Locking-Free Mixed Finite Element Methods and Their Spurious Hourglassing Patterns

Hourglassing‐ and locking‐free mesh distortion insensitive Petrov–Galerkin EAS element for large deformation solid mechanics

“…Furthermore, the too soft bending behavior of H1/P0E6T (see Section 5.2) is resembled by the comparably soft behavior (low eigenvalue) of mode 7. The 3D results described for H1/P0E6T and H1U/F12 are inline with the observation by Hille et al 59 for the elasto-plastic material. Both elements show hourglassing in such simulations.…”

“…The element is implemented as described by Pfefferkorn and Betsch 16 and closely related to the improved EAS version by Simo et al 5 H1/P0 (Q1/P0): Mixed pressure element by Simo et al 80 based on a Hu–Washizu functional with elementwise constant pressure field. H1/P0E6T (Q1/P0E2T): Combination of H1/P0 and H1/E9T which was proposed by Armero 54 for 2D plane strain problems (Q1/P0E2T). A 3D extension (H1/P0E6T) has recently been proposed by Hille et al 59 H1/S18 (Q1/S5): Assumed stress element as proposed by Pian and Sumihara 74 and Pian and Tong 86 for linear elasticity in 2D and 3D, respectively. An extension to nonlinear problems can, for example, be found in the work of Viebahn et al 29 However, we use the inverse stress strain relation for a Neo‐Hookean material model proposed by Pfefferkorn et al 19 instead of the numeric procedure in aforementioned reference.…”

“…The final two examples in this work are elasto‐plastic necking simulations which are well‐known 59 to trigger hourglassing for standard elements. First we consider a plate subjected to plane strain conditions 2,10,14,47,54,89 with length $$$ 2L=53.334 $$$, width $$$ 2R=12.826 $$$ and thickness $$$ t=1 $$$.…”

“…Unfortunately though, this attempt could not cure hourglassing under tension. Other methods to suppress the spurious behavior use artificial hourglass stabilization with user‐defined parameters, 10,48,49 hourglass stabilization based on mixed methods 50‐52 or combine the EAS method with other mixed elements 53‐59 . However, none of these alternative approaches led to the desired result of a completely stable, locking‐free element without artificial stabilization.…”

“…Other methods to suppress the spurious behavior use artificial hourglass stabilization with user-defined parameters, 10,48,49 hourglass stabilization based on mixed methods [50][51][52] or combine the EAS method with other mixed elements. [53][54][55][56][57][58][59] However, none of these alternative approaches led to the desired result of a completely stable, locking-free element without artificial stabilization. A final avenue followed to overcome the hourglassing issue is enhancing different kinematic fields than the originally 2 proposed and most frequently employed enhancement of the deformation gradient (F-enhancement).…”