3. Mesh convergence metrics for crashworthiness

Mesh convergence in crashworthiness is not a single-number exercise. A model can show a stable global energy balance while still producing a 15% swing in a peak acceleration or a 20 mm shift in a toe-pan intrusion. The reason is that different response quantities are sensitive to different discretization errors: strain energy is dominated by the bulk of the deforming structure, intrusion is dominated by local bending and folding at a few elements, and acceleration peaks are dominated by contact stiffness and the highest-frequency modes the mesh can represent. A defensible acceptance decision therefore requires tracking a small set of metrics, each with its own convergence behavior and its own tolerance.

This section defines those metrics, explains what each one actually measures in a crash model, and shows how to extract and compare them across a mesh refinement study so that the acceptance decision rests on evidence rather than on a single global number.

Why one metric is never enough

Consider a full-frontal rigid-barrier impact of a body-in-white at 50 km/h. The internal energy absorbed by the front structure is a robust, globally integrated quantity: as the mesh refines, the total plastic dissipation converges quickly because it is an integral over the whole deforming volume, and local errors average out. Intrusion at the toe pan or the pedal box is a local displacement read at a specific node or a small set of nodes; it converges more slowly because it depends on the exact fold pattern of a few elements. The acceleration at the B-pillar base is a point measurement of a high-frequency signal; its peak value is extremely sensitive to element size, contact penalty stiffness, and time-step, and it may never converge in the strict sense — it will oscillate within a band as the mesh refines.

The practical consequence is that a mesh is "converged" only relative to the metrics you care about. If the acceptance criterion is occupant injury risk, the acceleration peak and its timing matter most. If the criterion is structural integrity or fuel-system intrusion, the intrusion and force–time history matter most. The convergence study must therefore be designed around the acceptance criteria, not around a generic "refine until nothing changes" rule.

The four core metrics

Strain energy (internal energy). In an explicit crash solver, the internal energy E_int is the sum of elastic strain energy and plastic dissipation. For a thin-walled front rail, the plastic dissipation dominates after the first few milliseconds. The convergence indicator is the total internal energy at the end of the event, plus the energy balance error (hourglass energy, contact energy, and the ratio of added mass to physical mass). A well-converged model typically shows hourglass energy below 5% of internal energy and added-mass ratio below 1–2%. If internal energy changes by less than about 3–5% between two refinement levels while the energy balance remains clean, the global energy response is considered converged.

Intrusion. Intrusion is a displacement measured at a defined location and direction — for example, the maximum rearward displacement of the toe pan relative to the rocker, or the pedal-box displacement along the vehicle X-axis. Because it is a local kinematic quantity, it is sensitive to element size in the folding zone. A common acceptance practice is to require the intrusion at the monitored location to change by less than 5% or less than 5 mm (whichever is larger) between successive refinement levels. The direction convention must be fixed before the study begins: define positive X as rearward, and report every intrusion value with that sign convention so that comparisons across runs are unambiguous.

Acceleration peaks. Acceleration is usually measured at a structural point (B-pillar base, sill, floor) and filtered according to the relevant standard — for example, SAE J211 Channel Frequency Class 60 (CFC 60) for occupant-relevant signals. The raw signal contains high-frequency content that the mesh cannot resolve consistently, so the peak must be evaluated on the filtered signal. Convergence is judged on the filtered peak magnitude and its time of occurrence. Because peaks are inherently noisy, a band criterion is appropriate: the filtered peak should fall within ±10% of the value from the next finer mesh, and the peak time within ±2 ms. If the peak keeps drifting outside that band, the mesh is not converged for this metric, and the cause is usually contact stiffness or a local element-size mismatch at the impact interface.

Force–time histories. The barrier contact force or the force transmitted through a specific section (for example, the section cut at the base of the A-pillar) is a time-history metric. Convergence is assessed by comparing the full curve, not just its peak: the initial slope, the timing of the first force drop, and the mean force level during the crush phase. A practical comparison uses the normalized integral difference between two curves, or a visual overlay with the envelope of the two curves. A force history that matches in peak but differs in the timing of the first fold will produce a different occupant deceleration pulse even if the peak values agree, so the timing of features matters as much as their magnitude.

Running the refinement study and reading the results

A workable study uses at least three mesh levels, typically with a target element size of 8 mm, 5 mm, and 3 mm in the front structure, while keeping the time-step consistent with the smallest element (the stable time-step scales with element size, so refining the mesh increases run time roughly with the cube of the refinement ratio). Keep the contact definitions, material cards, and boundary conditions identical across runs; change only the mesh. Extract the four metrics from each run and tabulate them.

The example values above are illustrative and are not measured results; they show the pattern a real study typically produces. Read them as follows. Internal energy moves from 42.1 to 43.6 kJ (3.6%) and then to 44.0 kJ (0.9%): the global energy response is converged by the medium mesh. Intrusion moves from 118 to 132 mm (11.9%) and then to 136 mm (3.0%): the coarse mesh is clearly inadequate, and the medium-to-fine change is within the 5% band, so intrusion is converged at the fine level. The acceleration peak moves 68 → 74 → 72 g: the coarse mesh under-predicts, and the medium-to-fine change is 2.7%, inside the ±10% band, so the peak is acceptable at the fine level. The first-fold time moves 21 → 18 → 18 ms: converged from medium onward.

The conclusion is not "use the fine mesh everywhere." It is that the medium mesh is adequate for global energy and force timing, while intrusion and acceleration require the fine mesh in the front structure. A common and cost-effective strategy is a hybrid: fine elements in the fold zones and at the contact interface, coarser elements in the rear structure where no significant deformation occurs. This preserves the convergence of the sensitive metrics while keeping the stable time-step and run time manageable.

Practical extraction notes

For strain energy, request the internal energy, hourglass energy, contact energy, and added-mass ratio as history variables and report their values at the end of the event, not at an arbitrary intermediate time. For intrusion, define the node set and the measurement direction before the study and keep them fixed; a node set that moves with the mesh refinement will not give comparable values. For acceleration, apply the filter in post-processing with the same filter implementation and order across all runs, and report both the peak magnitude and its time. For force histories, export the contact force or section force at a fixed output interval and overlay the curves from successive mesh levels; a difference in the timing of the first force drop is a reliable indicator that the fold pattern has not converged.

Finally, tie the acceptance decision to the purpose of the model. A concept-phase model used to rank design variants may accept the medium mesh if energy and force timing are converged. A model used to support a safety case or a regulatory submission should demonstrate convergence of the metrics that the regulation or the occupant criterion depends on — typically the filtered acceleration and the intrusion — at the finest level the program can afford, with the convergence evidence documented alongside the results.