Comparison of second-order serendipity and Lagrange tetrahedral elements for nonlinear explicit methods

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Highlights

  • 15-node tetrahedra are generally more accurate than HRZ Lumped 10-node ones.

  • Number of quadrature points for 10-node tetrahedra is about ¼ that of 15-node ones.

  • Stable time increment size can be much greater for 10-node than for 15-node tetrahedra.

  • HRZ Lumped 10-node tetrahedra can produce reasonable results with large cost savings.

  • Inclusion of both 10- and 15-node tetrahedra into explicit codes is recommended.

Abstract

This paper evaluates the performances of second-order finite elements for nodal lumped-mass explicit methods in nonlinear solid dynamics, with a particular emphasis on 10-node “serendipity” and 15-node “Lagrange” tetrahedral elements. Historically, many nonlinear explicit finite element codes have exclusively used first-order elements, until a fairly recent flurry of activity that has resulted in higher-order elements becoming available in explicit codes including the authors' in-house one, ParaAble, and the production software EPIC, IMPETUS, LS-DYNA, and Abaqus. A major attractiveness of tetrahedrons is their ease in meshing and higher-order elements can facilitate the avoidance of severe volumetric locking with unstructured C0 meshes, which are generally used with these codes for the discontinuities of inelasticity, contact, etc. They also can improve modeling of flexure and curved shapes as well as eliminate spurious modes without artificial stabilization. The inclusion of face and body centroid nodes with Lagrange interpolants, including the 15-node tetrahedron, has proven to provide robust overall performance with lumped-mass explicit methods and with contact. Nevertheless, versions of the 10-node tetrahedron have also emerged in lumped-mass explicit software. In contrast to hexahedrons, an important observation about tetrahedrons is that the 10-node serendipity version uses about four times fewer quadrature points and a larger time increment than their 15-node Lagrange counterpart, which could result in tremendous computational differences. Serendipity elements, however, notoriously do not nodal mass lump well and tetrahedron versions have not been rigorously evaluated/documented for their effectiveness, as will be done herein with comparisons of those using 15-node tetrahedrons. Using row-summation lumping for Lagrange elements and the ad hoc HRZ scheme for serendipity ones, performances are assessed in common benchmark problems and practical applications using various elastic and inelastic material models and involving large strains/deformations/rotations and severe distortions. Whereas the 10-node tetrahedrons were found to perform much better than their 20-node serendipity hexahedral counterparts, specifically with substantial computational reductions and reasonable predictions, they are not generally quite as accurate or robust as the 15-node tetrahedral elements. The results thus indicate benefits of including both 10- and 15-node tetrahedrons in an explicit code's element library.

Section snippets

Introduction and background

Classical finite element analysis remains a primary computational method of choice for most solid mechanics applications, and many newer enhanced methods are frequently used in combination with finite elements, such as conversion to meshfree interpolants or adaptive enrichment in regions of interest. Therefore, improved finite element capabilities continue to be important. Tetrahedral (Tet) finite elements are attractive because of their ease with automatic mesh generators, and the explicit

Governing equations

The basic form of the nonlinear equations is derived as that in standard texts, e.g., Refs. [[19], [20], [21],38,39]. The motion is described in a Lagrangian sense with reference to the original undeformed right-handed Cartesian coordinate system. As shown in Figure 3, the current position of any point, x, at time, t, is related to the undeformed position, X, using the displacements, u, from the undeformed configuration byx=X+u(X,t)

The following virtual work statement for large

Discrete equations of motion

Eq. (2) is spatially discretized into a mesh of individual elements with each possessing the following local interpolations for all X, u, and x, and using values of the respective vectors at each nodal location.{X,u,x}(η1,η2,η3)=I=1#NodeshI(η1,η2,η3){X˜I,u˜I,x˜I}where are the local natural element coordinates of the parent element (see Figure 1), hI is the interpolation (or “shape”) function for each node, the tilde (~) designates the quantity as a nodal vector, and the subscript I refers to

Evaluation example problems

The above element formulations have been implemented into the authors’ MPI-based parallel finite element code, ParaAble [52,53], and executed for the following suite of sample problems. The examples vary in their use of English, SI, and non-dimensional units, so as to be consistent with the analyses in the literature that are used for comparison. The purpose of this paper is to evaluate the lumped-mass explicit formulations of the 2 s-order tetrahedral element types in nonlinear applications.

