2. LS-DYNA explicit dynamics fundamentals for crashworthiness

Crashworthiness simulation is a transient, highly nonlinear structural problem: large deformations, self-contact, material failure, and strain-rate effects all occur within tens to hundreds of milliseconds. LS-DYNA is built around an explicit central-difference time integration scheme, which is why it dominates the crash and impact domain. This section explains the four pillars that determine whether a crash model is both physically meaningful and numerically stable: time integration, contact algorithms, hourglass control, and the stable time step.

Explicit Time Integration: The Central-Difference Core

LS-DYNA solves the semi-discrete equation of motion M·a = F_ext − F_int without assembling or factorizing a global stiffness matrix. At each step, nodal accelerations are obtained directly:

a_n = M⁻¹ · (F_ext,n − F_int,n)
v_(n+1/2) = v_(n−1/2) + a_n · Δt
u_(n+1)   = u_n + v_(n+1/2) · Δt

Because M is lumped (diagonal), the inverse is trivial and the update is fully vectorizable. The trade-off is conditional stability: the scheme is stable only if Δt stays below a critical value. This is the single most important operational fact for crash analysts — a crash model with a bad time step will either blow up or run for days.

Two practical consequences follow. First, the analysis is inherently transient; you cannot use it for static preload without special techniques (dynamic relaxation or implicit initialization). Second, energy balance becomes your primary health check: monitor GLSTAT for total, kinetic, internal, and hourglass energy throughout the run.

Contact Algorithms: Penalty vs. Kinematic

Crash models are dominated by contact — a bumper against a barrier, a rail folding onto itself, an occupant against an airbag. LS-DYNA offers several formulations, and choosing the wrong one is a common source of penetration or spurious energy.

The penalty method resists penetration with a spring of stiffness k = SLSFAC · k_segment, where SLSFAC (default 0.1) scales the segment stiffness. Increasing SLSFAC reduces penetration but lowers the stable time step, because the contact stiffness adds to the system's highest frequency. A pragmatic starting point for automotive crash is SLSFAC = 0.1 with SOFT = 1 for soft materials, then tune based on observed penetration depth in d3plot.

Hourglass Control: Taming Reduced Integration

LS-DYNA's default shell and solid elements use one-point (reduced) integration for speed. This makes them susceptible to hourglassing — zero-energy deformation modes that produce a characteristic zig-zag pattern with no corresponding strain energy. Left unchecked, hourglass modes can grow and corrupt results.

Control is applied through *HOURGLASS and the element formulation. The most common choices:

• IHQ = 1 (Flanagan–Belytschko): stiffness-based, default for shells; robust for crash.
• IHQ = 2 (Belytschko–Bindeman): stiffness-based, often better for large deformation.
• IHQ = 4: viscous form; use sparingly, can add artificial damping.
• ELFORM = 2 (Belytschko–Tsay) vs. ELFORM = 16 (fully integrated): fully integrated shells eliminate hourglassing but are ~2–3× slower and prone to shear locking.

The key diagnostic is the hourglass energy reported in GLSTAT and MATSUM. As a rule of thumb, hourglass energy should stay below 5% of peak internal energy; above 10% the solution is suspect. If hourglass energy is high, first refine the mesh, then consider switching to a fully integrated formulation in the affected region.

The Stable Time Step: The Governing Constraint

For a shell or solid element, the critical step is approximated by the Courant condition:

Δt_crit = L_c / c
c = sqrt(E / ρ)          (wave speed, elastic)
Δt = TSSFAC · Δt_crit     (TSSFAC default 0.9)

Here L_c is the characteristic element length, c the sound speed, E Young's modulus, and ρ density. The smallest element in the model dictates the global step. This is why a single tiny element — often a sliver created during meshing — can slow an entire crash run by orders of magnitude.

LS-DYNA reports the initial time step in the d3hsp file and the message file. Typical automotive steel (E = 210 GPa, ρ = 7.85e-9 tonne/mm³) gives c ≈ 5.2 mm/µs; a 5 mm shell element yields Δt_crit ≈ 0.96 µs, and with TSSFAC = 0.9, Δt ≈ 0.86 µs. A 100 ms crash event therefore needs roughly 116,000 steps — a useful sanity check when planning compute budget.

How the Four Pillars Interact

These mechanisms are not independent. Contact penalty stiffness raises the system's highest frequency and lowers Δt. Hourglass control adds stiffness that also affects stability. Mass scaling trades physical fidelity for step size. The SVG below summarizes the coupled relationships and the primary diagnostics for each.

Practical Verification Workflow

Before committing to a long crash run, verify the model in this order:

1. Check the initial time step in d3hsp. If it is far below your expectation, locate the smallest element with LS-PrePost's element quality tools and remesh.
2. Confirm contact definitions cover all expected interfaces. Run a short 5 ms simulation and inspect penetration in d3plot.
3. Monitor hourglass energy in GLSTAT. If it exceeds 5% of internal energy, refine the mesh or change IHQ.
4. Verify energy balance: total energy should be approximately conserved; kinetic energy should peak early and decay as deformation absorbs it.
5. Only then extend to the full event duration and extract crashworthiness metrics (intrusion, deceleration, energy absorption).

Mastering these four fundamentals — the explicit update, contact penalty formulation, hourglass suppression, and the Courant-limited step — is what separates a crash model that runs from one that predicts. Every subsequent optimization in this course depends on a stable, physically credible baseline simulation.