2. MADYMO solver architecture for impact problems

MADYMO is built around a single explicit time-integration core that drives two distinct structural representations: rigid multi-body (MB) systems and finite element (FE) models. Understanding how these two formulations share the same solver loop — and where they diverge in contact treatment and time-step control — is essential for building pedestrian impact models that are both stable and computationally tractable.

2.1 Explicit Time Integration: The Central Difference Loop

MADYMO advances the solution using the explicit central difference method. For each body or node, the equation of motion is solved directly without assembling or inverting a global stiffness matrix:

a(t) = M⁻¹ · (F_ext(t) − F_int(t))
v(t+Δt/2) = v(t−Δt/2) + a(t) · Δt
x(t+Δt) = x(t) + v(t+Δt/2) · Δt

Here M is the mass matrix (lumped and diagonal for FE, diagonal for MB), F_ext collects external loads such as gravity and contact forces, and F_int is the internal force from element deformation or joint constraints. Because the method is conditionally stable, the time step must satisfy the Courant–Friedrichs–Lewy (CFL) condition. For FE parts, the critical step is governed by the smallest element:

Δt_crit = L_min / c,   c = √(E/ρ)

where L_min is the smallest characteristic element length, E is Young's modulus, and ρ is density. A typical steel-like material with E = 210 GPa and ρ = 7800 kg/m³ gives a wave speed near 5190 m/s; a 5 mm element therefore limits Δt to roughly 0.96 µs. MADYMO reports the actual step in the .out file, and mass scaling is the standard remedy when a few small elements would otherwise dominate the cost.

2.2 Multi-Body versus Finite Element Formulations

The multi-body formulation treats each body as rigid, defined by mass, inertia tensor, and joint constraints. Deformation is concentrated in discrete force elements — springs, dampers, and contact interactions — attached between bodies. This makes MB ideal for the pedestrian's gross kinematics: the head, torso, pelvis, and limbs rotate and translate as rigid segments, and injury-relevant deformation is captured by calibrated contact springs at the head, chest, and knee.

The finite element formulation discretizes deformable continua into elements (shells, solids, membranes). It resolves stress and strain fields directly, which is necessary when the injury mechanism depends on local deformation — for example, skull fracture, rib bending, or soft-tissue compression. The trade-off is cost: FE models require far smaller time steps and more degrees of freedom than MB models.

In practice, pedestrian models are hybrid: the vehicle front-end and the pedestrian head or chest may be FE, while the rest of the body is MB. MADYMO couples them through shared interfaces, and the global time step is the minimum of all subsystems.

2.3 Contact Algorithms

Contact is where MB and FE diverge most. MB contact uses analytical geometry — ellipsoids, planes, cylinders — and computes penetration depth δ directly. The contact force follows a penalty formulation:

F_c = k · δ + c · δ̇

where k is contact stiffness and c is damping. Because MB surfaces are smooth and penetration is computed analytically, contact is cheap and robust. The stiffness must be high enough to prevent visible penetration but low enough to avoid forcing a tiny time step; a common starting point is k = 100–1000 N/mm for pedestrian–vehicle contacts, tuned against test data.

FE contact uses node-to-segment or segment-to-segment search. Each slave node is checked against master segments; penetration triggers a penalty force normal to the segment, plus a Coulomb friction force tangential to it. The friction coefficient μ is typically 0.3–0.6 for pedestrian–vehicle surfaces. FE contact is more accurate for complex geometry but requires careful master/slave assignment and can produce hourglassing or excessive penetration if stiffness is mis-set.

2.4 Coupling and Time-Step Control in a Hybrid Model

When MB and FE subsystems coexist, MADYMO advances them in the same explicit loop. The global time step is the minimum of the MB step (set by joint and contact stiffness) and the FE step (set by CFL). Contact between an MB ellipsoid and an FE segment is handled by projecting the ellipsoid onto the FE surface and applying the penalty force to the FE nodes.

A practical workflow for a pedestrian head-to-bonnet model:

1. Build the head as an FE sphere or ellipsoid mesh with skull and brain layers; assign E and ρ so the CFL step is acceptable.
2. Build the bonnet as FE shell elements with a representative material card.
3. Define the remaining body as MB with calibrated joint stiffness and contact ellipsoids.
4. Set the global Δt to the minimum of all subsystems; verify in the .out file.
5. Run a short 10 ms check, inspect contact forces and penetration, then extend to the full 120 ms impact window.

Verification criteria: the reported time step should be stable (no sudden drops), contact penetration should stay below roughly 10% of the local element size, and the head center-of-gravity acceleration should show a physically plausible pulse rather than a spike. If the step collapses mid-run, locate the smallest element or the stiffest contact and address it directly.

2.5 Practical Implications for Pedestrian Impact Models

The choice between MB and FE is not binary; it is a resource allocation decision. Use MB where the injury metric depends on gross kinematics — pelvis–bonnet interaction, leg–bumper contact, torso rotation. Use FE where the metric depends on local deformation — head injury criterion (HIC) from skull/brain response, chest deflection from rib bending, knee ligament strain from local shear.

Two configuration details matter repeatedly in practice. First, the contact stiffness of MB ellipsoids must be tuned against the FE or test data they replace; an untuned k will produce either excessive penetration or an artificially stiff pulse. Second, the FE mesh must be checked for the smallest element before the run; a single 1 mm element in an otherwise 5 mm mesh can cut the global time step by a factor of five. Both issues are visible in the .out file and should be resolved before interpreting any injury metric.

With the solver architecture understood — explicit loop, MB and FE branches, contact penalty formulations, and global time-step control — the next step is to apply these principles to a concrete pedestrian model, which is the subject of the following chapter.