3.2.6.11. ExplicitBathe

integrator ExplicitBathe $p $compute_critical_timestep?

Note

  • The method is second-order accurate and explicit.

  • Only mass matrix is assembled on RHS (no tangent matrix required).

  • Critical time step is approximately 2× larger than central difference method.

  • For stability: \(\Delta t \leq \frac{2}{\omega_{max}}\)

  • Typical p values: 0.54-0.95 for optimal numerical damping.

3.2.6.11.1. Theory

The ExplicitBathe method is a two-step explicit integration scheme with built-in numerical damping. The method performs two sub-steps per time step:

  1. First step: \(t \rightarrow t + p\Delta t\)

  2. Second step: \(t + p\Delta t \rightarrow t + \Delta t\)

The integration coefficients are computed from the damping parameter p:

\[ \begin{align}\begin{aligned}q_1 = \frac{1 - 2p}{2p(1-p)}\\q_2 = 0.5 - p \cdot q_1\\q_0 = -q_1 - q_2 + 0.5\end{aligned}\end{align} \]

The method offers enhanced stability compared to standard central difference, with a critical time step approximately twice as large. The parameter p controls numerical damping, with higher values providing more damping but reduced accuracy.

Example

  1. Tcl Code

integrator ExplicitBathe 0.54

  1. Python Code

integrator('ExplicitBathe', 0.54)
[Noh2013]

Noh, G., & Bathe, K.J. (2013). “An explicit time integration scheme for the analysis of wave propagations.” Computers & Structures, 129, 178-193.

Code Developed by: |jaabell|