Conforming radial point interpolation method for spatial shell structures on the stress-resultant shell theory

Abstract

The implementation of the conforming radial point interpolation method (CRPIM) for spatial thick shell structures is presented in this paper. The formulation of the discrete system equations is derived from a stress-resultant geometrically exact theory of shear flexible shells based on the Cosserat surface. A discrete singularity-free mapping between the five degrees of freedom of the Cosserat surface and the normal formulation with six degrees of freedom is constructed by exploiting the geometry connection between the orthogonal group and the unit sphere. A radial basis function is used in both the construction of shape functions based on arbitrarily distributed nodes as well as in the surface approximation of general spatial shell geometries. The major advantage of the CRPIM is that the shape functions possess a delta function property and the interpolation function obtained passes through all the scattered points in the influence domain. Thus, essential boundary conditions can be easily imposed, as in finite element method. A range of shape parameters is studied to examine the performance of CRPIM for shells, and optimal values are proposed. The phenomena of shear locking and membrane locking are illustrated by presenting the membrane and shear energies as fractions of the total energy. Several benchmark problems for shells are analyzed to demonstrate the validity and efficiency of the present CRPIM. The convergence rate of the results using a Gaussian (EXP) radial basis is relatively high compared to those using a multi-quadric (MQ) radial basis for the shell problems.

Developments in the application of the generalized finite element method to thick shell problems

“…Considering the kinematic model represented by (27), the displacement field can be written in matrix form as…”

“…In the GFEM approach, a unique continuous mapping for each cloud (in this case a patch of elements) is necessary in order to define the approximation space on the curved shell. A possible strategy consists of using an approximation of the reference surface defined as in [24,27]. In the present paper, however, this concept is substituted by a local approximation based on pseudo-tangent planes.…”

“…Among the most recent works we should mention the paper of Liu et al [27], who approximate the stress field in shear deformation model for shells utilizing a geometrically exact formulation. The basis functions in the radial direction utilized for the stresses are also utilized to approximate the reference surface.…”

The uselessness of the Fast Gauss Transform for summing Gaussian radial basis function series

“…The FGT and TGT are illuminating to analyze because both are particularly tailored to Gaussians. Gaussian RBFs are useful examples because (i) they are popular [45,51,52,54,12,13,30,34,37,49,59,3], 1 widely used and converge exponentially fast for smooth functions f ðxÞ and (ii) have only short-range interactions, allowing us to focus on that part of summing RBFs where Fast Summations have difficulties. (As we shall explain later, long-range interactions, for those RBF species that have them, are well accelerated by some Fast Summations.…”

A smoothed Hermite radial point interpolation method for thin plate analysis

“…The PIM and RPIM shape functions using local nodes have the Kronecker delta function properties; the essential boundary conditions can be easily imposed for PIM and RPIM models. A conforming RPIM for thick shell structures was proposed in [33] and the PIM was extended for spatial general shells structures [34]. Liu et al [35] proposed the conforming radial point interpolation method for static and free vibration analysis of plates using stabilized conforming nodal integration technique.…”