2. PAM-CRASH solver architecture for crashworthiness

Rear impact and whiplash simulation in PAM-CRASH is built on a solver architecture that differs fundamentally from implicit FEA codes. Understanding how the explicit solver advances time, how contact is detected and enforced, and how material models respond to high strain rates is essential before any rear-impact model can be trusted. This section examines the three pillars of that architecture and how they interact in a typical rear-end collision simulation.

Explicit Time Integration: The Central Difference Core

PAM-CRASH uses the explicit central difference method to advance the solution in time. The equations of motion are solved without assembling a global stiffness matrix or performing matrix inversion. At each time step, nodal accelerations are computed from the net force (external minus internal), then integrated to obtain velocities and displacements:

a_n = M⁻¹ (F_ext - F_int)
v_(n+1/2) = v_(n-1/2) + a_n · Δt
d_(n+1) = d_n + v_(n+1/2) · Δt

Here M is the lumped mass matrix, F_int is the internal force vector from element stresses, and F_ext includes contact forces, gravity, and applied loads. Because M is diagonal, the inversion is trivial and the method is perfectly parallelizable — a critical advantage for the large models typical of full-vehicle rear-impact simulations.

The stability of this scheme is conditional. The time step must satisfy the Courant–Friedrichs–Lewy (CFL) condition:

Δt ≤ Δt_crit = 2 / ω_max ≈ L_min / c

where L_min is the smallest element characteristic length and c is the wave speed in the material. For steel with c ≈ 5000 m/s and a 5 mm element, Δt_crit ≈ 1 μs. PAM-CRASH computes this automatically and applies a mass-scaling option when the smallest elements would otherwise force an impractically small step. In rear-impact models, the seat structure, head restraint, and neck elements often govern the critical step, so mesh quality in these regions directly controls simulation cost.

Contact Algorithms: Penalty, Kinematic, and Type-Specific Formulations

Rear impact involves multiple simultaneous contact interfaces: the impacting vehicle against the rear bumper and body, the seat back against the occupant's torso, the head restraint against the head, and the occupant against the seat belt. PAM-CRASH provides several contact algorithms, each suited to different interface conditions.

Penalty contact is the default for most crashworthiness applications. When a slave node penetrates a master segment, a restoring force proportional to the penetration depth is applied:

F_contact = k · δ

where k is the interface stiffness and δ is the penetration. The stiffness is computed from the element properties of the contacting surfaces. Penalty contact is robust and energy-conserving when the stiffness is well-chosen, but excessive stiffness causes numerical instability, while insufficient stiffness allows visible penetration.

Kinematic contact enforces non-penetration by directly modifying nodal velocities and accelerations. It is more accurate for hard contacts but can introduce energy errors and is less forgiving of poor mesh alignment. In rear-impact models, kinematic contact is often used for the head restraint–head interface where penetration would distort the injury metrics.

Type-specific contacts in PAM-CRASH include:

• Type 1 (node-to-surface): General purpose, used for vehicle-to-vehicle and occupant-to-interior contacts.
• Type 7 (surface-to-surface): Preferred for large sliding interfaces such as the seat back against the torso.
• Type 13 (tied contact): Bonds dissimilar meshes, useful for joining seat frame components modeled with different element sizes.
• Type 24 (self-contact): Handles folding and self-intersection in thin-walled structures during collapse.

Contact thickness and friction coefficients are critical parameters. For the head restraint–head interface, a friction coefficient of 0.2–0.4 is typical, depending on the trim material. For vehicle-to-vehicle contact, 0.3–0.5 is common. These values should be documented and justified, as they directly affect occupant kinematics.

Material Models for Rear Impact

Rear impact engages a range of materials: high-strength steel in the bumper and rear rails, aluminum in some body structures, foam in the seat and head restraint, and biological tissue in the neck. PAM-CRASH provides constitutive models for each.

Elastoplastic metals (Type 2, 24, 36): The rear rails and bumper beam are typically modeled with MAT_TYPE 24 (piecewise linear plasticity) or MAT_TYPE 36 (strain-rate-dependent plasticity). Type 36 uses the Cowper–Symonds model:

σ_y = σ_0 [1 + (ε̇ / D)^(1/p)]

where σ_0 is the quasi-static yield stress, ε̇ is the strain rate, and D and p are material constants. For mild steel, D ≈ 40 s⁻¹ and p ≈ 5; for high-strength steel, D ≈ 3200 s⁻¹ and p ≈ 5. Rear-impact strain rates in the rails typically reach 10–100 s⁻¹, so rate sensitivity must be included.

Foam (Type 50, 57): Seat foam and head restraint padding are modeled with crushable foam models. MAT_TYPE 50 uses a user-defined stress–strain curve in compression, with optional tension cut-off and rate effects. The foam's energy absorption is characterized by the plateau stress and densification strain. For a typical polyurethane seat foam, the plateau stress is 0.1–0.3 MPa and densification begins at 60–70% strain.

Biological tissue (Type 91, 92): The neck is often represented by a rigid body or a deformable model with viscoelastic properties. MAT_TYPE 91 (soft tissue) uses a Mooney–Rivlin or Ogden formulation, while MAT_TYPE 92 (viscoelastic) adds time-dependent relaxation. For whiplash simulation, the neck model must capture the flexion–extension response that produces the characteristic S-shaped curvature.

Integrated Workflow for a Rear-Impact Model

A typical rear-impact simulation in PAM-CRASH follows this sequence:

1. Define the vehicle model with shell elements for the body-in-white and solid elements for the bumper foam.
2. Assign material models: Type 24 or 36 for steel, Type 50 for foam, Type 91/92 for the neck.
3. Define contact interfaces: Type 7 for vehicle-to-vehicle, Type 1 for occupant-to-seat, Type 13 for tied connections.
4. Apply the rear-impact pulse as a prescribed velocity or acceleration to the impacting vehicle or a rigid wall.
5. Set the termination time to 150–300 ms, sufficient to capture the full occupant kinematic response.
6. Request output at 1–2 ms intervals for occupant kinematics and 0.1 ms for contact forces.

The solver's explicit architecture means that the entire simulation is driven by the smallest element and the stiffest contact. A single poorly shaped element in the neck or a misaligned contact segment can reduce the time step by an order of magnitude. Mesh quality control and contact initialization checks are therefore not optional — they are prerequisites for a meaningful rear-impact result.

The architecture described here is not a collection of independent features. The time step, contact stiffness, and material stiffness are coupled: a stiff contact on a soft foam forces a smaller time step, while a soft contact on a stiff metal allows penetration. In rear-impact simulation, where the head–neck system is sensitive to small changes in contact force and timing, these couplings must be understood and controlled. The next section applies this architecture to the specific problem of building a validated rear-impact model.