Materials Science · Additive manufacturing and multiscale characterisation

Process models, data and validation

Useful background: Processing routes and microstructural evolution; Measuring and checking material claims. Each worked case explains the quantities it uses.

Back to the chapter

After reading, you can

Validate process predictions against quantities actually measured.
Derive and use one deliberately reduced thermal model, keeping its assumptions separate from a complete additive-manufacturing simulation.
Distinguish numerical verification, parameter fitting, an independent validation comparison and evidence for a particular intended use.
Match a model output to the location, time and averaging represented by a measurement.
Interpret held-out residuals with explicit error bounds, and identify additional evidence needed to distinguish competing explanations.

A process model represents selected physics

A model may predict temperature, melt-pool shape, solidification or stress. Inputs include material properties, heat sources and boundary conditions. A visually realistic output does not show that these representations match the machine and specimen.

Imagine that a simulated thermal history and a measured curve have the same peak. The model may still cool too quickly, place the hot region incorrectly or average over a different area. Agreement needs a defined quantity: what is predicted, at which location and time, under which conditions, and how the instrument turns the physical state into a reported reading.

Additive manufacturing creates material through a sequence of local processing events. An energy-input description, heat transport, changing geometry and contact with surrounding material help determine temperature history. Solidification, later reheating and mechanical constraint can then influence microstructure and residual stress, stress remaining without an external applied load. A model may represent only part of that chain.

A thermal model can be useful without resolving fluid motion or predicting stress, provided its claimed use respects those omissions. Conversely, resolving more equations does not guarantee that heat input, properties or boundary conditions are right. Begin with the decision and the observable, a quantity that can be compared with evidence.

Build a small model whose accounting can be checked

Consider an invented solid coupon after a heating source has been switched off. Assume its internal temperature is spatially uniform, the surrounding bath stays at T_b = 300 K, and no phase change, continuing heat generation or mechanical work contributes. Let C be its constant total heat capacity in J/K and G its constant thermal conductance to the bath in W/K. These are whole-system quantities, including the stated contact; G is not the material thermal conductivity.

Define excess temperature θ = T − T_b in K. The stored excess thermal energy in this model is Cθ. Heat leaves at rate Gθ. Conservation of energy therefore gives C dθ/dt = −Gθ. The sign makes a positive excess temperature decrease. Dividing by C gives dθ/dt = −θ/τ, with time constant τ = C/G in seconds because (J/K)/(J/s/K) = s.

Over a short time dt, the fractional change is dθ/θ = −dt/τ. Adding those fractional changes gives ln(θ/θ₀) = −t/τ, so θ(t) = θ₀ exp(−t/τ). The exponential is dimensionless, and T(t) = T_b + θ(t). At t = 0 the formula returns the imposed θ₀; as t grows it approaches the bath. The heat lost from 0 to t is C[θ₀ − θ(t)], consistent with the reduction in stored energy.

Use C = 0.20 J/K, G = 0.10 W/K and T(0) = 400 K as teaching inputs, so θ₀ = 100 K and τ = 2.00 s. They are not measured properties or settings for a real machine. At 2.00 s, θ = 100 exp(−1) = 36.7879 K and T = 336.7879 K. The elapsed time above a chosen 350 K threshold is 2 ln(100/50) = 1.3863 s, provided the observation starts at t = 0 and this single monotonic cooling event is the defined event.

A laser melt pool generally has strong gradients, moving heat input, fluid flow, changing phases and repeated heating. This one-temperature coupon is a reduced model for learning how to test a prediction; it is not a replacement for those process models. A uniform-temperature assumption needs evidence about internal versus external heat-transfer times, and constant C or G can fail when state or contact changes.

Check the calculation before fitting the physics

Numerical verification asks whether an implementation solves the chosen mathematical model with adequate accuracy. Validation asks how well its predictions agree with relevant observations for the stated use. A perfectly calculated wrong model can fail validation; a reasonable model calculated inaccurately can also disagree. Neither question is answered by an attractive plot.

Our cooling formula provides an exact reference for its own differential equation. A simple time-step method uses θ_(n+1) = θ_n[1 − Δt/τ], where Δt is the step duration. With τ = 2 s and Δt = 1 s, two steps predict θ(2 s) = 25 K, far below the exact 36.7879 K. Four half-second steps give 31.6406 K. Reducing the step has improved this calculation, but does not yet establish an error small enough for a chosen comparison. Check further against the exact reference; do not fit G to hide a large discretisation error.

