3.2.6.12. ExplicitDifferenceStatic

integrator ExplicitDifferenceStatic

Note

  • Uses leap-frog integration with velocities at half time steps.

  • FLAC-style local non-viscous damping (α = 0.59) for pseudo-static analysis.

  • Only mass matrix is required (no tangent matrix assembly).

  • Velocity sign memory prevents chatter near zero velocity.

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

  • Suitable for quasi-static and dynamic problems with adaptive damping.

3.2.6.12.1. Theory

The ExplicitDifferenceStatic method is an explicit difference scheme with FLAC-style local non-viscous damping. The method uses a leap-frog approach where:

  • Velocities are defined at half time steps: \(v_{n+1/2}\)

  • Displacements are defined at full time steps: \(d_n\)

The damping force uses an adaptive FLAC-style approach:

\[F_{damping} = -\alpha \cdot |F_{unbalanced}| \cdot sign(v)\]

where α = 0.59 by default. The method includes velocity sign memory with a deadband (1e-4) to prevent numerical chatter when velocities approach zero.

The central difference equations are:

\[ \begin{align}\begin{aligned}v_{n+1/2} = v_{n-1/2} + \Delta t \cdot a_n\\d_{n+1} = d_n + \Delta t \cdot v_{n+1/2}\end{aligned}\end{align} \]

This integrator is particularly effective for quasi-static analysis and problems where adaptive local damping is beneficial, such as rock mechanics and soil-structure interaction problems.

Example

  1. Tcl Code

integrator ExplicitDifferenceStatic

  1. Python Code

integrator('ExplicitDifferenceStatic')
[Cundall1987]

Cundall, P.A. (1987). “Distinct Element Models of Rock and Soil Structure.” In Analytical and Computational Methods in Engineering Rock Mechanics, 129-163.

Code Developed by: |jaabell|