PSI - Issue 25

Corrado Groth et al. / Procedia Structural Integrity 25 (2020) 136–148 C. Groth et al. / Structural Integrity Procedia 00 (2019) 000–000

139

4

The passage of the RBF through source points and the imposing orthogonality conditions for the polynomial terms:

N i = 1

s ( x k i ) = g i , 1 ≤ i ≤ N

γ i p ( x k i ) = 0

and

(2)

for all the polynomials p of degree less or equal to polynomial h. A single interpolant exists if the basis is conditionally positive definite (Micchelli (1986)). If the degree is m ≤ 2 (Beckert and Wendland (2001)) a linear polynomial can be used:

h ( x ) = β 1 + β 2 x 1 + β 3 x 2 + ... + β n + 1 x n

(3)

The system 2 built to calculate coe ffi cients and weights can be easily written in matrix form for an easy implementa tion: M P P T 0 γ β = g 0 (4)

Where g is is the vector of known terms for each source point and M is the interpolation matrix with the radial function transformed distances between source points:

M i j = φ ( x k i − x k j ) , 1 ≤ i ≤ N , 1 ≤ j ≤ N

(5)

P is the constraint matrix resulting from the orthogonality conditions:

   

P =    

1 x k 1 y k 1 z k 1 1 x k 2 y k 2 z k 2 .. . .. . .. . 1 x k N y k N z k N

(6)

The system 4 is solved considering as known terms the three components of the deformation field. Once the RBF weights and polynomial coe ffi cients of the system have been obtained, displacement values for the three directions can be obtained at a given x point as:   S x ( x ) = N i = 1 γ x i φ ( x − x k i ) + β x 1 + β x 2 x 1 + β x 3 x 2 + β x 4 x n S y ( x ) = N i = 1 γ y i φ ( x − x k i ) + β y 1 + β y 2 x 1 + β y 3 x 2 + β y 4 x n S z ( x ) = N i = 1 γ z i φ ( x − x k i ) + β z 1 + β z 2 x 1 + β z 3 x 2 + β z 4 x n (7) 2.2. Crack Propagation

The propagation of a crack with a Multiple Degrees of Freedom (MDOF) model is a complex task and di ff erent mathematical aspects need to be managed. The first topic concerns the values of displacement, related to the flaw