2. Multibody and finite element formulations in MADYMO
MADYMO is a hybrid solver: it can represent a single occupant safety model as a mixture of rigid multibody (MB) chains and deformable finite element (FE) meshes, solved in one time-integration loop. Choosing the right formulation for each subsystem — and deciding how the two domains exchange loads — is the single most consequential modelling decision in a restraint-design study. This section explains what each formulation actually computes, how coupling is implemented, and how to trade fidelity against runtime without invalidating the result.
What each formulation actually solves
The multibody formulation treats bodies as rigid, with mass and inertia concentrated at a centre of gravity. Motion is described by a small set of generalized coordinates (typically 6 DOF per free body), and joints — spherical, revolute, translational, universal, bracket, belt — impose kinematic constraints between bodies. Contact is handled by ellipsoid surfaces: each body carries one or more ellipsoids, and penetration depth between two ellipsoids generates a penalty force through a user-supplied force–penetration characteristic. Because the state vector is tiny, MB models integrate very fast and are numerically robust even with stiff belt and contact forces.
The FE formulation discretizes a deformable body into solid, shell, or membrane elements with nodal coordinates as unknowns. It captures local bending, buckling, wrinkling, and through-thickness stress that a rigid ellipsoid cannot represent. The cost is a much larger system: an airbag mesh with 20,000 shell elements and 20,000 nodes contributes roughly 120,000 DOF, and explicit time stepping is limited by the smallest element's stable time increment.
Coupling strategies
MADYMO couples the two domains through contact interfaces and kinematic attachments. Three patterns dominate practice:
- MB-to-FE contact (ellipsoid vs. mesh): a rigid ellipsoid (e.g., the head) penetrates an FE airbag or FE instrument panel. The solver computes penetration of the ellipsoid surface into the FE mesh, distributes nodal forces over the contacted elements, and applies the reaction to the MB body. This is the standard head-to-airbag and knee-to-bolster path.
- FE-to-FE contact: two deformable meshes interact, e.g., an FE airbag against an FE seat cushion. Requires consistent element size at the interface and careful contact-thickness definition.
- Kinematic attachment (FE node tied to MB body): a belt segment modelled as FE shells is attached to an MB dummy's shoulder or pelvis. The attachment transfers displacement and force between the FE nodes and the MB body without contact search.
Coupling is not free. Each interface adds contact-search cost and can introduce high-frequency noise if the FE mesh is coarse relative to the MB ellipsoid. A practical check: the ellipsoid's contact patch should span at least three to four elements; otherwise the force distribution becomes lumpy and the head acceleration trace shows spurious spikes.
Fidelity trade-offs and when they matter
The choice is not binary. A common production model uses MB for the dummy, FE for the airbag, and MB ellipsoids for the seat and floor. This hybrid runs in minutes and is adequate for restraint timing, belt-force, and head-travel studies. Switching the knee bolster from MB to FE adds local crush detail but can multiply runtime by 3–10× and requires a validated material card (e.g., LS-DYNA *MAT_PLASTIC_KINEMATIC with strain-rate scaling) — otherwise the extra fidelity is fictional.
| Subsystem | Typical choice | Why | Runtime impact |
|---|---|---|---|
| Occupant dummy | MB | Rigid segments, joint kinematics dominate | Baseline |
| Airbag | FE (shell/membrane) | Deployment, folding, venting are deformation-driven | High |
| Seat cushion | MB or FE | FE if penetration/rebound matters | Medium |
| Belt | MB (1D) or FE | FE for belt-to-airbag interaction | Low–Medium |
| Steering column | MB (force characteristic) | Collapse is a prescribed force–displacement curve | Low |
Practical verification
Before trusting a hybrid model, run three checks. First, an energy balance: total energy (kinetic + internal + contact) should be conserved to within a few percent; a growing internal energy with no deformation indicates a contact penalty problem. Second, a mesh-sensitivity check: refine the FE airbag by 1.5× and confirm head acceleration peaks shift by less than ~5%. Third, a time-step audit: if the explicit step is dominated by a single tiny element, the model is wasting runtime — remesh or use mass scaling deliberately, and document it.
In short, MB buys speed and robustness; FE buys deformation physics. Couple them at the interfaces where load transfer is physically meaningful, verify energy and mesh sensitivity, and let the study's question — not the solver's capability — decide the fidelity level.