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Amortized Posteriors for Estimation of Material Constitutive Parameters from Multimodal Measurements on Small Punch Tests

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper claims that a bottom-surface DIC displacement field added to the early force–displacement response of a small punch test sharply tightens Bayesian inference of Young's modulus and yield strength, shrinking 95% credible…

desk verdict A competent, transparent GP-CFM demonstration whose DIC-driven posterior contraction is plausible but rests on one specimen and an unvalidated FE model. read the letter →

arxiv 2607.24534 v1 pith:G44B4IOO submitted 2026-07-27 cond-mat.stat-mech cond-mat.mtrl-sci

Mohammad Ali Seyed Mahmoud , Aditya Venkatraman , Raj Mahat , Samantha Mitra , Surya R. Kalidindi This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.mtrl-sci
keywords AmortizedBayesianinferenceConditionalflowmatchingSmallpunchtestDigitalimagecorrelationConstitutiveparameteridentificationGaussianprocesssurrogateLikelihood-freeSimulation-basedcalibration
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that adding a full-field displacement measurement to the global force–displacement response of a small punch test (SPT) materially improves Bayesian identification of Young's modulus $E$ and yield strength $\sigma_y$ from the early loading portion. The authors build a likelihood-free, amortized inference pipeline in which Gaussian process surrogates replace finite element simulations and a Conditional Flow Matching model learns the posterior $p(E,\sigma_y \mid \text{features})$ from synthetic parameter–observation pairs. Applied to an AA6111-T4 aluminum specimen, force–displacement features alone give broad posteriors; including three principal-component scores of the bottom-surface DIC displacement field at the stiffness-peak displacement $D_{peak}$ shrinks the 95% HPD width from 39.8 to 9.8 GPa for $E$ and from 57.0 to 22.7 MPa for $\sigma_y$. The posterior medians land at 70.03 GPa and 156.34 MPa, close to independently measured tensile values of 70 GPa and 157 MPa. The broader significance is a template for calibrating constitutive parameters from multimodal mechanical data without constructing a joint likelihood.

What carries the argument

The central mechanism is a two-stage amortized likelihood-free posterior estimator. A Gaussian process surrogate maps the three inputs $(E,\sigma_y,t)$ to five compact features, and a Conditional Flow Matching network transports a standard normal reference distribution to the conditional posterior $p(E,\sigma_y \mid y,t)$ by regressing a velocity field along a linear interpolation between reference and target samples. The flow is trained on 210 finite element simulations augmented each epoch with 1000 GP-generated pairs that include both surrogate interpolation variance and feature-level measurement noise. Because the CFM conditioning vector can mix modalities directly, no joint likelihood or explicit cross-modality covariance across the force–displacement and DIC measurements is required; at inference, 20,000 posterior samples are generated by one ODE integration per specimen.

What would settle it

Run the same pipeline on one or more additional materials with independently measured tensile properties but different $E$ and $\sigma_y$ combinations, and check whether the 95% HPD intervals from the DIC-conditioned posterior contain the tensile reference values in repeated blind tests; a systematic miss rate well above 5% would indicate finite element model bias contracted into an overconfident posterior. A complementary check is to compare finite element predicted and DIC-measured displacement fields at stages beyond $D_{peak}$, where the elastic-perfectly-plastic model is expected to fail, to expose the sign and magnitude of model-form error.

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Extended reading notes

Core claim

The central claim is that the spatial displacement field measured by stereoscopic DIC on the specimen's bottom surface carries information about elastic stiffness and yield strength that the global force–displacement curve alone cannot resolve in the early SPT response. In the paper's framework, each simulation is reduced to five features: the coefficient $A$ of the power-law fit $F=AD^{1.15}$, the truncation displacement $D_{peak}$, and the first three PCA scores of the out-of-plane displacement field at $D_{peak}$. Conditioning the CFM posterior on all five features instead of only the two force–displacement features reduces the 95% credible interval for $E$ by a factor of 4.1 and for $\sigma_y$ by a factor of 2.5, while moving the posterior medians onto the independent tensile reference values. The paper also argues that this inference is statistically calibrated on held-out simulations and internally consistent with both measured modalities, while stating explicitly that these checks do not validate the finite element model as an independent representation of the experiment.

