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Slope stability analysis and stabilisation
Published in Yung Ming Cheng, Chi Wai Law, Leilei Liu, Analysis, Design and Construction of Foundations, 2021
Yung Ming Cheng, Chi Wai Law, Leilei Liu
Different meshfree methods have been proposed and evolved after the development of the first meshfree methods (the SPH by Gingold and Monaghan 1977 and Lucy 1977). These include the diffuse element method (DEM) by Nayrole et al. (1992), the element free Galerkin method (EFG) by Belytschko et al. (1994), the reproducing kernel particle method (RKPM) by Liu et al. (1995), the partition of unity finite element method (PUFEM) by Babuska and Melenk (1997), the H–P clouds by Duarte and Oden (1996), the moving least-square reproducing kernel method (MLSRK) by Liu et al. (1997), the meshless local boundary integral equation method (LBIE) by Zhu and Atluri (1998), the meshless local Petrov-Galerkin method (MLPG) by Atluri et al. (2002), the meshless point collocation methods by Aluru (2000), the meshless finite point method by Oñate and Idelsohn (1998) and more. Broadly speaking, the meshless method can be grouped by the definition of the shape functions and/or the minimisation method of the approximation. The minimisation may be via a strong form as in the point collocation approach (with no associated mesh) or via a weak form as in the Galerkin method, which actually requires an auxiliary mesh or cell structure.
Predicting in-silico structural response of dental restorations using meshless methods
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
G.A.R. Caldas, J. Belinha, R.M. Natal Jorge
As mentioned in the work of Belinha (Jorge Belinha, 2014), the first meshless approximation method was the Smoothed-Particle Hydrodynamics (SPH), developed for astronomy. This was the origin of the Reproducing Kernel Particle Method (RKPM). One of the oldest methods is the Diffuse Element Method (DEM). This method uses the Moving Least Square approximants (MLS), proposed by Lancaster and Salkauskas, in the construction of the approximation function. Belytschko evolved DEM by developing one of the most popular meshless methods, the Element Free Galerkin Method (EFGM) (Dinis, Natal Jorge, & Belinha, 2007) (Jorge Belinha, 2014).
A review on development and applications of element-free galerkin methods in computational fluid dynamics
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2020
Wah Yen Tey, Yutaka Asako, Khai Ching Ng, Wei-Haur Lam
The work of EFG is inspired by the work of Nayroles et al. [1] who introduced Diffuse Element Method (DEM). In DEM, weighted residual method (WRM) is applied to construct the shape function, instead of local polynomial-based guess function as used in FEM. Belytschko and his co-workers [26, 39, 40] improvised DEM by replacing WRM with Moving Least Squares (MLS) method, devising special treatment on essential boundary condition and including the derivatives of field variables into the calculation. Henceforth, DEM has been transmuted into EFG method. According to Belytschko et al. [26], EFG method is able to deal with steep local field gradient in linear elastic fracture problem, even with higher computational speed. Despite its efficiency for simulating deforming solids, the strategy in distributing nodes across domain still plays the pivotal role to ensure accuracy of the results [26, 40]. In this section, the construction of MLS method and its implementation strategy of imposing essential boundary condition will be discussed.
A new meshless “fragile points method” and a local variational iteration method for general transient heat conduction in anisotropic nonhomogeneous media. Part I: Theory and implementation
Published in Numerical Heat Transfer, Part B: Fundamentals, 2020
Yue Guan, Rade Grujicic, Xuechuan Wang, Leiting Dong, Satya N. Atluri
There are also meshless methods which are based on weak-forms or variational principles. The Diffuse Element Method (DEM) [19] was initially introduced as a generalization of the FEM by removing some limitations related to the trial functions and mesh generations. The original formulation fails in passing the patch test. Nevertheless, Krongauz and Belytschko [20] established an improved DEM based on Petrov-Galerkin formulation (PG DEM) which satisfies the patch test but resulting in asymmetric matrices. Based on the DEM, the Element-Free Galerkin (EFG) method was carried out by Belytschko et al. [21] in which the shape functions are developed by Moving Least Squares (MLS) or Radial Basis Function (RBF) approximations. The MLS approximation does not have delta-function properties and hence imposition of essential boundary conditions necessaires special treatments [22]. In addition, the integration of the Galerkin functional in EFG requires back-ground meshes. Furthermore, the EFG is not necessarily objective when the back-ground meshes are rotated. After that, a Local Boundary Integral Equation (LBIE) Method is introduced by Zhu et al. [23]. While analogous to the BEM, the LBIE method circumvents the problem of global fundamental solutions in nonhomogeneous anisotropic materials in BEM. It uses local fundamental solutions (assuming locally homogeneous material properties) or Heaviside functions as test functions. Finally, Atluri and Zhu [24] proposed the Meshless Local Petrov-Galerkin (MLPG) approach. Compared with the EFG method, the MLPG approach employs the local Petrov-Galerkin weak formulation instead of the global Galerkin weak formulation. The MLPG method is a truly meshless method and has shown its capability and accuracy in 2D and 3D transient heat conduction analysis involving anisotropy, nonhomogeneity and temperature-dependent material properties [5, 25]. Yet the MLPG method still has its limitations: the matrices are asymmetric; the numerical integration in the Petrov-Galerkin weak-form over local subdomains is expensive, as a result of the complex shape functions; and (the same as the EFG method) the essential boundary conditions cannot be imposed directly if MLS approximation is used as trial functions. A modified collocation method has to be applied to enforce the essential boundary conditions [22].