The Dimensionless Check: Auditing Units at the Seams of a Simulation Workflow

A model converged on the first pass. The stress contour sat comfortably below yield, the displacement plot looked like every other displacement plot, and the report went to the customer with a green check in the summary table. Three weeks later a prototype bracket cracked at roughly 60 percent of the predicted load. The geometry was right. The mesh was fine. The boundary conditions matched the test fixture. What went wrong was that a density value arrived from a supplier spreadsheet in pounds-mass per cubic inch and was entered into a field the solver interpreted as kilograms per cubic meter. The number was accepted without complaint. It always is.

Unit errors do not announce themselves. They produce smooth, converged, professional-looking results that happen to be wrong by a factor of 27,680. That is the specific cruelty of the problem: the software is doing exactly what it was told, and nothing in the output stream flags the fact that it was told in the wrong language.

A Number Without a Unit Is a Rumor

Most CAE tools are strict about syntax and permissive about semantics. They will reject a malformed decimal, a stray character, a node that belongs to no element. They will not reject a modulus entered in psi when the interface expects MPa, because both are real numbers and the solver has no opinion about your intentions.

Some codes sidestep this by being unitless by design. Explicit dynamics solvers in particular often ask the user to declare a consistent set: millimeters, tonnes, seconds, and newtons, or inches, slugs, and pounds-force. That freedom is efficient for experienced analysts and dangerous for everyone else, because consistency is a property of the whole set, not of any single entry. One field in the wrong system and the set is no longer consistent, even though each individual value looks plausible.

Other tools carry an internal unit system and convert on input. Those are safer but not safe. They convert what you type, not what you meant, and they cannot know that the spreadsheet column labeled density was populated in lbm/in³ while the header says kg/m³.

Three Seams Where Units Slip

The first seam is CAD to CAE. Native CAD files usually carry units explicitly; neutral formats often do not. An STL file has no unit field at all, which means the importer applies an assumption, usually millimeters. A part modeled in inches arrives at one twenty-fifth of its intended size, and if the analyst scales it back by eye, the scale factor may or may not propagate to every derived quantity.

The second seam is supplier to material card. This is the most common failure point in US supply chains, where a domestic job shop quotes in ksi and thousandths while the prime contractor's simulation group works in SI. A yield strength of 36 ksi is 248 MPa. A 5 mil shim is 0.127 mm, not 5 mm. An inch-pound torque spec converted with a rounded factor of 0.113 instead of 0.112985 introduces a small but real bias that compounds across a joint model.

The third seam is test to model. Test data arrives in whatever the rig recorded: strain in microstrain, acceleration in g, force in lbf, temperature in degrees Fahrenheit. Thermal expansion coefficients are a frequent casualty, since the difference between 1.2e-5 per degree Fahrenheit and 2.3e-5 per kelvin is a factor of 1.8 disguised as a similar-looking number. Specific heat in Btu/(lbm·°F) and thermal conductivity in Btu/(hr·ft·°F) have no intuitive feel for most analysts trained in SI, so they get copied without scrutiny.

The Mass-Weight Confusion, Quantified

The most expensive single mistake in US simulation practice is the conflation of pound-mass and pound-force. The two share a name, differ by a factor of 32.174 in the wrong context, and appear side by side in the same documents without comment.

Consider a bracket modeled at 2 lbm when the intent was 2 kg. The model is 4.4 times heavier than the hardware. For a structure dominated by mass, natural frequencies scale with the inverse square root of mass, so the first mode comes out roughly 2.1 times lower than reality. An analyst comparing that prediction to a modal test would see a discrepancy large enough to trigger a redesign of the mount, when the actual fix was a single unit field.

The related trap is g-coupling. In inch-pound-second systems, the gravitational constant of 386.09 in/s² belongs in the equations wherever mass and force meet. Drop it, or apply it twice, and you get errors of 32.2 in either direction. In SI the constant is 1 and vanishes from view, which is precisely why analysts who move between systems forget it exists.

A Unit Audit You Can Run in an Afternoon

Start with a written unit declaration for the model, one page, signed by the analyst and referenced in the report. It should state the consistent set for every input category: length, mass, force, time, temperature, stress, density, and any thermal properties. For unitless solvers, this document is not paperwork, it is the model definition.

Next, audit the boundary values rather than the interior ones. Check the three or four numbers that set the physical scale: total mass, applied load, stiffness of the softest component, and the dominant excitation frequency. If the total mass of the assembly is within a few percent of the bill of materials, the density chain is probably intact. If it is off by 4.4 or 27,680, you have found the seam in minutes instead of weeks.

Then run a deliberate scale check. Multiply one input by a known factor and confirm the output responds as theory predicts. Doubling mass should halve the square root of stiffness over mass. Doubling load should double stress. If the response does not track, the model is not dimensionally coherent regardless of how the numbers were labeled.

Finally, keep a conversion ledger with full-precision factors, not rounded ones, and cite the source. NIST publishes the exact definitions: 25.4 millimeters per inch, 4.4482216 newtons per pound-force, 6894.757 pascals per psi. Rounding 0.112985 to 0.113 is harmless once and corrosive across a hundred entries.

Guardrails That Survive Turnover

The real risk is not the analyst who knows the trap. It is the engineer hired in 2027 who inherits a template, a material library, and a folder of supplier PDFs with no indication of which system any of them used. Unit discipline has to live in the tooling: solver templates that carry declared units and refuse ambiguous ones, material libraries with the unit embedded in the field name rather than the column header, and import scripts that check overall bounding box dimensions against the drawing before anything is meshed.

None of this is exotic. It is the same discipline the shop floor applies when it checks a first article against a drawing, and it costs a fraction of a rebuild. The solver will never ask whether the number you gave it was in inches or millimeters. That question has to be asked upstream, in writing, by someone who owns the answer.