Interdisciplinary Applied Mathematics

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{[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).

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