PSI - Issue 54

J.V. Araújo dos Santos et al. / Procedia Structural Integrity 54 (2024) 575–584 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

577

3

(− ) 0 (− ) 1 … (− ) −1 (− ) ⋮ ⋮ ⋮ ⋮ ⋮ (−1) 0 (−1) 1 … (−1) −1 (−1) (0) 0 (0) 1 … (0) −1 (0) (1) 0 (1) 1 … (1) −1 (1) ⋮ ⋮ ⋮ ⋮ ⋮ ( ) 0 ( ) 1 … ( ) −1 ( ) ]

=

(2.7)

[

From Equation (2.1), one sees that at = 0 =0 the polynomial takes a value equal to the 0 coefficient and thus we just need the first row of , ℎ 1, , being the smoothed value of ( 0 ) obtained from Equation (2.3) for =0 ( 0 )= ∑ ℎ 1, ( 0 − ) =− (2.8) The computation of derivative approximations can be described based on the previous formulation of Savitzky Golay smoothing filters. Taking into consideration the differentiation rules of a polynomial of degree , its th order derivative can be written as ( ) ( ) = ∑ (∏( + ) =1 ) + − =0 (2.9) with ≤ . At = 0 =0 , the equation above becomes ( ) (0) = (∏ =1 ) = ! (2.10) Equation (2.10) states that the th order derivative of the polynomial at the central point of the data set is equal to the factorial of the derivative order multiplied by the coefficient . Thus, one obtains the discrete convolution ( ) ( 0 )= ∑ ℎ +1, ( 0 − ) =− (2.11) for the computation of the derivative approximation, where ℎ +1, is now given as the multiplication of line +1 of matrix by the factorial of the derivative order. Ratzlaff and Johnson (1989) and Kuo et al. (1991) developed two distinct approaches to extend the Savitzky Golay smoothing method to data sets with two variables. The first approach is equivalent to a sum of two one dimensional polynomials, ( , ) = ( ) + ( ) (2.12) whereas the second relies on the product of one-dimensional polynomials, ( , ) = ( ) ( ) (2.13) According to conclusions of a study by Nikitas and Pappa-Louisi (2000), the approach in Equation (2.12), when applied with the moving window technique, gives better smoothing results than the one based on Equation (2.13). Therefore, in the present work, we applied the Ratzlaff and Johnson (1989) approach to smooth the full data set of