Introduction to Finite Element, Boundary Element, and Meshless Methods: With Applications to Heat Transfer and Fluid Flow
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The finite element method (FEM) and the boundary element method (BEM) have been developed to a mature stage such that they are now utilized routinely to model complex multi-physics problems. Meshless methods are a relative newcomer to the field of computational methods. The term “Meshless Methods” refers to the class of numerical techniques that rely on either global or localized interpolation on non-ordered spatial point distributions. As such, there has been much interest in the development of these techniques as they have the hope of reducing the effort devoted to model preparation. The approach finds its origin in classical spectral or pseudo-spectral methods that are based on global orthogonal functions such as Legendre or Chebyshev polynomials requiring a regular nodal point distribution. In contrast, Meshless methods use a nodal or point distribution that is not required to be uniform or regular due to the fact that most such techniques rely on global radial-basis functions (RBF). However, global RBF-based Meshless methods have some drawbacks including poor conditioning of the ensuing algebraic set of equations which can be addressed to some extent by domain decomposition and appropriate pre-conditioning. Moreover, care must be taken in the evaluation of derivatives in global RBF-based Meshless methods. Although, very promising, these techniques can also be computationally intensive. Recently, localized collocation Meshless methods have been proposed to address many of these issues.