Comparison of second-order serendipity and Lagrange tetrahedral elements for nonlinear explicit methods
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 by
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.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:
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10-node tetrahedra are much superior to 20-node hexahedra with HRZ lumping
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15-node tetrahedra are generally more accurate, robust, and expensive, than 10-node ones
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10-node tetrahedra can produce reasonable results with tremendous cost savings
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Computational costs are frequently 1/10 or less of the 15-node ones
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Predictions are
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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
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