Formability classifier for a TV back panel part with machine learning

Abstract

This study proposes a machine learning-based methodology for evaluating the formability of sheet metals. An XGBoost (eXtreme Gradient Boosting) machine learning classifier is developed to classify the formability of the TV back panel based on the forming limit curve (FLC). The input to the XGBoost model is the blank thickness and cross-sectional dimensions of the screw holes, AC (Alternating Current), and AV (Audio Visual) terminals on the TV back panel. The training dataset is generated using finite element simulations and verified through experimental strain measurements. The trained classification model maps the panel geometry to one of three formability classes: safe, marginal, and cracked. Strain values below the FLC are classified as safe, those within 5% margin of the FLC are classified as marginal, and those above are classified as cracked. The statistical accuracy and performance of the classifier are quantified using the confusion matrix and multiclass Receiver Operating Characteristic (ROC) curve, respectively. Furthermore, in order to demonstrate the practical viability of the proposed methodology, the punch radius of the screw holes is optimized using Brent's method in a Java environment. Remarkably, the optimization process is completed swiftly, taking only 3.11 s. Hence, the results demonstrate that formability for a new design can be improved based on the predictions of the machine learning model.

Appendix 1

A. Tree Boosting

• (I) Decision Tree

Decision tree learning is a supervised learning approach that utilizes if-else or true–false feature questions to predict a category in a classification problem or continuous numeric value in a regression problem. For a given dataset $\mathcal{D}={\{\left({\mathbf{x}}_{i}, {y}_{i}\right)\}}_{i=1}^{n}$, a tree model is given by,

$$f\left({\mathbf{x}}_{i}\right)=\sum_{j=1}^{T}{w}_{j}\mathrm{I}\left({\mathbf{x}}_{i}\in {R}_{j}\right)$$

(6)

where ${w}_{j}$ is the score (prediction) of the $j$-th leaf, referred to as weight in the region ${R}_{j}$, $\mathrm{I}$ is the set of indices of data points assigned to the $j$-th leaf, and T is the total number of leaves in the tree (see Fig. 1).

• (II) Boosting

Boosting is a class of machine learning algorithms that iteratively combines multiple base learners to form a prediction model. The base learners are generally weak but provide accurate predictions when combined in an ensemble hence the term 'boosting.' Given the base learner to be decision trees with K trees, the predicted output ${\widehat{y}}_{i}$ corresponding to the input vector ${\mathbf{x}}_{i}$ of the i-th instance is given by,

$${\widehat{y}}_{i}=\sum_{k=1}^{K}{f}_{k}\left({\mathbf{x}}_{i}\right),{f}_{k}\in \mathcal{F}$$

(7)

where ${f}_{k}$ is the output of the k-th tree, and $\mathcal{F}$ is a set of all possible classification and regression tree (CART) functions. Tree boosting learns by iteratively adding ${f}_{t}\left({x}_{i}\right)$ to base learners, such that it minimizes the following objective function,

$$\begin{array}{c}{\mathcal{L}}^{\left(t\right)}=\sum\limits_{\mathcal{D}}l\left({y}_{i},{{\widehat{y}}_{i}}^{\left(t\right)}\right)\\ =\sum\limits_{\mathcal{D}}l\left({y}_{i},{{\widehat{y}}_{i}}^{\left(t-1\right)}+{f}_{t}\left({\mathbf{x}}_{i}\right)\right)\end{array}$$

(8)

where,

$${\widehat{y}}_{i}^{\left(t\right)}=\sum_{k=1}^{t}{f}_{k}\left({x}_{i}\right)={\widehat{y}}_{i}^{\left(t-1\right)}+{f}_{t}\left({x}_{i}\right)$$

(9)

