PSI - Issue 8

M.E. Biancolini et al. / Procedia Structural Integrity 8 (2018) 433–443 Biancolini et al. / Structural Integrity Procedia 00 (2017) 000 – 000

435

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ζ ∗ Dimensionless curvilinear abscissa ζ ∗ = ζ⁄ℎ 1.1. Background on RBF

RBF were introduced in the early 60s to manage problems of multidimensional interpolation (Davis (1963)). If data are given in the form of scattered scalar values at a series of point in the space ℝ , an interpolator is introduced in order to have an approximating smooth function in the same space. At a location its value is   1 ( ) N i i s       i k x x x (1) The above summation is extended to all the points where data are given, called source points. The point at which the value is retrieved is a target point. is the so-called radial basis function, namely a defined scalar function of the Euclidean distance between source and target points. are weights of the radial basis, for their computation a linear system of equations, whose order is equal to the number of source points introduced, needs to be solved. Typical RBF are shown in Table 1, if = ‖ − ‖ . Table 1. Most common Radial Basis Functions Inverse quadric (IQ) − 2 Sometimes, the radial basis interpolator has to be slightly modified in order to guarantee the existence and the uniqueness of the solution. This is obtained with a polynomial part ℎ that is added to the form presented in (1).   1 ( ) ) ( N i i s h        i k x x x x (2) The degree of the polynomial depends on the kind of RBF adopted. If are the given values at the source points , a radial basis fit exists if the coefficients and the weights of the polynomial can be found such that the following conditions are satisfied: ( ) ( ) 0 i s g h   i i k k x x 1 i N   (3) Gaussian (GS) RBF Column A ( t ) , log( ) , √1 + 2 Spline type (Rn) Thin plate spline Multiquadric (MQ) Inverse multiquadric (IMQ) √1 1 + 2 1+ 1 2

A condition of orthogonality is also required

N

  i k x



i p 

0



(4)

i

1

