4. Time-step convergence in explicit integration

In an explicit dynamics analysis, the solution is advanced one increment at a time using the central-difference operator. Unlike implicit solvers, which solve a global equilibrium system at every increment and are unconditionally stable for linear problems, the explicit method is conditionally stable: the time increment must remain below a critical value or the solution diverges. For crash evidence work this is not a numerical nicety — it directly determines whether the simulation result can be accepted as evidence at all. A run that completes but contains spurious high-frequency oscillation, hourglassing, or non-physical energy growth is worse than a run that fails, because it can silently corrupt the conclusions drawn from the model.

This section explains where the critical time step comes from, how mass scaling changes it, and how to verify that the chosen time step has actually converged.

Where the critical time step comes from

For a single degree-of-freedom system with mass m and stiffness k, the central-difference scheme is stable when the increment satisfies Δt ≤ 2/ω_n, where ω_n = √(k/m) is the natural frequency. In a finite element mesh, the highest natural frequency is governed by the stiffest, lightest element — typically the smallest element in the model. The standard estimate used by explicit solvers is:

Δt_cr = 2 / ω_max ≈ L_e / c_d

where:
  L_e = characteristic element length (shortest edge or minimum
        distance across the element, depending on element type)
  c_d = dilatational wave speed = sqrt(E / ρ)   (for 1-D bars)
        c_d = sqrt(E(1-ν) / (ρ(1+ν)(1-2ν)))     (for 3-D solids)

The practical consequence is that Δt_cr scales linearly with element size and inversely with wave speed. Halving the smallest element length halves the stable increment and doubles the cost of the run. This is why mesh quality control in crash models is inseparable from time-step control: a handful of tiny elements created during geometry cleanup can dominate the entire run time.

Most explicit codes apply a stability scale factor (often 0.9 by default, sometimes 0.6–0.8 for severe contact or highly nonlinear materials) so that the working increment is:

Δt = SF · Δt_cr,   SF ≈ 0.9

The scale factor absorbs the fact that the analytical estimate ignores contact stiffness, hourglass control, and material nonlinearity, all of which can lower the true stability limit.

Mass scaling: what it does and what it costs

Because Δt_cr ∝ √(m/k), increasing mass in the smallest elements raises the critical time step. Mass scaling adds non-physical mass to selected elements (or to the whole model) to achieve a target increment. Two variants are common:

• Selective mass scaling — only elements whose Δt_cr falls below a target are scaled, leaving the rest of the model untouched. This is the preferred approach for crash models because it limits the added inertia to the small elements that would otherwise control the step.
• Global mass scaling — a uniform factor applied to the entire model. Simple, but it changes the total mass and therefore the inertial response everywhere, which is rarely acceptable for evidence-grade work.

The cost of mass scaling is added inertia. If the scaled mass is a significant fraction of the physical mass in a region that undergoes high acceleration, the model will respond too slowly and under-predict peak deceleration. A common acceptance rule of thumb is to keep the added mass below 1–5% of the total model mass, and to check that the added mass is not concentrated in the components whose response is being evaluated.

Stability limits beyond the element estimate

The element-based estimate is necessary but not sufficient. Several mechanisms can lower the effective stability limit or destabilize an otherwise stable run:

• Contact stiffness — penalty contact introduces very stiff springs at interfaces; if the penalty stiffness is high relative to element stiffness, the contact can control the step. Softened penalty or a contact-aware scale factor is often required.
• Hourglass control — reduced-integration elements need artificial stiffness to suppress hourglass modes; excessive hourglass stiffness raises ω_max and can itself destabilize the run.
• Material nonlinearity — the wave speed depends on the tangent modulus; softening materials reduce c_d, but stiffening (e.g., strain-rate hardening) can raise it locally.
• Element distortion — as elements compress or shear, their characteristic length shrinks, and the solver may reduce Δt mid-run. A run that starts stable can become unstable after severe deformation.

The practical check is to monitor the solver's reported time step and the ratio of current Δt to Δt_cr throughout the run. A sudden drop in Δt usually signals a collapsing element or a contact event, not a modelling error per se, but it must be understood before the result is accepted.

Verifying time-step convergence

A single run at one time step proves nothing about convergence. The acceptance procedure is to run the same model at two or three increments — for example, the baseline Δt, then Δt/2, then Δt/4 — and compare the quantities that matter for the evidence: peak deceleration, intrusion depth, energy balance, and the time history of the key response. Convergence is demonstrated when the difference between successive refinements falls within a stated tolerance (commonly 2–5% for peak acceleration, tighter for displacement).

Two practical points are often missed. First, refining Δt alone does not fix a mesh that is too coarse to resolve the deformation; time-step convergence and mesh convergence must be checked together. Second, the energy balance is a powerful diagnostic: in a stable explicit run, the ratio of artificial energy (hourglass, damping, contact) to total energy should stay small — typically below 5%. A rising artificial-energy fraction is a sign that the step is too large, the hourglass control is too stiff, or the mass scaling is too aggressive.

When these checks are documented, the time-step settings become part of the evidence record rather than an invisible solver default. That is the difference between a simulation that happens to finish and one that can be defended.