{[1 x y] m = 3 (nonsingular cloud),
[ 1 x] or [1 y ] m = 2 (singular cloud).
For a point t, the unknown coefficient vector bt is computed by minimizing
NP
Jt = ^2 wi(xt,yt) [p^(xi,yi)bt — щ ,
i=1
where NP is the number of nodes, wi(xt,yt) is the weighting function centered at (xt,yt) and evaluated at node i, whose coordinates are (xi,yi), and Ui is a nodal parameter. Once the unknown coefficient vector (bt) is computed, the approximation for the unknown u(x,y) in equation (14.12) is defined. Numerical integrations in the boundary cloud method are implemented using the standard cell structure and Gaussian quadrature.
The application of the boundary cloud method to the Stokes equations (see Chapter 2 for a discussion on the Stokes equations) is now discussed. The boundary integral equation of the Stokes equations without body forces can be written as (Phan et al., 2002)
Cik(x)ui(x) = [Uik(x, y)&ij(y) — Wijk(x, y)ui(y)]nj(y)dS, (14.13)
dB
where i,j, к = 1, 2; cik is the corner tensor, nj is the unit outward normal at dB; ui is velocity; Uik and Wijk are the kernel tensors; aij = -pSij +
n(ui,j + Uji) is the stress tensor, and x, y are the source point and field
point, respectively. For 2-D problems, dB is the boundary curve defining the body B. The kernel tensors in equation (14.13) for 2-D Stokes flow are given by
Uik Wijk
1
4np
1
[Sik ln(r)
r,ir,k ],
—r,ir,jrk, nr
where Sik is the Kronecker delta function, r = ||y — x||, r,i = ri/r, and ri = Xi (y) — Xi (x).
