3. LS-DYNA explicit solver fundamentals for airbag deployment

Curtain airbag simulation sits at the intersection of three demanding requirements: a highly nonlinear gas-inflated membrane, a rapidly changing contact topology against the occupant and trim, and a time scale measured in tens of milliseconds. LS-DYNA's explicit solver is the standard tool for this class of problem, but its behavior is governed by a small set of numerical principles that must be understood before any keyword deck can be trusted. This section covers the time-integration scheme, the contact algorithms that dominate curtain airbag response, and the stability controls that keep a run alive long enough to produce usable results.

Explicit time integration: the central difference scheme

LS-DYNA's explicit solver advances the solution using the central difference method. At each time step, nodal accelerations are computed from the assembled force vector, then integrated to velocity and displacement:

a_n = M^-1 (F_ext - F_int)
v_(n+1/2) = v_(n-1/2) + a_n * dt
x_(n+1)   = x_n + v_(n+1/2) * dt

Here M is the lumped mass matrix, F_ext is the external load (including contact and pressure forces), and F_int is the internal element force. Because M is lumped and diagonal, no matrix inversion or linear solve is required — each step is a sequence of vector operations. This is why explicit dynamics scales well to millions of elements and why it is the natural choice for airbag deployment, where the physics is dominated by wave propagation and short-duration transients rather than by long-term equilibrium.

The trade-off is conditional stability. The scheme is stable only if the time step satisfies the Courant–Friedrichs–Lewy (CFL) condition:

dt <= dt_crit = 2 / omega_max

where omega_max is the highest natural frequency of the discretized system. In practice LS-DYNA estimates this from the smallest element characteristic length and the material wave speed:

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

For a typical airbag fabric with E = 5 GPa and rho = 800 kg/m^3, the wave speed is roughly 2500 m/s. A 2 mm shell element therefore gives dt_crit ≈ 0.8 µs. A 50 ms deployment event then requires on the order of 60,000–100,000 time steps. This is the fundamental cost driver of explicit airbag simulation, and it explains why mesh quality and mass scaling decisions have such a large effect on runtime.

Contact algorithms for curtain airbag interaction

Curtain airbags interact with the occupant, the seat, the roof rail, and the vehicle interior through contact. LS-DYNA offers several contact formulations, and the choice matters for both accuracy and stability.

Penalty-based contact is the default for most airbag applications. A penalty stiffness is computed from the slave segment area, the material bulk modulus, and a scale factor. When penetration occurs, a restoring force proportional to the penetration depth is applied. The penalty method is robust and energy-conserving in the limit of small penetration, but it allows a small amount of interpenetration that must be monitored.

Soft constraint contact (SOFT=1 or SOFT=2) is often preferred for airbag-to-occupant contact because it reduces the stiffness discontinuity that occurs when a node first enters a segment. This is particularly important for the head-to-airbag interface, where a hard penalty can produce spurious stress waves and premature element failure.

Single surface contact (*CONTACT_AUTOMATIC_SINGLE_SURFACE) is used for the airbag's self-contact, which is essential for capturing folding and unfolding behavior. The automatic option handles the initial penetration check and is generally more forgiving of imperfect initial geometry.

For curtain airbags specifically, the following contact definitions are typical:

• *CONTACT_AUTOMATIC_SINGLE_SURFACE for airbag self-contact, with SOFT=1 and SBOPT=3 for segment-based processing.
• *CONTACT_AUTOMATIC_SURFACE_TO_SURFACE for airbag-to-head and airbag-to-torso, with a friction coefficient of 0.2–0.3.
• *CONTACT_AUTOMATIC_NODES_TO_SURFACE for airbag-to-trim or airbag-to-roof-rail, where the airbag is the slave and the rigid trim is the master.

Contact thickness and penetration depth should be checked in the d3hsp and messag files. A persistent penetration greater than 10% of the element thickness usually indicates that the penalty stiffness is too low or that the mesh is too coarse at the contact interface.

Stability controls: mass scaling, time step, and hourglass

Three stability controls dominate explicit airbag runs: mass scaling, time-step control, and hourglass control.

Mass scaling artificially increases the density of small elements to raise dt_crit. For airbag fabric, a selective mass scaling with a target time step of 1–2 µs is common. The added mass must be monitored: if the total added mass exceeds 1–2% of the model mass, inertial effects can corrupt the deployment kinematics. LS-DYNA reports this in the d3hsp file under "added mass."

Time-step control is managed through *CONTROL_TIMESTEP. The DTINIT parameter sets the initial step, while TSSFAC (typically 0.9) provides a safety factor. For airbag runs, ISDO=1 enables automatic time-step calculation based on the smallest element. The DTMIN parameter sets a floor; if the step falls below it, the run terminates. Setting DTMIN too low wastes runtime, while setting it too high causes premature termination.

Hourglass control is required for reduced-integration shell elements, which are standard for airbag fabric. The default IHQ=4 (Flanagan–Belytschko) with a coefficient of 0.03–0.05 is typical. Excessive hourglass energy — more than 5% of internal energy — indicates that the mesh is too coarse or that the loading is producing non-physical deformation modes.

Putting it together: a reproducible baseline run

To make these principles concrete, consider a baseline curtain airbag model with the following inputs: a 2 mm shell mesh for the airbag fabric, E = 5 GPa, rho = 800 kg/m^3, and a target deployment duration of 50 ms. The expected dt_crit is approximately 0.8 µs, so the run will require about 62,500 steps. With selective mass scaling to a target step of 1.5 µs, the step count drops to roughly 33,000, but added mass must be checked.

After the run, the following outputs should be inspected:

• d3hsp: initial time step, added mass percentage, and hourglass energy summary.
• messag: contact penetration warnings and any DTMIN or negative volume messages.
• glstat: total energy, kinetic energy, internal energy, and hourglass energy over time.
• matsum: energy distribution across materials, useful for detecting unrealistic fabric behavior.

Typical failure symptoms and their remedies: if the run terminates with a negative volume error, reduce the time step or refine the mesh in the region of large deformation. If added mass exceeds 2%, reduce the mass-scaling target or refine the smallest elements. If hourglass energy exceeds 5% of internal energy, switch to a finer mesh or increase the hourglass coefficient. If contact penetration is excessive, increase the penalty scale factor or switch to SOFT=2.

Understanding these three pillars — explicit time integration, contact algorithms, and stability controls — provides the foundation for every subsequent modeling decision in this course. The next sections build on this baseline to address airbag folding, gas inflation modeling, and occupant interaction.