Interdisciplinary Applied Mathematics

{[1 x y]    m = 3 (nonsingular cloud),

[ 1 x] or [1 y ]    m = 2 (singular cloud).

For a point t, the unknown coefficient vector b_t is computed by minimizing

NP

Jt = ^2 wi(xt,yt) [p^(xi,yi)bt — щ ,

i=1

where NP is the number of nodes, w_i(x_t,y_t) is the weighting function centered at (x_t,y_t) and evaluated at node i, whose coordinates are (x_i,y_i), and U_i is a nodal parameter. Once the unknown coefficient vector (b_t) 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) — Wi_jk(x, y)ui(y)]nj(y)dS,    (14.13)

dB

where i,j, к = 1, 2; c_ik is the corner tensor, nj is the unit outward normal at    dB; u_i is    velocity;    U_ik    and    W_ij_k    are    the    kernel tensors;    a_ij    =    -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

where S_ik is the Kronecker delta function, r = ||y — x||, r,_i = r_i/r, and ri = Xi (y) — Xi (x).

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