2. PAM-CRASH explicit solver fundamentals

PAM-CRASH is an explicit finite element solver widely used in automotive crashworthiness, and battery pack impact simulation is one of its most demanding applications. A traction battery enclosure combines thin stamped steel or aluminum panels, bolted joints, welded seams, foam or mastic fillers, and dense cell modules. During a side pole impact or a bottom intrusion event, the structure undergoes large deformation, self-contact, and localized buckling within tens of milliseconds. Understanding how the explicit solver advances the solution in time, how it detects and enforces contact, and how it suppresses spurious zero-energy deformation modes is therefore essential before any credible battery safety result can be produced.

Explicit Time Integration and the Stability Limit

PAM-CRASH advances the equations of motion with an explicit central-difference scheme. For each degree of freedom, the acceleration at time step n is obtained directly from the assembled force vector:

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

Because the mass matrix M is lumped (diagonal), no system of equations is solved and no global stiffness matrix is factorized. Each step is cheap, but the step size is bounded by the Courant–Friedrichs–Lewy condition. For a shell or solid element, the critical step is approximately:

dt_crit = 2 / omega_max  ~  L_e / c
c = sqrt(E / rho)          (bar wave speed)
dt = TSSFAC * dt_crit      (TSSFAC typically 0.9)

Here L_e is the characteristic element length, E the Young's modulus, and ρ the density. In a battery pack model, the smallest elements usually sit on weld nuggets, bolt shanks, or thin cooling-channel walls. A single 1.5 mm element in aluminum (E ≈ 70 GPa, ρ ≈ 2700 kg/m³, c ≈ 5090 m/s) gives dt_crit ≈ 0.29 µs. If the rest of the model is meshed at 5 mm, the global step is dragged down by that one element, and a 50 ms impact would require roughly 170,000 steps. Two standard remedies are mass scaling and selective mesh refinement. Mass scaling adds non-physical mass to the smallest elements so that their stable step matches the target; the added mass must be monitored, because it changes inertia and can artificially reduce intrusion. A practical rule is to keep total added mass below 1–2 % of the model mass and to verify that the kinetic energy of scaled regions remains negligible.

Contact Algorithms for Battery Enclosure Impact

Contact in PAM-CRASH is penalty-based. When a slave node penetrates a master segment, a restoring force proportional to the penetration depth is applied:

F_contact = k_pen * delta
k_pen = alpha * K_master * A_master / V_master

The penalty stiffness k_pen is scaled by a factor α (often 0.1–1.0) relative to the master segment stiffness. Too small an α allows visible penetration; too large an α drives the contact frequency up and forces a smaller stable time step. PAM-CRASH offers several contact types, and choosing the right one matters more than tuning the penalty:

• Node-to-surface (NTS) — the classic option for shell-to-shell and shell-to-solid contact. Use it for the enclosure outer panel against a rigid pole or barrier.
• Surface-to-surface (STS) — better force distribution for large flat or curved surfaces, such as the battery tray bottom against a ground obstacle.
• Single surface — detects self-contact within one part set. Essential for folded or crushed cell modules and for the enclosure walls that buckle inward.
• Tied contact — used for spot welds, adhesive beads, and bolted flanges where nodes must remain connected until a failure criterion is reached.

For a battery pack, a typical contact deck defines a single-surface contact over all deformable parts, an NTS contact between the pack and the rigid impactor, and tied contacts at weld and adhesive locations. Friction is usually Coulomb with a static coefficient around 0.2–0.3 for painted steel and 0.4–0.6 for aluminum-to-aluminum. Contact thickness must be set explicitly for shells, because shell elements have no physical thickness in the contact search; a value of 0.5–1.0 times the actual sheet thickness is common.

Hourglass Control

Reduced-integration shell and solid elements are the default in crash models because they are fast, but they possess zero-energy deformation modes called hourglass modes. These modes produce no strain at the integration point and therefore no resisting force, yet they can grow and pollute the displacement field. PAM-CRASH provides several hourglass control formulations:

The hourglass coefficient HQ controls the magnitude of the artificial resistance. Values between 0.03 and 0.10 are typical. Too low and hourglass energy grows; too high and the structure becomes artificially stiff, reducing intrusion and masking real failure. The standard check is to compare hourglass energy with internal energy: for a well-controlled model, hourglass energy should stay below 5 % of internal energy throughout the event. PAM-CRASH reports this in the energy summary, and it should be plotted alongside kinetic and internal energy for every battery impact run.

Putting the Three Together: A Practical Battery Pack Run

Consider a 60 kWh pack with an aluminum tray, a steel lid, and four cell modules, subjected to a 25 km/h side pole impact. A workable setup proceeds as follows. Mesh the tray and lid at 4–5 mm with reduced-integration shells, refine the pole contact region to 2 mm, and keep weld and bolt details at 1.5–2 mm. Set TSSFAC to 0.9 and apply mass scaling only to elements below 2 mm, targeting a 1 µs global step. Define a single-surface contact over all deformable parts, an NTS contact between the pole and the tray, and tied contacts at the 180 spot welds and the adhesive bead around the lid flange. Activate hourglass Type 4 with HQ = 0.05.

After the run, inspect three things before looking at intrusion. First, the energy plot: internal energy should rise as kinetic energy falls, hourglass energy should remain under 5 % of internal energy, and total energy should be nearly constant. Second, the contact penetration plot: maximum penetration should stay below the shell contact thickness. Third, the added-mass report: total added mass should be under 2 % of the model mass. If hourglass energy exceeds 10 %, refine the mesh in the affected region or raise HQ slightly; if penetration is excessive, increase the penalty scale or contact thickness; if the step collapses, find the smallest element and either remesh or scale its mass. Only when these checks pass is the predicted intrusion and cell deformation meaningful for a safety assessment.