2. Full-order restraint simulation baseline

Before any reduced-order model (ROM) can be trusted, it must be validated against a credible full-order model (FOM). In restraint system simulation, the FOM is almost always an explicit finite element (FE) model built in a solver such as LS-DYNA, Radioss, or Abaqus/Explicit. This chapter establishes what that baseline actually contains, how each subsystem is discretized and coupled, and where the computational cost concentrates — the bottlenecks that justify ROM development in later chapters.

Why explicit integration defines the baseline

Restraint events are short, highly nonlinear, and dominated by contact and inertia: a frontal impact develops over roughly 80–120 ms, airbag deployment takes 25–40 ms, and belt loading peaks within 60 ms. Explicit time integration is the natural choice because it avoids forming and factorizing a global stiffness matrix. The central-difference scheme advances the solution with a critical time step governed by the smallest element characteristic length and the material wave speed:

Δt_crit ≈ L_e / c,   c = sqrt(E / ρ)   (solid elements, 1D approximation)
Δt_crit = L_e / c_d,  c_d = sqrt(E / ρ(1 − ν²))   (shell elements, plane stress)

In practice, solvers apply a safety factor of 0.9 and may add mass scaling to raise the stable step. A typical occupant-restraint model runs with Δt between 0.5 and 1.5 µs, so a 120 ms event requires on the order of 10⁵ time steps. Every one of those steps touches every element, which is why model size translates almost linearly into wall-clock cost.

Occupant model: the Hybrid III 50th percentile dummy

The occupant is usually a validated dummy FE model — commonly the Hybrid III 50th percentile male, or a THOR or human body model for research work. These models are rigid-body assemblies with deformable contact surfaces: the head, neck, thorax, abdomen, pelvis, and limbs are connected by joints with defined stiffness and damping. Mesh density is deliberately high in regions that contact the restraint: the chest skin, the face, and the knees. A typical dummy model contains 60,000–150,000 elements and 200–400 contact definitions. The dummy is positioned with a prescribed H-point and a seat-belt routing that matches the physical sled setup; small positioning errors (a few millimeters at the shoulder) can shift chest deflection by several percent, so the baseline must document the positioning procedure, not just the geometry.

Seat belt model: 1D elements with 2D/3D contact

Belts are modeled as seatbelt elements — a specialized 1D element type that carries tension only, with a defined cross-sectional area and a uniaxial material law. The belt path is defined by a sequence of nodes threaded through sliprings at the retractor, D-ring, buckle, and anchor points. Friction at the D-ring and through the latch is modeled with a coefficient of friction (typically 0.1–0.3) applied at the slipring contact. The retractor is represented by a discrete element with a force-limiting torsion bar or a load-limiter curve, and the pretensioner by a time-dependent applied retraction.

Two modeling decisions dominate belt fidelity. First, the belt must contact the dummy through a surface-to-surface or node-to-surface contact with a penalty stiffness tuned to avoid penetration without introducing spurious oscillation. Second, belt slack and routing must reproduce the physical build: a 20 mm slack difference changes the time of belt engagement and therefore the chest acceleration peak. The belt model itself is cheap — a few hundred elements — but its contact with the dummy and seat is not.

Airbag model: the dominant cost center

The airbag is where the FOM becomes expensive. A driver airbag is modeled as a folded fabric shell (membrane elements) that must unfold, inflate, and vent. The folding is generated by a separate pre-processor and imported as an initial geometry; the inflator is represented by a tank-test mass-flow curve or a pyrotechnic gas model feeding a uniform-pressure (UP) or control-volume (CV) chamber. The CV approach solves a thermodynamic state equation each step:

P = (m_g R T) / V + P_atm,   with energy balance
d(m_g c_v T)/dt = ṁ_in h_in − ṁ_out h_out − P dV/dt + Q̇

Fabric leakage is modeled through porosity (fabric permeability) and explicit vent holes. The computational burden comes from three sources: the fine membrane mesh needed to resolve wrinkles and contact (often 30,000–80,000 shell elements), the self-contact of the folding bag during deployment, and the bag-to-occupant contact that must remain stable as the bag inflates against the dummy. Airbag self-contact alone can consume 40–60% of total solver time in a frontal model.

Coupling and the full event

The three subsystems are coupled through contact and through the seat and steering wheel, which are themselves deformable or rigid bodies. The complete model typically includes the seat frame, cushion foam, steering column, and knee bolster. A representative frontal baseline might total 500,000–1,500,000 elements, 1,000–3,000 contacts, and run 8–48 hours on 32–128 cores depending on mesh density and contact settings.

Where the cost actually goes

The cost distribution above is representative rather than universal, but the ranking is robust across frontal restraint models: contact dominates. Explicit contact requires a broad-phase search (bucket sorting or bounding-volume hierarchies) and a narrow-phase penalty or projection solve for every candidate pair, every time step. When the airbag is folded, its self-contact pairs are dense and change rapidly during deployment, which is precisely the phase where the time step is also smallest. The result is a double penalty: more contact work per step, and more steps per millisecond.

A second bottleneck is the time step itself. Thin fabric shells and foam cushions force Δt down. Mass scaling can recover some of this, but it changes the dynamics and must be verified against an unscaled run for the outputs of interest — chest deflection and belt force are sensitive to added mass in the occupant, so scaling is usually applied only to the bag and seat structure.

What the baseline must deliver

A defensible FOM baseline is not just a deck that runs. It must produce a documented set of outputs that the ROM will later be required to reproduce: chest deflection time history, belt force at the retractor and buckle, head acceleration (resultant and HIC), airbag internal pressure, and occupant kinematics at defined checkpoints. Each output should be accompanied by the mesh, contact, and material settings that produced it, plus a note on sensitivity — for example, that a 10% change in bag porosity shifts peak pressure by a measurable amount. This documentation is what makes the ROM credible: the ROM is only as good as the baseline it is trained against, and a baseline whose contact settings were never verified is a weak reference.

The full-order model therefore serves two roles: it is the physical reference for restraint performance, and it is the training and validation target for every reduced-order surrogate that follows. Understanding exactly which subsystems drive its cost — airbag self-contact, fine occupant contact surfaces, and the small stable time step — tells us where a ROM can legitimately substitute a cheaper representation and where it cannot.