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Predicting the stress distribution in the mandible bone due to the insertion of implants: A meshless method study
Published in J. Belinha, R.M. Natal Jorge, J.C. Reis Campos, Mário A.P. Vaz, João Manuel, R.S. Tavares, Biodental Engineering V, 2019
H.I.G. Gomes, J. Belinha, R.M. Natal Jorge
To solve this problem, several interpolating meshless methods have been developed in recent years. Some of these methods are the Point Interpolation Method (PIM) (G. R. Liu & Gu, 2001), Point Assembly Method (G. R. Liu, 2002), Meshless Finite Element Method (MFEM) (Calvo, Pin, Idelsohn, & Eugenio, 2003), Natural Neighbour Finite Element Method (NNFEM) (Traversoni, 1994) or the Natural Element Method (NEM) (Braun & Sambridge, 1995). Through evolution of the PIM, which used the original polynomial base function, it was possible to develop the RPIM (Wang & Liu, 2002). This method uses a Radial Basis Function (RBF) to construct the shape functions, combined with the polynomial base function. Recently, through the combination of NEM and RPIM, it was possible to develop the NNRPIM (Dinis, Jorge, & Belinha, 2007).
Meshless methods of analysis
Published in Ken P. Chong, Arthur P. Boresi, Sunil Saigal, James D. Lee, Numerical Methods in Mechanics of Materials, 2017
Ken P. Chong, Arthur P. Boresi, Sunil Saigal, James D. Lee
A series of meshless methods for analysis have been introduced in the literature, mostly in the late 1990s. The development of these methods may have been motivated by the shortcomings arising in the difficulty of use of the finite element methods by analysts. Some of the shortcomings were outlined in Section 6.1. Methods in the category of meshless methods include smooth particle hydrodynamics (SPH) (Swegle et al., 1994), the element-free Galerkin (EFG) method (Belytschko et al., 1994a; Lu et al., 1994), reproducing kernel particle methods (RKPMs) (Liu, 1995; Liu and Chen, 1995; Liu et al., 1996), partition of unity methods (Melenk and Babuska, 1996), h-p clouds (Duarte and Oden, 1996), the meshless local Petrov–Galerkin (MLPG) method (Atluri, 2004; Atluri and Zhu, 1998, 2000a), the natural element method (Sukumar et al., 1998), the natural neighbor Galerkin methods (Sukumar et al., 2001), and the method of finite spheres (De and Bathe, 2000, 2001). Except for the SPH methods, these methods are all based, in principle, on the finite element method. Several additional meshless methods based on the boundary element method have also been proposed. These include boundary contour methods (Mukherjee et al., 1997; Nagarajan et al., 1994, 1996), the boundary node method (Mukherjee and Mukherjee, 1997a; Chati and Mukherjee, 2000; Chati et al., 1999), and the local boundary integral equation method (Atluri et al., 2000). Most of these methods are of recent origin and are still undergoing development. Some applications of meshless methods in modeling life-cycle engineering appeared in a book by Chong et al. (2002). A popular meshless method is the EFG method. It has been developed largely in the context of elastostatics. Chapters 4 and 6 have both been presented for use in elastostatics analysis. Continuing in that vein, the EFG method is also presented in this chapter for linear elastic analysis of solid bodies.
Flexible GMRES solver for interpolating MLPG analysis of heat conduction
Published in Numerical Heat Transfer, Part B: Fundamentals, 2022
Abhishek Kumar Singh, Krishna Mohan Singh
Smooth particle hydrodynamics (SPH) method [1] is the first mehfree method reported in the literature. After that, various meshless methods such as diffuse element method (DEM) [2], element free Galerkin (EFG) [3], reproducing kernel particle method (RKPM) [4], partition of unity method (PUM) [5], natural element method (NEM) [6], and meshless local Petrov–Galerkin (MLPG) [7] came into the picture. MLPG method is a truly meshless method, as there is no need for background mesh. Also, the Petrov–Galerkin formulation gives an autonomy in the selection of trial (shape) and test functions. A comprehensive discussion on the MLPG formulation and its possible variations are given in Atluri [8].