Level‐wise strain recovery and error estimation for natural element hierarchical plate models

Abstract

Differing from the finite element method, most of mesh-free methods are restricted to two-dimensional (2D) problems owing to the difficulty in constructing three-dimensional (3D) grids. But, for 3D elastic structures, this limitation could be effectively overcome when hierarchical models are employed. In the hierarchical models of elastic structures, only the in-plane displacement field is needed to be approximated because the thickness-wise displacement field is assumed by polynomials. In this context, this paper is concerned with the reliable numerical implementation of hierarchical models for plate-like structures using 2D natural element method (NEM). However, one critical problem in implementing the hierarchical model is to secure the continuity in strain and stress fields. To resolve this problem, this paper presents a level-wise strain recovery method. In addition, a recovery-based error estimation is introduced to predict the modeling error and the numerical approximation error.The proposed methods are illustrated and validated through the numerical experiments. It is confirmed that the level-wise strain recovery successfully and CPU-time effectively provides the continuous stress resultants and the proposed error estimator accurately predicts the modeling and approximation errors.

Large Deflection Geometrically Nonlinear Bending of Porous Nanocomposite Cylindrical Panels on Elastic Foundation

“…A good agreement with Shen and Wang [45] is seen for the load-central deflection plots, but the present method produces the load-bending moment curves lower than those of Shen and Wang [45]. To obtain better numerical results, the iteration compensation parameter β(n) = √ 1 + 4n/400 was introduced to the shear elastic modulus G e f f in Equation ( 23) (see also Equation (A10) in Appendix A) and the strains and stresses are enhanced by applying the strain recovery scheme [32] and Reddy and Liu's HSDT, given in Appendix A. The introduction of β(n) was motivated that the variation in stiffness matrix K NL in Equation (A8) along the nonlinear iteration requires the adaption of K e L,s in Equation (A6), which was derived according to the MITC3+ shell element concept.…”

“…Laplace interpolation functions φ I (ζ 1 , ζ 2 ) in Figure 3 are C 0 -continuous functions, so their second derivatives are not defined. Thus, in this study, the derivative terms w 0,xx , w 0,xy , w 0,yy , ϑ x,x , ϑ x,y and ϑ y,y in Equation (A2) are alternatively calculated using the strain recovered derivatives [32]. For example,…”

“…In this context, this study aims at the scrutinizing investigation of the nonlinear bending responses of FG-GPLRC porous cylindrical panels resting on an elastic foundation by developing a reliable and effective meshfree-based 2D nonlinear numerical method. The large deflection of cylindrical panel is expressed by the FSDT together with the von Kármán geometry nonlinearity, but the approximated stresses are enhanced by applying the strain recovery [32] and the Reddy's TSDT. The equivalent elastic properties of FG-GPLRC cylindrical panels are determined by applying the Halpin-Tsai homogenization model, and those are modified by introducing three different porosity distributions.…”