Parameter calibration selects model values using designated data. Suppose an invented calibration record gives θ(0) = 100 K and θ(2 s) = 36.7879 K, with the exact unrounded value defined as 100 exp(−1). Then τ = −2/ln[θ(2)/θ(0)] = 2 s. Reproducing that same endpoint is expected after fitting it; it is not an independent validation result.

Only the ratio C/G has been identified by this cooling curve. C = 0.20 J/K with G = 0.10 W/K and C = 0.40 J/K with G = 0.20 W/K produce the same τ and the same temperatures for the same initial state. An independently determined C, or another observable sensitive to stored energy, is needed to infer G separately. Parameter values can be nonunique even when the entire fitted curve agrees.

Choose independent builds or conditions before looking at their results. Repeated frames from one cooling event share material, mounting, sensor calibration and history; randomly splitting those frames does not create independent builds. Preserve which data fitted parameters, which selected the model, and which were reserved for its evaluation. Once a reserved case is used to change the model, further independent evidence is needed to evaluate the changed candidate.

Original uniform-coupon model, not AM measurements. Excess temperature 100 exp(−t/2) K is compared on common time and temperature axes with explicit Euler nodes joined by lines. At 2 s the exact excess is 36.7879 K; steps 1 s and 0.5 s give 25 K and 31.6406 K. Different line styles and circle/square nodes distinguish the numerical approximations. Decreasing their error does not validate the assumed physical model.

Predict what the instrument actually observes

A numerical temperature field and a sensor reading can represent different quantities. A sensor can average over an area and exposure interval, lag behind a changing temperature, or need a conversion from signal to temperature. Its position and clock must align with the model. Apply the corresponding measurement representation to the predicted field before comparing the two.

For a deliberately simple example, a calibrated sensor reports the equal-weight average of two equal-area surface patches at one instant. Candidate A predicts 360 K and 340 K; candidate B predicts 350 K on both patches. Both predict the same sensor reading, 350 K. That agreement cannot identify the maximum temperature or prove spatial uniformity: their maxima are 360 K and 350 K. Resolving the two patches would discriminate between them. The average is imposed by this example, not a universal camera response law.

Infrared cameras respond to radiation. In a named 2020 AM-Bench thermography study, the reported radiance temperature required effective surface emissivity to infer physical temperature. A process-temperature prediction and a radiance-temperature image therefore could not simply be treated as identical. This is a measurement-model issue, distinct from improving the heat solver. Our invented coupon readings below are already calibrated physical temperatures; no radiometric conversion or real emissivity value is assumed for them.

Definitions of derived features also matter. Time above a threshold can mean the longest continuous interval or the sum of several intervals; these differ after reheating. Cooling rate can mean an instantaneous derivative or a specified temperature drop divided by the later-minus-earlier crossing time. State the choice, threshold, observation range and treatment of unobserved events. A benchmark comparison must use the same observable and extraction rule on both sides.

Read a discrepancy without inventing its cause

Freeze the analytically evaluated τ = 2 s model and use independent fictional coupons starting at 400 K in a 300 K bath. The table gives measured physical temperatures at specified instants. B repeats the intended mounting condition; C has a changed mounting condition. The readings are invented, and the two readings on each coupon are correlated through that coupon and its measurement setup.

Define residual r = measured − predicted in K. At 2 s the prediction is 336.7879 K; at 4 s it is 313.5335 K. For this teaching comparison, stipulate an absolute measurement error bound of 1.0 K and an independently established numerical prediction error bound of 0.2 K at each instant. The initial temperature, bath and fitted parameters are treated as exact to isolate the comparison; this is not a complete uncertainty budget for a real process.

If model structure and parameters describe the true response, the triangle inequality bounds |r| by 1.0 + 0.2 = 1.2 K. This is a worst-case bounded-error screen, not a statistical confidence test or an acceptance threshold for manufactured parts. The numerical term is an approximation error, not a freely adjustable allowance for missing physics.

B has residuals +0.2121 K and +0.4665 K, compatible with that screen. C has +13.2121 K and +11.4665 K, which the stipulated errors cannot explain. The frozen candidate therefore does not describe this changed-condition record within the supplied bounds. Passing B supports only this limited tested use; it does not establish every temperature, microstructure or stress prediction.

The residual does not uniquely reveal a physical cause. A changed contact conductance, a different initial state, a sensor offset or a wrongly registered clock can all alter a comparison. In this constructed C record, θ halves from 100 to 50 K in 2 s and then to 25 K in another 2 s, which is compatible with a larger τ = 2/ln 2 = 2.8854 s. That fit makes changed thermal loss one possible explanation; it does not independently prove that the mounting changed G. Check the actual initial state, sensor calibration, timing and contact evidence.

