Multi-objective optimization for minimizing weldline and cycle time using variable injection velocity and variable pressure profile in plastic injection molding

Abstract

Since weldline in plastic injection molding (PIM) has an influence on not only the strength but also the surface appearance of a plastic product, it is preferable to reduce it as much as possible for high product quality. High mold temperature will be valid to the weldline reduction, but long cycle time is required for cooling down the plastic product. Generally, short cycle time is required for high productivity and the cycle time should be minimized. The trade-off between product quality and productivity is always observed in the PIM. In this paper, the weldline reduction and the cycle time are considered for the product quality and the productivity. In particular, the variable injection velocity and the variable pressure profile are adopted for simultaneously improving them. Numerical simulation in the PIM is so intensive that a sequential approximate optimization (SAO) using a radial basis function (RBF) network is adopted to determine the optimal injection velocity and pressure profile with a small number of simulations. The trade-off between the weldline and the cycle time is clarified through the numerical result, and the experiment (GL100, Sodick) is carried out to confirm the validity of the proposed approach.

Appendix

1.1 Radial basis function network

The RBF network is a three-layer feed-forward network. Let {x_j, y_j}(j = 1, 2, ⋯, m) be the training data, where m represents the number of sampling points. The response surface is given by

$$ \hat{y}\left(\boldsymbol{x}\right)={\sum}_{j=1}^m{w}_jK\left(\boldsymbol{x},{\boldsymbol{x}}_j\right) $$

(A1)

where K(x, x_j) is the jth basis function and w_j denotes the weight of the jth basis function. Gaussian kernel given by Eq. (A2) is generally used in this paper.

$$ K\left(\boldsymbol{x},{\boldsymbol{x}}_j\right)=\exp \left(-\frac{{\left(\boldsymbol{x}-{\boldsymbol{x}}_j\right)}^T\left(\boldsymbol{x}-{\boldsymbol{x}}_j\right)}{r_j^2}\right) $$

(A2)

In Eq. (A2), x_j represents the jth sampling point and r_j is the width of the jth basis function. The response y_j is calculated at the sampling point x_j. In the RBF network, the following equation is minimized:

$$ {\sum}_{j=1}^m{\left({y}_j-\hat{y}\left({\boldsymbol{x}}_j\right)\right)}^2+{\sum}_{j=1}^m{\lambda}_j{w}_j^2\to \min $$

(A3)

where the second term is introduced for the purpose of the regularization. It is recommended that λ_j in Eq. (A3) is sufficient small value (e.g., λ_j = 1.0 × 10⁻²). The necessary condition of Eq. (A3) results in the following equation:

$$ \boldsymbol{w}={\left({\boldsymbol{H}}^T\boldsymbol{H}+\boldsymbol{\varLambda} \right)}^{-1}{\boldsymbol{H}}^T\boldsymbol{y} $$

(A4)

where H, Λ, and y are given as follows:

$$ \boldsymbol{H}=\left[\begin{array}{cccc}K\left({\boldsymbol{x}}_1,{\boldsymbol{x}}_1\right)& K\left({\boldsymbol{x}}_1,{\boldsymbol{x}}_2\right)& \cdots & K\left({\boldsymbol{x}}_1,{\boldsymbol{x}}_m\right)\\ {}K\left({\boldsymbol{x}}_2,{\boldsymbol{x}}_1\right)& K\left({\boldsymbol{x}}_2,{\boldsymbol{x}}_2\right)& \cdots & K\left({\boldsymbol{x}}_2,{\boldsymbol{x}}_m\right)\\ {}\vdots & \vdots & \ddots & \vdots \\ {}K\left({\boldsymbol{x}}_m,{\boldsymbol{x}}_1\right)& K\left({\boldsymbol{x}}_m,{\boldsymbol{x}}_2\right)& \cdots & K\left({\boldsymbol{x}}_m,{\boldsymbol{x}}_m\right)\end{array}\right] $$

(A5)

$$ \boldsymbol{\varLambda} =\left[\begin{array}{cccc}{\lambda}_1& 0& \cdots & 0\\ {}0& {\lambda}_2& \cdots & 0\\ {}\vdots & \vdots & \ddots & \vdots \\ {}0& 0& 0& {\lambda}_m\end{array}\right] $$

(A6)

$$ \boldsymbol{y}={\left({y}_1,{y}_2,\cdots, {y}_m\right)}^T $$

(A7)

The width in the Gaussian kernel plays an important role for a good approximation. We have proposed the following simple estimate about the width [22]:

$$ {r}_j=\frac{d_{j,\max }}{\sqrt{n}\sqrt[n]{m-1}}\kern0.5em j=1,2,\cdots, m $$

(A8)

where r_j denotes the width of the jth Gaussian kernel and d_j,max denotes the maximum distance between the jth sampling point and the other sampling points. Equation (A8) is applied to each Gaussian kernel individually and can deal with the non-uniform distribution of sampling points.