p-Adaptive C k generalized finite element method for arbitrary polygonal clouds

Abstract

A p-Adaptive Generalized Finite Element Method (GFEM) based on a Partition of Unity (POU) of arbitrary smoothness degree is presented. The shape functions are built from the product of a Shepard POU and enrichment functions. Shepard functions have a smoothness degree directly related to the weighting functions adopted in their definition. Here the weighting functions are obtained from boolean R-functions which allow the construction of C k approximations, with k arbitrarily large, defined over a polygonal patch of elements, named cloud. The Element Residual Method is used to obtain error indicators by taking into account the typical nodal enrichment scheme of the method. This procedure is enhanced by using approximations with a high degree of smoothness as it eliminates the discontinuity of the stress field in the interior of each cloud. Adaptive analysis of plane elasticity problems are presented, and the performance of the technique is investigated.

Effects of the smoothness of partitions of unity on the quality of representation of singular enrichments for GFEM/XFEM stress approximations around brittle cracks

“…The first seven digits of the constant in (10) agrees with the value given by [60] in which only seven digits were provided. 3 The crack representation was performed using only branch functions in order to reduce sources of blending/transition effects. The authors are also concerned with transition effects which manifest through anomalous reductions in the rates of convergence due to partially enriched elements.…”

“…In other words, quasi-singular stiffness matrices can be obtained with C k -GFEM when higher-degree polynomial enrichments, with the same nature as in (5), are uniformly applied on the domain, even though linear dependence theoretically does not occur when the PoU functions are not polynomial [24,46,45,48,75]. However, such intense conditioning deterioration related to the polynomial enrichment does not occur when the enrichment is localized, as in adaptive procedures [2,3].…”

“…The extrinsic manner of adding algebraic enrichments to an ansatz space is a robust and efficient choice for addressing special information on the features of the solution being sought. The procedure enables adaptive polynomial refinements as different degrees p of functions can be used according to error oriented algorithms (see [2] [3], for instance). Additionally, any special function can be used, which may avoid changing the characteristic dimension h of the elements to accommodate localized features such as singularities, boundary layers and material interfaces or represent boundary evolutions without changing the mesh.…”

A C k continuous generalized finite element formulation applied to laminated Kirchhoff plate model

“…Aiming at removing this limitation, Duarte et al [18] used the so-called boolean R-function of Shapiro [41]. Latter, Barros et al [6] discusses this procedure for linear elasticity problems. The arbitrary continuity is based on the type of selected edge functions and on the value of a parameter of a boolean function.…”

On error estimator and adaptivity in the meshless Galerkin boundary node method

“…The subject of error estimates for meshless methods and a consequent adaptive analysis is central to the effective application of meshless algorithms for practical engineering computation. In recent years, a large amount of work has been performed concerning adaptive analysis based on a posteriori error estimation for domain-type meshless methods such as the h-p meshless method [15], the GFEM [16], the EFG method [17][18][19][20], the RKPM [21][22][23][24][25], the FPM [26] and the PIM [27,28]. Significant progress has been achieved in the theory and implementation of the adaptive procedures for these meshless methods.…”