4. Gas dynamics and inflator modeling

An airbag deploys because a pyrotechnic or hybrid inflator releases a controlled mass of hot gas into a fabric cushion within tens of milliseconds. The pressure that this gas exerts on the cushion wall — and therefore on the occupant — is the single most important quantity linking the inflator to the contact simulation. If the gas model is wrong, the contact pressure history is wrong, and every downstream injury metric (HIC, chest deflection, neck force) becomes unreliable. This chapter examines the three modeling families used in production CAE workflows: the uniform-pressure (lumped) model, the control-volume (CV) model, and particle-based methods. The goal is to understand what each method actually computes, when it is appropriate, and how to set it up so that the predicted contact pressure matches the physical inflation event.

Why the Gas Model Dominates Contact Pressure

In an airbag simulation, the cushion is a membrane structure with negligible bending stiffness. Its shape is determined almost entirely by the internal pressure field and the external contact constraints. The contact pressure on the occupant is therefore a direct function of the internal gas state. Three physical effects must be captured:

• Mass inflow from the inflator, which is a function of propellant burn rate or stored-gas release.
• Energy balance — hot gas cools as it expands and mixes with entrained ambient air through the vent holes and fabric porosity.
• Volume change — the cushion volume grows as the bag unfolds, and the pressure depends on how fast that volume grows relative to the mass inflow.

The uniform-pressure model treats the entire cushion as a single thermodynamic state. The CV model divides the cushion into multiple connected control volumes, each with its own state. Particle methods discretize the gas itself into discrete parcels that carry mass, momentum, and energy. The choice among them is a trade-off between fidelity and cost, and it must be made before the contact simulation is run.

Uniform-Pressure (Lumped) Model

The uniform-pressure model is the workhorse of occupant-safety CAE. It assumes that pressure, temperature, and density are spatially uniform inside the cushion at every instant. The governing equations are the ideal-gas law and the first law of thermodynamics applied to a single control volume:

P(t) = m(t) * R * T(t) / V(t)
dT/dt = (1/(m * cv)) * [ m_in * cp * T_in - m_out * cp * T - P * dV/dt + Q_loss ]

Here m(t) is the instantaneous gas mass, V(t) is the cushion volume computed from the membrane geometry, T_in is the inflator gas temperature, and Q_loss accounts for heat transfer to the fabric. The inflator is represented by a mass-flow-rate curve m_in(t) and a temperature curve T_in(t), both obtained from tank tests or from a detailed inflator model.

In LS-DYNA this is the *AIRBAG_SIMPLE_PRESSURE_VOLUME or *AIRBAG_WANG_NEFSKE keyword; in PAM-CRASH it is the GAS model. The key input is the LOAD_CURVE for mass flow and the LOAD_CURVE for temperature. A typical driver-side inflator might deliver 0.8–1.2 kg/s peak mass flow over 30–50 ms, with a gas temperature of 800–1200 K at the inflator exit.

Control-Volume (CV) Model

The CV model partitions the cushion into n interconnected chambers. Each chamber has its own mass, temperature, and pressure, and the chambers exchange mass and energy through orifices that represent fabric porosity, seams, and vent holes. The governing equations become a system of coupled ODEs:

For chamber i:
dm_i/dt = sum_j (m_ji) - sum_k (m_ik) + m_inflator_i
d(m_i * cv * T_i)/dt = sum_j (m_ji * cp * T_j) - sum_k (m_ik * cp * T_i) + Q_i - P_i * dV_i/dt
P_i = m_i * R * T_i / V_i

The flow between chambers is typically modeled with an orifice equation:

m_ij = C_d * A_ij * sqrt(2 * rho * (P_i - P_j))   for P_i > P_j

where C_d is the discharge coefficient (typically 0.6–0.8 for fabric orifices) and A_ij is the effective flow area. This model captures the pressure gradient between the inflator-side chamber and the occupant-side chamber, which is critical for predicting the initial contact pressure peak.

In practice, a driver-side bag is often modeled with 2–4 chambers: one for the inflator plenum, one for the main cushion, and one or two for the outer regions near the occupant. The chamber boundaries are defined by the fabric geometry and the seam lines. The CV model is implemented in LS-DYNA as *AIRBAG_MULTIPLE_PRESSURE_VOLUME and in PAM-CRASH as the multi-chamber gas model.

