A Linearly Conforming Point Interpolation Method (Lc-Pim) for 2d Solid Mechanics Problems

Abstract

A linearly conforming point interpolation method (LC-PIM) is developed for 2D solid problems. In this method, shape functions are generated using the polynomial basis functions and a scheme for the selection of local supporting nodes based on background cells is suggested, which can always ensure the moment matrix is invertible as long as there are no coincide nodes. Galerkin weak form is adopted for creating discretized system equations, and a nodal integration scheme with strain smoothing operation is used to perform the numerical integration. The present LC-PIM can guarantee linear exactness and monotonic convergence for the numerical results. Numerical examples are used to examine the present method in terms of accuracy, convergence, and efficiency. Compared with the finite element method (FEM) using linear triangle elements and the radial point interpolation method (RPIM) using Gauss integration, the LC-PIM can achieve higher convergence rate and better efficiency.

Free vibration analysis of laminated composite plates based on FSDT using one-dimensional IRBFN method

“…In the PIM and RPIM, the compatibility characteristic is not ensured so the field function approximated could be discontinuous when nodes enter or leave the moving support domain. Liu et al suggested a linearly conforming point interpolation method (RC-PIM) [20] with a simple scheme for local supporting node selection, and a linearly conforming radial point interpolation method (RC RPIM) [21] to overcome the singular moment matrix issue and ensure the compatibility of the displacement.…”

A Cell-Based Smoothed Point Interpolation Method for Axisymmetric Problems

“…The second and third terms are calculated adopting an approach similar to that proposed in [1] for the calculation of the � matrix. The last term is calculated over the smoothing domains using the standard Gauss integration approach.…”

“…A wide class of efficient smoothed point interpolation methods (SPIMs) have been recently developed [1,2] by incorporating strain smoothing operation [2] to point interpolation methods (PIMs). In SPIMs, instead of using a compatible strain field, a smoothed strain field is constructed through a smoothing operation performed over smoothing domains.…”

A First Course in Finite Elements

“…The gradient smoothing was also used in the well-known widely used finite volume method (FVM) [32], the so-called quasi-conforming elements [33], and for the discretization of differential operator based on nodes [34]. It has been applied to resolve the material instabilities [14] and spatial instability in nodal integrated meshfree methods [15], and recently obtaining upper bound solution in meshfree point interpolation methods [18,19].…”

“…Proven types of smoothing domains are the cell-based smoothing domains that have been used in the smoothed FEM (or SFEM) [6][7][8] and cell-based smoothed point interpolation method (CS-PIM) [28]; node-based as used in the NS-PIM [16][17][18][19][20] and NS-FEM [24]; edge-based as in ES-PIM [26] and ES-FEM [25]; face-based FS-FEM [27]; as well as partial smoothing used in the α FEM [31]. Fig.…”