Elimination of hourglass distortions by means of the Lagrangian-Gauss-point-mass method for compressible fluid flows

Staggered Lagrangian schemes on quadrilaterals or hexahedrons with constant stress have been troubled by hourglass distortions for many years. Hourglass viscosities have been often used to suppress hourglass distortions resulting in the nonconservative total energy, even so computations often halt abnormally because of spurious large grid distortions. It is difficult to discern the spurious distortions from the physical in multidimensional problems, so we want to get the scheme eliminating hourglass distortions. This paper gives a new Lagrangian scheme for compressible flows to eliminates the grid hourglass motions by means of the Lagrangian Gauss-point Mass (LGM) method. In the LGM method, pressures are varied in a zone similar to the subzonal pressure method, but there are several important differences between them. Numerical examples show that the LGM method can eliminate the hourglass motions for many problems, and it has the good property of preserving the conservation of total energy nearly in the two-dimensional plane geometry. Because of the area-weighted scheme, it cannot preserve the total energy but improves it to a great extent. The procedure of the LGM method is simple to be implemented into the existing 2D finite element codes on quadrilaterals, and it can be extended to the 3D finite element codes on hexahedrons without theoretical difficulties.