A hybrid-stress element based on Hamilton principle

Abstract

A novel hybrid-stress finite element method is proposed for constructing simple 4-node quadrilateral plane elements, and the new element is denoted as HH4-3β here. Firstly, the theoretical basis of the traditional hybrid-stress elements, i.e., the Hellinger-Reissner variational principle, is replaced by the Hamilton variational principle, in which the number of the stress variables is reduced from 3 to 2. Secondly, three stress parameters and corresponding trial functions are introduced into the system equations. Thirdly, the displacement fields of the conventional bilinear isoparametric element are employed in the new models. Finally, from the stationary condition, the stress parameters can be expressed in terms of the displacement parameters, and thus the new element stiffness matrices can be obtained. Since the required number of stress variables in the Hamilton variational principle is less than that in the Hellinger-Reissner variational principle, and no additional incompatible displacement modes are considered, the new hybrid-stress element is simpler than the traditional ones. Furthermore, in order to improve the accuracy of the stress solutions, two enhanced post-processing schemes are also proposed for element HH4-3β. Numerical examples show that the proposed model exhibits great improvements in both displacement and stress solutions, implying that the proposed technique is an effective way for developing simple finite element models with high performance.

High‐performance geometric nonlinear analysis with the unsymmetric 4‐node, 8‐DOF plane element US‐ATFQ4

“…In addition to the aforementioned incompatible and reduced integration schemes, some other techniques, including the hybrid stress method, [7][8][9][10][11][12] the quasi-conforming method, 13 the generalized conforming method, 2 the improved enhanced strain method, [14][15][16] the B-bar method, 17 the quadrilateral area coordinate method, [18][19][20][21][22][23][24][25][26][27] etc, can be found in related references. Although these models can improve the performance more or less, the sensitivity problem to mesh distortion has never been overcome from the outset.…”

Shape-Free Finite Element Method: Another Way between Mesh and Mesh-Free Methods

“…Obviously, new techniques must be developed for solving above problems. But at the same time, do not forget that some existing achievements, such as the new natural coordinate methods [30][31][32][33][34][35][36][37][38], generalized conforming method [2,23], assumed strain method [17], could play important roles in the constructions of the new shape-free finite elements, especially for low-order high-performance models.…”

“…On the other hand, during the history of the FEM itself, numerous efforts have been also made for improving performance and robustness of the traditional finite element models, such as the hybrid stress method proposed by 2 Mathematical Problems in Engineering Pian et al [12][13][14], the incompatible displacement modes proposed by Wilson et al [15] and Taylor et al [16], the enhanced strain method proposed by Simo and Rifai [17], the stabilization method proposed by Belytschko and Bacharch [18], the selectively reduced integration scheme proposed by Hughes [19], the assumed strain formulations proposed by MacNeal [20] and Piltner and Taylor [21], the quasiconforming element method proposed by Tang et al [22], the generalized conforming method proposed by Long and Huang [23], the Alpha finite element method ( FEM) [24] and the smoothed finite element method (S-FEM) [25,26] proposed by Liu et al, the new spline finite element method [27] proposed by Chen et al, the FE-mesh-free element method proposed by Rajendran et al [28,29], the new natural coordinate methods proposed by Long et al [30][31][32][33][34][35][36][37], and the Hamilton hybrid tress element method proposed by Cen et al [38]. These works made great contributions on the finite element method.…”

Several Treatments on Nonconforming Element Failed in the Strict Patch Test

“…Then, after substituting (16) into (11), the additional strain matrix B of element AGQ6-I in (11) can be rewritten in terms of the isoparametric coordinates. It is obvious that the additional strain matrix B of element AGQ6-I is different with that of element Q6.…”

“…Many efforts have been made for developing 4-node nonconforming plane elements without any problem in convergence, such as the element QM6 proposed by Taylor et al [10], QP6 by Wachspress [11], NQ6 by Pian and Wu [12], the 2 Mathematical Problems in Engineering generalized conforming element GC-Q6 by Long and Huang [13], the quasiconforming element QC6 by Chen and Tang [14], the hybrid-stress element P-S by Pian and Sumihara [15], and the Hamilton hybrid-stress element HH4 by Cen et al [16]. All these elements can pass the strict form of patch test and possess much better performance than usual 4-node conforming isoparametric element Q4.…”