2. Finite element occupant simulations

Before a machine learning surrogate can be trained to predict occupant responses, we must understand precisely what it is being trained to approximate. The ground-truth generator in almost every occupant-safety workflow is an explicit finite element (FE) crash simulation — typically LS-DYNA, PAM-CRASH, or Radioss — in which a human body model (HBM) interacts with a restraint system inside a deforming vehicle structure. This section dissects that generator: how the model is assembled, where the nonlinearities originate, and why a single frontal-impact run can consume hours of wall-clock time on hundreds of CPU cores. These three aspects are not academic background; they determine which input parameters a surrogate can realistically learn, how much training data is affordable, and where surrogate error is likely to concentrate.

Model architecture: a four-subsystem assembly

A restraint-system occupant model is best understood as four coupled subsystems sharing a common kinematic frame. The vehicle structure is represented by a coarse shell-element body-in-white with a prescribed crash pulse applied at the rocker or tunnel — the pulse is a time-history of deceleration, not a force, and it defines the inertial environment the occupant experiences. The restraint system comprises the seat (foam modeled with low-density solid elements, frame with shells), the seatbelt (1D seatbelt elements routed through sliprings and retractors, with a force-limiting load limiter and pretensioner), and the airbag (a control-volume or corpuscular-particle gas model bounded by a fabric membrane). The occupant is a validated HBM — THOR, Hybrid III, or a detailed human body model such as GHBMC — positioned by a pre-simulation settling step so that contact with the seat is in equilibrium. The contact and constraint layer ties everything together: penalty-based surface-to-surface contacts between occupant and seat, belt-to-occupant friction contacts, and rigid-body joints at the pelvis, spine, and neck.

The critical architectural point is that these subsystems are not independent. The belt routes over the occupant's chest, so belt geometry depends on occupant posture; the airbag inflates against the occupant, so bag pressure depends on occupant position; the seat foam compresses under occupant load, which changes belt routing again. This two-way coupling is what makes the model expensive and what makes surrogate modeling non-trivial: a change in one input (say, seatbelt anchor height) propagates through contact and kinematics to every output channel.

Where the nonlinearities come from

An FE occupant model is nonlinear in three distinct senses, and each has a different consequence for surrogate design.

Material nonlinearity. Seat foam is a low-density polyurethane that exhibits a pronounced plateau in its stress–strain curve: it compresses at nearly constant stress for 50–70% strain before densifying. Belt webbing is a woven fabric whose force–strain response stiffens progressively as the weave locks. Airbag fabric is a coated nylon with anisotropic, rate-dependent behavior. None of these can be linearized without losing the very load-limiting behavior that defines occupant injury risk.

Geometric and contact nonlinearity. The belt must slide through sliprings and across the occupant's chest; the contact status (open/closed) changes thousands of times per run. Large rotations of the occupant's torso and head mean that small-strain assumptions fail. Contact is the dominant source of both cost and discontinuity: a belt node that slips off the shoulder in one design and stays on in another produces a step change in chest deflection, not a smooth trend. This is precisely the kind of non-smooth input–output mapping that a surrogate must capture, and it is why purely linear regression surrogates fail on restraint problems.

Boundary-condition and loading nonlinearity. The crash pulse is prescribed, but the occupant's response to it is not: the pretensioner fires at a trigger time (often 10–20 ms), the load limiter pays out webbing at a threshold force (typically 3–4 kN), and the airbag inflates over a 20–30 ms window. Each of these is a threshold event, so the output is a piecewise function of the input parameters. A surrogate trained on a coarse design-of-experiments grid can easily miss the threshold crossing between two samples.

Computational bottlenecks

The explicit solver advances in time steps set by the smallest element and the material wave speed: Δt ≤ L_min / c. A 5 mm foam solid with c ≈ 50 m/s gives Δt ≈ 100 µs, but a 1 mm belt shell with c ≈ 2000 m/s gives Δt ≈ 0.5 µs. The belt therefore dictates the global step size, and a 120 ms event requires roughly 240,000 steps. Each step touches every element and resolves every contact pair, so cost scales with model size and contact count, not with the number of design variables.

Three bottlenecks dominate in practice. First, contact search: the occupant–seat and belt–occupant interfaces involve tens of thousands of candidate node–segment pairs, and penalty stiffness must be tuned to avoid penetration without introducing spurious high-frequency noise. Second, airbag gas dynamics: a control-volume model is cheap but a corpuscular-particle (CPM) model, needed for out-of-position studies, adds tens of thousands of particles and their collisions. Third, foam element distortion: highly compressed solid elements reduce the stable time step mid-run, so the solver may cut Δt dynamically, extending wall-clock time unpredictably.

What this means for the surrogate

The architecture tells us the input space: belt anchor coordinates, load-limiter force, pretensioner trigger time, airbag vent size and inflator mass flow, seat friction, and occupant stature. The nonlinearity analysis tells us the output space is non-smooth and threshold-driven, so the surrogate must be trained with samples dense enough near threshold events and validated on held-out designs that cross those thresholds. The bottleneck analysis tells us the sample budget is the binding constraint, so every design decision — parameter ranges, output channels, fidelity level — should be made with the cost of one FE run in mind. A surrogate that predicts HIC and chest deflection to within 5% using 200 well-placed samples is worth far more than one that needs 2,000 random samples to reach the same accuracy.

The table above is a working reference for the rest of the course: when a surrogate underperforms on a particular output channel, the first diagnostic question is which subsystem's nonlinearity that channel is most sensitive to, and whether the training samples resolved the relevant threshold. That diagnostic habit — mapping error back to physics — is what separates a surrogate that is merely accurate on its test set from one that is trustworthy in restraint-system design.