Load-bearing premise

The load-bearing premise is that the finite element model—with its elastic-perfectly-plastic constitutive law, fixed Poisson ratio and friction coefficient, and chosen contact and boundary conditions—faithfully represents the real small punch test in the early response regime, so the synthetic training distribution is not systematically biased against the experimental measurement.

Editorial extensions

If this is right

  • If the central claim holds, bottom-surface DIC at $D_{peak}$ is a practical way to break the $E$–$\sigma_y$ coupling in the early SPT response, enabling joint identification without a separate tensile test for the elastic modulus.
  • The amortized estimator makes per-specimen inference nearly instantaneous after training, so the same pipeline can be applied to many specimens once the finite element and training costs are paid.
  • The compact five-feature representation lets a three-input GP surrogate be trained on 210 simulations, suggesting that similar identifiability gains could be obtained for other miniaturized tests with modest simulation budgets.
  • Posterior predictive checks show the multimodal posterior remains consistent with the measured force–displacement curve and displacement field, with the $D_{peak}$ offset reduced from 0.30 to 0.07 µm.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The decisive open question is finite element model fidelity: if the elastic-perfectly-plastic model, fixed Poisson ratio, fixed friction coefficient, and contact setup are biased relative to the real test, the contracted DIC-conditioned posterior could be confidently wrong rather than merely more precise; the paper explicitly flags that its checks are internal consistency checks, not independent v
  • Because the PCA basis and $D_{peak}$ protocol are trained on simulations of one geometry and material family, the same features may not transfer to hardening materials or different specimen geometries without retraining; an obvious test is a blind calibration on a second alloy with known tensile properties.
  • One could probe the information content of the DIC field by ablating individual PC scores or by conditioning on fields at earlier or later displacement stages; the paper compares only the full three-PC set against no DIC.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript proposes an amortized, likelihood-free Bayesian inference framework that combines Gaussian process (GP) surrogates with Conditional Flow Matching (CFM) to estimate Young's modulus E and yield strength sigma_y from the early force-displacement response and the bottom-surface DIC displacement field of a small punch test (SPT). The pipeline is trained on 300 Abaqus finite element simulations of an elastic-perfectly-plastic SPT, using a compact five-feature representation: two F-D features (a power-law coefficient A and the truncation displacement D_peak) and three PCA scores of the DIC displacement field at D_peak. The paper reports that F-D features alone give broad 95% HPD intervals of 39.8 GPa for E and 57.0 MPa for sigma_y, while adding the three DIC PC scores contracts these to 9.8 GPa and 22.7 MPa, and shifts the posterior medians (70.03 GPa, 156.34 MPa) close to independently measured tensile reference values (70 GPa, 157 MPa). The GP surrogate is assessed on 90 held-out FE simulations with mean NMAE 0.99% for the reconstructed F-D curves and worst-case field MAE 0.0046 micron, and the F-D+DIC posterior is assessed with simulation-based calibration and posterior predictive checks. The paper explicitly acknowledges that these checks do not independently validate the FE model against experiment.

Significance. If the central claim holds, the framework is a useful methodological contribution: it addresses a real bottleneck in multimodal constitutive calibration, namely the difficulty of specifying a joint likelihood across modalities with different units, dimensionalities, and noise structures, and it avoids repeated MCMC at inference time. The GP-CFM combination is well matched to the problem, and the compact five-feature representation is a sensible way to make the surrogate tractable with 300 FE simulations. The paper has genuine strengths: the GP surrogate is tested on fully held-out FE simulations; SBC is reported with empirical coverage curves and rank histograms; the DIC displacement accuracy is characterized; and the authors are unusually candid about the limits of their validation. However, the experimental demonstration rests on an FE model that is not independently validated, and the comparison between the two inference scenarios is missing a calibration check for the F-D-only baseline. These gaps limit the strength of the headline claim that DIC materially improves identifiability of E and sigma_y from the early SPT response.

