Incompatible Modes for Volumetric Shell Elements in Explicit Time Integration

Abstract

Explicit time integration is characterized by small time steps and efficient global operations, leading to a domination of element processing regarding CPU‐time, in particular, as diagonal mass matrices are used. The Enhanced Assumed Strain (EAS) approach requires local operations in order to condense‐out additional degrees of freedom. The method of Incompatible Modes (IM) is presented as an alternative, where incompatible degrees of freedom are included in the global equations, which leads to efficient element formulations. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Selective mass scaling for distorted solid-shell elements in explicit dynamics: optimal scaling factor and stable time step estimate

“…However, these elements exhibit an inherently bad aspect ratio due to the small thickness compared with the in‐plane dimensions. For this reason, they are preferably used in implicit dynamics approaches, to avoid the small critical time step produced by the high maximum eigenfrequency, and only recently, their usage in explicit dynamics approaches seems to have been attracting a specific attention (see, e.g., ). Systematic studies on the computational effectiveness of solid‐shell elements in explicit dynamics have also been recently presented in .…”

“…Numerical tests, both in small and large displacements and rotations, using a state-of-the-art solidshell element taken from the literature, confirm the effectiveness and accuracy of the proposed approach. 702 G. COCCHETTI, M. PAGANI AND U. PEREGO explicit dynamics approaches seems to have been attracting a specific attention (see, e.g., [7,[9][10][11]). Systematic studies on the computational effectiveness of solid-shell elements in explicit dynamics have also been recently presented in [12,13].Because solid-shell elements do not make use of rotational DOFS, direct application of the selective mass scaling proposed in [2, 3] is not possible.…”

Enhanced low‐order displacement finite elements using incompatible mass scaling and static condensation in explicit time integration

“…These concepts are bound to a static elimination procedure of internal parameters which may result in a considerable expansion of the computational costs in the case of an explicit time integration scheme. In order to avoid the static elimination procedure inertia is considered for the incompatible parameters as suggested by Mattern et al [3,4]. This step allows integrating the incompatible parameters directly in time, but since rather large stiffnesses are related to the incompatible parameters -particularly for the volumetric mode enhancements -the highest eigenfrequency may be increased which would lead to a decrease of the time step size.…”

“…Contrary to this the expression on the right in equation (2) needs to be solved in each step via linearization only with respect to α n which appears as a bottleneck concerning computational efficiency when using the EAS scheme in explicit time integration, see [4]. Following [1] the discretization of the enhanced strain field in equation (1) can be reproduced by a displacement field enhanced by incompatible modes (IM) which reproduce the enhanced strain interpolation identically in the formWith equation (3) it is possible to introduce inertia for the IMs into the Hu-Washizu functional as proposed in [3]. Following then the procedure described for the classical EAS approach and varying with respect toû andũ results in the equation systemM Dün + f int (û n ,ũ n ) = f ext nMü +f (û n ,ũ n ) = 0 .Compared to equation (2, right) the expression on the right of equation (4) allows to solve for the additional parameters represented now byũ n in the fashion of an explicit time integration scheme as the dynamic equilibrium on the left where, compared to equation (2,left) α is exchanged forũ.…”

“…With equation (3) it is possible to introduce inertia for the IMs into the Hu-Washizu functional as proposed in [3]. Following then the procedure described for the classical EAS approach and varying with respect toû andũ results in the equation system…”