3. Solution verification in crashworthiness

Verification answers a narrower question than validation: not "does the model match the physical test?" but "is the numerical solution of the equations we wrote down accurate enough to trust?" In occupant safety work this distinction matters because restraint models — airbags, seatbelts, pretensioners, dummy FE meshes — routinely produce plausible-looking animations that are numerically under-resolved. A dummy chest deflection curve can look smooth while the underlying shell mesh is too coarse to capture rib bending, or while a reduced-integration element is quietly dissipating energy through hourglassing. Solution verification is the discipline that catches these failures before they contaminate a design decision.

This chapter treats four verification activities as one connected workflow: mesh convergence, time-step sensitivity, energy balance auditing, and hourglass control assessment. They are not independent checkboxes. A mesh refinement changes the stable time step; a smaller time step changes how much hourglass energy accumulates per unit of simulated time; and the energy audit is the instrument that reveals whether either change actually improved fidelity. The figure below shows how these four checks feed one another and where each one terminates in a pass/fail decision.

Mesh convergence: refining until the answer stops moving

Mesh convergence is the practice of refining characteristic element size until the quantities of interest (QOIs) change by less than a stated tolerance. For an occupant model, the QOIs are not stress contours but the injury-relevant scalars: peak chest deflection, chest deflection rate, HIC₃₆, peak belt force, and head acceleration resultant. Choose these before you start refining, because a mesh that converges chest deflection may still be under-resolved for belt force.

A practical procedure for a driver-side frontal sled model: start with a baseline characteristic length of 8 mm on the dummy thorax shell and 10 mm on the seat structure, then refine to 5 mm, 3.5 mm, and 2.5 mm. Keep the contact thickness and the belt shell mesh consistent with the dummy surface, otherwise contact penetration will dominate the error. At each level, record the QOIs in a table and compute the relative change between successive levels. A common acceptance criterion is that the change in each QOI falls below 5% between the two finest levels; if the change is still 12% from 3.5 mm to 2.5 mm, the model is not converged and the 2.5 mm result should not be used for design decisions.

Two cautions are worth stating explicitly. First, convergence is QOI-specific and load-case-specific: a mesh converged for a 50 km/h full-frontal impact is not automatically converged for an oblique pole test, where local bending of the ribcage and shoulder loading dominate. Second, refinement interacts with the time step. Halving the element size halves the stable time step for that part, so the 2.5 mm model may run four times slower than the 5 mm model. Budget the refinement study accordingly, and consider running the coarse levels with mass scaling only for the convergence trend, never for the final reported result.

Time-step sensitivity and mass scaling

LS-DYNA computes the stable time step from the element characteristic length and the material sound speed, Δt = L/c, scaled by the time-step safety factor TSSFAC (default 0.9 for solid elements). For a steel seat frame with c ≈ 5000 m/s and L = 5 mm, Δt ≈ 1 µs; a 120 ms event therefore needs roughly 120,000 cycles. Time-step sensitivity asks whether the chosen Δt is small enough that the solution is not changing with further reduction. Run the same model at TSSFAC 0.9, 0.6, and 0.45 and compare contact force histories and QOIs. If peak contact force changes by more than a few percent between 0.9 and 0.6, the contact algorithm is being stepped too coarsely and the smaller factor should be adopted.

Mass scaling is the usual reason a model appears to run "fast enough." Selective mass scaling adds nonphysical mass to small elements so their stable time step rises to a target value. The verification obligation is to quantify it: the added mass reported in the glstat output must stay below about 1% of the total model mass, and it must not be concentrated in parts that carry the QOIs. Adding mass to a 2 mm belt shell or a thin dummy rib will change the inertia of exactly the component you are trying to verify. A defensible workflow is to run the baseline without mass scaling, confirm the QOIs, then introduce scaling only if the runtime is prohibitive and re-confirm that the QOIs shift by less than the convergence tolerance.

Energy balance audits

The energy audit is the single most informative verification check in an explicit crash model, because it detects errors that no contour plot will show. LS-DYNA writes the global energy summary to glstat and the per-part hourglass and internal energy to matsum; sliding interface energy is in sleout. The audit has four parts:

• Total energy drift. The sum of kinetic, internal, sliding, hourglass, and damping energy should remain approximately constant after the initial impact, apart from energy removed by prescribed motion or damping. A drift above roughly 5% of the initial kinetic energy indicates a numerical problem, often excessive hourglassing or contact energy generation.
• Hourglass-to-internal-energy ratio. For each part, hourglass energy should stay below about 5% of that part's peak internal energy. A dummy thorax foam or a belt element exceeding this threshold is a red flag that the reduced-integration formulation is not controlling the zero-energy modes.
• Sliding energy. Sliding energy must be non-negative and bounded; large negative values point to penetration or an inconsistent contact definition, while unbounded growth suggests contact chattering.
• Added mass. Report the mass-scaling percentage from glstat alongside the energy results, since scaling changes the kinetic energy budget itself.

Hourglass control assessment

Reduced-integration shell and solid elements (ELFORM 1 and its solid counterparts) are the default in crash models because they are fast, but they admit zero-energy deformation modes. Hourglass control suppresses these modes either through viscous damping (IHQ 1–3) or stiffness-based control (IHQ 4–5). The choice is not cosmetic: viscous control can add significant artificial damping that lowers peak forces, while stiffness control is more physically consistent but can stiffen the response if the coefficient is too high.

Assess hourglass control by running a controlled comparison on the parts that matter most for occupant loading — the dummy ribcage, the belt webbing, and the seat foam. Compare ELFORM 1 with IHQ 4 against a fully integrated formulation (ELFORM 2 for shells, ELFORM -1 or -2 for solids) on the same mesh. If the QOIs agree within the convergence tolerance and the hourglass energy ratio stays below 5%, the reduced-integration model with its chosen control is verified. If the fully integrated model produces a noticeably different peak chest deflection, the reduced-integration result is being contaminated and the hourglass coefficient or formulation must be revised. Note that fully integrated elements are much stiffer in bending and much more expensive, so they serve here as a verification reference, not necessarily as the production model.

A compact verification record

The following table is a reusable record for a single load case. Fill one row per refinement or configuration and keep it with the model deck so the verification is reproducible.

The numbers above are illustrative and represent a plausible trend, not measured results from a specific model. The pattern to look for is the one they encode: successive refinement reduces the QOI change until it falls below tolerance, the time-step check confirms the baseline Δt is not the limiting error, and the fully integrated reference confirms that hourglass control is not distorting the answer. When all three conditions hold and the energy audit is clean, the model is verified for that load case and can proceed to validation against physical test data. If any condition fails, the feedback path in the figure applies: refine, reduce the time step, or change the hourglass formulation, then re-run the audit before drawing any conclusion about occupant protection performance.