2. MADYMO solver architecture for side impact

Side impact simulation in MADYMO is not a single monolithic solver. It is a hybrid computational environment in which rigid multi-body (MB) dynamics, explicit finite element (FE) structural analysis, and their coupling coexist within one time-stepping loop. Understanding how these three formulation families are organized, how they exchange data, and how the integrator advances them is the prerequisite for building a side impact model that is both stable and physically credible. This section examines the solver architecture from the inside out: the formulation types, the time integration scheme, and the contact algorithms that dominate runtime and accuracy in lateral loading.

Formulation Families and Their Roles

MADYMO distinguishes between three formulation classes, each suited to a different physical regime:

• Multi-body (MB) systems — rigid bodies connected by kinematic joints (revolute, translational, spherical, universal, free) and restrained by force-based elements such as springs, dampers, and contact interactions. MB bodies carry mass and inertia but do not deform. They are computationally cheap and are the natural choice for the occupant skeleton, dummy limbs, and any component whose deformation is negligible compared to its rigid-body motion.
• Finite element (FE) structures — deformable continua discretized into shell, solid, beam, or membrane elements. In side impact, FE is used for the door trim, B-pillar, door inner panel, armrest, seat frame, and the deformable barrier face. FE captures local buckling, folding, and energy absorption that MB cannot represent.
• Coupled MB–FE systems — the two formulations interact through contact interfaces and kinematic constraints. A typical side impact model couples an MB dummy (or human body model) against FE door and pillar structures, with the FE barrier striking the FE door outer panel.

The choice is not arbitrary. A component should be modeled as FE when its deformation mode governs the injury outcome — for example, the door trim that the pelvis and thorax strike. It should be modeled as MB when its motion is predominantly rigid and its internal deformation is secondary — for example, the dummy's upper arm or the seat cushion frame. Over-modeling with FE increases cost without improving the injury prediction; under-modeling with MB loses the force–deflection characteristic that determines chest deflection.

Time Integration

MADYMO uses an explicit central-difference time integration scheme for both MB and FE subsystems. The equations of motion are advanced in the form:

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

where M is the mass matrix, a the nodal or body acceleration, F_ext external loads, F_int internal (element) forces, and F_contact contact forces. The scheme is conditionally stable: the time step must satisfy the Courant–Friedrichs–Lewy (CFL) condition, which for shell elements is approximately

Δt ≤ L_min / c,   c = sqrt(E / ρ)

with L_min the smallest element characteristic length, E the Young's modulus, and ρ the density. For a typical steel door panel with E = 210 GPa, ρ = 7850 kg/m³, and a 5 mm shell, the wave speed is about 5170 m/s and the stable step is roughly 1 µs. MADYMO reports the actual critical step at initialization; if the requested step exceeds it, the solver either reduces the step automatically or terminates with a stability error.

Two practical consequences follow. First, mass scaling is often applied to small or stiff elements to raise the critical step, but it must be limited — a common guideline is to keep added mass below 1–2% of the total model mass, otherwise inertia effects contaminate the force–deflection response. Second, the MB subsystem is integrated with the same global step, so a very small FE step forces the entire model to run slowly. This is why side impact models frequently use a subcycling or mixed-step strategy, where the FE domain advances at its critical step while the MB domain advances at a larger step, with contact forces exchanged at the finer interval.

Contact Algorithms

Contact is the dominant nonlinearity in side impact. MADYMO provides several contact formulations, and selecting the right one is as important as mesh quality:

All contact types use a penalty formulation by default: when penetration δ occurs, a restoring force F = k · δ is applied, where k is the contact stiffness. Too low a stiffness allows visible interpenetration and under-predicts force; too high a stiffness drives the stable time step down and can cause high-frequency noise. MADYMO allows the contact stiffness to be scaled, and a common starting point is to set it so that the maximum expected penetration is a small fraction of the element thickness — for a 1 mm trim panel, keeping penetration below 0.1 mm is a reasonable target.

Friction is defined per contact pair with a Coulomb model. In side impact, the door trim–thorax interface typically uses a friction coefficient around 0.3–0.5, while the barrier–door interface may be higher to represent the gripping effect of the barrier face. These values should be treated as calibration parameters, not universal constants.

Practical Implications for Model Setup

The architecture has direct consequences for how a side impact model should be built and debugged. Three checks are worth performing before any production run:

1. Verify the critical time step. Run a short initialization (or a 1 ms dummy run) and read the reported stable step. If it is far below the intended step, locate the smallest element and either remesh or apply controlled mass scaling. A model that silently runs at 0.2 µs instead of 1 µs costs five times more than necessary.
2. Check contact penetration. Request contact force and penetration output for the primary interfaces (pelvis–trim, thorax–trim, barrier–door). Penetration exceeding roughly 10% of the local element thickness indicates the penalty stiffness is too low; oscillating contact force with high-frequency content indicates it is too high.
3. Confirm energy balance. The sum of kinetic, internal, and contact energy should track the external work within a few percent. A growing energy residual usually points to unstable contact or an excessively scaled mass.

In summary, the MADYMO solver for side impact is best understood as a coupled explicit system: MB provides efficient rigid-body kinematics, FE provides deformable structural response, contact transfers force between them, and a single conditionally stable integrator advances the whole model. Mastery of this architecture — particularly the time step and contact stiffness — is what separates a model that runs from a model that predicts.