Read the latest blogs from BlackRock Engineering
When running dynamic simulations in Finite Element Analysis (FEA), the stability and accuracy of your results often hinge on one critical factor: the timestep size. Among the various methods for determining appropriate timesteps, understanding Rayleigh's time offers powerful insights into numerical stability. This article explores what Rayleigh's time is, why it matters, and how to leverage it for successful FEA simulations.
Rayleigh's time (sometimes called Rayleigh's damping time) refers to a characteristic time constant derived from the Rayleigh damping model used in structural dynamics. In the context of FEA, it represents the time required for a system's energy to be reduced to 1/e (approximately 36.8%) of its initial value due to damping effects.
The concept originates from Lord Rayleigh's work on the theory of sound and vibrations, where he introduced a damping model that combines mass-proportional and stiffness-proportional damping components:
C = αM + βK
Where:
In explicit dynamic FEA solvers, the timestep size directly affects both computational efficiency and solution stability. If your timestep is too large, the solution can become numerically unstable and "blow up." If too small, you waste valuable computational resources.
This is where Rayleigh's time becomes crucial. The critical timestep for stability in explicit dynamics is directly related to Rayleigh's time through the following relationship:
Δt ≤ 2/ωmax
Where ωmax is the maximum eigenfrequency of the system. For systems with Rayleigh damping, this can be expressed as:
Δt ≤ 2/(ωmax + α/2 + βωmax²/2)
When Rayleigh damping is present, the critical timestep becomes even more restrictive than in undamped systems, particularly when stiffness-proportional damping (β) is significant.
The fundamental reason we care about Rayleigh's time in FEA is to maintain numerical stability. Explicit time integration schemes (like the central difference method commonly used in FEA) have a conditional stability limit. Exceeding this limit by using too large a timestep leads to error amplification and eventual solution divergence.
Understanding Rayleigh's time helps engineers strike the optimal balance between:
Different elements in your FEA model have different stable timestep requirements. Rayleigh's time considerations allow you to:
For most FEA practitioners, the following approach works well when determining appropriate timesteps:
Estimate the highest eigenfrequency (ωmax) in your system, often related to the smallest element size.
Determine your Rayleigh damping coefficients (α and β) based on your material properties and frequency range of interest.
Calculate the critical timestep using the formula:
Δtcritical = 2/(ωmax + α/2 + βωmax²/2)
Apply a safety factor (typically 0.8-0.9) to ensure stability:
Δt = safety_factor × Δtcritical
Many commercial FEA packages will automatically calculate this for you, but understanding the underlying principles helps you make better modeling decisions.
In complex models, very high-frequency modes may exist that severely restrict timestep size. These often contribute little to the overall structural response but can dramatically increase computational cost.
Solution: Consider:
Inaccurate estimates of Rayleigh damping parameters can lead to either overly conservative timesteps or numerical instability.
Solution:
Material nonlinearities can change the effective stiffness during simulation, potentially invalidating your initial timestep calculation.
Solution:
Different FEA packages handle Rayleigh's time and timestep calculations differently:
Abaqus:
Ansys:
LS-DYNA:
In some applications, damping properties may change during the simulation. This can be particularly relevant in:
When damping properties change, Rayleigh's time also changes, potentially requiring timestep adjustments during simulation.
When coupling mechanical analysis with thermal, electromagnetic, or fluid dynamics analysis, additional timestep constraints may apply. Each physics domain has its own characteristic time scales, and coupling between them introduces new considerations for stability.
Rayleigh's time provides a robust framework for understanding how damping affects timestep requirements in FEA. By properly accounting for it in your simulations, you can achieve the optimal balance between computational efficiency and numerical stability.
For practicing engineers, this translates to:
As computational methods continue to advance, the fundamental principles of Rayleigh's time remain essential knowledge for anyone working with dynamic finite element analysis.