These shape functions are locally supported, because only a set of field nodes in a small local domain are used in the construction and the shape function is not used or regarded as zero outside the local domain. In this chapter definition of base and shape functions and various techniques for meshfree shape function constructions are discussed. According to the formulation procedure, meshfree methods fall into three categories: meshfree weak form methods (like: EFG, MLPG, LRPIM,…), meshfree strong form methods (like: SPH, Collocation method,…) and meshfree weak-strong form methods based on the combination of both weak form and strong form (like: MWS method).These three categories and their limitations, applications, advantages and other descriptions will be introduced.In seeking for an approximate solution to the problem governed by PDEs and boundary conditions, one first needs to approximate the unknown field function using shape (trial or base) functions before any formulation procedure can be applied to establish the discretized system equations. Meshfree methods have been proposed and achieved remarkable progress over the past few years.
The individual elements are connected together by atopological map called a mesh. Inthe FEM, a continuum with a complicated shape is divided into elements, finite elements. This infant method uses a set of scattered nodes, called field nodes, to establish the problem domain and boundaries, which do notrequire any priori information on the relationship between the nodes for the interpolation or approximation of the unknown functions of field variables. One of the most important tools in the field of numerical methods that has been developed newly is meshfree or meshless methods.Ī meshfree method is a method used to establish system algebraic equations for the whole domain of problem without using a predefined mesh for the domain discretization. Therefore using numerical procedure to obtain approximate solutions is inevitable. Unfortunately, we can only obtain exact ones for limited practical problems because most of these problems are complex.
Obtaining exact solutions for these equations is ideal. We know phenomena in nature, whether physical, geological, mechanical, electrical, or biological, can often be describe by means of algebraic, differential, or integral equations. In this chapter we will describe about numerical simulation with meshfree methods.
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