Probabilistic calibration of crystal plasticity material models with synthetic global and local data

Abstract: Crystal plasticity models connect macroscopic deformation with the physics of microscale slip in polycrystalline materials. These models can be calibrated using global stress-strain curves, but the resulting parametrization is often not unique: multiple parametrizations can predict the same global behavior but different local, grain-scale behavior. Using local data for calibration can mitigate uniqueness issues, but expensive specialized experiments like high-energy X-ray diffraction (HEDM) are typically required to gather the data. The computational expense of full-field simulations also often prevents uncertainty quantification with sampling-based calibration algorithms like Markov chain Monte Carlo. This study presents a two-stage calibration procedure that combines global and local data and balances the efficiency of a surrogate model with the accuracy of full-field crystal plasticity simulations. The procedure quantifies uncertainty using Bayesian inference with an efficient, parallelized sequential Monte Carlo algorithm. Calibrations are completed using synthetic data with a microstructure representative of Inconel 718 to assess uncertainty and accuracy of the parameters relative to a known ground truth. Global data comes from the uniaxial stress-strain curve, while local data comes from grain-average stresses, reflecting typical outputs of HEDM experiments. Additional calibrations with limited and noisy local data demonstrate robustness of the procedure and identify the most important features of the data. Overall, the results demonstrate the computational efficiency of the two-stage procedure and the value of local data for reducing parameter uncertainty. In addition, joint distributions of the calibrated parameters highlight key considerations in choosing constitutive models and calibration data, including challenges resulting from correlated parameters.

• The paper presents a two-stage probabilistic calibration method for crystal plasticity models using synthetic global and local data.
• The approach combines a neural network surrogate with a full-field crystal plasticity model to reduce uncertainty in key parameters such as the initial critical resolved shear stress (CRSS) and the rate sensitivity.
• The study found that sufficient local data could significantly reduce uncertainty in certain parameters, but identifying hardening parameters still posed challenges due to correlations.

Motivation and problem statement

Crystal plasticity (CP) models link macroscopic deformation of polycrystalline metals to slip at the grain scale, but their calibration is fraught with two well-known difficulties. First, calibration against a global uniaxial stress-strain curve alone is frequently non-unique: multiple parameter sets reproduce the same mean response while predicting substantially different local fields such as intragranular stress concentrations and fatigue indicator parameters. Second, resolving this non-uniqueness requires local data—typically grain-average stresses from high-energy X-ray diffraction microscopy (HEDM)—which is expensive to acquire and computationally costly to use in calibration, since full-field CP simulations make sampling-based Bayesian inference with Markov chain Monte Carlo (MCMC) generally intractable.

This study by Pribe and co-workers addresses both problems with a two-stage probabilistic calibration procedure. The first stage uses an inexpensive neural network surrogate trained on global mean-stress data to infer preliminary posteriors via sequential Monte Carlo (SMC). A kernel density estimate (KDE) of these posteriors then serves as an informative prior for the second stage, which employs the full-field CP model with both global and local (grain-average stress) data. SMC's embarrassingly parallel structure makes the second-stage Bayesian inference tractable on a compute cluster. All calibrations use synthetic data generated from a CP simulation with known ground-truth parameters representative of Inconel 718, enabling direct assessment of posterior accuracy and uncertainty.

Crystal plasticity model

The simulations use the small-strain Fourier-Galerkin spectral scheme implemented in NASA Langley's open-source Materialite code. The constitutive model is deliberately simple: a viscoplastic power-law flow rule over the twelve $\{111\}\langle 1\bar{1}0\rangle$ octahedral slip systems of FCC crystals, with a basic hardening/dynamic recovery law similar in form to Armstrong-Frederick-type models. Four material parameters are calibrated: the inverse rate sensitivity $m$, initial CRSS $g_0$, hardening coefficient $H$, and the ratio $g_0/g_1$ (the saturation stress $g_1 = H/H_d$). Notably, the authors replace the dynamic recovery coefficient $H_d$ with $g_0/g_1$ as a calibration parameter because independent priors on $H$ and $H_d$ can produce extreme, non-physical softening that destabilizes the solver; the ratio has intuitive interpretation (hardening below unity, softening above). The noise standard deviation $m$0 is also calibrated, giving five parameters total.

The microstructure derives from the open-source Inconel 718 dataset of Stinville et al., simplified by coarsening to 5 µm voxels, merging annealing twins into parent grains, and extracting a $m$1-voxel region surrounded by an elastic-perfectly-plastic buffer material. Uniaxial strain-controlled loading is applied to 1% mean strain.

The synthetic measurement data consist of 13 mean-stress values (global) plus loading-direction grain-average stresses for 32 interior grains at those same time points (416 local points), all polluted with Gaussian noise of 5 MPa standard deviation. Grain selection criteria—at least 5 voxels from any edge and volume ≥ 20 voxels—mimic HEDM experimental practice.

Sequential Monte Carlo and the two-stage procedure

SMC evolves a population of weighted particles from the prior through likelihood-annealed targets $m$2 toward the posterior, combining importance sampling, resampling, and short per-particle MCMC runs. Because particles evolve independently, likelihood evaluations parallelize across cores—a decisive advantage over serial MCMC. An adaptive scheme selects each $m$3 to maintain a target effective sample size.

In stage one, a multilayer perceptron surrogate maps the four material parameters to principal components of the 13 mean stresses (trained on 1000 Latin hypercube samples; four PCs explain >99.9% of variance; test RMSE ≈ 5.4 MPa, about 0.6% of the ~900 MPa yield stress). Stage-one calibration required only about 10 seconds on a laptop. In stage two, run on 644 CPU cores with 1288 particles, the full CP model was evaluated against both global and local data: over 100,000 CP simulations completed in about 90 hours walltime (~2400 days of aggregate CPU time)—a computation that would be impossible without SMC's parallelism.

