Numerical investigation and process parameters optimization in three-dimensional multi-stage hot forging for minimizing flash and equivalent strain

Abstract

Multi-stage hot forging is widely used to produce complex forged products in inFdustry. Flash after the forging should always be minimized for the material saving, and it is important to determine the preform shape minimizing the flash. The process parameters in multi-stage hot forging such as the billet temperature and the stroke have also an influence on the product quality as well as the flash, but they are still determined by a trial-and-error method. In this paper, multi-objective design optimization in three-dimensional multi-stage hot forging is numerically performed so as to minimize both the flash and the distribution of equivalent strain. The preform shape and the process parameters such as the billet temperature, the die temperature and the stroke are optimized. The three-dimensional numerical simulation is computationally so expensive that sequential approximate optimization that response surface is repeatedly constructed and optimized is adopted to identify the pareto-frontier between the flash and the distribution of equivalent strain. It is found from the numerical result that 60% reduction of flash can be achieved, compared to the conventional preform shape. In addition, the total forging energy considering the multi-stage hot forging is reduced. It is confirmed through the numerical result that the proposed approach is valid to determine the optimal preform shape and process parameters in multi-stage hot forging.

Appendix

Let m be the number of sampling points, and the training data is given by {x_j, y_j}($j=\mathrm{1,2},\cdots ,m$). The response surface using the RBF network is constructed by Eq. (A1).

$$\widetilde{y}(x)={\sum }_{j=1}^{m}{\alpha }_{j}K(x,{x}_{j})$$

(A1)

where $K(x,{x}_{j})$ in Eq. (A1) is the j-th basis function, and ${\alpha }_{j}$ denotes the weight of the j-th basis function. The Gaussian kernel given by Eq. (A2) is widely used as the basis function.

$$K(x,{x}_{j})=\mathrm{exp}(-\frac{{(x-{x}_{j})}^{T}(x-{x}_{j})}{{r}_{j}^{2}})$$

(A2)

where r_j is the width of the j-th basis function, which is determined by the following equation.

$$\begin{array}{cc}{r}_{j}=\frac{{d}_{j,\mathrm{max}}}{\sqrt{n}\sqrt[n]{m-1}}& j=\mathrm{1,2},\cdots ,m\end{array}$$

(A3)

where d_j,max denotes the maximum distance between the j-th sampling point and the other sampling points. The weight vector $\alpha ={({\alpha }_{1},{\alpha }_{2},\cdots ,{\alpha }_{m})}^{T}$ is determined by solving the following equation [24].

$$\alpha ={({K}^{T}K+\Lambda )}^{-1}{K}^{T}y$$

(A4)

where $K$, $\Lambda$, and $y$ in Eq. (A4) are given as follows:

$$K=\left[\begin{array}{cccc}K({x}_{1},{x}_{1})& K({x}_{1},{x}_{2})& \cdots & K({x}_{1},{x}_{m})\\ K({x}_{2},{x}_{1})& K({x}_{2},{x}_{2})& \cdots & K({x}_{2},{x}_{m})\\ \vdots & \vdots & \ddots & \vdots \\ K({x}_{m},{x}_{1})& K({x}_{m},{x}_{2})& \cdots & K({x}_{m},{x}_{m})\end{array}\right]$$

(A5)

$$\Lambda =\left[\begin{array}{cccc}{\lambda }_{1}& 0& \cdots & 0\\ 0& {\lambda }_{2}& \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0& 0& 0& {\lambda }_{m}\end{array}\right]$$

(A6)

$$y={({y}_{1},{y}_{2},\cdots ,{y}_{m})}^{T}$$

(A7)

In Eq. (A6), ${\lambda }_{j}$ is a sufficiently small value for the regularization (e.g. ${\lambda }_{j}=1.0\times {10}^{-3}$).