The Collapse Deformation Prediction Model of Wide 7075 Al-Alloy Intermediate Slabs Based on Particle Swarm Optimization and Support Vector Regression During the Hot Rolling Process

Abstract

Aiming at the collapse deformation problem of ultra-wide Al-alloy intermediate slabs on the finishing mill run-in table during the hot rolling process, the collapse deformation prediction model of ultra-wide 7075 Al-alloy intermediate slabs based on particle swarm optimization and support vector regression (PSO-SVR) was proposed. The 7075 Aluminum alloy (Al-alloy) constitutive relationship in a temperature range of 330-370 °C was established by the Gleeble thermal compression experiment. According to the actual transporting process, the thermomechanical coupling finite element model (FEM) of 7075 Al-alloy intermediate slabs was established. It was demonstrated that width, temperature, and transportation time were positively related to contact length, while thickness was negatively related to contact length. The PSO-SVR collapse deformation prediction model was proposed by combining simulation data and the PSO-SVR algorithm. The PSO-SVR prediction model constructed was applied and verified by production test in the “1 + 3” hot rolling production line of an Al-alloy production line. The results show that the prediction model can achieve 100% prediction accuracy to predict whether the slab can be produced normally, and the maximum error of the contact length was 10.5%, it can provide guidance for the high-quality and stable production of ultra-wide Al-alloy.

Appendices

Appendix 1: QR Model

The quadratic multiple regression method is a conventional quantitative analysis method with good statistical properties. Its general form expression is shown in Eq 6, including a constant term, a linear term, a cross-product term, and a square term, the input variables x, respectively, correspond to the width, thickness, outlet temperature, and time in this study, the output variables y correspond to the contact length L.

$$y = k_{0} + \sum\limits_{i = 1}^{m} {k_{i} x_{i} } + \sum\limits_{j = 1}^{m} {k_{ii} x_{i}^{2} } + \sum\limits_{i = 1}^{i < j} {k_{ij} x_{i} } x_{j} \quad i = 1,2, \ldots ,m;j = 1,2, \ldots ,m$$

(6)

Equation (6) can be expanded as shown below:

$$\begin{gathered} L_{1} = a_{1} W + a_{2} H + a_{3} TP + a_{4} TM + a_{5} W*H + a_{6} W*TP + a_{7} W*TM + a_{8} H*TP + \hfill \\ a_{9} H*TM + a_{10} TP*TM + a_{11} W^{2} + a_{12} H^{2} + a_{13} TP^{2} + a_{14} TM^{2} + a_{15} \hfill \\ \end{gathered}$$

(7)

The values of parameters $a_{1} \sim a_{15}$ obtained by the QR model are substituted into Eq 8 as follows:

$$\begin{gathered} L_{1} = - 21.8167W + 1301.4374H - 202.8378TP - 26.4571TM - 0.4401W*H + 0.0686W*TP + 0.0089W*TM - \hfill \\ 2.9936H*TP - 0.3904H*TM + 0.0609TP*TM + 0.0050W^{2} + 9.6038H^{2} + 0.2333TP^{2} + 0.0039TM^{2} + 44090.3405 \hfill \\ \end{gathered}$$

(8)

Appendix 2: SVR Model

For nonlinear SVR, the low-dimensional data were first mapped to the high-dimensional space, the linearly separable hyperplane was found in the high-dimensional space, and then, the hyperplane of the high-dimensional space was mapped back to the low-dimensional space to achieve SVR.

SVR constructed the hyperplane through the training set so that all samples are closest to the hyperplane $\omega x_{i} + b = 0$. Assuming that the training samples are $D = \{ (x_{1} ,y_{1} ),(x_{2} ,y_{2} ), \ldots ,(x_{m} ,y_{m} )\}$, where $x_{i} \in R^{{\text{n}}}$, $y_{i} \in R^{{\text{n}}}$,$i = 1,2 \ldots ,m$. A regression model $f(x) = \omega x + b$ was built to make $f(x)$ and y as close as possible. A sensitive factor $\varepsilon$ is introduced into the SVR model, assuming that the maximum deviation between $f(x)$ and y is tolerated, and the loss is calculated only if the absolute value of the difference between $f(x)$ and y is greater than the sensitive factor. $\omega$ and b are parameters to be, respectively, determined. From this, the SVR transforms the convexity optimization problem into Eq 9:

$$\mathop {\min }\limits_{\omega ,b} \left\{ {\frac{1}{2}\left\| \omega \right\|^{2} + C\sum\limits_{i = 1}^{m} {L(f(x_{i} ) - y_{i} )} } \right\}$$

(9)

where, C is the penalty factor and $L( \cdot )$ is the loss function, whose expression is as follows:

$$L(f(x_{i} ) - y_{i} ) = \left\{ {\begin{array}{*{20}l} 0 \hfill & {f(x_{i} ) - y_{i} \le \varepsilon } \hfill \\ {\left| {f(x_{i} ) - y_{i} } \right| - \varepsilon } \hfill & {f(x_{i} ) - y_{i} > \varepsilon } \hfill \\ \end{array} } \right.$$

(10)