$D$ is the size of the training set, ${\widehat{y}}_{i}^{(t)}$ and ${y}_{i}$ are the predicted and target values at the $t$-th iteration, respectively, and the $l$ term is a differentiable convex loss function that measures the difference between ${\widehat{y}}_{i}^{(t)}$ and ${y}_{i}$. In general settings, boosting utilizes a second-order taylor approximation of the loss function, and the objective function is re-written as follows,

$${\widetilde{\mathcal{L}}}^{\left(t\right)}=\sum_{\mathcal{D}}\left[{g}_{i}{f}_{t}\left({\mathbf{x}}_{i}\right)+\frac{1}{2}{h}_{i}{f}_{t}^{2}\left({\mathbf{x}}_{i}\right)\right]$$

(10)

where ${g}_{i}$ and ${h}_{i}$ are the gradient and hessian, respectively, defined as,

$${g}_{i}={\partial }_{{\widehat{y}}^{\left(t-1\right)}}l\left({y}_{i},{\widehat{y}}^{\left(t-1\right)}\right);{h}_{i}={\partial }_{{\widehat{y}}^{\left(t-1\right)}}^{2}l\left({y}_{i},{\widehat{y}}^{\left(t-1\right)}\right)$$

(11)

• (III) XGBoost method

XGBoost employs Newton tree boosting to fit additive tree models. In Newton boosting, the base learners are the tree models and for a given dataset with $n$ examples $\mathcal{D}={\{\left({\mathbf{x}}_{i}, {y}_{i}\right)\}}_{i=1}^{n}$, a tree model from Equation 6 is given by,

$${f}_{t}\left({\mathbf{x}}_{i}\right)=\sum_{j=1}^{T}{w}_{j}\mathrm{I}\left({\mathbf{x}}_{i}\in {R}_{j}\right)$$

(12)

XGBoost learns by iteratively adding ${f}_{t}\left({\mathbf{x}}_{i}\right)$ to base learners, such that it minimizes the following objective function (as described in Equation 10),

$$\begin{array}{c}{\widetilde{\mathcal{L}}}^{\left(t\right)}=\sum\limits_{\mathcal{D}}\sum\limits_{j=1}^{T}\left[{g}_{i}{w}_{j}+\frac{1}{2}{h}_{i}{w}_{j}^{2}\right]\\ =\sum\limits_{j=1}^{T}\left[{G}_{j}{w}_{j}+\frac{1}{2}{H}_{j}{w}_{j}^{2}\right]\end{array}$$

(13)

where

$${G}_{i}=\sum_{\mathcal{D}}{g}_{i};{H}_{i}=\sum_{\mathcal{D}}{h}_{i}$$

At each iteration, XGBoost learns the leaf weights and tree structure through the following steps:

• 1. For each leaf $j$, the optimization problem in Equation 13 is quadratic in ${w}_{j}$. Hence, the optimal leaf weight or prediction ${w}_{j}^{*}$ for a proposed (fixed) tree structure is obtained by setting $d{\widetilde{\mathcal{L}}}^{(t)}/d{w}_{j}=0$ as follows,

  $${w}_{j}^{*}=-\frac{{G}_{j}}{{H}_{j}},j=1,\dots ,T.$$

  (14)

• 2. Learning the tree structure involves searching for splits of internal nodes. To compute the optimal split, the objective reduction following Equations 13 and 14 is,

  $${obj}^{*}=-\frac{1}{2}\sum_{j=1}^{T}\frac{{G}_{j}^{2}}{{H}_{j}}$$

  (15)

  Equivalently, the splits are determined that maximizes the gain given by,

  $${\text{Gain}}=\frac{1}{2}\left[\frac{{G}_{L}^{2}}{{H}_{L}}+\frac{{G}_{R}^{2}}{{H}_{R}}-\frac{{G}^{2}}{H}\right]$$

  (16)

  Here, the terms with subscripts $L$ and $R$ correspond to the left and right leaf scores, respectively, and the last term is the score of the original node. The nodes with negative gain are pruned out in bottom-up order.

• 3. The final leaf weights ${\widehat{w}}_{j}$ are computed following Equation 14 for the learned tree structure.