A two‐step Taylor‐Galerkin formulation for fast dynamics

A two‐step Taylor‐Galerkin formulation for fast dynamics Purpose – The purpose of this paper is to present a new stabilised low‐order finite element methodology for large strain fast dynamics. Design/methodology/approach – The numerical technique describing the motion is formulated upon the mixed set of first‐order hyperbolic conservation laws already presented by Lee et al. (2013) where the main variables are the linear momentum, the deformation gradient tensor and the total energy. The mixed formulation is discretised using the standard explicit two‐step Taylor‐Galerkin (2TG) approach, which has been successfully employed in computational fluid dynamics (CFD). Unfortunately, the results display non‐physical spurious (or hourglassing) modes, leading to the breakdown of the numerical scheme. For this reason, the 2TG methodology is further improved by means of two ingredients, namely a curl‐free projection of the deformation gradient tensor and the inclusion of an additional stiffness stabilisation term. Findings – A series of numerical examples are carried out drawing key comparisons between the proposed formulation and some other recently published numerical techniques. Originality/value – Both velocities (or displacements) and stresses display the same rate of convergence, which proves ideal in the case of industrial applications where low‐order discretisations tend to be preferred. The enhancements introduced in this paper enable the use of linear triangular (or bilinear quadrilateral) elements in two dimensional nearly incompressible dynamics applications without locking difficulties. In addition, an artificial viscosity term has been added into the formulation to eliminate the appearance of spurious oscillations in the vicinity of sharp spatial gradients induced by shocks. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Engineering Computations Emerald Publishing

A two‐step Taylor‐Galerkin formulation for fast dynamics

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Publisher
Emerald Publishing
Copyright
Copyright © 2014 Emerald Group Publishing Limited. All rights reserved.
ISSN
0264-4401
DOI
10.1108/EC-12-2012-0319
Publisher site
See Article on Publisher Site

Abstract

Purpose – The purpose of this paper is to present a new stabilised low‐order finite element methodology for large strain fast dynamics. Design/methodology/approach – The numerical technique describing the motion is formulated upon the mixed set of first‐order hyperbolic conservation laws already presented by Lee et al. (2013) where the main variables are the linear momentum, the deformation gradient tensor and the total energy. The mixed formulation is discretised using the standard explicit two‐step Taylor‐Galerkin (2TG) approach, which has been successfully employed in computational fluid dynamics (CFD). Unfortunately, the results display non‐physical spurious (or hourglassing) modes, leading to the breakdown of the numerical scheme. For this reason, the 2TG methodology is further improved by means of two ingredients, namely a curl‐free projection of the deformation gradient tensor and the inclusion of an additional stiffness stabilisation term. Findings – A series of numerical examples are carried out drawing key comparisons between the proposed formulation and some other recently published numerical techniques. Originality/value – Both velocities (or displacements) and stresses display the same rate of convergence, which proves ideal in the case of industrial applications where low‐order discretisations tend to be preferred. The enhancements introduced in this paper enable the use of linear triangular (or bilinear quadrilateral) elements in two dimensional nearly incompressible dynamics applications without locking difficulties. In addition, an artificial viscosity term has been added into the formulation to eliminate the appearance of spurious oscillations in the vicinity of sharp spatial gradients induced by shocks.

Journal

Engineering ComputationsEmerald Publishing

Published: Apr 28, 2014

Keywords: Finite element method; Conservation laws; Fast dynamics; Low order; Riemann solver; Taylor‐Galerkin

References

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