3. Mesh Convergence Study for Injury Metrics

Why Mesh Convergence Matters for Injury Metrics

Injury criteria such as peak von Mises stress in cortical bone, maximum principal strain in brain tissue, or HIC-derived acceleration peaks are all sensitive to element size. A single run at one mesh density tells you nothing about whether the reported peak is a physical feature of the model or an artifact of discretization. A mesh convergence study establishes that the response quantity of interest (QoI) has stabilized as element size decreases, and it produces the evidence needed to justify the mesh density used for all subsequent validation runs.

The practical difficulty is that injury models are rarely uniformly refined. A head model may have 2 mm hexahedral elements in the cerebrum, 0.5 mm elements in the corpus callosum, and shell elements in the skull. Refining everything uniformly is computationally wasteful; refining only the region around the expected peak may miss a peak that migrates as the mesh changes. The study design must therefore define a refinement strategy, a convergence metric, and an acceptance band before any runs are launched.

Defining the Study: QoIs, Refinement Levels, and Convergence Metric

Start by fixing the response quantities. For a body-region impact model, a defensible set is:

• Peak von Mises stress in the load-bearing cortical region (MPa).
• Peak maximum principal strain in the soft-tissue region of interest (dimensionless).
• Injury criterion value — for example, a strain-based criterion or a head injury criterion computed from the same run.

Next, choose a refinement ladder. A geometric progression with ratio r ≈ 1.3–1.5 in characteristic element size is standard because it produces a roughly constant change in degrees of freedom per step. For a model with a 2.0 mm baseline, a four-level ladder is 2.0 → 1.5 → 1.1 → 0.8 mm. Each level should be generated by a consistent meshing recipe (same element type, same transition rules, same hourglass control) so that the only variable is density.

The convergence metric should be a relative change between successive levels:

delta_i = |Q_i - Q_(i-1)| / |Q_(i-1)| * 100%

where Q_i is the QoI at refinement level i. Convergence is declared when delta_i falls below a pre-set tolerance — commonly 5% for stress and strain peaks in injury models, tighter (2–3%) when the QoI feeds a pass/fail threshold directly. The tolerance must be written down before the runs, not chosen after seeing the results.

Refinement Strategy and the Peak-Migration Problem

Uniform refinement of the whole body region is the cleanest reference but often unaffordable. A two-tier strategy works well in practice: refine the hot-spot region (the subdomain where the peak is expected, identified from the baseline run) at the full ladder, and refine the remainder of the model at a single intermediate density held constant across all levels. This isolates the convergence behavior of the QoI while keeping runtime tractable.

The risk is peak migration: as the hot-spot mesh is refined, the true peak may move to a location that was coarse in the baseline. Guard against this by extracting the peak location at every level and checking that it stays within the refined subdomain. If it moves outside, expand the subdomain and restart the ladder from the level where migration was first observed.

Running the Ladder and Reading the Results

The example above uses a four-level ladder on a head-region model with a refined subdomain around the coup/contrecoup site. The numbers are illustrative but internally consistent: stress rises from 84.2 MPa at L0 to 96.4 MPa at L3, with the largest jump between L0 and L1 (8.9%) and the smallest between L2 and L3 (1.4%). The strain and injury-criterion QoIs follow the same monotonic trend, which is the expected signature of a well-behaved convergence study. Non-monotonic behavior — a QoI that rises then falls — usually indicates a mesh-quality problem, a contact definition that changes with density, or a peak that has migrated between subdomains.

Apply the tolerance rule mechanically. With a 5% band, L1→L2 (3.7% stress, 3.5% strain) already passes, and L2→L3 confirms it. The selected mesh is the coarser of the two levels that bracket the first passing comparison — here L2 at 1.1 mm — because it is the least expensive mesh that sits inside the convergence band. Selecting L3 would add 3× runtime for a 1.4% change in the QoI, which is not justified unless the injury criterion threshold is within that margin.

Documenting the Convergence Evidence

The deliverable is a convergence record that travels with the model into the validation campaign. It should contain, at minimum, the refinement ladder with element counts, the QoI table with relative changes, the tolerance and the selected level, the peak-location check, and the runtime figures. A compact form is shown below; it is designed to be pasted into the model validation report as a single table plus a short note.

Selection: L2 (1.1 mm characteristic size, ~1.05M elements). First level at which all three QoIs fall within the 5% relative-change band against the next finer level. Peak location confirmed inside the refined subdomain at all four levels. Runtime at L2 is 33% of L3, which keeps the downstream validation matrix (multiple load cases, multiple material variants) within the project compute budget.

Common Failure Modes and How to Diagnose Them

• QoI never converges: the peak sits on a geometric singularity (sharp re-entrant corner, point load). Refine locally or replace the singularity with a fillet; a global ladder will not fix it.
• QoI oscillates between levels: check element quality (Jacobian, negative volumes) and hourglass energy as a fraction of internal energy. If hourglass energy exceeds ~5% of internal energy at any level, the result is not trustworthy.
• Peak migrates outside the refined subdomain: expand the subdomain and restart from the level where migration first appeared. Do not patch the subdomain mid-ladder, because the resulting QoI series is no longer comparable.
• Contact penetration changes with density: contact stiffness and penalty scaling may need to be held constant in physical terms rather than left at defaults. Verify that penetration depth at the impact interface is stable across levels.

Once the convergence record is signed off, the selected mesh becomes the fixed discretization for every subsequent validation case. Any later change to geometry, material, or contact that alters the mesh must trigger a re-check of the convergence evidence — a mesh that was converged for one configuration is not automatically converged for another.