2. LS-DYNA fundamentals for airbag simulation

Side airbag simulation sits at the intersection of three LS-DYNA pillars: explicit time integration, nonlinear material behavior, and penalty-based contact. Before building a side airbag model, you need a working mental model of how LS-DYNA advances a solution in time, why the stable time step is so small, and how contact forces are generated between the folded bag, the seat, the door trim, and the occupant. This section establishes those fundamentals with concrete numbers and configuration details you will reuse in later chapters.

Why Explicit Dynamics for Airbag Deployment

An airbag deployment is a short, violent, highly nonlinear event. The bag inflates from a folded state to full deployment in roughly 20–40 ms, the fabric undergoes large strain, and the occupant interacts with the bag over a similar window. Implicit solvers require assembling and inverting a global stiffness matrix at every iteration, which is expensive and prone to convergence failure when contact status changes rapidly. LS-DYNA's explicit solver avoids matrix inversion entirely: it computes nodal accelerations from the current force state and marches forward in time. This makes it well suited to problems dominated by wave propagation, contact, and large deformation.

The trade-off is conditional stability. The explicit central-difference scheme is stable only if the time step satisfies the Courant–Friedrichs–Lewy (CFL) condition. For a solid element, the critical step is approximately:

Δt_crit = L_e / c
c = sqrt(E / ρ)          (elastic bar wave speed)
c = sqrt((E/ρ) · (1-ν) / ((1+ν)(1-2ν)))   (3D solid, ν ≠ 0)

where L_e is the characteristic element length, E is Young's modulus, ρ is density, and ν is Poisson's ratio. For a typical airbag fabric modeled with shell elements, E ≈ 500 MPa, ρ ≈ 800 kg/m³, and a 3 mm element gives c ≈ 790 m/s and Δt_crit ≈ 3.8 × 10⁻⁶ s. A 40 ms event therefore requires on the order of 10,000–15,000 time steps. This is the fundamental cost driver, and it explains why mesh quality directly controls runtime.

LS-DYNA reports the minimum time step and the controlling element in d3hsp and in the message file. Always check the first few hundred cycles: if the step drops abruptly, a single distorted or tiny element is usually responsible.

Time Integration and Mass Scaling

LS-DYNA uses the central-difference method with a lumped mass matrix. Nodal accelerations are obtained from:

a_n = M⁻¹ (F_ext - F_int)

Velocities and displacements are then updated explicitly. Because the mass matrix is diagonal, no linear solve is needed. The practical consequence is that the stable step is governed by the smallest element in the model. Two standard remedies exist:

  • Mass scaling (*CONTROL_TIMESTEP with DT2MS negative): adds mass to small elements so their step matches a target. Use it sparingly on airbag fabric, because added mass changes inertia and can alter deployment timing.
  • Selective mass scaling (*CONTROL_TIMESTEP with DT2MS positive and MS1ST): scales only elements below the target, leaving the rest untouched.

For a side airbag model, a common target is DT2MS = -1.0e-6 s, meaning elements with a step below 1 µs receive added mass. The added-mass percentage is reported in the message file; keep it below about 1–2% of total model mass to avoid distorting the physics.

Contact Algorithms for Airbag–Occupant Interaction

Contact in LS-DYNA is penalty-based by default. When a slave node penetrates a master segment, a penalty force proportional to penetration depth is applied:

F_penalty = k · δ

where k is the contact stiffness and δ is the penetration. The stiffness is computed from the slave and master element properties. For airbag applications, three contact types dominate:

Contact Type Typical Use Key Card
Single Surface Bag self-contact during folding and unfolding *CONTACT_AUTOMATIC_SINGLE_SURFACE
Surface to Surface Bag against seat, door trim, or occupant *CONTACT_AUTOMATIC_SURFACE_TO_SURFACE
Airbag Single Surface Dedicated fabric self-contact with leakage control *CONTACT_AIRBAG_SINGLE_SURFACE

For side airbags, *CONTACT_AIRBAG_SINGLE_SURFACE is usually the best choice for the bag itself because it handles the highly folded geometry and includes options for fabric permeability and leakage. The occupant is typically modeled with *CONTACT_AUTOMATIC_SURFACE_TO_SURFACE against the bag, with the occupant as the slave side and the bag as the master.

