Interdisciplinary Applied Mathematics

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w(z) = {


0,


Z > dn


where dmi is the support size of a cloud (see Figure 14.9), and c is the dilation parameter, which is taken as dmi/2. In two dimensions, the kernel function is constructed as


ф^ — xi, y — yi) = ф^ — xi)ф(у — yI).


Also, C(x,y,xk — s,yk — t) is the correction function, which is given by


C(x, y, xk — s,yk — t) = pT(xk — s,yk — t)c(x, y),    (14.11)


PT = {Pi ,p2,… ,pm} is an m x 1 vector of basis functions. In two dimensions, a quadratic basis vector is given by


pT = [1, xk    — s,yk    — t, (xk    —    s)2, (xk    — s)(yk — t),    (yk    —    t)2] , m =    6.


Finally, c(x, y) is an m x 1 vector of unknown correction function coefficients. The correction function coefficients are computed by satisfying the consistency conditions, i.e.,


/ PT (xk—s,yk —t)c(x,y)ф(xk s, yk —t)pi(s,t)dsdt = Pi(x,y), i = 1,…,m. Jn


, In discrete form, the above equation can be written as


NP


^2 PT (xk — xi, yk — yi )c(x,y)ф(xk — xi ,yk — yi )pi (xi ,yi )AVi = Pi(x,y),


i=1


i = 1,…,m,


where NP is the number of points in the domain and AVi is the nodal volume of node I (the nodal volumes can be simply set to unity; see (Aluru and Li, 2001; Aluru, 2000), for a more detailed discussion on nodal volumes). The above equation can be written in matrix form as

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