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 smooth particle hydrodynamic (SPH) method, corrected gradient methods, and the moving least squares (MLS) approximation. The discrete equations are derived from a collocation scheme or a Galerkin method. Special attention is paid to the treatment of essential boundary conditions. A brief review of radial basis functions is given because they play a significant role in meshfree methods. Finally, different approaches for modelling discontinuities in meshfree methods are described.

A meshless quasicontinuum method based on local maximum-entropy interpolation

“…, n. In the special case of equally spaced repatoms at distances L, symmetry again results in a deformed configuration in which all repatoms are equally spaced at distances (1 + ε)L. Therefore, all lattice sites, repatoms and sampling atoms are equally spaced at the correct distances. Consequently, as discussed before, the energy of each approximated lattice site is computed correctly and, since the summation rule is exact for a constant field, see our discussion of ( 35), we correctly recover (57). Therefore, for uniformly spaced repatoms, effective elastic moduli are reproduced exactly:…”

“…Here, we introduce a new meshless QC method that is based on local maximum-entropy (or max-ent for short) interpolation schemes in a fully nonlocal energy-based formulation with efficient quadrature-type summation rules. While meshless QC methods have been proposed in the past, see [40,41], those (i) made use of the Cauchy-Born rule and (ii) did not accurately bridge from full atomistics to the continuum, among other reasons because of inherent deficiencies of the employed smoothed-particle interpolation rules: traditional meshfree particle methods (see [57] for a comprehensive review) do not satisfy a weak Kronecker property on the boundary, which becomes crucial when enforcing essential boundary conditions. More importantly, the nonlocal support of meshfree shape functions has prevented all previous meshless QC techniques from recovering full atomistic resolution where shape functions should approach an affine interpolation to converge to the exact description.…”

NURBS-Enhanced Finite Element Method (NEFEM)

“…As discussed in the previous section, since NEFEM uses polynomials to approximate the solution, the difficulties in numerical integration are only restricted to the definition of a proper numerical quadrature in the curved element I e = W −1 ( e ) or its corresponding curved face. This, as will be observed below, reduces 64 R. SEVILLA, S. FERNÁNDEZ-MÉNDEZ AND A. HUERTA the complexity in the accurate evaluation of integrals, which are not as costly as in standard mesh-free methods [21] or in isogeometric analysis [12].…”

A comparison of parallel implementation of explicit DG and central difference method

“…Many researchers for example Belytschko [6], Dokainish and Subbaraj [7], Ou and Fulton [8] to mention some, have proposed alternative formulations and investigated the properties of the most frequently used method for explicit dynamic analysis; the central di erence method. Below, two equivalent formulations, one obtained from the Newmark implicit algorithm by inserting limit values of the parameters ( = 1=2; ÿ = 0) and one based on pure central di erence approximation of accelerations, formulated in incremental displacements are presented.…”