-   Minimize : Warpage W ¼ f ðTm ; Ti ; Pp ; tp; ti Þ ð3Þ

Cooling time tc ¼ f ðTm; Ti ; Pp ; tp; ti Þ

Residual stress S ¼ f ðTm; Ti ; Pp ; tp; ti Þ

-  Subject  to : 45 6 Tm  6 60;      220 6 T1  6 260 ð4Þ

60 6 Pp 6 90;    4 6 tp 6 7;    1:5 6 ti 6 3:0

Table 4

Comparison of optimization results among different optimization   methods。

No。 Variable Lower range Upper range Optimum RSM Optimum RBF

1 Tm 45 60 45 45

2 Ti 220 260 220 220

3 ti 1。5 2。5 2。5 2。5

4 Pp 60 90 60 60

5 tp 4 7 7 7

Outputs (responses)

RSM model

RBF model

1 Total  defection (mm) 2。58 2。59

2 Cooling  time (s) 11。53 11。57

3 Residual stress (MPa) 23。7 23。6

Fig。 12。 Plot of fitness of RSM  metamodel。

Fig。 13。 History of design variables and objective functions using SQP algorithm。

To solve the above optimization problem, metamodel-based optimization method is applied。 Eighty-one experiments are carried out based on L81 orthogonal DOE method。 Both of RSM and RBF metamodels are used to approximate the relation- ship between inputs and outputs。 Fig。 12 shows that R-squared values of fitness of responses, including cooling time and warpage, are close to 1。 The fidelity of the RSM and RBF metamodels is adequate, thus these mathematical models well cap- ture the behavior of the  responses。

Sequential quadratic programing (SQP) algorithm is applied to solve the multi-objective optimization problem after the approximate objective functions had been modeled。 The history of design variables and objective functions is depicted in Fig。 13, and Table 5 shows the optimization results。 The optimum values of process parameters tend to reach their margin。

Table 5

Comparison results between predicted optimum and actual   optimum。

Response (outputs) RSM RBF

Predicted