major comments (3)
  1. [Section 4.3 / Table 5] The SBC calibration is reported only for the F-D+DIC estimator, and the paper states that a parallel assessment of the F-D-only baseline is not included. Because the headline quantitative claim is a comparison of posterior widths between the two scenarios (Table 4), the F-D-only posterior should be subjected to the same empirical-coverage and rank-based diagnostics on the same 90 held-out cases. Without this, one cannot determine whether the broad F-D-only posterior and the reported contraction factors of 4.1 and 2.5 reflect genuine identifiability differences or partly an artifact of the F-D-only CFM's calibration, and the comparison is incomplete.
  2. [Section 4.4 / Figure 13] The posterior-mean DIC field comparison is not an independent predictive check: the reconstructed field is built from the posterior mean of the PC scores that were used as conditioning inputs, so the good qualitative agreement largely checks internal centering of the CFM rather than the ability of the FE model to predict the measured field. A stronger and more informative check would compare the FE-predicted displacement field at the posterior median, or at the tensile reference parameters, directly with the experimental DIC field, and report the spatial distribution of residuals. As written, this check does not provide evidence about FE model fidelity.
  3. [Sections 3.1, 3.5, and 5] The central experimental claim, that adding the DIC field contracts the posterior by factors of 4.1 and 2.5 and shifts the medians to the tensile reference, rests on the FE model in Section 3.1 being an unbiased description of the real SPT in the early-response regime. The paper correctly states in Section 5 that the checks 'do not constitute an independent validation of the FE model,' but the abstract and conclusions still present the experimental case as a demonstration of the modality benefit. With one SPT specimen and one tensile reference, the proximity of the multimodal posterior medians (E=70.03 GPa, sigma_y=156.34 MPa) to the tensile values (70 GPa, 157 MPa) could be coincidental if the FE model is biased at the operating point. The authors should add an independent FE-model validation at the reference parameters (comparing simulated F-D and DIC fields with the measured ones, including sensitivity to the fixed Poisson ratio and friction coefficient), or explicitly reframe the experimental component as an illustrative proof-of-concept rather than a demonstration.
minor comments (5)
  1. [Section 2.2.3 / Table 2] The rigid-body validation of the DIC system was performed at imposed displacements of 0.5-1.5 mm, while the SPT out-of-plane displacement field has a peak of about 3 microns; the reported accuracy at the millimeter scale is not directly informative at the operating scale. A validation step at the micron scale, or a discussion of how the 0.06 micron noise floor was established at that scale, would strengthen the feature-noise model.
  2. [Section 4.3] The rank histograms are presented without uncertainty bands or a quantitative uniformity test. With N=90 held-out cases, bin-to-bin variation is expected to be substantial, and a flat-looking histogram is a weak check; adding pointwise credible bands or a formal SBC rank test would make the diagnostic more interpretable.
  3. [Section 3.2.1] The power-law exponent n is reported to vary only between 1.14 and 1.16 and is then fixed at 1.15. Since the extracted feature A depends on the chosen exponent, the paper should report how sensitive A is to n within this narrow range, or justify that the resulting variation is negligible relative to the feature-level noise and GP surrogate error.
  4. [Section 4.4 / Table 6] The posterior predictive offset for D_peak is reduced from 0.30 to 0.07 microns when DIC features are added, but the measured D_peak is itself obtained from a smoothing-differentiation-extraction protocol. Reporting the uncertainty of the extracted experimental D_peak under that protocol would help assess whether the remaining offset is significant.
  5. [Data availability statement] The data availability statement says 'Data will be made available on request.' For a computational framework whose reproducibility depends on the FE dataset, GP surrogate, PCA basis, and trained CFM weights, providing the code and trained models in a public repository would substantially increase the value of the paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the DIC posterior contraction is a genuine held-out-model result, the tensile reference is external, and the closed-loop SBC/PPC checks are explicitly labeled as internal-consistency checks, not independent FE validation.

