PSI - Issue 8

N. Di Domenico et al. / Procedia Structural Integrity 8 (2018) 422–432

426

N. Di Domenico et al. / Structural Integrity Procedia 00 (2017) 000–000

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Table 1: Common RBF with global and local support.

Compactly supported RBF

Abbreviation

φ ( ζ )

Wendland C 0 Wendland C 2 Wendland C 4

(1 − ζ ) 2

C 0 C 2 C4

(1 − ζ ) 4 (4 ζ + 1) 2 + 6 ζ + 1) 3 ζ

(1 − ζ ) 6 ( 35

Globally supported RBF

Abbreviation

φ ( ζ )

r n , n odd r n log ( r ), n even

Polyharmonic spline

PHS

r 2 log ( r ) a 2 + ( r ) 2 1 √ a 2 + ( r ) 2 1 + ( r ) 2

Thin plate spline

TPS

Multiquadric biharmonics

MQB IMQB

Inverse multiquadric biharmonics

Quadric biharmonics

QB

1 1 + ( r ) 2 e − r 2

Inverse quadric biharmonics

IQB

Gaussian biharmonics

GS

An interpolation exists if coe ffi cients and weights of the polynomial that allow to guarantee the exact value at source points during interpolation can be found. In this case the polynomial contribution should be zero. It is then:

N i = 1

s ( x k i ) = g i , 1 ≤ i ≤ N

γ i p ( x k i ) = 0

(9)

and

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

(10)

The system built to calculate coe ffi cients and weights can be easily written in matrix form for an easy implementation:

M P

P T 0

γ β

=

g 0

(11)

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

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

(12)