Invented held-out temperatures; each coupon begins at 400 K in a 300 K bath. Model parameters remain frozen.
Time after source off (s)	Model (K)	B · common mounting (K)	C · changed mounting (K)
2.00	336.7879	337.0	350.0
4.00	313.5335	314.0	325.0

Top: invented model fields A 360/340 K and B 350/350 K share an equal-patch mean 350 K but have different maxima. Filled circles and open squares identify them. Bottom is a separate held-out cooling comparison: measured-minus-predicted residuals for coupon B are 0.2121/0.4665 K at 2/4 s; coupon C gives 13.2121/11.4665 K. Common axes and a labelled ±1.2 K supplied bounded-error region show B compatible and C outside. The symbol letters refer to their own panels; neither comparison proves a unique cause or every model output.

Keep validation targets separate

Compare temperature with temperature evidence, stress with stress evidence and structure with structural evidence. A fit to one melt-pool image does not validate every downstream property. Reserve independent cases to examine transfer beyond fitted inputs.

A thermal-history comparison tests the chosen thermal observable. Residual elastic strain, released-part deflection and a direction-specific grain-size distribution need their own material state, location, averaging and measurement evidence. Inferring stress from diffraction also requires a reference lattice spacing and an appropriate mechanical interpretation; an image alone does not supply that inference.

The historical AM-Bench 2022 programme defined separate challenges for thermal features, residual elastic strain, part deflection, microstructure and phase evolution. The lesson is the separation of targets and methods, not that one benchmark certifies a whole model. A model may reproduce a mean feature while missing its spatial distribution; inspect residuals by location, layer, direction and condition as appropriate.

For an intended use, decide which errors would change the conclusion. A model useful for estimating a cooling interval may still be unsuitable for ranking local stress peaks. Report the tested conditions, fitted and reserved data, observable definition, known errors, discrepancies and permitted use. Agreement can support a bounded prediction; it cannot supply a material qualification, manufacturing recipe or unlimited extrapolation.

Worked examples

Work through the case

A model reproduces final shape. Does that establish correct residual-stress prediction?

Shape and stress are different outputs.
Different internal histories can yield similar shape.
Compare stress predictions with suitable measurements.
A thermal model may fit a heat-input parameter to shape while representing restraint or inelastic response incorrectly. Preserve that shape comparison, then reserve suitable strain/stress observations with specified components, positions, material state and uncertainty. A new fit to stress becomes calibration of the changed candidate, not retrospective validation of the original.

No. Each consequential output needs relevant evidence.

Fit a time constant, then test a reserved coupon

Use the calibration endpoint to find τ. With parameters frozen, decide what the 2 s reading on B can establish under the supplied error bounds.

Take θ₀ =400−300=100 K and the exact calibration θ₂=100 exp(−1) K. The ratio is exp(−1), giving τ=−2/ln[exp(−1)]=2 s.
The frozen model predicts T₂=300+100 exp(−1)=336.7879 K for B. Its reading 337.0 K gives r=+0.2121 K.
The largest allowed combined discrepancy is 1.0+0.2=1.2 K by the triangle inequality. The residual is within that screen.
B supports consistency of this specific temperature observation with the candidate and bounds. It does not separately identify C and G, validate stress or prove an untested build.

τ=2 s. The independent B observation is compatible with the stated temperature comparison; the fitted endpoint alone was not independent evidence.

Keep a consequential mismatch visible

C reads 350 K at 2 s and 325 K at 4 s. Quantify the mismatch and examine the proposed explanation “the conductance is lower”.

Subtract frozen predictions:350−336.7879=13.2121 K and 325−313.5335=11.4665 K. Both exceed 1.2 K, so the stipulated measurement/numerical bounds cannot explain them.
The excess readings are 50 K and 25 K. Their factor 1/2 over 2 s is compatible with τ=2/ln 2=2.8854 s, including the initial 100 K.
If the heat capacity independently remains 0.20 J/K and the same lumped/contact form is valid, this fitted time constant corresponds to G=C/τ=0.0693147 W/K.
This calculation is a conditional candidate explanation. Independent contact/sensor/state evidence is needed; using coupon C’s readings to fit a new G makes those readings calibration evidence and requires another reserved comparison.

The frozen model fails the supplied screen on C. Lower G is compatible under added assumptions, but the residual alone does not identify the cause.

An equal sensor output can conceal different fields

A sensor averages two equal-area patches. Model A gives 360/340 K and model B gives 350/350 K. The measured average is 350 K. Can that reading decide the peak or the thermal field?

