Interdisciplinary Applied Mathematics

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where Q is the domain, Гд is the portion of the boundary on which Dirich-let    boundary    conditions    are specified,    Г^    is    the    portion    of    the    boundary

on which Neumann boundary conditions are specified, and L, G, and H are the differential, Dirichlet, and Neumann operators, respectively. The boundary of the domain is given by Г = Гд U Г^. After the meshless approximation functions are constructed, for each interior node, the point collocation technique simply substitutes the approximated unknown into the governing equations. For nodes with prescribed boundary conditions the approximate solution or the derivative of the approximate solution is substituted into the given Dirichlet or Neumann-type boundary conditions, respectively. Therefore, the discretized governing equations are given by

L(ua) = f (x, y)    for    points    in    Q,

G(ua )= g(x,y)    for    points    on    Гд,

H(ua) = h(x,y)    for    points    on    Г^.

In the following we present an example of meshless simulation.

Flow in a Driven Cavity: As an application of the finite cloud method to fluids, a typical flow solution using the incompressible Navier-Stokes equations and no-slip condition is presented here. We consider a square cavity of dimensions 1 x 1 mm with the top wall moving at constant velocity corresponding to Re = 3.2. The two-dimensional Navier-Stokes equations are discretized in collocation fashion. Two different node distributions are employed as shown in Figure 14.10. In the first one, 961 points are distributed uniformly in    the    cavity.    In    the    second    case, 961    points    are distributed    ran

domly in the cavity. The corresponding velocity vectors are shown in Figure 14.11. Examination of velocity profiles at different locations shows negligible differences in the two solutions.

x position (mm)

FIGURE 14.11. Velocity vectors of flow in a driven cavity: Uniform (left) and random (right) point distribution.

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