Particle-Based Methods

Particle methods — most commonly Corpuscular Particle Method (CPM) in LS-DYNA and Discrete Element Method (DEM) variants — represent the gas as a large number of discrete particles. Each particle carries mass, velocity, and internal energy. Particles move ballistically inside the cushion, collide with the fabric, and exchange momentum and energy with the membrane. The pressure on the fabric is computed from the momentum flux of particles hitting the surface.

The CPM in LS-DYNA is activated with *AIRBAG_PARTICLE. The key inputs are:

• Number of particles — typically 50,000 to 500,000 for a driver bag. Too few particles produce noisy pressure; too many increase runtime linearly.
• Particle injection curve — the mass-flow-rate history, same as the uniform model.
• Particle temperature — the initial gas temperature at injection.
• Fabric permeability — the leakage model for particles escaping through the fabric.

The advantage of CPM is that it naturally captures non-uniform pressure during the filling phase, the jetting effect from the inflator, and the local pressure peaks where the occupant first contacts the bag. The disadvantage is computational cost: a CPM simulation can be 5–20 times more expensive than a uniform-pressure simulation for the same mesh.

Choosing the Right Method

The decision is driven by the physical question being asked. The following table summarizes the trade-offs for a typical driver-side frontal impact simulation.

Worked Example: Contact Pressure Prediction for a Driver Bag

Consider a 60 L driver-side airbag with a hybrid inflator. The tank-test mass-flow curve peaks at 1.0 kg/s at 15 ms and decays to zero by 45 ms. The gas temperature at the inflator exit is 1000 K. The cushion has two vent holes of 30 mm diameter each, and the fabric permeability is 5 L/dm²/min at 10 kPa.

Step 1 — Build the uniform-pressure baseline. Define the mass-flow and temperature curves, set the vent hole area to 2 × π × (0.015)² = 1.41 × 10⁻³ m², and run the simulation. The predicted peak internal pressure is typically 180–220 kPa (gauge) at 25–30 ms. The contact pressure on the occupant chest, measured at the chest-deflection sensor location, peaks at 35–45 kPa.

Step 2 — Refine with a 3-chamber CV model. Split the bag into an inflator plenum (5 L), a central chamber (35 L), and an outer chamber (20 L). The orifice between plenum and central chamber is the diffuser exit area; the orifice between central and outer chamber is the fabric porosity area. Re-run. The peak contact pressure typically increases by 10–20% because the outer chamber pressure lags the central chamber, creating a sharper pressure rise when the occupant contacts the bag.

Step 3 — Validate with CPM. Run the same case with 200,000 particles. The CPM result should show a pressure distribution on the bag surface that is non-uniform during filling, with a high-pressure jet region near the inflator exit. The contact pressure peak may be 15–30% higher than the CV result if the occupant contacts the bag during the jetting phase.

Step 4 — Compare against test data. The validation metric is the contact pressure history at the chest and the bag internal pressure at the inflator plenum. A typical acceptance criterion is that the peak contact pressure must be within ±15% of the test measurement, and the time-to-peak must be within ±3 ms. If the uniform model over-predicts the peak by more than 30%, the CV or CPM model is required.

Practical Setup Checklist

• Verify that the mass-flow curve integrates to the total gas mass specified by the inflator supplier. A 60 L bag at 200 kPa and 300 K contains roughly 0.14 kg of gas; the inflator must deliver at least this mass plus venting losses.
• Check that the vent hole area in the model matches the physical vent hole diameter. A 10% error in vent area produces a 5–10% error in peak pressure.
• For CV models, ensure that the chamber volumes sum to the total bag volume at full inflation. A mismatch indicates a geometry or seam definition error.
• For CPM, run a particle-count convergence study before using the results for injury assessment.
• Always compare the predicted internal pressure against a tank test or a sled test before trusting the contact pressure prediction.

The workflow above is iterative: start with the uniform-pressure model to establish a baseline, refine with CV if the pressure gradient matters, and validate with CPM when local pressure peaks or jetting effects are critical. The contact pressure prediction is only as good as the gas model that feeds it, and the validation against test data is the final arbiter of whether the model is fit for purpose.