AAE550_Class37_2011 - AAE 550 Class 37 Fall 2011 1 Class...

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1 AAE 550 Class 37 Fall 2011 1 Class 37: Approximation Techniques II - Response Surfaces Response surface coefficients – Least squares regression Design of experiments – Full-Factorial Design (FFD) – Central-Composite Design (CCD) – Face-centered CCD – Box and Behnken design – Saturated D-optimal design 2 Response Surface Coefficients ( 29 0 1 1 ˆ n n n i i ij i j i i j i f a a x a x x = = = = + + ∑∑ x Once a response surface model (here, a quadratic equation) is chosen, coefficients must be determined ( a 0 , a i , a ij are unknown) Note the index convention used here
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2 AAE 550 Class 37 Fall 2011 3 Response Surface Coefficients Use some form of linear regression to find coefficients Run several “exact” (or “actual”) analyses, f ( x i ) f = X a + ε 4 Response Surface Coefficients ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 1 2 1 1 2 2 1 1 1 1 1 1 2 2 1 2 1 1 2 2 2 2 2 2 2 2 2 2 1 2 1 1 0 1 2 11 12 22 1 2 1 2 2 2 1 1 1 dp dp dp dp dp dp dp dp n n n n n n n n a x x x x x x x x x x x a a a a f f x x x x x x x f a = + ε ε ε x x x Two-variable example ( x 1 ) i = first variable of “data point” i ( x 2 ) i = second variable of “data point” i n dp = number of “data points”
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3 AAE 550 Class 37 Fall 2011 5 Response Surface Coefficients Regression finds a to minimize error minimize ( 29 ( 29 2 2 1 1 dp dp n n i i i i L f Xa = = = ε = - this is a “least-squares” problem 6 Response Surface Coefficients Least squares solution [ ] f X X X a T T 1 - = Generalized least squares – Solve in Matlab using either: a = X\f (simplistic and easy) or lsqlin (provides additional info)
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This note was uploaded on 02/14/2012 for the course IE 530 taught by Professor Ravindran during the Spring '97 term at Purdue University-West Lafayette.

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AAE550_Class37_2011 - AAE 550 Class 37 Fall 2011 1 Class...

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