2. PAM-CRASH solver architecture for explicit crash analysis

PAM-CRASH is an explicit finite element solver whose architecture is organized around three tightly coupled subsystems: the time integration scheme that advances the solution, the contact algorithms that enforce non-penetration between bodies, and the material models that convert strain history into stress. For pedestrian protection work, all three must be tuned together — a correct head impact response depends on stable explicit stepping, a contact definition that captures the scalp–hood–inner-panel load path, and a strain-rate-sensitive material law for the hood and bumper fascia.

2.1 Explicit Time Integration and the Stability Limit

PAM-CRASH uses the central difference method, an explicit scheme in which displacements at time t+Δt are computed from the equilibrium at time t. No stiffness matrix is assembled or inverted, so each increment is cheap, but the scheme is only conditionally stable. The critical step is governed by the Courant–Friedrichs–Lewy condition:

Δt_crit = 2 / ω_max  ≈  L_min / c
c = sqrt(E / ρ)          (wave speed in the stiffest element)

Here L_min is the smallest element characteristic length, E the Young's modulus, and ρ the density. In a typical full-vehicle pedestrian model, the smallest shell elements sit on the hood inner panel or the bumper beam, with L_min around 3–5 mm. For steel (E ≈ 210 GPa, ρ ≈ 7.85 × 10⁻⁹ t/mm³), the wave speed is roughly 5.2 mm/µs, giving Δt_crit ≈ 1.2 µs. PAM-CRASH applies a safety factor (default 0.9) and reports the actual step in the output listing.

Two practical consequences follow. First, mass scaling is the standard remedy when a handful of tiny elements force an unnecessarily small step. A targeted scaling — for example, raising the density of elements below 4 mm so that the global step rises from 0.8 µs to 1.2 µs — is acceptable if the added kinetic energy stays below about 1 % of the internal energy. Second, the analyst must monitor the added mass and time step blocks in the listing; a sudden drop in Δt mid-run usually signals element distortion, not a modeling error in the contact definition.

2.2 Contact Algorithms for Pedestrian Impact

Pedestrian loading is dominated by surface-to-surface interaction: the headform against the hood, the upper legform against the bumper and grille, and the lower legform against the bumper beam. PAM-CRASH offers penalty-based contact, where a restoring force proportional to penetration depth resists interpenetration. The penalty stiffness is computed automatically from the element stiffness, but the user controls the search and the friction.

Three contact types matter here:

• Node-to-surface (NTS) — used when a rigid or coarse body (the headform) strikes a deformable shell mesh (the hood). It is inexpensive and robust for the head impactor.
• Surface-to-surface (STS) — preferred for the legform–bumper interface, where both bodies deform and a distributed pressure is needed.
• Self-contact — required once the hood inner panel folds onto itself or the bumper fascia contacts the beam during deep intrusion.

Friction is defined with a Coulomb coefficient; for painted steel against the headform skin, values between 0.2 and 0.3 are typical. The contact thickness must be set deliberately: if the shell thickness offset is ignored, the headform will appear to penetrate the hood by half the shell thickness, inflating the HIC value. A common check is to request the CONTACT FORCE output and confirm that the peak force occurs at the expected instant of first contact.

2.3 Material Models for Pedestrian-Relevant Components

Pedestrian protection involves three material families, each with a distinct constitutive requirement.

Steel grades used in hoods are strain-rate sensitive. The Johnson–Cook flow stress is:

σ = (A + B ε^n) · (1 + C ln(ε̇/ε̇₀))

where A is the quasi-static yield stress, B and n describe hardening, and C scales the rate effect. For a typical bake-hardened steel, A ≈ 180 MPa, B ≈ 400 MPa, n ≈ 0.5, and C ≈ 0.015. At the strain rates reached during a 40 km/h head impact (10²–10³ s⁻¹), the rate term raises the flow stress by 10–20 %, which directly reduces hood deformation and therefore lowers HIC.

For the headform itself, a linear elastic model with E ≈ 1.5 GPa and a density of 1.4 × 10⁻⁹ t/mm³ reproduces the mass and stiffness of the EEVC impactor. The aluminum honeycomb core is often represented by a crushable foam law with a plateau stress of about 0.5 MPa.

2.4 How the Three Subsystems Interact

The architecture is not a sequence of independent modules; it is a feedback loop. The time step determines how often contact is searched and how often stress is updated. A coarse step can miss the first contact increment, producing a spurious force spike. Conversely, an overly stiff penalty can drive the local element distortion up, forcing the global step down and stalling the run. Material softening after yield reduces the wave speed, which actually relaxes the stability limit — but element deletion removes stiffness entirely and can cause an abrupt step change.

The diagram above shows the coupling explicitly. The time integration block feeds the contact search; the contact forces feed back into the element distortion, which in turn modifies the stable step. Material softening and failure sit inside that loop. When a run terminates unexpectedly, the correct diagnostic order is: check the DT history, then the contact initialization summary, then the material failure flags — not the other way around.

2.5 A Reproducible Configuration Check

Before launching a pedestrian head impact run, the following minimal set of inputs should be confirmed. This is not a full deck, but it captures the parameters that most often cause a run to fail or produce a non-physical HIC.

# Time integration
DT_INIT   = 1.0e-6      # s, initial step
DT_MIN    = 1.0e-8      # s, abort threshold
MASS_SCALING = ON       # target DT = 1.2e-6 s, added mass < 1%

# Contact
CONTACT 1 = NTS
  SLAVE  = HEADFORM_NODES
  MASTER = HOOD_OUTER_SHELL
  THICKNESS = ON        # respect shell offset
  FRICTION  = 0.25

# Material
MAT 103, ID=1001        # hood outer, Johnson-Cook
  A = 180 MPa, B = 400 MPa, n = 0.5, C = 0.015
  FAILURE_STRAIN = 0.35

After the run, three checks confirm the configuration is sound. First, the energy balance: total energy should stay within 5 % of the initial kinetic energy, and hourglass energy should remain below 5 % of internal energy. Second, the contact force history should show a single clean peak for a single impact, not a series of spikes. Third, the headform acceleration should be smooth enough to integrate into HIC without numerical noise; if the acceleration curve is jagged, the contact penalty is too stiff or the time step is too large.

These three subsystems — integration, contact, and material — form the operational core of PAM-CRASH for pedestrian protection. Understanding how they interact is what separates a run that converges from one that produces a plausible-looking but physically wrong HIC.