Fig。 7。 Fitness plot of RSM metamodel with 36 design  points。

Fig。 8。  Fitness plot of RBF metamodel with 36 experiments。

Fig。 9。 History plot of optimization process using direct GA optimization  method。

optimum point is not always guaranteed。 The evidence is that the value of objective function of GA method is 2。93 meanwhile this value continues to be improved when applying gradient-based optimization method subsequently (2。93 compared to 2。72, Table 2)。 The starting point for searching the optimum point of gradient-based optimization method is the optimum point obtained by GA。

When we applied the direct gradient-based optimization method using sequential quadratic programming, the conver- gence is reached very fast, just after 50 iterations。 However, the optimum result is worse than other methods。 The evidence is that the value of the warpage is 3。54 mm。 The history plot of the optimization process using direct gradient-based approach is shown in Fig。 10。 Trying other starting points may improve the final optimum result but the simulation cost will increase。 The efficiency of the optimization method depends on the number of simulations and on the fidelity of optimum point。

Table 3 compares the number of simulations of the optimization methods that have been used。 Direct GA optimization meth- od requires a lot of simulations compared to other methods。 The combination of coarse GA and gradient-based fine search has a moderate number of simulations gives the best optimum solution。 Gradient-based methods require less iteration, but they are easy being trapped in a local minimum。 RBF and ANN metamodel-based optimization methods reduce the number of simulations; however, the error at the optimization point is high。 We found that using rough GA search followed by a gra- dient-based optimization technique is a good choice that ensures an expected optimum point for the high nonlinear re- sponse problem。

4。2。 Case study 2: low nonlinear response   problem

This case study investigates the proposed optimization methods when the behavior of objective functions is low nonlin- ear。 Multi-objective optimization is considered instead of single objective optimization。 Warpage, cooling time, and residual stress were minimized  simultaneously。

Fig。 10。  History plot of optimization process using direct gradient-based method。

Table 3

The number of simulations of different optimization   method。

Optimization methods GA (direct) Gradient-based GA then gradient RBF (metamodel) NN (metamodel)

Number  of  simulations (runs) 200 (predefined) 50 (approximate) 123a  (approximate)

60 (predefined) 60 (predefined)

a    100 runs for  GA

Fig。 11。  A  deep tray  made by  polypropylene  material。

The molded part is a deep tray with 2。5 mm thickness as shown in Fig。 11。 Due to the geometrical of the molded part, the response of the warpage is a low nonlinear function。 Molded material is polypropylene, and mold material is P20 steel。 Injec- tion molding machine is a default machine selected from the database of injection molding software。 Five design variables and their ranges are shown in Table  4。

The optimization problem is stated as   follows: