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Title: A velocity/stress mixed stabilized nodal finite element for elastodynamics: Analysis and computations with strongly and weakly enforced boundary conditions

Abstract

A new nodal mixed finite element is proposed for the simulation of linear elastodynamics and wave propagation problems in time domain. Our method is based on equal-order interpolation discrete spaces for both the velocity (or displacement) and stress (or strain) tensor variables. The mixed form is derived using either the velocity/stress or velocity/strain pair of unknowns, the latter being instrumental in extensions of the method to nonlinear mechanics. The proposed approach works equally well on hexahedral or tetrahedral grids and, for this reason, it is suitable for time-domain engineering applications in complex geometry. The peculiarity of the proposed approach is the use of the rate form of the stress update equation, which yields a set of governing equations with the structure of a non-dissipative space/time Friedrichs’ system. We complement standard traction boundary conditions for the stress with strongly and weakly enforced boundary conditions for the velocity (or displacement). Weakly enforced boundary conditions are particularly suitable when considering complex geometrical shapes, because they do not require dedicated data structures for the imposition of the boundary degrees of freedom, but, rather, they utilize the structure of the variational formulation. Here we also show how the framework of weakly enforced boundary conditions canmore » be used to develop variational forms for multi-domain simulations of heterogeneous media. A complete analysis including stability and convergence proofs is included, in the case of a space–time variational approach. A series of computational tests are used to demonstrate and verify the performance of the proposed approach.« less

Authors:
ORCiD logo [1];  [1];  [2]
+ Show Author Affiliations
  1. Duke University, Durham, NC (United States)
  2. University of Texas at El Paso, TX (United States)
Publication Date:
Research Org.:
Duke Univ., Durham, NC (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); ExxonMobil Upstream Research Company; US Department of the Navy, Office of Naval Research (ONR)
OSTI Identifier:
1538113
Alternate Identifier(s):
OSTI ID: 1549632
Grant/Contract Number:  
SC0012169; N00014-14-1-0311
Resource Type:
Accepted Manuscript
Journal Name:
Computer Methods in Applied Mechanics and Engineering
Additional Journal Information:
Journal Volume: 325; Journal Issue: C; Journal ID: ISSN 0045-7825
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
42 ENGINEERING; weak boundary conditions; wave equation; nodal stress or nodal strain; elastodynamics; stabilized methods; variational multiscale analysis

Citation Formats

Scovazzi, G., Song, T., and Zeng, X. A velocity/stress mixed stabilized nodal finite element for elastodynamics: Analysis and computations with strongly and weakly enforced boundary conditions. United States: N. p., 2017. Web. doi:10.1016/j.cma.2017.07.018.
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Scovazzi, G., Song, T., & Zeng, X. A velocity/stress mixed stabilized nodal finite element for elastodynamics: Analysis and computations with strongly and weakly enforced boundary conditions. United States. https://doi.org/10.1016/j.cma.2017.07.018
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Scovazzi, G., Song, T., and Zeng, X. Tue . "A velocity/stress mixed stabilized nodal finite element for elastodynamics: Analysis and computations with strongly and weakly enforced boundary conditions". United States. https://doi.org/10.1016/j.cma.2017.07.018. https://www.osti.gov/servlets/purl/1538113.
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@article{osti_1538113,
title = {A velocity/stress mixed stabilized nodal finite element for elastodynamics: Analysis and computations with strongly and weakly enforced boundary conditions},
author = {Scovazzi, G. and Song, T. and Zeng, X.},
abstractNote = {A new nodal mixed finite element is proposed for the simulation of linear elastodynamics and wave propagation problems in time domain. Our method is based on equal-order interpolation discrete spaces for both the velocity (or displacement) and stress (or strain) tensor variables. The mixed form is derived using either the velocity/stress or velocity/strain pair of unknowns, the latter being instrumental in extensions of the method to nonlinear mechanics. The proposed approach works equally well on hexahedral or tetrahedral grids and, for this reason, it is suitable for time-domain engineering applications in complex geometry. The peculiarity of the proposed approach is the use of the rate form of the stress update equation, which yields a set of governing equations with the structure of a non-dissipative space/time Friedrichs’ system. We complement standard traction boundary conditions for the stress with strongly and weakly enforced boundary conditions for the velocity (or displacement). Weakly enforced boundary conditions are particularly suitable when considering complex geometrical shapes, because they do not require dedicated data structures for the imposition of the boundary degrees of freedom, but, rather, they utilize the structure of the variational formulation. Here we also show how the framework of weakly enforced boundary conditions can be used to develop variational forms for multi-domain simulations of heterogeneous media. A complete analysis including stability and convergence proofs is included, in the case of a space–time variational approach. A series of computational tests are used to demonstrate and verify the performance of the proposed approach.},
doi = {10.1016/j.cma.2017.07.018},
journal = {Computer Methods in Applied Mechanics and Engineering},
number = C,
volume = 325,
place = {United States},
year = {Tue Jul 25 00:00:00 EDT 2017},
month = {Tue Jul 25 00:00:00 EDT 2017}
}
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Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity: Motivation, formulation, and numerical examples
journal, August 2008

  • Eyck, Alex Ten; Celiker, Fatih; Lew, Adrian
  • Computer Methods in Applied Mechanics and Engineering, Vol. 197, Issue 45-48
  • DOI: 10.1016/j.cma.2008.02.020

Discontinuous Galerkin methods for non-linear elasticity
journal, January 2006

  • Ten Eyck, A.; Lew, A.
  • International Journal for Numerical Methods in Engineering, Vol. 67, Issue 9
  • DOI: 10.1002/nme.1667

Development of a cell centred upwind finite volume algorithm for a new conservation law formulation in structural dynamics
journal, March 2013

Approximation des problèmes aux limites non homogènes pour des opérateurs non linéaires
journal, June 1970

On stability and convergence of projection methods based on pressure Poisson equation
journal, May 1998

Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity: Analytical estimates
journal, June 2008

  • Eyck, Alex Ten; Celiker, Fatih; Lew, Adrian
  • Computer Methods in Applied Mechanics and Engineering, Vol. 197, Issue 33-40
  • DOI: 10.1016/j.cma.2008.02.022

A stabilized finite element formulation for finite deformation elastoplasticity in geomechanics
journal, April 2009

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