Interdisciplinary Applied Mathematics

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TABLE 14.2. A summary of meshless methods developed for domain simulation. See (Belytschko et al., 1996; Li and Liu, 2002; Atluri, 2002; Li, 2003), for references to the various methods listed here.


Point


collocation


Cell integration Galerkin


Local domain integration Galerkin


Moving


least-


squares


Finite point method


Element-


free Galerkin method, Partition of unity finite element method


Meshless local Petrov-Galerkin method, Method of finite spheres


Fixed least-squares


Generalized finite difference method, h-p meshless cloud method, Finite point method


Diffuse element method


Reproducing


kernel


Finite cloud method


Reproducing kernel particle method


Fixed kernel


Finite cloud method


Radial basis


Many


techniques


Many


techniques

• Node о Star point □ Cloud О Cloud


14.9, an approximation ua(x, y) to an unknown u(x, y) is given by


ua(x,y)=    C(x,y,Xk — s,yk — t)4>(xk — s,yk — t)u(s,t) dsdt, (14.10)


n


where ф is the kernel function centered at (xk,yk), which is usually taken


as a cubic spline or a Gaussian function. For the results shown here, ф is taken as a modified Gaussian function, i.e.,


ф(х — xi)


w(x — Xi)


1 — w(x — xi) + e’


where e is a small number that is used to avoid the singularity of ф(x — xI); typically, e is chosen to be 10~5. Also, w(x — xI) is a normalized Gaussian function given by


l-e-^mi/c)2    ’ Z — “ml

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