A sensitivity analysis of the training data reveals why uniqueness is problematic: the mean stresses are strongly correlated with $m$4 but only weakly correlated with $m$5, $m$6, and $m$7, foreshadowing identifiability failures for the hardening parameters under global data alone.

Posterior results and the value of local data

After stage one (surrogate + global data), $m$8 and $m$9 are reasonably calibrated near ground truth, but $g_0$0 and $g_0$1 span essentially their entire prior ranges, and $g_0$2 is biased upward—the noise term compensating for surrogate-model discrepancy. After stage two, uncertainty in $g_0$3 and $g_0$4 contracts sharply (e.g., the 95% credible interval length for $g_0$5 shrinks from 5.27 to 0.48; for $g_0$6 from 23.0 MPa to 2.63 MPa), softening samples are entirely eliminated ($g_0$7 across all samples), and the $g_0$8 bias disappears.

However, the posteriors for $g_0$9 and $H$0 remain bimodal with a strong correlation structure resembling a "banana-shaped" distribution. Even with both global and local data, point estimates of these hardening parameters are unreliable, although the ground truth remains within the support of the joint posterior. This is arguably the paper's most consequential finding: parameter correlations can render unique identification of hardening parameters difficult or impossible regardless of calibration methodology, implying that constitutive model selection and experiment design should prioritize identifiability rather than relying solely on more or better data.

An additional observation concerns the rate sensitivity: local data significantly reduces uncertainty in $H$1 despite all measurements being taken at a single applied strain rate, which the authors attribute to grains experiencing different effective local strain rates.

Robustness to realistic local-data constraints

Two variants probe practical limitations of HEDM data:

• Reduced grain stresses: local data available at only four strain levels (128 instead of 416 points), mirroring the AFRL additive manufacturing modeling challenge.
• Increased noise: local noise doubled to 10 MPa, with separate global and local noise parameters inferred ($H$2).

Both cases preserve the sharply reduced uncertainty in $H$3 and $H$4. The increased-noise case introduces no bias, whereas the reduced-quantity case biases $H$5 and $H$6 slightly downward. The authors conclude that the quantity of local measurements may matter more than their precision—a claim they correctly flag as requiring further study with realistically characterized noise. Both degraded cases introduce notable low-side bias in $H$7, though the ground truth remains supported jointly with $H$8.

Comparison with one-stage calibrations

Three one-stage baselines clarify the trade-offs:

Approach	Key outcome	Cost
CP model, mean stresses only	Large uncertainty in $H$9, $g_0/g_1$0; hardening parameters unidentified	Highest relative cost
CP model, mean + grain-average stresses	Nearly identical posteriors to two-stage result	22 SMC steps, ~7.5 days
Surrogate on mean + grain-average stresses	Comparable uncertainty but significant bias in $g_0/g_1$1, $g_0/g_1$2, $g_0/g_1$3, $g_0/g_1$4	Negligible evaluation cost

The one-stage full-CP calibration with all data took roughly twice the walltime of the two-stage procedure (3.75 vs. 7.5 days) despite only ~22% more SMC steps, because informed priors eliminate computationally expensive, non-physical early-stage samples (in the first 72 hours, the two-stage second stage completed 14 SMC steps versus 8 for the one-stage run). Meanwhile, the surrogate-based calibration demonstrates the central pitfall of surrogates: test-set RMSE of ~6 MPa gave no warning of the substantial posterior bias, underscoring that average predictive accuracy does not guarantee unbiased inference. The two-stage design—surrogate for efficiency, full model for final inference—is therefore not merely convenient but necessary for accuracy.

Limitations and open questions

The paper is explicit about several caveats. All results rest on synthetic data generated by the same model form used for calibration, so model discrepancy is absent by construction; applying the Kennedy-O'Hagan discrepancy framework would likely exacerbate identifiability challenges. The microstructure was simplified (twins merged, buffer region introduced) in ways that real HEDM workflows may not permit, particularly for heavily twinned or columnar-grained additive-manufacturing microstructures. Only 1% strain is considered, and the authors concede this may simply be too shallow into the hardening regime to identify $g_0/g_1$5 and $g_0/g_1$6—though prior work with genetic-algorithm calibration using strains out to 40% found similar correlations, suggesting the difficulty may be intrinsic to the constitutive form rather than the strain range. Surrogate bias evidently propagated into the second-stage posterior (the spurious low-$g_0/g_1$7 mode), indicating the second-stage data were insufficient to fully overcome an overconfident prior. Algorithmically, SMC struggled with the correlated $g_0/g_1$8–$g_0/g_1$9 joint distribution; kernel-adaptive proposals or autodifferentiation-enabled gradient information could improve sampling, and little hyperparameter tuning was performed. Finally, the claim that local-data quantity matters more than precision rests on a single noise-magnitude comparison and awaits systematic investigation.

Conclusion

This work demonstrates a computationally tractable route to fully Bayesian, uncertainty-quantified calibration of full-field crystal plasticity models by combining a surrogate-informed first stage, a full-field second stage, and parallelized sequential Monte Carlo. The procedure reduces walltime by roughly half relative to single-stage inference while mitigating surrogate-induced bias, and it quantifies how strongly local grain-average data constrains the initial CRSS and rate sensitivity. Its equally important negative results—that correlated hardening parameters remain unidentifiable even with abundant local data, and that surrogate error can silently bias posteriors—argue for identifiability-aware constitutive model selection and physics-based parametrizations as complements to purely statistical calibration strategies.