoZRoA36ezsROVYjDkwwBtiN8wfvfGkQrASj8ctKg0P4ejwnCtXruBIcbz6Eyh+v9/6lAoRkcs59WpAtBEbGxtbW1tv3rypNtmbtM6bmpqkqEE7taurq6qqqre3F81WePPNNx89eqQeXnH9+nVE95deegnrx44dw/Lu3bupR/4Pb4VEob6+fmho6Lvvvjtx4gS2HD16VD7aGp7Z0dGBIOrz+dSm0vn8889lNzweD0I72vrmQ/j1119lpbu7u7q6+q233goEAliXjdDT04Maa2trU2UiIjJQmYBzaKd+seKUMYCAdir22TxGD41UHIUBArl6eAXawXgHWZfOD2nr66FBLNu1T5Gxh3hDKVro7+9HvLRDi1mOTuvhkHrTjwAV0hGCvwFtn6Uo6wIPod7KYFwkEdF6cF4GIDESgUEuBLAYWm8rcilgLsHYTEK7Pi2QYzckChLv9VEQlYMtac8X6GGv8DQ79P+D5CKqsHL6Q79FYCNo9SlPQyVLUYM6QRJgzqiIiMjB4wDQwpNAmPup7hKSXVWFPCGwGWIbwiTeTesVENI7oo/3+NC0J9EN8EJziC0VfW+HkOa+/rgkZdHXiQwLMLwQpBvAKWkiEdFGcvCswB6P5/Tp01hJO1bcVhYXF9EMlUiWL7x2bGxMRsVrXn/9dSyvXbsmRREOhxHIZawAyIeaRx4Y3LlzBy+8ePGiKpdUKBRKJpPaDIlCxj18o7sA8ueff8ZSXyc//vgjluarQvBH0tfXFwgEyn6aZCKifDk4A9AsLS2pNbt68OABloZgjJiUy7C7W7duoRVreKbX68XGSCSiXfAmY+X0Scbc3ByWjY2NUswEsR9vZQi6pTI6OoqWvWFKY+nb+P7776UI9+7dw1J/0SMexVFo2Y/eO++8g+Vnn30mRSIiEo7JAORSdcOY8OXlZSyzBrmS++mnn7B88cUXpShyjElXr15NmyjIRu0eQnJpgL4qMjWL9ZCFhMNhvJXFPAQbRnbG0NsBiOs1NTVId7R2vCHeY7tFbwfyCSRGoVBIy5aIiAgckwE8efIE4aGjo0Pf6g0EAvhxl0aenUn7VT8XEAISdj5r7jIxMSFj3IZMpOfj9u3bqSeqPoDnn39eiiDxEmESK3i+bDS4fv06lm+88YYUS0t25tGjR3KAeoj3eEjSHRxOMpnUV+bjx4+xfPnll7EMBoOGNBHwZ4OXFHbHRSKisqXGA9geAmFnZyciARxJ3T4HO48tdriAzZpczwbYbaH11eOg1JMyaF25mZAFeaZ5pKFcGoDPwidmuhxA9sQmdShh3gIOBE+TQX/6a/xkuB/gcNJe+4fDx0sMoymJiFxuM/5L/bo6Blp4T548wcqePXvs0HedVab724LX61VrGWgHa0HeBJ+yZcsWw4nwtBs1aEwjb0DUnJ+fV5tKyqKihBwLdhuNfsO3n3ajXlVVFZb2HzJCRLRxJBEoAJpi0u4EtK4Ku9KdSkimGXDJlXJHTJdKEhHZWTwex++z9H3mAhHZ7/fn1adb4DgANNfefffd06dP4y3kRPWhQ4eGMpxsLqFQKLTZRObNVc+w1NXVpV6j09zcnOludc5y//59LHfu3ClFC2hh46ht+P3mTu4hZJ5KmYjIhhBldu/ejd9nbeZ7/A4Hg0H8FEskqqqqQoTCRnkUZmdnl5eX8SrzWKiMJBHIl5xj1nINrCD7MM/HYgcyeY52Dhj5ijQH054wNpMz5dqhzczMSM9HGfR5SD1kPRBUlJyhz7HG7En+Yh19CETkEtJBa5inFVtAgpF0D6CIH2dDox+vwsYc+zsL7AOQO9DL