Implementation of one-point quadrature in a finite element CFD code

The use of one-point quadrature integration greatly enhances the computational efficiency of finite element solutions. The procedure is particularly useful when applied to explicit schemes. One-point quadrature integration may, however, produce spurious modes if applied without hourglass stabilization. This paper discusses the application of an efficient hourglass control method to the Galerkin formulation of the conservation of momentum equations. Bilinear shape functions are used for both pressure and velocities, and integration is performed using a one-point quadrature. Hourglass control terms can increase the order of integration up to two-point quadrature. This approach allows a highly vectorized formulation for velocity equations. The proposed hourglass control method has been previously applied to three-dimensional solid elements and its effectiveness and computational efficiency have been established in solid mechanics modelling. Stability problems associated with explicit solvers are overcome by modifying the diffusivity matrix using a balancing tensor. In this work, details of the finite element approach and solution methodology are presented, and the hourglass correction terms corresponding to the diffusion and convection terms in the momentum equation are constructed.