An anisotropic, superconvergent nonconforming plate finite element

Abstract

The classical finite element convergence analysis relies on the following regularity condition: there exists a constant c independent of the element K and the mesh such that h K / K c, where h K and K are diameters of K and the biggest ball contained in K, respectively. In this paper, we construct a new, nonconforming rectangular plate element by the double set parameter method. We prove the convergence of this element without the above regularity condition. The key in our proof is to obtain the O(h 2 ) consistency error. We also prove the superconvergence of this element for narrow rectangular meshes. Results of our numerical tests agree well with our analysis.

Seeing is believing: atomic force microscopy imaging for nanomaterial research

“…These three modes are mainly used for height imaging. Furthermore, based on the three primary AFM imaging modes, some secondary imaging modes have been developed such as deection imaging, 37 phase imaging, 38 lateral force imaging, 39 magnetic force gradient imaging, 40 torsional resonance imaging, 41 conductive AFM, 42 electric eld gradient distribution imaging, 43 surface potential imaging, 44 electrical carrier concentration imaging, 45 force modulation imaging, 46 surface thermal imaging, 47 and scanning spreading resistance imaging. 48 In general, the applied imaging operation mode depends on the surface characteristics of interest and the hardness/stickness of the sample.…”

New High Accuracy Analysis of a Double Set Parameter Nonconforming Element for the Clamped Kirchhoff Plate Unilaterally Constrained by an Elastic Obstacle

“…Compared with the conforming FEMs studied in [1,11], C 0 continuous and discontinuous nonconforming (i.e., C 0 and non-C 0 nonconforming) FEMs were discussed in [10], and convergence analyses were established. On the other hand, the high accuracy analysis of nonconforming FEMs has been an active research area in practical computations, and many high accuracy results have been derived for variational inequalities and the boundary value problem [12][13][14][15][16][17][18][19][20]. However, for the plate obstacle problem (4) with a rigid obstacle, the exact solution u only belongs to H 3 (Ω) ∩ C 2 (Ω) instead of H 4 loc (Ω) [21,22]; thus, the lack of H 4 regularity makes it impossible to develop high accuracy analysis.…”

“…The first set is chosen to meet the convergence requirements according to the generalized patch test [25] or F-E-M-test [26], while the second set is selected to be simple so that the total number of unknowns in the resulting discrete system is small. Up to now, several nonconforming double set parameter plate elements have been successfully applied to deal with the two-sided displacement obstacle problem of the clamped plate [8], plate bending problems [16,27], the linear elasticity problem [28], the fourth-order elliptic singular perturbation problem [29][30][31] and so on. In this paper, a non-C 0 nonconforming double set parameter plate element is employed for the elastic obstacle problem (2).…”