# Interdisciplinary Applied Mathematics

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Extensive research has been conducted in the area of meshless methods; see (Belytschko    et    al., 1996;    Li and    Liu,    2002;    Atluri,    2002;    Li,    2003),    for

an overview. Broadly defined, meshless methods contain two key steps:

Construction of meshless approximation functions and their derivatives, and

Meshless discretization of the governing partial differential equations.

Least-squares (Lancaster and Salkauskas, 1981), kernel based (Monaghan, 1992), and radial basis function (Hardy, 1971; Kansa, 1990a; Kansa, 1990b) approaches are three techniques that have gained considerable attention for construction of meshless approximation functions (see (Jin et al., 2001), for a detailed discussion on least-squares and kernel approximations). The meshless discretization can be categorized into three classes:

1. Cell integration (Belytschko et al., 1994),

2. Local point integration (Atluri, 2002), and

3. Point collocation (Liszka et al., 1996; Aluru, 2000).

Both interior and exterior domain problems (using a boundary-only formulation such as the boundary-integral formulation) encountered in microsystems have been solved with meshless methods. In this section we provide a brief overview of the application of meshless methods for interior (or domain) and boundary-only problems.

###### 14-2.1 Domain Simulation

A summary of the various meshless techniques that have been developed for domain simulation is provided in Table 14.2. Here we outline the key steps in the finite cloud method and show its application to some examples.

The meshless finite cloud method uses a fixed kernel technique to construct the interpolation functions and a point collocation technique to discretize the governing partial differential equations. In a two-dimensional fixed kernel approach, given a scattered set of points as shown in Figure

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