Hourglass Subcycling Approach for Explicit Time Integration of Finite Elements

Abstract

Explicit methods are widely used in finite element analysis as efficient ways to solve differential equations. The efficiency of explicit methods relies on the economical evaluation of internal forces at each time step. The greatest efficiency can be provided by one-point quadrature. However, instability arises because of the shortcomings in the use of one-point quadrature. The instability is called hourglass mode, or spurious singular mode. An effective method to control the instability is to add “hourglass stiffness” to an element integrated by one-point quadrature. Explicit methods often require a very small time step to ensure stability. Thus, for complex problem with refined meshes, a very large number of timesteps will be required to complete the analysis. Minimizing the number of operations per time step can provide significant improvement on efficiency of the methods. Since hourglass terms typically require more computational operations than one-point quadrature terms, we are very interested in reducing the number of operations on hourglass control. In addition, considerable approximation is involved with hourglass control, and hence overall accuracy may not be seriously affected by relaxing the precision of the temporal integration of the hourglass force. Consequently, there is a possibility of trading some accuracy of the hourglass control for computational efficiency. A subcycling approach is applied to the hourglass portion of explicit methods. Namely, instead of updating hourglass forces every time step, we update hourglass forces every two steps. The proposed approach is examined with the use of mass-spring models. The applicability to more complex models is demonstrated on a 3-D model with the subcycling approach implemented into an explicit finite element code. Efficiency, stability and accuracy are discussed as important issues of the proposed approach. The mass-spring models and finite element implementation show that a beating instability can be introduced by the subcycling approach, and additional restriction is placed on the stable time step for the central difference operator. However, sufficient damping can restore the usual stability conditions. Thus, the proposed subcycling approach is seen to be highly advantageous where damping can be used, and it can cut computation time by 30% or more without significantly affecting the overall accuracy of the solution.