4. Multibody and finite element formulations in MADYMO

MADYMO is a hybrid solver: it can run pure multibody (MB) models, pure finite element (FE) models, or a coupled MB–FE model in a single simulation. Sled test reconstruction almost always benefits from the coupled approach, because the occupant, restraint system, and instrumentation are naturally suited to different formulations. Choosing the right body type, joint, and contact model for each subsystem is the single most influential modelling decision in the whole workflow — it determines runtime, numerical stability, and whether the correlation with measured dummy responses is physically meaningful or merely curve-fitted.

Formulation trade-offs at a glance

Multibody bodies are rigid by definition. Each body has six degrees of freedom, and its inertia is fully described by mass, centre of gravity, and an inertia tensor. Contacts between MB bodies are resolved analytically through ellipsoids, planes, and cylinders, which makes them extremely fast and robust. FE bodies, by contrast, are discretised meshes whose deformation is computed from element formulations; they capture belt webbing stretch, airbag inflation, foam crushing, and sheet-metal folding, but they cost orders of magnitude more CPU time and require careful time-step control.

The practical rule for sled reconstruction is: use MB for anything whose deformation is negligible compared to its rigid-body motion, and use FE only where deformation or membrane behaviour carries the injury-relevant signal. A Hybrid III dummy is a classic MB object — its segments are rigid and its joints are calibrated. A seat foam, a deploying airbag, and a three-point belt with pretensioner and load limiter are classic FE objects.

Selecting body types

In MADYMO's XML input, body type is declared through the element used. A BODY.RIGID is a six-DOF rigid body; a BODY.FE references a mesh and material; a BODY.FLEX is a modal body useful for components that vibrate but do not need full FE. For sled work, the following allocation is typical and defensible:

• Sled buck and seat frame: BODY.RIGID. The buck is driven by a measured acceleration pulse applied via MOTION.ACC or a prescribed MOTION.JOINT. Making the buck rigid removes thousands of elements that contribute nothing to occupant response.
• Seat cushion and back foam: BODY.FE with a low-density foam material (e.g., MATERIAL.FOAM or MATERIAL.HYPERELASTIC). Foam compression is a first-order contributor to chest and pelvis loading, so it must deform.
• Dummy: BODY.RIGID segments connected by joints. MADYMO ships validated dummy databases; do not remodel the dummy unless the test uses a non-standard configuration.
• Belt webbing: BODY.FE shell elements with a MATERIAL.FABRIC or orthotropic membrane. Webbing stretch and belt–occupant sliding are essential for belt force correlation.
• Airbag: BODY.FE shell with a gas model. For most sled reconstructions, a uniform-pressure gas model is sufficient; CFD coupling is reserved for out-of-position or venting-sensitive cases.

Joint selection and calibration

Joints define the kinematic constraints between MB bodies. MADYMO offers JOINT.BRACKET, JOINT.REVOLUTE, JOINT.SPHERICAL, JOINT.TRANSLATIONAL, JOINT.UNIVERSAL, and JOINT.FREE. For a Hybrid III dummy, the neck is a JOINT.BRACKET with a moment–rotation curve, the spine is a series of JOINT.BRACKET elements, and the hip is a JOINT.UNIVERSAL. The critical point is that joint stiffness and damping must come from the dummy certification data, not from arbitrary tuning.

When a joint is used to represent a real mechanism — for example, a seat recliner or a belt retractor — the joint's CHARACTERISTIC (force or moment versus displacement or rotation) should be derived from component test data. A retractor pretensioner, for instance, is modelled as a JOINT.TRANSLATIONAL with a prescribed stroke and a force limit; the load limiter is a FORCE_ELEMENT with a plateau force.

Contact models

Contact in MADYMO is declared through CONTACT elements. The three families relevant to sled reconstruction are:

Friction coefficients are not free parameters. For dummy–seat contact, a value around 0.3–0.5 is typical for fabric on foam; for belt–dummy, 0.2–0.3 is common. These should be documented and, where possible, verified against component-level tests. Penalty stiffness controls penetration: too low and the dummy sinks into the seat; too high and the time step collapses. A practical starting point is to set penalty stiffness so that maximum penetration stays below 2–3 mm at peak load.

Coupling MB and FE domains

Coupling is where most sled models fail. MADYMO transfers forces and kinematics between MB and FE domains through contact definitions and, for tied interfaces, through COUPLING elements. Two rules keep the model stable:

1. Match time steps. The FE time step is governed by the smallest element and the material wave speed. If the MB solver runs at a larger step, the coupling becomes explicit and can inject energy. Use TIME_STEP control to keep the FE step at or below the Courant limit.
2. Align contact surfaces. MB ellipsoids must be positioned so that they do not initially penetrate FE surfaces. A small initial gap (0.5–1 mm) is preferable to an initial overlap.

Worked example: driver-side sled reconstruction

Consider a 50th-percentile Hybrid III driver on a rigid buck with a foam seat, three-point belt with pretensioner and load limiter, and a driver airbag. A defensible formulation is:

<BODY.RIGID id="buck" />
<BODY.RIGID id="seat_frame" />
<BODY.FE id="seat_foam" mesh="foam.msh" material="foam_mat" />
<BODY.RIGID id="dummy" />   <!-- Hybrid III database -->
<BODY.FE id="belt" mesh="belt.msh" material="fabric_mat" />
<BODY.FE id="airbag" mesh="airbag.msh" material="fabric_mat" />

<JOINT.BRACKET id="neck" body1="head" body2="thorax" characteristic="neck_curve" />
<JOINT.TRANSLATIONAL id="retractor" characteristic="pretension_stroke" />
<FORCE_ELEMENT id="load_limiter" force="3500" />

<CONTACT id="dummy_seat" type="MB_FE" friction="0.4" penalty="1e6" />
<CONTACT id="belt_dummy" type="FE_FE" friction="0.25" />
<CONTACT id="airbag_dummy" type="FE_FE" friction="0.1" />

After running, compare the simulated head resultant acceleration, chest deflection, and belt force against the measured channels. A well-formulated model should reproduce the timing of peak head acceleration within 5–10 ms and the peak chest deflection within 10–15%. If the belt force is too high, check the load limiter plateau and the retractor stroke; if the head acceleration is too low, check the airbag pressure and the dummy–airbag contact friction.

Summary of selection logic

The formulation choice is driven by one question: does the deformation of this component carry information that the correlation targets depend on? If yes, use FE. If no, use MB. Joints should mirror the real mechanism's degrees of freedom and use measured stiffness and damping. Contacts should use documented friction coefficients and penalty stiffness tuned to keep penetration small without destabilising the time step. When these three decisions are made consistently, the MB–FE coupled model becomes both fast enough to iterate and accurate enough to correlate.