On the tensor viscosity based on Gauss quadrature: A comparison of robustness, efficiency, and connection with hourglass control

Abstract

From the numerical quadrature perspective, we study the tensor viscosity for the staggered Lagrangian hydrodynamics. Although this approach is analogous to the tensor viscosities in [5, Campbell&Shashkov2001] and [14, Kolev&Rieben2009], we gain some novel insights into such tensor viscosity. The instability originates from the singularity of the Jacobian determinant, and there's a naturally trade-off between efficiency and robustness of the tensor viscosity. Based on the four-point Gauss quadrature, a novel tensor viscosity is proposed which overcomes the instability in some degenerate cases of the quadrilateral grid. On the other hand, the proposed viscosity improves the numerical precision due to the exactness of the tensor product Gauss quadrature. A more compact tensor viscosity form is presented by the reduced integration technique, which is identical to the compact tensor viscosity in [29, Wendroff2010]. By analyzing the discrepancy between the two viscosities, we clarify why the tensor viscosity based on four-point quadrature can suppress hourglass motion. Furthermore, we demonstrate that the tensor viscosity proposed in [18, Lipnikov&Shashkov2010] is equivalent to the combination of the compact tensor viscosity and Flanagan-Belytschko type anti-hourglass force in a quadrilateral grid. The numerical experiments illustrate the efficiency of the proposed tensor viscosities. By varying the hourglass control strategies, we verify the propositions of the existing tensor viscosities.