Two parameters deserve attention. SOFT controls the stiffness formulation: SOFT = 0 uses the default penalty, SOFT = 1 uses a softer formulation that reduces numerical noise, and SOFT = 2 is a segment-based penalty often used for fabric. FS and FD are the static and dynamic friction coefficients; for fabric against fabric, values around FS = 0.2 and FD = 0.2 are reasonable starting points, while fabric against trim may use FS = 0.3.

Contact penetration is the most common source of non-physical results in airbag models. If the bag passes through the occupant or the seat, check the contact definition first, then the mesh density. A coarse master surface with large segments will allow deep penetration before the penalty force develops.

Putting It Together: A Minimal Side Airbag Setup

The following example illustrates the essential cards for a side airbag deployment against a simplified occupant. Parameters are representative and intended for teaching, not as validated values.

*CONTROL_TIMESTEP
$#  DTINIT   TSSFAC    ISDO    TSLIMT     DT2MS      LCTM     ERODE    MS1ST
        0.0      0.9       0       0.0  -1.0e-6         0         0         1

*CONTACT_AIRBAG_SINGLE_SURFACE
$#    SSID      MSID     SSTYP     MSTYP     SBOXID    MBOXID       SPR       MPR
         1         0         2         0         0         0         0         0
$#      FS        FD        DC        VC       VDC    PENCHK        BT        DT
       0.2       0.2       0.0       0.0       0.0         0       0.0       0.0
$#    SFS       SFM       SST       MST      SFST      SFMT       FSF       VSF
       1.0       1.0       0.0       0.0       1.0       1.0       1.0       1.0
$#  SOFT    SOFSCL    LCIDAB    MAXPAR     SBOPT     DEPTH     BSORT    FRCFRQ
         2       0.1         0       0.0         2         2         0         1

*CONTACT_AUTOMATIC_SURFACE_TO_SURFACE
$#    SSID      MSID     SSTYP     MSTYP     SBOXID    MBOXID       SPR       MPR
         2         1         2         2         0         0         0         0
$#      FS        FD        DC        VC       VDC    PENCHK        BT        DT
       0.3       0.3       0.0       0.0       0.0         0       0.0       0.0
$#  SOFT    SOFSCL    LCIDAB    MAXPAR     SBOPT     DEPTH     BSORT    FRCFRQ
         1       0.1         0       0.0         2         2         0         1

Here SSID = 1 is the airbag part set and SSID = 2 is the occupant. The airbag uses SOFT = 2 for segment-based penalty, while the occupant contact uses SOFT = 1 to reduce noise. DEPTH = 2 activates the optional depth-based penetration check, which is useful when the bag folds back on itself.

After running, inspect the message file for the added-mass percentage and the minimum time step. Then plot the bag internal pressure and the occupant chest deflection. A physically reasonable side airbag deployment shows pressure rising to a peak within 10–20 ms and chest deflection peaking shortly after. If pressure spikes unrealistically or the bag fails to unfold, revisit the contact definitions and the folding geometry before adjusting material parameters.

Explicit Time Integration and Contact Force Chain in a Side Airbag Model Explicit Central Difference a_n = M⁻¹ (F_ext − F_int) v_{n+1/2} = v_{n−1/2} + a_n · Δt u_{n+1} = u_n + v_{n+1/2} · Δt Lumped mass → no matrix solve CFL Stability Limit Δt_crit = L_e / c c = √(E / ρ) Example: L_e = 3 mm, c ≈ 790 m/s Δt_crit ≈ 3.8 µs Penalty Contact F_penalty = k · δ k from slave/master properties SOFT = 2 for fabric self-contact FS / FD control friction Internal forces from contact and element stresses feed back into the next time step Folded Airbag Shell elements, fabric *CONTACT_AIRBAG_SINGLE_SURFACE Occupant Dummy / human body model *CONTACT_AUTOMATIC_SURFACE_TO_SURFACE Seat / Door Trim Rigid or deformable Master surface for bag contact Outputs: bag pressure, chest deflection, contact force Check message file for added mass and min time step

The diagram above summarizes the chain: the explicit solver advances nodal kinematics, the CFL condition sets the step size, and penalty contact generates the forces that couple the airbag, occupant, and surrounding structure. Each block feeds the next, and the contact forces return to the solver as internal forces for the following cycle. Keeping this loop in mind will help you diagnose the common failure modes—excessive mass scaling, contact penetration, and unstable time steps—as you build and run side airbag models in the chapters ahead.

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