3。2。 Optimization method for optimizing molding parameters using   metamodels

A framework for optimizing molding parameters using metamodel-based optimization approach is comprised of two components: a CAE component and an integration controller as shown in Fig。 4。 The CAE component is responsible for sim- ulating and calculating the values of the objective functions。 The integration controller reads the results obtained from CAE simulation and stores them in an output file, controls the number of simulation and organizes the combination of process parameters for every run based on a chosen DOE technique。 The number of simulations is predetermined by the certain DOE strategy。 All the tasks in the framework should be automatic using application programming interface (API) language。

Fig。 3。 Schematic procedure of the direct simulation-based optimization for optimizing injection molding process parameters。

Fig。 4。 A framework for automated simulation applied to metamodel-based optimization method。

Fig。 5。  Schematic  procedure  of  simulation-based  and  metamodel-based  optimization  strategy  for  optimizing process parameters。

After the last simulation is completed, a metamodel in the form of explicit equation is built, and we can optimize the process parameters based on the  metamodel。

Table 1

Comparison between direct optimization and metamodel-based optimization   methods。

Simulation cost and number of iterations Response nonlinearity Molded  part geometry Accuracy  of  optimization result

Direct optimization

Metamodel-based High if design variables increases

Lower compared to  direct High nonlinearity Moderate Not appropriate for complex molded part

Suitable  for  complex molded High if the local optimum is avoided

High when using proper model

optimization optimization nonlinearity part and vice versa

The schematic procedure of simulation-based in conjunction with metamodeling techniques is presented in Fig。 5 in which the selection of metamodel type should be elaborate。 The popular metamodels are RSM, ANN, RBF, and Kriging model。 Second order RSM is suitable for low or moderate nonlinear responses。 It requires less number of simulations compared to other models。 RBF, ANN, and Kriging model are different from RSM and RBF because they interpolate the sample data points, and their response surface is not as smooth as those of RSM。 RBF, ANN, and Kriging model are suitable for high nonlinear problems, but they require an adequate set of sample data obtained from experiment or simulation。

After selecting the metamodel type, DOE or space sampling technique is an important step when using metamodel-based optimization approach。 The common DOE or space sampling techniques include full factorial, D-optimal design, central com- posite design, orthogonal array, Latin hypercube, and optimal Latin hypercube。 The right choice of sampling technique can be referenced from the work of Wang and Shan [4]。 After running a predefined number of simulations according to the DOE strategy, the approximation process is carried out, and the metamodel is then built。 The accuracy or the goodness of fit of the metamodel is often assessed by four error measures: averages absolute error, maximum error, root mean square error, and R-squared。 If the metamodel is adequate, the optimization process is then performed based on this model。 Otherwise, it is necessary to improve or change the metamodel type。 Any optimization techniques (gradient-based optimization tech- niques or non-gradient based optimization techniques) can be used to solve optimization problem。 Because the objective and constraint functions are in the form of explicit equation, the computing cost of optimization is ignored compared to the total simulation  cost。