4G3weDwPHz58//33pWgrlZWVWGqj7nft2iXB4OrVq7LFmpxX1q6p83q9H3/8MVa++uor2eJcWa+OQyJZX18/PT0tJ9EdTf5i5QbNRES2hV/mkydPIsCbrwNH8JI4i8A0MjKC5qj5Kie8qqmp6e23387l+ucCMwAEQrSqe3t78+htKJE///wTS/1l8du3b8dSTl5khcwG+ZR+MF11dTWW+r4Xh4pEIlhajEbE9/vFF19MTk7K3ItERLTe0LRGXD979qwqr0CTvaenRxVSamtrsTTfS3ZwcBABTmuiWyh8PoBTp05hiba1zWPh/Py8IYTPzs5iKX3g1pBDoR4NU83I9D6vvvqqFMsYYn/WqxWc4tlnn8UyHA5LkYjIni5fvowGtmFqVAs7duxQaytkGjS8jypnVkgGgLjo8/kuXLggV2ZjX2W7DWFX0dI1hPAvv/wSy/b2dilakFxBP28P3jAQCGDF/tMQWculg6ic5P7PiYioVCYmJpLJpMxvZg2/4WjSoH2bdk73Y8eO4X2yXmKddwaAT21ubkbLGA1E5AF9fX0IsVlHiWvDF62t+WhzQwhHdWBPUGtpT7GYSXNfziDgwPHdNDQ0oFqR+jg9ovz+++9qjYiI7EEu0ZJxS9aGh4exvHHjRtpJUF544QUs5SZqFvLOANra2iT8y6e+9957WGYdVYfny8hDa4aTHMWTEP7BBx9IhnHo0KGtW7eOj4+PjIzIE6zJbL579+7Fa5955pmTJ082NTUtLCzkkj0QERHlRUYrb9myRYqZoDk6ODjo9/tbWlrUptUqKiqwlGFwFvLLAEKhkDSgtaQDK3V1dcgJ1rVXWeK3NfXU1X755RcsVX6RgkPIVGVmkUhEPxvD0tISUodMU+ylpXbOBtQOraehoSH1YZaam5vVC3TUY2tEvSkRkQNZR5loNCoXC2RtM2cdpZdfBiD3oTl+/LgUhWQD69qrrCKwJfXU1ZCvyAX9FlBHSAtUQUfOoMhMMpkgEUPYg0zXRKidswG1Q+sJf47qwyxNTk6qF+iox9aIelMiovKCWNPY2Ojz+XLsybaWXwYg148Z0pOHDx9iqb9Zu1lJxgFICK+vr5eiWSKRwCfW1NSMjo6qTTpyBsVwS18NYn9VVdWHH36I9enp6b17967tzhMREekhZnV0dCBmrUn4h/wygJjpGno0oLER7ey0gxE0JRkHICE8U3cKQjjykh07dsgcf2ZyPiZTZiPjNWZnZ7HPODq8SY5TDNlE2VzmR0RUZjKN4e/u7k4mk2m7UdPKOmI9vwxArqHXn/K/desWlh999JEUbeXevXtY6m+Zr9fQ0DA/P+/z+WRSBbO5uTksMyUQBw4cQBam5T2Z3oRsogxmcCKismcx0wxarWNjY6Ojo/r2tpyJVgWdf/75B8ust33JLwOQTu+BgQFJAoLBYG9vb45X1m2waDSKysLKX3/9JVsMrDstcGhItbCSKXKgDa0dNZ6DdKHIifNQpcj7hD7H0guFQvhQOWNSX1+P7149sJ6wM9IdIhnVxsulZrJ6/PgxlrlMA0VEVCoWV/Eh/m7btu3ixYvNOkePHlUPrybXwWW/qlC633M3Pj6uja3DiuHWBRtpYWEh011t4vG47KHAPqsH0kFUAFVIkbsyaAz3XTDAo6gHpEGqXCgcjnY+Iu0n4mDxED4Ij8ZiMTwZfw3qsYLI92hxZyB8udp3rUFdbfCXnrVmciG1Z/iiiYhsBT9x+GHHD68qr5BfsLTS3vAM75BLgLDRYPW8ICrIwatyEVIJQIGBYa3CvwZHZP7uhfwFaCFQ7m9kEb+zwlEX+Q4byaJmciHVlfafChGRfSCg4McKzTxVzp/Ex1x+7vI7C2AfHR0daq10EqnpET2pezSpTcWRMw6ZLl7wer34wrSTF8vLy0jxihnQJyecsk4aZQfWNZMLVBeW5gm0iYhs5dNPP8Vvu9zXvjC9vb14B5mvz5ojMwAZ+JBpDH++Cj613N3djeXNmzelmHZSgbw8ePAAy8OHD0vRQjQaHRwcLDLzkBNOWSeNsoPcayYTmRvK+ppVIqKSQzPvxo0bY2NjhcWUYDAYDod/+OEH67FuinQFOIj0bxR/FiAWi/n9fulvAazk1UUsd0VCmoZXgeRr6rFC4X3wJjg0Vc4AT0B+V/zJ+HhqtEQxXesbJseasYAaA1UgIrK38fFx/GQhsuQ++AnPRCDDq3I/t+uwPgC01zs6OhAP8pqa18LOnTv/DeB+f9arJszwqsrKSlnHCoqyXrDp6Wl8eXJoi4uLd9JdEioTQqH1X/z1F8gQEf4jkUjBvSAbJpeasYBKSyaThltEEhHZVktLC36c8cPV1tamNmXT0NCAQPbHH3/kcXZYZQIOgYSopqZG1p24/9ZwOEeOHEEe19raKsP0EKH1CWAsFkMgXMOBh9J1YX2thB1krRlrV65cwUtKeN0KEZENOSmCymB4rX8D6yDrZUDOayAkI8LJKFAJz/r+HIQ9ffhHOCxyJD8+CB+xhinFesilZqyh3pA55Z4xEBG5gWPOAiQSifb2dvz0m/s38JCMFXe0u3fvYhkIBPx+v0zlqJ1iEKFQKBKJ4AkyIxDIlEfFwAchrOKd7XwiIGvNWMPfBurN5/PlNC6GiMg1HJMBDA8PLy0tHTx4MDU13L9kO1YGBgZkcmJHi6buLnjp0iVtiIPhxgSIYSpt0ynmakBx/vz5ZDI5NTWlyvaTtWasyd/G2bNnpUhERGIzoohatbdm003lw+EwlnJWuL29vfiRcaVVW1sbi8Xi8bjWVEUrv66ubn5+XorrB3X7MEWVbaaYmkkkErt378bfxlrN2UBEVD6kKelETt9/PYQ3HAuyGVVeGfTQ39+vyutJPsueA+WKrBm/379t2zYZPUBERHpOnROwzMzOzmK5b98+KYJM1Xfw4EEpriuv14uAev78eRuOBiimZnA4ly9f7uvry3qLTCIiF2IGYAv379/HUh/V5FR3Q0MDlhsQmM+cOYO28vDwsCrbRjE1093dvX///p6eHlUmIiIdp2YAMjoMyuAqAJCo9txzz0kRtm7diiVawOfOnRsYGJCN68fj8Vy7di0QCBQ/t/HaKrhmhoaGYrGYNmczEREZOGYkoB6ifldXl6wjHgwODjq9m7e5uRkHoo++SHF6e3ux8dSpUy0tLWrrOpMPnZycVGUbKKxmEolEW1sbwj+vACQiysSRGQAREREVieMAiIiI3IgZABERkRsxAyAiInIjZgBERERuxAyAiIjIjZgBEBERuREzACIiIjdiBkBERORGzACIiIjciBkAERGRGzEDICIiciNmAERERG7EDICIiMiNmAEQERG5ETMAIiIiN2IGQERE5EbMAIiIiNyIGQAREZEbMQMgIiJyI2YAREREbsQMgIiIyI2YARAREbnPpk3/AwjQ7KsSB+GAAAAAAElFTkSuQmCC)
Typical values for α, TG , and recommended TC (if indicated) for the selected materials are reported in Table 1. The resulting range values for P1 and P2 are reported in Table 2.