Apply the actual stipulated sensor representation: A gives(360+340)/2=350 K; B gives(350+350)/2=350 K.
A has a 10 K deviation from the mean in each patch and maximum 360 K; B has zero deviation and maximum 350 K. The reported mean is the same.
The single average cannot distinguish those fields. A smaller averaging region or separately resolved patch readings would add discriminatory information, with its own calibration and spatial registration.

Both match the mean; their peak predictions differ by 10 K. Agreement with an average does not validate an unobserved peak.

Try the reasoning

Can sensor calibration affect a fitted model?
Show the answer and reasoning
Yes.
The calibration influences the target data. A shared positive temperature offset can shift many points together; treating those points as independent random errors understates that effect. Retain calibration and measurement-model evidence alongside fitted parameters.
Why retain independent builds?
Show the answer and reasoning
They test transfer beyond calibration cases.
Fitted agreement can hide overfitting. Frames from one build share history and system errors. A held-out build still needs conditions within the claimed domain and a comparison with the same observable.
A new ideal coupon has C=0.30 J/K, G=0.10 W/K, bath 290 K and initial 350 K. Predict temperature at 3 s and heat lost by then. Are these constants identified by that temperature alone?
Show the answer and reasoning
τ=3 s; T(3)=312.0728 K; heat lost=11.3782 J. A temperature trajectory alone identifies the ratio, not both constants.
θ₀=60 K, so θ₃=60/e=22.0728 K. Energy lost=C(60−22.0728)=11.3782 J. For example doubling both C and G leaves τ and all temperatures unchanged but doubles the stored/lost energy.
For the original τ = 2 s example, do two 1 s numerical steps at t=2 s meet the stipulated 0.2 K numerical bound? Does fitting G until its answer matches one reading verify that implementation?
Show the answer and reasoning
No. It predicts 325 K instead of 336.7879 K, an error of−11.7879 K; fitting cannot establish numerical accuracy.
The step multiplier is 1−1/2=1/2;100×(1/2)²=25 K excess. Verification must compare the computed solution with a reliable reference or convergence evidence for the chosen equation, independently of tuning parameters to measurements.
A reserved reading gives r=+2.5 K, with supplied absolute measurement bound 1.5 K and numerical bound 0.4 K; all other inputs are stipulated exact. Can those errors alone explain it? What if another reading gives r=+1.0 K?
Show the answer and reasoning
2.5 K exceeds the 1.9 K bound;1.0 K is compatible with it. Compatibility does not prove the model.
The worst-case sum is 1.5+0.4=1.9 K. An exceedance rejects agreement under this complete stipulated error description; it does not distinguish model form from a wrong error description or unrecognised setup mistake. A smaller residual can still conceal compensating errors.
Two modelled patches are 370 K and 330 K; the sensor again reports their equal average. Calculate its predicted reading and explain why a 350 K measurement cannot validate a 360 K peak claim.
Show the answer and reasoning
The predicted average is 350 K, while the modelled peak is 370 K. The mean does not observe or establish a 360 K peak.
Average=(370+330)/2=350 K. Infinitely many pairs share this mean; the peak needs resolved spatial evidence or justified additional structure. A reported comparison must name whether it concerns mean, maximum or another feature.
One observed temperature record is above a chosen threshold for 0.4 s, falls below it, then is above it again for 0.3 s. Find the longest continuous and cumulative durations. Is an output of 0.7 s equivalent to a benchmark target defined as the longest continuous interval?
Show the answer and reasoning
Longest continuous duration 0.4 s; cumulative duration 0.7 s. They are different observables.
The longest is max(0.4,0.3); the cumulative duration adds the disjoint intervals. Agreement needs a common rule, time window and handling of censored/unobserved events. Neither definition is universally preferable for every physical question.
A separately supplied sensor-check record predicts physical temperatures 337 K and 314 K but reports 347 K and 324 K. What simple measurement explanation is compatible, and does this prove the process model or its contact conductance?
Show the answer and reasoning
A constant+10 K sensor offset is compatible with both residuals. It proves neither the process model nor the conductance.
Both differences are 10 K. A calibration check using independent reference conditions could test the offset; another process change can also imitate a pattern over limited points. Correcting an assumed offset and refitting on the same data is not a new independent validation.

What to carry forward

Match prediction and validation quantities.
Separate fitting from independent checks.
A verified numerical solution can still represent incomplete physics; fitting parameters can hide either kind of error.
A sensor average, local value, radiation-derived reading and inferred material quantity need matching comparison definitions.
A bounded held-out discrepancy can rule out a stated error explanation while leaving several physical or measurement causes possible.
Report the tested domain and intended use; agreement for one observable does not validate every downstream property.

References

Teaching resources and research supporting this lesson’s explanations and examples.