PSI - Issue 12

Stefano Porziani et al. / Procedia Structural Integrity 12 (2018) 416–428 S. Porziani et al. / Structural Integrity Procedia 00 (2018) 000–000

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Table 1. Most common RBFs. RBF type

Equation

r n , n odd

Spline type (Rn) Thin plate spline Multiquadric (MQ)

r n log ( r ) , n even

√ 1 + r 2

1 √ 1 1 + r 2 e − r 2

Inverse multiquadric (IMQ)

1 + r 2

Inverse quadric (IQ)

Gaussian (GS)

The degree of polynomial h depends on the RBF type adopted for the interpolation problem. The weights γ i and the coe ffi cient of the polynomial can be found if the following conditions are satisfied:

s ( x k i ) = g i h ( x k i ) = 0

1 ≤ i ≤ N

(3)

In (3) g i are the given values at source points x k i . A condition of orthogonality is also required:

N i = 1

γ i p x k i = 0

(4)

for all polynomials p with a degree less or equal than that of polynomial h . A unique interpolator exists if the basis functions is a conditionally positive definite function. If a linear polynomial is chosen in a 3D space

h ( x ) = β 1 + β 2 x + β 3 y + β 4 z

(5)

a non-singular square system can be obtained as follows:

M P

P T 0

γ β

=

g 0

(6)

M is the interpolation matrix

M i , j = ϕ x k i − x k j

1 ≤ i , j ≤ N

(7)

P is a constraint matrix that arises in the system to balance the polynomial contribution and contains a columns of “1” and the coordinates of source points in following columns: