Table 1 Steepest ascent design and results

in the response y。

Then, Eq。 (4) can be expressed by matrix:

According   to   the   method   of   central composite

design,  thirty  experiments  must  be  carried  out  for the

  

 

central composite design with four factors and five levels。

The  central  composite  test  table  and  the experimental

results are listed in Table 2。

4Results and discussion

In statistics, response surface methodology (RSM) [17] is used to explore the relationship between explanatory variables and response variables。 The relationship can be constructed by polynomial functions and further displayed in the graphics。

Usually, a two-order polynomial is used to formulate the relationship between the explanatory variables and the response variables。 The response variable y can be defined as follows:

Matrix B with unknown coefficients can  be obtained from the formulation:

B=(XTX)−1XTY (6)

When polynomial coefficients are substituted into Eq。 (4), the value of y can be predicted at any x。

4。1Mathematical modeling

Before establishing the regression model, the singular points should be deleted。 The regression response surface models of the fracture function Df, the wrinkling function Dw and thickness varying Dt are expressed as

Df=26042。18746+31。46078x1−86。05939x2−43。04582x3−

224。31387x4+0。065050x1x2−0。12953x1x3−

where xi  represents independent variables; b0, bi, bii     and

0。57679x1x4+0。094554x2x3−0。081526x2x4+

bij represent the unknown coefficients needed to be defined; ε represent the variables noise or error  observed

0。54418x3x4+0。11137 x1  +0。057776 x2 +7。50434×

Table 2 Central composite test table and experimental results

Experiment No。 x1 x2 x3 x4/105 Df Dw Dt

1 3 385 435 0。8 0。037031 27。5719 33。2243

2 3 385 445 0。8 0。00364 16。2145 65。7158

3 2 380 430 1。1 8。10194 16。4318 53。1864

4 2 390 430 1。1 0。011397 10。824 7202。7

5 3