PSI - Issue 8

N. Di Domenico et al. / Procedia Structural Integrity 8 (2018) 422–432 N. Di Domenico et al. / Structural Integrity Procedia 00 (2017) 000–000

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4

For a static case the di ff erential equation of motion:

¨ η + ω 2 η = N

(4)

obtained from (1) becomes:

ω 2 η = N

(5)

In which the modal forces N are obtained by integrating the structural loads weighted by the eigenvectors. By taking into account also the viscous damping, equation (1) becomes:

[ M ] ¨ q + [ C ] ˙ q [ K ] q = Q

(6)

and the modal coordinate, function of time, can be written as (Meirovitch (1975)):

sin ( ω d t ) +

ξ ( t ) = e ζω n t ξ

1 m ω d

˙ ξ 0 + ζω n ξ 0 ω d

t

e − b ( t − τ )

0 cos ( ω d t ) +

2 m f ( τ ) sin ( ω d ( t − τ )) dx

(7)

0

2.2. Radial Basis Functions

RBF are mathematical functions able to interpolate, on a distance basis, scalar information known only at discrete points (source points). The quality and the behavior of the interpolation depends both from the function and from the kind of chosen basis function: RBF can be indeed classified depending on the kind of support they guarantee (local or global), meaning the domain in which the function is not zero valued (De Boer et al. (2007)). Some of the most common functions are shown in table 1. RBF can be defined in an n dimension space and are function of the distance that, in the case of morphing, can be assumed as the euclidean norm of the distance between two points in the space. A linear system of order equal to the number of points used (Buhmann (2000)) must be solved in order to find system coe ffi cients. Once the coe ffi cients have been found the displacement of a given node of the mesh, being it inside (interpolation) or outside the domain (extrapolation) , can be calculated as the superimposition of the radial contribution of each source point. It is then possible to define at known points the displacement in the space and to retrieve the value at mesh nodes, obtaining a mesh deformation that leaves unaltered grid topology (Beckert and Wendland (2001), Biancolini (2012)). The interpolation function is composed by the basis φ and by the polynomial term h with a degree that depends on the kind of the chosen basis. This latter contribution, in particular, is added to assure uniqueness of the problem and polynomial precision, allowing to recover exactly rigid body translations. If N is the total number of source points it can be written:

N i = 1

s ( x ) =

γ i φ ( x − x k i ) + h ( x )

(8)