Interdisciplinary Applied Mathematics

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where A and B are 2NP x 2NP matrices, and й and т are 2NP vectors. After applying the boundary conditions, the unknowns on the boundary nodes can be calculated.

As an    example,    the    results    for    a    step    flow are presented    here.    The    ge

ometry of the channel is shown in Figure 14.13. The velocity at the inlet is uniformly distributed; the value of velocity u is 1 unit/s. The point distribution on    the    boundary is    shown    in Figure    14.14.    The    velocity    at    the

interior nodes is computed by using the boundary integral equations. The computed x-component of the velocity across line AB and at the outflow are shown in Figure 14.15. The results from the boundary cloud method match well with the results obtained from the finite element method.

In summary, boundary-only formulations are more efficient for linear problems with a known Green’s function(s), since they eliminate the need to discretize the entire domain. Meshless boundary-only formulations further improve the efficiency by eliminating the need for a mesh (a scattered set of    points    is    used to discretize    the    boundary)    on    the    boundary.    Typ

ically, the meshless boundary-only formulation is a factor of two slower than the boundary element method. However, the error obtained with the meshless boundary-only formulation is lower than the error obtained with the boundary element method for the same point distribution. Meshless boundary-only formulations are relatively new approaches compared to meshless domain formulations. As a result, a complete mathematical analysis of the boundary-only formulations is not yet available. The extension of boundary element methods or meshless boundary-only methods to unsteady methods can be quite involved.

FIGURE 14.14. Boundary cloud method: Scattered point distribution along the boundary of the step channel example.

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