Meshfree Methods

Abstract

The aim of this chapter is to provide an in‐depth presentation and survey of meshfree particle methods. Several particle approximations are reviewed; the SPH method, corrected gradient methods, the moving least‐squares (MLS) approximation, and maximum‐entropy approximations. The discrete equations are derived from a collocation scheme or a Galerkin method. Special attention is paid to the treatment of essential boundary conditions. Finally, different approaches for modeling discontinuities in meshfree methods are described, including peridynamics and phase‐fields.

Multi-Scale Overlapping Domain Decomposition to Consider Elasto-Plastic Local Buckling Effects in the Analysis of Pipes

“…() =d() dz  . The nonlinear strains can be written as (21) in which N  is the nonlinear axial strain and N  is considered to vanish. Similar to linear strains, the nonlinear strain vector in Eq.…”

“…[6][7][8], Generalized-FEM [9][10][11] and Multi-scale methods [12][13][14][15][16][17], which improve the efficiency and accuracy of the numerical results by refining the model only where required and without changing the global simpler model of the whole structure. Common to these numerical methods is that the partition of unity concept is exploited to allow overlapping decompositions of the analysis domain so that a local enrichment can be seamlessly incorporated [18][19][20][21]. In various types of problems which naturally give rise to multiple scales in the deformation fields, such as crack propagation e.g., [22], or localized damage problems e.g., [23] multiscale numerical analysis techniques have been effectively used.…”

Modelling of Cutting Fibrous Composite Materials: Current Practice

“…Remeshing is time consuming, can be complicated for 3D problems and for every iteration, studied quantities should be projected on the new mesh leading to gradual accumulation of error [10]. Moreover, FEM is not well suited for modelling discontinuities if they do not coincide with elements' boundaries [11].…”

“…This eliminates part or whole of the meshing process [12]. Some advantages in using meshfree methods in machining problems are (i) the ability to simulate large deformations and discontinuities without the need for remeshing, (ii) the flexibility in adding or removing nodes without worry about their relation to neighbouring nodes [12], (iii) better integration with CAD/CAE/CAM software [10], (iv) elimination of separation criteria and arbitrary contact conditions [13]. Meshfree methods include: smoothed particle hydrodynamics, finite pointset method, element-free Galerkin, reproducing kernel particle, moving least square interpolations and constrained natural element method.…”

On meshless and nodel-based numerical simulations of forming processes

“…Reviews on SPH, DEM, RKPM and EFG which are the typical 'diffuse' meshless methods are given in Li and Liu [80], Huerta et al [64], Duarte [45], Babuŝka et al [13] and Belytschko et al [16]. An overview of developments for NEM is given by Cueto et al [37].…”