2. PAM-CRASH solver architecture for multi-impact simulation

Multi-impact vehicle design requires a solver that can carry a structure through several sequential collision events without losing the physical state accumulated in earlier impacts. PAM-CRASH is built around an explicit finite element core, and understanding how that core integrates time, resolves contact, suppresses hourglassing, and advances the time step is essential before any multi-event load case can be trusted. This section examines the four architectural pillars that govern sequential impact simulation: explicit time integration, contact algorithms, hourglass control, and time-step management.

Explicit Time Integration as the Backbone

PAM-CRASH advances the solution with the central difference method. For each node, acceleration is obtained from the assembled force balance, then velocity and displacement are updated explicitly:

a_n = M^-1 (F_ext - F_int)
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, no system of equations is solved and no global stiffness matrix is inverted. Each increment is cheap, which is what makes millions of cycles feasible for crash events lasting 50–150 ms. The trade-off is conditional stability: the scheme is stable only while the time step stays below the Courant limit. In a multi-impact sequence, this means the solver must re-evaluate stability after every event, because elements that were crushed, folded, or deleted in impact one change the critical step for impact two.

Internal forces are computed from element stresses, which are updated from the strain rate. PAM-CRASH supports a wide material library — elastoplastic metals with strain-rate sensitivity (Cowper–Symonds, Johnson–Cook), composites with progressive failure, foams, and spot-weld or adhesive connections. For sequential impacts, the material state (equivalent plastic strain, damage, failure flags) must persist across events; restart files written at the end of each event carry this state forward.

Contact Algorithms

Contact is where multi-impact models most often fail. PAM-CRASH offers penalty-based and kinematic contact, with the penalty method being the default for vehicle crashworthiness. A penetration between a slave node and a master segment generates a restoring force proportional to the penetration depth and a contact stiffness:

F_contact = k_pen * penetration
k_pen = alpha * K_element * A / V

where alpha is a scale factor (typically 0.1–1.0), K_element is the element bulk stiffness, and A/V is a characteristic segment ratio. Too low a penalty stiffness allows visible interpenetration; too high a value drives the stable time step down and can trigger instabilities.

For sequential impacts, contact definitions must be reviewed event by event. A barrier that contacts the front rail in event one may be irrelevant in event two, while a new contact pair (e.g., the deformed front structure against the occupant compartment) becomes active. PAM-CRASH contact types relevant here include:

• Node-to-surface — efficient for localized contact such as a pole or barrier edge.
• Surface-to-surface — preferred for large, evolving contact patches between vehicle subsystems.
• Self-contact — essential when a crushed structure folds onto itself, common after a first impact.
• Tied and tied-with-failure — used for spot welds and adhesive joints that may rupture in a later event.

Friction is applied through a Coulomb model with a static and dynamic coefficient. In multi-impact sequences, friction coefficients should be checked against the surface condition of each event (painted metal, bare metal, or contact with a deformable barrier face).

Hourglass Control

Reduced-integration shell and solid elements are standard in crash models because they are fast, but they are susceptible to zero-energy hourglass modes — non-physical deformations that produce no strain energy. PAM-CRASH provides several hourglass control formulations, including viscous (Flanagan–Belytschko), stiffness-based, and combined forms. The control force is proportional to a coefficient QH and the element's characteristic stiffness:

F_hg = QH * k_char * (hourglass mode amplitude)

Typical QH values range from 0.03 to 0.10 for shells. The diagnostic to watch is the hourglass energy reported in the energy balance: it should remain below roughly 5% of the total internal energy. In a multi-impact run, hourglass energy can accumulate silently across events, so the energy summary must be inspected after each impact, not only at the end. If hourglass energy grows after event two, the likely causes are excessive QH reduction, a poorly refined mesh in a fold region, or a contact definition that lets elements pass through each other.

Time-Step Management Across Sequential Events

The stable time step is governed by the smallest element in the model:

dt_crit = L_min / c
c = sqrt(E / rho)

where L_min is the smallest element characteristic length and c is the wave speed. PAM-CRASH applies a scale factor (default 0.9) for safety. Two mechanisms keep the step manageable: mass scaling and selective mass scaling. Mass scaling adds non-physical mass to small elements to raise their stable step; the added mass must be monitored so it does not exceed about 1–2% of the total model mass, otherwise inertia effects corrupt the physics.

For sequential impacts, the critical step changes between events. After a severe first impact, elements may be compressed to a fraction of their original length, reducing L_min and forcing a smaller step. Conversely, element deletion of fully failed elements can remove the smallest elements and allow a larger step. The practical workflow is to run each event as a separate restart, inspect the reported time step and mass-scaling ratio at the start of the next event, and adjust scaling or mesh if the step has collapsed.

Sequential Event Workflow in Practice

A representative multi-impact study — a 40% offset frontal impact followed by a side pole impact — illustrates how the architecture interacts. Event one runs from 0 to 120 ms with the front structure absorbing energy. At 120 ms, a restart file is written containing nodal coordinates, element stresses, plastic strains, damage variables, and contact state. Event two is initialized from that restart, with the pole barrier repositioned and a new contact pair defined between the pole and the deformed side structure. The time step at the start of event two is typically smaller than at the start of event one because the crushed front elements have shortened. Mass scaling may need to be re-tuned, and hourglass energy must be re-checked because the fold patterns from event one can excite new hourglass modes.

The energy balance is the single most useful diagnostic across the whole sequence. Kinetic energy, internal energy, contact energy, hourglass energy, and external work should sum to a near-constant total. A drift in this balance after event two almost always points to a contact or hourglass problem introduced by the restart, not to the integration scheme itself.

The architecture is therefore not a fixed pipeline but a set of coupled decisions. Explicit integration sets the cost per increment; contact determines whether forces are transferred correctly between bodies; hourglass control protects the reduced-integration elements from non-physical modes; and time-step management keeps the whole sequence stable. In a multi-impact study, each of these must be re-validated at every restart, because the state carried forward from one event changes the conditions under which the next event runs.