2. PAM-CRASH solver architecture for crashworthiness

PAM-CRASH is an explicit finite element solver whose architecture is organized around three tightly coupled subsystems: the explicit time-integration engine, the contact and impact interface, and the material model library. For full vehicle frontal impact simulation, these three subsystems determine whether the model remains stable, whether load paths are transferred correctly through the body-in-white, and whether the predicted deformation matches physical crash behavior. Understanding how they interact is more important than memorizing individual keywords, because most frontal impact failures—negative volume errors, contact penetration, or non-physical material response—originate at the boundary between these subsystems.

Explicit Time Integration and the Stability Limit

PAM-CRASH advances the solution using the central difference method. The semi-discrete equation of motion is solved for nodal accelerations, which are integrated to velocity and displacement:

M · a(t) = F_ext(t) − F_int(t)
a(t) = M⁻¹ · (F_ext − F_int)
v(t+Δt/2) = v(t−Δt/2) + a(t) · Δt
u(t+Δt) = u(t) + v(t+Δt/2) · Δt

Because the method is conditionally stable, the time step must satisfy the Courant–Friedrichs–Lewy condition. For shell-dominated vehicle structures, the critical step is governed by the smallest element and the local wave speed:

Δt_crit = L_min / c,   c = sqrt(E / ρ)

where L_min is the characteristic element length, E is Young's modulus, and ρ is density. In a typical frontal impact model, a 5 mm shell element in steel (E ≈ 210 GPa, ρ ≈ 7.85 × 10⁻⁹ t/mm³) yields a wave speed near 5170 mm/ms and a critical step around 9.7 × 10⁻⁴ ms. PAM-CRASH applies a scale factor (commonly 0.9) and reports the actual step in the output listing. If a single distorted element drives the step down, total runtime increases proportionally; mass scaling is the standard remedy, but added mass must be monitored so that it does not exceed roughly 1–2% of the total model mass in the regions of interest.

Two integration controls deserve attention in frontal impact work. First, the termination time must cover the full crash pulse—typically 100–150 ms for a 50 km/h full frontal barrier test. Second, output intervals should be fine enough to capture peak deceleration (often every 0.1–0.5 ms) without generating unmanageable result files. The solver writes a status file at each output interval; monitoring kinetic energy, internal energy, and the energy ratio during the run is the primary check that the explicit solution remains physically meaningful.

Contact Algorithms for Structural Interfaces

Frontal impact simulation is dominated by contact: the barrier against the bumper and front rails, the tire against the ground, the engine block against the firewall, and self-contact of folded sheet metal. PAM-CRASH provides penalty-based contact, where a penetration produces a restoring force proportional to the penetration depth and the interface stiffness. The interface stiffness is estimated from element properties:

k = f · (K · A²) / V

with K the bulk modulus, A the segment area, V the element volume, and f a scale factor. Too low a stiffness allows visible penetration; too high a stiffness drives the stable time step down or causes contact chatter.

For vehicle crashworthiness, the relevant contact types are:

• Node-to-surface (NTS) — used for the barrier and rigid walls, where a rigid master surface contacts deformable slave nodes.
• Surface-to-surface (STS) — preferred for large self-contact areas such as the folded front rails and the hood, because it distributes contact forces over segments and reduces penetration.
• Single surface — automatically detects self-contact within a defined part set; essential for sheet metal that folds onto itself.
• Tied contact — used at spot welds and adhesive joints, where nodes are constrained to a master surface to transfer load without sliding.

Friction is defined by a Coulomb coefficient, typically 0.1–0.3 for steel-to-steel contact. A practical check after a frontal run is to plot contact energy and the number of active contacts; a sudden spike in contact energy usually indicates a penetrations or an incorrectly defined master/slave pair.

Material Model Library for Frontal Impact

The material library in PAM-CRASH is organized by constitutive behavior, and frontal impact work relies on a small subset of models. The most common is the elastoplastic model with isotropic hardening, often referenced as MAT 1 or MAT 2 in the PAM-CRASH material cards. It requires density, Young's modulus, Poisson's ratio, yield stress, and a hardening curve. For high-strength steel rails, a strain-rate-dependent formulation (such as MAT 103 or the Johnson–Cook model) is preferred because yield stress increases significantly at the strain rates encountered in a crash—typically 10² to 10³ s⁻¹.

Spot welds are represented either by tied contact or by dedicated weld elements with a failure criterion based on normal and shear force. Adhesive joints use cohesive elements with a traction–separation law. For the bumper foam or energy-absorbing inserts, a crushable foam model with a yield surface defined by hydrostatic pressure and deviatoric stress is appropriate. Composite hoods or battery enclosures require progressive damage models with fiber and matrix failure criteria.

A practical material card for a DP780 front rail might specify:

MAT 103 (elastic-plastic, strain-rate dependent)
  ρ  = 7.85e-9 t/mm³
  E  = 210000 MPa
  ν  = 0.30
  σ_y = 550 MPa (quasi-static)
  Hardening curve: ε_p = 0.00 → 550 MPa
                   ε_p = 0.05 → 720 MPa
                   ε_p = 0.15 → 850 MPa
  Strain-rate factor: 1.0 at 0.001 s⁻¹
                      1.15 at 1 s⁻¹
                      1.30 at 100 s⁻¹

These values are illustrative and must be replaced by coupon-test data for the actual material grade. The key point is that the hardening curve and rate factor directly control the folding pattern of the front rail, which in turn determines the deceleration pulse transmitted to the occupant cell.

Verifying Solver Behavior in a Frontal Run

After a frontal impact run, three checks establish whether the solver architecture is functioning correctly. First, the energy ratio—total energy divided by initial kinetic energy—should remain close to 1.0; a drift above 5% indicates hourglassing or contact energy leakage. Second, the time step history should be stable; a sudden drop suggests element distortion or a contact stiffness problem. Third, the contact penetration plot should show no penetration exceeding half the shell thickness. If penetration is excessive, increase the contact stiffness scale factor or switch from NTS to STS contact. If the time step collapses, inspect the smallest elements and apply selective mass scaling rather than global scaling, which would artificially alter the inertia of the occupant cell.

The solver architecture is therefore not a set of independent options but a coupled system: the time step depends on element size and material stiffness, contact stiffness depends on element properties, and material response determines how contact forces redistribute through the structure. Tuning one subsystem without monitoring the others is the most common source of non-physical frontal impact predictions.