Table 1
Materials and properties overview [21][31-32].
|
Material
|
α (K-1)
|
TG [°C]
|
TC [°C]
|
ABS
|
60*10-6
|
95
|
75
|
PLA
|
68*10-6
|
62
|
25 #
|
PEEK
|
98*10-6
|
143
|
120
|
PA12
|
95*10-6
|
70
|
25 #
|
# TC value not suggested and hence room temperature is considered
|
Table 2
Range of P1 and P2 values.
|
Parameter
|
Range
|
P1 [%]
|
60 ÷98
|
P2 [°C]
|
20 ÷ 45
|
Different P1-P2 combinations were generated by sampling according to a Sobol sequence [33] the design space constituted by the range values for P1 and P2. A database consisting of one thousand (1,000) P1-P2 data point combinations were selected. The sampling was performed in MatLab using the code developed by Bessa et al. [34]. Parts warpage were then computed with equation (2).
2.3 Machine Learning for printable/non-printable part classification
Support Vector Machine (SVM) algorithms were developed in MatLab to create a map of printable/non-printable configurations based on P1-P2 combinations that satisfy the condition of part warpages below a user-defined threshold. Bayesian ML method called scalable variational Gaussian process (SVGP) was chosen as the most adequate for this classification application [35].
Different test cases were investigated performing the proposed methodology for different warpage thresholds, as reported in Table 3. The part length L, the printed layer thickness Δh, and the part height h were kept constant to 100 mm, 0.254 mm, and 1.5 mm, respectively.
Table 3
Test cases overview for classification.
|
Length
L [mm]
|
Layer thickness
Δh [mm]
|
Part height
h [mm]
|
Warpage threshold
δ [mm]
|
100
|
0.254
|
1.5
|
1.5
|
2
|
2.5
|
2.4 Machine Learning for regression in the classified space design
Linear Regression, Support Vector Machine, and Gaussian Process were the model classes analysed for the prediction of the part warpage for the P1-P2 combinations that were classified as printable. The warpage threshold was randomly set to 2 mm. Firstly, the regression models were built via the MatLab “Regression Learner” application for the lowest part height (1.5 mm) and the best model was then applied to the other two-part heights, namely 3.5 and 5.5 mm, as shown in Table 4.
Cross-validation of 10 folds for the training data was used to protect from overfitting. The comparison of classification performance was performed in terms of the root mean squared error (RMSE) and the training time.
Table 4
Test cases overview for regression.
|
Part Length
L [mm]
|
Layer thickness
Δh [mm]
|
Warpage threshold
δ [mm]
|
Part height
h [mm]
|
100
|
0.254
|
2
|
1.5
|
3.5
|
5.5
|
2.5 Validation and Application
The regression model was then validated against Equation 2 in terms of the resulting warpage values for different part heights. In Table 5 the considered variables for the regression model validation are reported.
Table 5
Parameters considered for the regression model validation.
|
Part Length
L [mm]
|
Layer thickness
Δh [mm]
|
Part height
h [mm]
|
Chamber Temperature
TC [°C]
|
100
|
0.254
|
1.5-7.5
|
75
|
Lastly, the validated model was applied for evaluating the influence of the chamber temperature on the warpage of the part. More details are provided in the corresponding results section.