Introducing the slack variables $\xi_{i}$ and $\hat{\xi }_{i}$ into Eq 6, the following equations can be obtained:

$$\left\{ \begin{gathered} \mathop {\min }\limits_{\omega ,b} \left\{ {\frac{1}{2}\left\| \omega \right\|^{2} + C\sum\limits_{i = 1}^{m} {(\xi_{i} - \hat{\xi }_{i} )} } \right\} \hfill \\ s.t.\quad f(x_{i} ) - y_{i} \le \varepsilon + \xi_{i} \hfill \\ \, f(x_{i} ) - y_{i} \le \varepsilon + \xi_{i} \hfill \\ \xi_{i} \ge 0,\hat{\xi }_{i} \ge 0,i = 1,2, \ldots ,m \hfill \\ \end{gathered} \right.$$

(11)

Introducing the Lagrange multipliers $u_{i} \ge 0$, $\hat{u}_{i} \ge 0$, $\alpha \ge 0$ and $\hat{\alpha } \ge 0$ constructing the Lagrange function as shown below:

$$\begin{gathered} L(\omega ,b,\alpha ,\hat{\alpha },\xi ,\hat{\xi },u,\hat{u}) = \frac{1}{2}\left\| \omega \right\|^{2} + C\sum\limits_{i = 1}^{m} {(\xi_{i} + \hat{\xi }_{i} )} - \sum\limits_{i = 1}^{m} {u_{i} \xi_{i} - \sum\limits_{i = 1}^{m} {\hat{u}_{i} \hat{\xi }_{i} } } + \hfill \\ \sum\limits_{i = 1}^{m} {\alpha_{i} (} f(x_{i} ) - y_{i} - \varepsilon - \xi_{i} ) + \sum\limits_{i = 1}^{m} {\hat{\alpha }_{i} (f(x_{i} ) - y_{i} - \varepsilon + \hat{\xi }_{i} )} \hfill \\ \end{gathered}$$

(12)

To find the partial derivative of $\omega$, $b$, and $\xi$, using the duality of the Lagrange function, the optimization function is transformed into:

$$\left\{ \begin{gathered} \max \left[ { - \frac{1}{2}\sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{n} {(\hat{\alpha }_{i} - \alpha_{i} )(\hat{\alpha }_{j} - \alpha_{j} )K(x_{i} ,x_{j} ) + \sum\limits_{i = 1}^{n} {(\hat{\alpha }_{i} - \alpha )y_{i} - \sum\limits_{i = 1}^{n} {(\hat{\alpha }_{j} - \alpha_{j} )\varepsilon } } } } } \right] \hfill \\ s.t.\left\{ \begin{gathered} \sum\limits_{j = 1}^{n} {(\hat{\alpha }_{i} - \alpha_{i} ) = 0} \hfill \\ 0 \le \hat{\alpha }_{i} ,\alpha_{i} \le C \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} \right.$$

(13)

The final solution of SVR is follows:

$$f(x) = \sum\limits_{i = 1}^{m} {((\hat{\alpha }_{i} - \alpha_{i} )K(x_{{\text{i}}} ,x_{j} ) + b)}$$

(14)

where, $K(x_{i} ,x_{j} ) = \phi (x_{i} )^{T} \phi (x_{j} )$ is the kernel function, the Gaussian kernel function $K\left( {x_{i} ,x_{j} } \right) = \exp \left( { - \frac{{\left\| {x_{i} - x_{j} } \right\|^{2} }}{{2\sigma^{2} }}} \right),\sigma > 0$ is selected in this study.

Appendix 3: PSO Algorithm

Inspired by the foraging behavior of bird groups in nature, Eberhart and Kennedy jointly proposed the PSO algorithm. The PSO algorithm uses the information interaction between the individuals in the group and the optimal individual to guide the particles in the entire group keeping the individual diversity information while moving toward the most optimal individual. The velocity v(t + 1) of the particle at the next moment was determined by the current velocity v(t), its optimal position pbest, and the global optimal position gbest. Then, the current position moved to a new position. With the continuous deepening of the iteration, the entire particle group was driven by the “leader” to complete the search for the optimal solution in the decision space.

Assuming that in the decision space, the particle swarm size is N, the current iteration number is t, and the position of the i $(i = 1,2, \ldots ,N)$ particle in the population is $x_{i} (t) = \left[ {x_{i,1} (t),x_{i,2} (t), \ldots ,x_{i,d} (t)} \right]$. The velocity of the i-th particle is $v_{i} (t) = \left[ {v_{i,1} (t),v_{i,2} (t), \ldots ,v_{i,d} (t)} \right]$. The historical optimal position of all particles in the whole population in the d-th dimension is $gbest_{d}$, the historical optimal position of particle i is $pbest_{i,d}$, and after the next iteration of particle i, the velocity and position update formulas are shown in Eq 15 and 16.

$$v_{i,d} (t + 1) = wv_{i,d} (t) + c_{1} r_{1} (pbest_{i,d} (t) - x_{i,d} (t)) + c_{2} r_{2} (gbest_{d} (t) - x_{i,d} (t))$$

(15)

$$x_{i,d} (t + 1) = x_{i,d} (t) + v_{i,d} (t + 1)$$

(16)

where r₁ and r₂ are random numbers distributed in the [0,1] interval to increase the randomness of the algorithm, c₁ and c₂ are the acceleration constant in [0, 2], w is the inertia weight, which is responsible for adjusting the influence of the last speed on the current speed.