full rationale

The derivation chain is self-contained and the central comparative claim is not circular. In Section 4.2, the two CFM models are trained on FE-generated parameter–feature pairs, and the DIC PC scores used for conditioning come from independently measured displacement fields projected onto a PCA basis fit on FE outputs. The tensile reference values are used only for comparison and are not exposed to the inference, as stated in Section 2.1: 'These values are used only as reference properties for comparison with the inferred posteriors; these were not exposed in any manner to the Bayesian inference protocol.' The GP surrogate is evaluated on 90 held-out FE simulations in Section 4.1, and SBC in Section 4.3 uses held-out parameter sets, so the surrogate and estimator are tested out-of-sample at the FE level. The only self-referential element is that SBC and PPC draw synthetic observations from the same GP-based observation model used in CFM training; however, the paper explicitly discloses this limitation in Section 3.5: 'Because the synthetic observations are generated from the same GP-based observation model used during CFM training, SBC assesses the calibration of the CFM estimator under that surrogate-based generator. It cannot detect GP bias relative to the FE outputs,' and Section 5 concedes that these checks 'do not constitute an independent validation of the FE model.' This is an acknowledged validation limitation rather than a hidden circular step: no parameter is fitted to the quantity later presented as a prediction, and no result is forced by definition. Self-citations to prior work (e.g., reference [20] for meshing strategy and sequential-updating observations) are methodological precedents, not load-bearing uniqueness arguments. Consequently, no circular step is present.

Assumptions & free parameters 6 free parameters · 10 assumptions · 0 invented entities

The central inference is a simulation-based posterior estimator. It rests on FE model fidelity, fixed noise variances, and several feature-compression choices. The ledger records the GP and CFM weights, the noise variance constants, the PCA truncation, the fixed power-law exponent, and the D_peak extraction protocol as the main fitted or chosen quantities, and records the FE and statistical modeling assumptions separately.

free parameters (6)
  • GP hyperparameters (40 total) = Not reported; estimated by maximizing marginal log-likelihood
    Five single-output GPs, each with mean weights and bias, three ARD lengthscales, and kernel variance, fitted on 210 FE simulations. These determine the GP predictive means and variances used to generate CFM training data.
  • CFM VelocityNet weights = Not reported; trained for 2000 epochs with Adam
    The neural velocity field is the posterior estimator. Its weights are fitted on the 210 base FE pairs plus 1000 GP-augmented synthetic pairs per epoch, so the learned posterior depends on them.
  • Feature-level noise variances nu_k = Fixed constants; DIC from 0.06 micron noise floor, F-D from sensor uncertainties, minimum floor 0.01 in normalized units
    Used in Eq. 13 and Eq. 17 to inject measurement noise into synthetic training observations. The widths of the learned posteriors depend on these constants, and the conservative floor is an author choice.
  • Number of retained PCA components = 3 components (more than 99% variance)
    Section 3.2.2. The 139,129-dimensional DIC field is compressed to three PC scores; this truncation discards spatial variation that could in principle carry parameter information.
  • Power-law exponent n = 1.15
    Section 3.2.1. The exponent is fixed at 1.15 after observing a narrow range of 1.14 to 1.16 across simulations, so each F-D curve is represented by the fitted coefficient A and D_peak.
  • D_peak extraction protocol = Savitzky-Golay order 3, window 11, lower cutoff 0.5 microns
    Section 3.1. Smoothing and cutoff choices affect D_peak, which is used both as a truncation point and as a conditioning feature for posterior inference.
assumptions (10)
  • domain assumption Elastic-perfectly-plastic constitutive model with von Mises yield and associated flow applies in the early SPT regime.
    Invoked in Section 3.1. This deliberately excludes hardening and restricts inference to E and sigma_y before through-thickness plastic coalescence; hardening could matter if it activates before D_peak.
  • domain assumption The Abaqus FE model faithfully represents the experimental SPT geometry, contact, clamping, and friction.
    Sections 2 and 3.1. The paper states in Section 5 that consistency checks do not constitute independent validation of the FE model, so this fidelity assumption is load-bearing for the contracted posterior.
  • domain assumption Poisson ratio 0.30 and friction coefficient 0.10 are acceptable fixed constants.
    Section 3.1 reports less than 1% sensitivity in FE outputs when nu varies over 0.27-0.33 and mu over 0.01-0.10, making this a mild assumption.
  • domain assumption Uniform priors on E (60-200 GPa) and sigma_y (120-500 MPa) cover the plausible material range.
    Section 3.4, Eq. 12. The prior bounds coincide with the training ranges, so the posterior support is constrained by design and cannot extend outside the simulated domain.
  • domain assumption Feature-level measurement noise is independent, Gaussian, and fixed at specified variances.
    Eq. 13 and Section 3.4. This diagonal noise model is used only as a stochastic training-data generator, and cross-modality noise correlations are not modeled.
  • domain assumption The GP surrogate accurately emulates the FE feature map.
    Supported by held-out test metrics, mean NMAE 0.99% and worst-case field MAE 0.0046 microns, but the GP is still an approximation and is used to generate augmented CFM training pairs.
  • domain assumption Three PCA components capture the informative DIC displacement variation.
    Section 3.2.2. The three retained components account for more than 99% of training variance; spatial details beyond the third component are discarded.
  • ad hoc to paper D_peak from the maximum stiffness is a reliable proxy for through-thickness plastic coalescence and a valid truncation point.
    Section 3.1. The stiffness-maximum criterion is observed in simulations rather than derived, and the paper notes that a fixed-displacement cutoff produced degraded estimates.
  • domain assumption The power-law form F = A D^1.15 with fixed exponent represents the truncated F-D curves.
    Section 3.2.1. Mean R^2 is 0.99 and the exponent varies only from 1.14 to 1.16, so fixing the exponent introduces a small but nonzero approximation.
  • standard math Conditional flow matching learns the conditional posterior from the synthetic training distribution.
    Section 3.4. The amortized posterior estimator inherits the standard convergence and consistency properties of flow matching as a simulation-based inference method.

