Coupled Model Correlation, Sensitivity & Uncertainty

Coupled Models Have More Ways to Match the Wrong Answer

A final displacement or temperature can agree with test even when errors in different disciplines cancel. An overpredicted electrical loss combined with an overpredicted convection coefficient may produce the correct temperature for the wrong reasons; an incorrect force distribution can be masked by an incorrect damping value. Correlation should therefore be hierarchical. Compare independent observables within each discipline and at each coupling interface before tuning the final structural response. The objective is not to minimise one residual but to demonstrate that the physical chain is credible.

Define the Evidence Hierarchy Before Calibration

Useful electromagnetic evidence may include current, voltage, resistance, torque, flux or force. Thermal evidence includes power balance, temperature at several locations and transient time constants. Structural evidence includes modal frequencies/shapes, static displacement, strain, support reaction, vibration order and clearance. Map each parameter to the observables it can physically influence. This prevents a thermal boundary coefficient from being tuned to compensate for electromagnetic loss error or a damping value from being adjusted to fix an incorrect force amplitude.

Use Sensitivity to Identify What Actually Matters

Perturb uncertain inputs over credible ranges and rank their effect on the engineering acceptance quantity. Candidate parameters include conductivity, magnetic B–H data, contact resistance, convection, thermal contact, CTE, modulus, joint stiffness, bearing stiffness, damping, preload, geometric gap and duty cycle. Local derivative methods can be efficient for smooth problems; design-of-experiments or global methods are more robust when interactions or nonlinearity are strong. The purpose is to direct evidence gathering toward parameters capable of changing the decision.

Separate Aleatory Variation From Knowledge Uncertainty

Manufacturing tolerance and unit-to-unit variation are different from lack of knowledge about a material curve or boundary condition. The first may need to be represented statistically or by tolerance cases; the second may be reduced by better data or testing. Combining both into one arbitrary safety factor obscures what the analysis knows. Even when a full probabilistic study is unnecessary, explicitly labelling variation versus epistemic uncertainty improves review and supports rational margins.

Correlation Should Respect Measurement Uncertainty

Sensors have accuracy, bandwidth, placement and installation effects. Thermocouples can disturb small thermal systems; strain gauges measure local surface strain at a finite footprint; accelerometers add mass and can miss high-frequency content; current probes and torque transducers have their own limits. Define the measurement uncertainty and compare like-with-like quantities. A model prediction inside the experimental uncertainty band may be sufficiently correlated even if it does not pass exactly through the measured central value.

Parameter Identification Needs Physical Bounds

Automated optimisation can tune convection, contact conductance, damping and stiffness to match data, but a mathematically excellent fit may require physically implausible values. Constrain parameters using independent evidence and preserve correlations between them where known. Avoid identifying too many parameters from too few observables. If several parameter combinations fit equally well, the model is non-identifiable and should not claim precise calibrated values; report the family of plausible models or gather additional measurements.

Propagate Uncertainty Across Interfaces

Uncertainty does not reset when a field is mapped. Variation in electromagnetic force amplitude becomes uncertainty in structural response; uncertainty in loss distribution and cooling becomes uncertainty in temperature and hence thermal stress. A practical approach is to propagate a small set of bounding or sampled coupled cases chosen from the sensitivity analysis. When the receiving model is linear, response surfaces can reduce cost; near contact changes, saturation or resonance, direct coupled cases are safer because the response can be strongly nonlinear.

Use Model Discrepancy, Not Just Parameter Uncertainty

Some errors come from the model form itself: homogenised windings, simplified contact, omitted leakage paths, linear material properties, one-way coupling or reduced support geometry. Parameter sweeps cannot fully represent these omissions. Compare alternative model forms for assumptions that could affect the decision. The difference between a detailed and reduced load transfer, for example, can be treated as model-form evidence and incorporated into the reported uncertainty or conservatism.

Validation Should Be Relevant to the Intended Use

A model correlated at low current and room temperature is not automatically validated for a high-current hot fault if saturation, resistance or contact state changes. Define the validation domain in terms of the variables that matter: current, frequency, temperature, speed, gap, duty and structural response range. Extrapolation beyond that domain should be explicit and supported by sensitivity or additional evidence. The model credibility argument should be proportional to the consequence of the engineering decision.

Use Prediction Error Metrics That Reflect the Intended Engineering Decision

Correlation quality should be judged using metrics relevant to the model use. Frequency error and MAC are suitable for modal models; amplitude and phase error matter for harmonic response; transient rise time and decay constant matter for thermal models; integrated force or torque matters for electromagnetic transfer. For a clearance decision, error in relative displacement may matter more than local stress correlation. Establish acceptance bands for these observables before tuning and report residual bias after correlation. If the remaining model error is comparable with the available design margin, carry that discrepancy into the substantiation rather than declaring the model validated solely because trends look similar.

Engineering Outcome

A credible uncertainty and correlation programme makes the coupled model harder to fool. It shows which parameters drive the decision, calibrates only quantities with independent evidence, propagates uncertainty through each interface and distinguishes parameter variation from model-form limitations. The result is not merely a closer test match; it is a defensible statement of what the multiphysics model can and cannot predict.

Do not tune the final structural response first. Correlate upstream physics and interfaces so agreement cannot be produced by compensating errors.

Key takeaways

• Correlate hierarchically across electromagnetic, thermal and structural observables.
• Use sensitivity to focus testing on uncertainties capable of changing the decision.
• Report model-form limitations separately from uncertain parameter values.