2. PAM-CRASH solver architecture for full vehicle crash

Full vehicle crash simulation in PAM-CRASH is built on an explicit finite element core. Understanding how that core advances time, resolves contact, suppresses spurious deformation modes, and manages the stable time step is what separates a model that runs from a model that produces trustworthy sign-off data. This section walks through the four architectural pillars that dominate runtime behavior and result quality: explicit time integration, contact algorithms, hourglass control, and mass scaling.

Explicit Time Integration

PAM-CRASH advances the solution with the central difference scheme. For each node, acceleration at time step n is obtained from the assembled force balance:

a_n = M^-1 * (F_ext,n - F_int,n)
v_(n+1/2) = v_(n-1/2) + a_n * dt
x_(n+1) = x_n + v_(n+1/2) * dt

Here M is the lumped mass matrix, F_ext is the external load vector (barrier, gravity, belt forces), and F_int is the internal force vector from element stresses and contact. Because M is diagonal, no matrix inversion is required — this is the reason explicit solvers scale so well on large vehicle models with millions of elements.

The scheme is conditionally stable. The critical time step follows the Courant–Friedrichs–Lewy (CFL) condition:

dt_crit = 2 / omega_max  ≈  L_min / c
c = sqrt(E / rho)

where L_min is the smallest element characteristic length, c is the acoustic wave speed, E is Young's modulus, and rho is density. In a typical steel body-in-white, c is roughly 5000 m/s. A 5 mm shell element therefore yields dt_crit near 1 microsecond. PAM-CRASH applies a safety factor (default 0.9) and reports the actual step in the output listing. If the step collapses unexpectedly, the cause is almost always a single distorted or tiny element, not the global mesh.

Contact Algorithms

Contact in a full vehicle model is dominated by shell-to-shell interaction: the barrier against the bumper, the occupant against the airbag, and structure against itself during folding. PAM-CRASH provides penalty-based and kinematic contact treatments. The penalty method is the workhorse because it conserves momentum and handles arbitrary self-contact.

For a node-to-segment penalty contact, the penetration g is measured along the segment normal. A contact force is applied:

F_contact = k_penalty * g
k_penalty = alpha * m_slave * c^2 / L^2

The stiffness scale factor alpha (often 0.1–1.0) controls how stiff the penalty spring is relative to the slave mass and element size. Too low and nodes tunnel through the barrier; too high and the contact stiffness drives the stable time step down, increasing cost. PAM-CRASH exposes this through contact type selection and the penalty scale parameter.

Three contact families matter most in vehicle crash:

• Single surface (self-contact): used for structure folding onto itself, e.g., the front rail crushing. No master/slave distinction; all segments are candidates.
• Surface-to-surface: used for barrier-to-vehicle and occupant-to-interior, where a defined master surface is more efficient.
• Tied and tied-with-failure: used for spot welds and adhesive joints, where nodes are constrained until a failure criterion releases them.

Friction is applied through a Coulomb model with a coefficient typically between 0.1 and 0.3 for painted steel. Contact thickness offsets must be set so that shell mid-surfaces do not falsely penetrate; a common value is half the physical sheet thickness.

Hourglass Control

Reduced-integration shell and solid elements are computationally efficient but admit zero-energy deformation modes called hourglass modes. These are non-physical and, if uncontrolled, grow until the mesh becomes a checkerboard pattern and the solution diverges. PAM-CRASH offers several hourglass control formulations:

The hourglass coefficient (often called QM or the hourglass control factor) is typically set between 0.03 and 0.10. The energy absorbed by hourglass control is reported in the energy balance. A practical acceptance rule: hourglass energy should stay below 5% of total internal energy. If it exceeds 10%, the mesh is too coarse in a region undergoing bending, or the coefficient is too low.

Mass Scaling Strategies

Mass scaling increases the density of small or stiff elements so that their local critical time step rises toward the global target. It is the primary tool for controlling runtime without remeshing. PAM-CRASH supports two modes:

• Fixed time step scaling: the solver computes the mass needed so every element meets a target dt_target. Simple and predictable.
• Selective scaling: only elements below a threshold are scaled, limiting added mass to problem regions.

The added mass fraction is reported as DM/M. For full vehicle sign-off, keep total added mass below 1–2% of the model mass, and never allow it to concentrate in a region whose inertia matters — the engine block, battery pack, or occupant. A useful workflow is to set dt_target to the 1st-percentile element step, scale selectively, then verify that the kinetic energy and peak intrusion of the scaled model match an unscaled reference run within a few percent.

Putting the Architecture Together

These four mechanisms are coupled. Contact stiffness and hourglass stiffness both enter the element-level force calculation and therefore influence the stable time step. Mass scaling changes the inertia term in the equation of motion. A disciplined setup sequence is:

1. Run a 10-cycle initialization and read the reported dt_crit and smallest element.
2. Fix any element whose step is more than 3× below the median — usually a sliver or a collapsed shell.
3. Set hourglass control to physical stabilization with coefficient 0.05 and confirm HG energy is small in the first 5 ms.
4. Apply selective mass scaling to a target step, then check DM/M and the region distribution.
5. Confirm the energy balance closes: kinetic + internal + hourglass + contact + damping ≈ initial kinetic energy plus external work.

When the energy balance closes and hourglass and added mass stay within the limits above, the explicit solution is behaving as intended. Deviations in any of these quantities are the earliest and most reliable indicators that a full vehicle model is not yet ready for sign-off.