The smooth piecewise polynomial particle shape functions corresponding to patch-wise non-uniformly spaced particles for meshfree particle methods

Abstract

In the previous papers (Kim et al. Submitted for publication, Oh et al. in press), for uniformly or locally non-uniformly distributed particles, we constructed highly regular piecewise polynomial particle shape functions that have the polynomial reproducing property of order k for any given integer k ≥ 0 and satisfy the Kronecker Delta Property. In this paper, in order to make these particle shape functions more useful in dealing with problems on complex geometries, we introduce smooth-piecewise-polynomial Reproducing Polynomial Particle shape functions, corresponding to the particles that are patch-wise non-uniformly distributed in a polygonal domain. In order to make these shape functions with compact supports, smooth flat-top partition of unity shape functions are constructed and multiplied to the shape functions. An error estimate of the interpolation associated with such flexible piecewise polynomial particle shape functions is proven. The onedimensional and the two-dimensional numerical results that support the theory are resented.

The reproducing singularity particle shape functions for problems containing singularities

“…This paper is a continuation of the previous paper [22] that is closely related to those element free methods: RKPM and RKEM. The Reproducing Kernel Particle Method (RKPM) [7,8,10,[13][14][15] is a meshfree method that yields highly accurate approximation to smooth functions by using the reproducing kernel particle (RKP) shape functions that can exactly interpolate the polynomials of a fixed degree.…”

“…Furthermore, in [22], by transforming these piecewise polynomial RPP shape functions via bilinear mappings, we construct piecewise polynomial particle shape functions, associated with patch-wise uniformly (or nonuniformly) distributed particles in a polygonal domain, that have the property of polynomial reproducing of a reduced order. However, elliptic boundary value problems on nonconvex domains (especially, cracked domains) contain singularities.…”

Patchwise Reproducing Polynomial Particle Method for Thick Plates: Bending, Free Vibration, and Buckling

“…Hence we do not need additional numerical scheme to impose essential boundary conditions. (See [37,38] for more details.) 2.1.…”

“…To overcome these difficulties, encountered in meshless methods, Oh et al introduced three closed-form partition of unity (PU) functions that have flat-top: (1) Convolution partition of unity [36] for any partition of a given domain; Using convolution partition of unity, Oh et al introduced several meshless methods that are called patchwise RPPM, adaptive RPPM, and RSPM (Reproducing Singularity Particle Method) in [32,35,36,38]. Note that RPPM is similar to RKPM [2,16,20,21,25,26,27,28].…”

Generalized Gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method

“…Oh et al [30] introduce several smooth-piecewise-polynomial regularized discontinuous functions that can be used in the X-FEM. Benvenuti et al [31] extended the method of Ventura [27] for regularized discontinuous enrichment functions.…”