Interdisciplinary Applied Mathematics

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Mc(x,y) = p(x, y),


NP


Mij = ^2Pj(xk — xi, yk — yiP(xk — xi, yk — yi)pi(xi, yi)AVi,


i =i


i,j = 1,,m.


From the above equation, the unknown correction function coefficients are computed as


c(x,y) = M1p(x,y).


Since M is a small matrix (6 x 6 matrix for a quadratic basis in 2-D, i.e., m = 6), the correction function coefficients can be computed using either a direct solver or any iterative solver. Substituting the correction function coefficients into equation (14.11) and employing a discrete approximation for equation (14.10), we obtain


NP


ua(x,y) = ^2 NI(x,y)ui,


I=1


where uI is the nodal parameter for node I, and NI(x, y) is the fixed kernel meshless interpolation function defined as


Ni(x, y) = pT(x, y)M-Tp(xfcxi, yk — yi)^(xfcxi, yk — yi)AVi.


The interpolation functions obtained from the above equation are multivalued. A unique set of interpolation functions can be constructed by computing Ni(xk,yk), I =1, 2,…, NP, when the kernel is centered at (xk, yk) (see (Aluru and Li, 2001), for more details). The derivatives of the unknown are approximated by


dua(x, y) dx


dua(x, y) dy


d2ua(x, y) dx2


d2ua(x, y)


dy2


d2ua(x, y) dxdy



dNi(x, y)


2^—hz—



i =1 NP



dx



sr-‘ dNj(x, y) 2^—^



i =1 NP



dy



d2Nj(x, y)


2A-^



i =1 NP



dx2

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