Interdisciplinary Applied Mathematics

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TABLE 14.3. A summary of various boundary-only meshless methods. See (Li, 2003), for references to the various methods listed here.


Boundary integral


collocation


Dual reciprocal collocation


Boundary local domain discretization


Moving


least-squares


Boundary node method


Local boundary integral equation method, Meshless local Petrov-Galerkin method, Hybrid boundary node method


Fixed


least-squares


Boundary cloud method


Boundary


point


interpolation


Boundary point interpolation method


Radial basis function approximation


Boundary Knot Method

complicated surfaces. In meshless boundary-only methods, the basic idea is to combine boundary-integral formulations with meshless approximation and discretization. A summary of the various boundary-only methods that have been    developed is    given    in    Table    14.3.    In    this section,    we    provide


an overview of the boundary cloud method method and its application to solving the Stokes equations.


The various least-squares and kernel approaches that have been discussed in the context of domain meshless methods can be applied to compute the approximation functions for boundary-only meshless methods. However, instead of Cartesian coordinates one needs to use the cyclic (for 2-D problems where the boundary is one-dimensional) or curvilinear coordinates (for 3-D problems where the boundary is two-dimensional) to overcome singularity issues in the moment matrix. The boundary cloud method uses a varying basis approach (Li and Aluru, 2003) and Cartesian coordinates to compute the approximate functions. In a varying basis approach, the unknown u(x,y) is approximated by


u(x,y) = PT (x,y)bt,    (14.12)


where p^ is the varying base interpolating polynomial and bt is the un-

known coefficient vector for point t. To construct the varying basis interpolation functions, clouds are classified into two types: singular and nonsingular. When all the points inside a cloud lie along a straight line, the cloud is defined as singular; otherwise, it is nonsingular. As shown in Figure 14.12, the cloud of point 1 is singular, and the cloud for point 2 is nonsingular, since the points do not lie along a straight line. Using a linear polynomial basis, the base interpolating polynomial is given by

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