Performance assessment and concluding remarks

  • The primary discovery was the value of using HRZ lumped-mass 10-node serendipity tetrahedral elements in many applications using explicit methods. Major observations are:

  • 10-node tetrahedra are much superior to 20-node hexahedra with HRZ lumping

  • 15-node tetrahedra are generally more accurate, robust, and expensive, than 10-node ones

  • 10-node tetrahedra can produce reasonable results with tremendous cost savings

    • -

      Computational costs are frequently 1/10 or less of the 15-node ones

    • -

      Predictions are

Author statement

Kent T. Danielson: Conceptualization, Methodology, Software, Formal analysis, Investigation, Data Curation, Writing - Original Draft, Writing - Review & Editing, Visualization, Supervision, Funding acquisition, Project administration. Robert S. Browning: Methodology, Software, Formal analysis, Investigation. Mark D. Adley: Methodology, Software, Formal analysis, Investigation.

Declaration of competing interest

None of the authors have a conflict of interest.

Acknowledgements

The authors dedicate this paper in memory of Dr. James Lowell O'Daniel, a great friend and colleague who contributed in the early development of this topic area—his tragic death is a loss for both the computational mechanics community and beyond. Permission to publish was granted by Director, Geotechnical and Structures Laboratory. The work was supported in part by grants of computer time from the DoD High Performance Computing Modernization Program at the ERDC DoD Supercomputing Resource

References (65)

  • R.S. Browning et al.

    Higher-order finite elements for lumped-mass explicit modeling of high-speed impacts

    Int. J. Impact Eng.

    (2020)
  • I. Fried et al.

    Finite element mass matrix lumping by numerical integration with no convergence rate loss

    Int. J. Solid Struct.

    (1975)
  • T. Belytschko et al.

    Mixed methods in time integration

    Comput. Methods Appl. Mech. Eng.

    (1979)
  • M.O. Neal et al.

    Explicit-explicit subcycling with non-integer time step ratios for structural dynamic systems

    Comput. Struct.

    (1989)
  • P. Keast

    Moderate-degree tetrahedral quadrature formulas

    Comput. Methods Appl. Mech. Eng.

    (1986)
  • K.S. Sunder et al.

    Integration points for triangles and tetrahedrons obtained from the Gaussian quadrature points for a line

    Comput. Struct.

    (1985)
  • K.T. Danielson et al.

    Nonlinear dynamic finite element analysis on parallel computers using FORTRAN 90 and MPI

    Adv. Eng. Software

    (1998)
  • T. Belytschko et al.

    Hourglass control in linear and nonlinear problems

    Comput. Methods Appl. Mech. Eng.

    (1984)
  • T. Belytschko et al.

    Assumed strain stabilization of the 4-node quadrilateral with 1-point quadrature for nonlinear problems

    Comput. Methods Appl. Mech. Eng.

    (1991)
  • T. Belytschko et al.

    Assumed strain stabilization of the eight node hexahedral element

    Comput. Methods Appl. Mech. Eng.

    (1993)
  • E. Zywicz et al.

    DYNA3D: A Nonlinear, Explicit, Three-Dimensional Finite Element Code for Solid and Structural Mechanics: Version 13.1

    (2013)
  • Abaqus 2019, Theory, Benchmarks, and Examples Manuals

    (2019)
  • S.W. Attaway et al.

    PRONTO 3D Users' Instructions: A Transient Dynamics Code for Nonlinear Structural Analysis, Report SAND98-1361

    (1998)
  • Presto User's Guide Version 4.16. Report SAND2010-3112

    (2012)
  • Johnson G.R., Beissel S.R., Gerlach C.A., Holmquist T.J., User instructions for the 2018 Version of the EPIC Code,...
  • LS-DYNA KEYWORD USER'S MANUAL VOLUME I, r:8124

    (January 2017)
  • H. Teng

    Recent advances on higher-order 27-node hexahedral element in LS-DYNA

  • S.R. Beissel et al.

    An Evaluation of Numerical Methods for High-Frequency Structural Response, SwRI Report 18.15199/008-2, Southwest Research Institute Contract Report for the U.S

    (October 2015)
  • S.R. Beissel et al.

    Modeling High-Frequency Wave Propagation in Solids Using Higher-Order Finite Elements in the EPIC Code, SwRI Report 18.19018/022, Southwest Research Institute Contract W56HZV-13-C-0047 with

    (November 2016)
  • S.R. Beissel et al.

    The Addition of Second-Order Hexahedral, Tetrahedral, and Wedge Elements into the EPIC Code, SwRI Report 18.22494/006, Southwest Research Institute Contract W15QKN-14—9-1001 with U.S

    (April 2018)
  • IMPETUS AFEA: SOLVER, User Documentation, Version 3.0 Beta, IMPETUS AFEA, Flekkefjord, Norway

    (August 3, 2012)
  • K.T. Danielson et al.

    Reliable second-order hexahedral elements for explicit methods in nonlinear solid dynamics

    Int. J. Numer. Methods Eng.

    (2011)
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