Meshfree Methods: Moving Beyond the Finite Element Method, Second Edition

By Grace Liu. 2009

527 citations recorded by the source corpus.

Subfields: Civil and Structural Engineering, Geophysics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/847c98917c064ce79cb9274265432f91368f221f

Citations in the corpus

• An edge-based smoothed finite element method (ES-FEM) for analyzing three-dimensional acoustic problems (2009, unknown): “The PIM shape functions are constructed based on a small set of nodes in a local support domain and possess the Delta function property, which allows straightforward imposition of essential boundary conditions [13].”
• EDGE-BASED SMOOTHED POINT INTERPOLATION METHODS (2008, unknown): “Two types of PIMs have been developed using different basis functions: the polynomial PIM, using polynomial basis functions [Liu and Gu (2001)]; and the radial PIM (RPIM), using radial basis functions [Wang and Liu (2002)].”
• A moving Kriging interpolation‐based meshless method for numerical simulation of Kirchhoff plate problems (2009, unknown): “Lagrange multipliers [4], penalty method [1] or coupling with FEM [6] and so on.”
• A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition (2014, unknown): “The final choice of trial functions is a set of radial basis functions (RBFs), which appeal to previous work on “mesh-free” methods [40].”
• Meshfree Methods: Progress Made after 20 Years (2017, unknown): “A good overview and comparison of the meshfree approximations discussed can be found in the literature [cf. (Belytschko et al. 1996b; Li and Liu 2007; Liu 2009)].”
• Smoothed Particle Hydrodynamics (SPH): an Overview and Recent Developments (2010, unknown): “This leads to insufficient “sampling” points for establishing equations, and can result in numerical instability problem [4].”
• Static and free vibration analysis of laminated composite plates using the conforming radial point interpolation method (2008, unknown): “, [1,2]), has shown some superiorities over traditional numerical methods.”
• On the essence and the evaluation of the shape functions for the smoothed finite element method (SFEM) (2009, unknown): “In these cases, we apply the coordinate transformation technique given in [14] (see, Figure 2): = (x−xe)cos +(y− ye)sin = −(x−xe)sin +(y− ye)cos (2)”
• Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC‐PIM) (2008, unknown): “A very detailed introduction on many types of mesh-free methods for different problems can be found in books by Liu [9] and Liu and Gu [10].”
• Accurate fracture modelling using meshless methods, the visibility criterion and level sets: Formulation and 2D modelling (2011, unknown): “Integration As previously highlighted in [42, 43], a large number of integration points are required in the EFGM to avoid the underestimation of the global weak form.”
• An n‐sided polygonal edge‐based smoothed finite element method (nES‐FEM) for solid mechanics (2010, unknown): “[1] to stabilize the solutions in the context of the meshfree method [2, 3] and then applied in the natural element method [4].”
• An efficient adaptive analysis procedure for certified solutions with exact bounds of strain energy for elasticity problems (2008, unknown): “Various numerical techniques have been therefore developed to obtain approximate solutions, such as the well-known finite difference method (FDM), widely used finite element method (FEM) and the newly developed meshfree methods [1], etc.”
• A face‐based smoothed finite element method (FS‐FEM) for 3D linear and geometrically non‐linear solid mechanics problems using 4‐node tetrahedral elements (2009, unknown): “In the other front of development, meshfree methods have been developed for various types of problems [4, 5].”
• A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate (2021, unknown): “Their mechanical behaviours have long been studied by various methods such as finite element method [2, 3], boundary element method [4, 5], meshfree method [6], isogeometric analysis [7], and numerical manifold method [8– 10].”