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Transient solid dynamics on linear tetrahedral finite elements using a variational multi-scale approach

A new tetrahedral finite element for transient dynamic computations in solids is presented. It utilizes the simplest possible finite element interpolations: Piece-wise linear continuous functions are used for displacements and pressures (P1/P1), while the deviatoric part of the stress tensor is evaluated with simple single-point quadrature formulas. This approach takes inspiration from previous work of the first author in the case of compressible fluid dynamics in Lagrangian coordinates [G. Scovazzi, “Lagrangian shock hydrodynamics on tetrahedral meshes: A stable and accurate variational multiscale approach,” J. Comp. Phys., 231(24), pp. 8029-8069, 2012]. The variational multiscale stabilization eliminates the pressure checkerboard instabilities affecting the numerical solution in the Stokes-type operator that arises in solid dynamics computations. The formulation is extended to visco-elastic solids. Extensive numerical tests are presented. Because of its simplicity, the proposed element could favorably impact complex geometry, fluid/structure interaction, and embedded discontinuity computations [G. Scovazzi, B. Carnes, X. Zeng, “A simple, stable, and accurate tetrahedral finite element for transient, nearly incompressible, linear and nonlinear elasticity: A dynamic variational multiscale approach,” Int. J. Num. Meth. Engr., (submitted), 2015].

Author(s):

Guglielmo Scovazzi    
Duke University
United States

Brian Carnes    
Sandia National Laboratories
United States

Xianyi Zeng    
Duke University
United States