#MESHFREE METHOD EXAMPLES SERIES#

These problems are solved with meshfree methods. To simulate some problems, the partial differential equations and boundary conditions for two dimensional solid mechanics and fluid mechanics problem and heat transfer problem especially thermodynamics of plates and shells will be given in sub-sections. One of these methods due to using meshfree method can be elected and they will be explained and compared. These two types of PIMs will be discussed in the following chapter.įor satisfying the boundary conditions, penalty method, direct method, lagrange multiplier method and direct interpolation method can be used to enforce essential boundary conditions. There are two types of PIM shape functions have been developed so far using different forms of basis functions Polynomial basis functions and radial basis functions (RBF) have often been used in meshfree methods. A scalar function defined in the problem domain that is represented by a set of scattered nodes will be shown.

#MESHFREE METHOD EXAMPLES SERIES#

Point interpolation method is one of the series representation methods for the function approximation, and useful for creating meshfree shape functions. The concept and kinds of support domain and determination of the dimension of the support domain will be described.Īfter introducing the concept of support domain, the point interpolation method (PIM) in detail will be discussed. Such a local domain is termed the support domain or influence domain. 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.

Meshfree method examples