An efficient meshless method based on a new shape function

Abstract

SUMMARYAlthough, the finite element method numerically is efficient it exhibits difficulties whenever the remeshing of the analysis domain must be performed. For such problems, utilizing meshless computation methods is very promising. But, their large computational cost, which arises from following a time-consuming process to find shape functions, is one of the most important factors limiting the use of these methods. In this paper, we introduce a direct approach, based on properties required for any shape function, to prepare the shape function and propose a new meshless method. The proposed method does not need any predetermined basis or weighting functions and can be performed faster and more efficiently. Another advantage of the introduced method is its capability to apply desirable changes to the shape function. Application of the proposed approach for electrostatic field computation and verification of the obtained results using theoretically known solution is also presented.

Unconditionally stable improved meshless methods for electromagnetic time-domain modeling

“…Most of the conventional meshless methods need to compute the inversion of a matrix, which is usually an expensive process, to obtain the shape functions (Liu and Gu, 2005). In proposed procedure which has been introduced in Razmjoo et al (2010aRazmjoo et al ( , b, 2011b, there is no need to do that; therefore, shape functions can be constructed faster. As mentioned, according to the data-fitting algorithm and PUM, a complete approach for constructing shape function of the meshless methods has been proposed, recently (Razmjoo et al, 2010a(Razmjoo et al, , 2011b.…”

“…In proposed procedure which has been introduced in Razmjoo et al (2010aRazmjoo et al ( , b, 2011b, there is no need to do that; therefore, shape functions can be constructed faster. As mentioned, according to the data-fitting algorithm and PUM, a complete approach for constructing shape function of the meshless methods has been proposed, recently (Razmjoo et al, 2010a(Razmjoo et al, , 2011b. A new weighting function is suggested so that shape function derivatives can be obtained easily in analytical forms (not numerical).…”

“…Unlike PUM, there is no need to use more than one unknown per each node to achieve acceptable accuracy. Another advantage of the proposed shape function, unlike the shape function introduced in Razmjoo et al (2010a), is that its derivatives can be obtained in analytical forms, simply, as have been given in Razmjoo et al (2011a, b). Since some of the shape functions used in the conventional meshless methods do not satisfy the Kronecker delta property (Belytschko et al, 1996), the imposition of essential boundary conditions is another problem in these methods.…”

“…In Das and Lowther (2011) and Lima et al (2012a, b), some recent developments of using RBFs in electromagnetic problems can be found. In Razmjoo et al (2010a) a special class of partition of unity method (PUM) is used and its efficiency in electrostatic problems, Cartesian and polar coordinates has been verified. The distinguished advantage of this method is in non-uniform node distribution domain where the shape functions are constructed according to nodes density.…”

Electromagnetic time domain modeling using an improved meshless method

“…NEM has been used with a weak form by Sukumar et al [4]. In [5] a special class of PUM is used and its efficiency in electrostatic problem, Cartesian and polar coordinates, has been verified. The distinguished advantage of this method is in nonuniform node distribution domain which the shape functions are constructed according to nodes density.…”