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Pith. "Pith review of Amortized Posteriors for Estimation of Material Constitutive Parameters from Multimodal Measurements on Small Punch Tests." pith.science (2026). https://pith.science/paper/G44B4IOO

@misc{pith2026260724534,
  author       = {Pith},
  title        = {Pith review of: Amortized Posteriors for Estimation of Material Constitutive Parameters from Multimodal Measurements on Small Punch Tests},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G44B4IOO}},
  note         = {Machine review of arXiv:2607.24534}
}
abstract

Bayesian calibration of material constitutive parameters from multimodal mechanical test data is often limited by the need to specify a joint likelihood across measurement modalities that differ in dimensionality, noise structure, and physical units. The resulting posteriors are often broad or strongly correlated, causing standard Markov Chain Monte Carlo (MCMC) samplers to mix poorly. Here, we present an amortized, likelihood-free framework that combines Gaussian process (GP) surrogates with Conditional Flow Matching (CFM) to learn conditional posteriors over constitutive parameters directly from synthetic multimodal parameter--observation pairs, avoiding hand-crafted likelihoods and repeated MCMC sampling. Once trained, the GP--CFM model generates posterior samples for each new specimen at negligible cost. The utility of this novel approach is demonstrated in this paper by estimating the values of Young's modulus and yield strength from the early portion of the force--displacement ($F$--$D$) curve and a Digital Image Correlation (DIC)-based displacement field measured in a Small Punch Test (SPT). It is observed that the $F$--$D$ data alone produce broad posteriors, consistent with limited parameter discrimination in the global response. Adding the DIC-measured displacement field was seen to contract the posteriors and shift them towards the independently measured tensile reference values. This work establishes a robust likelihood-free framework for the inference of material constitutive parameters from multimodal data, demonstrated through SPT--DIC integration.

Figures

Figures reproduced from arXiv: 2607.24534 by the authors.

Figure 1
Figure 1. Parameter coupling in the early SPT response. (a) Simulated early force–displacement responses for three [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Experimental setup: small punch test fixture with the direct-view stereoscopic DIC system. The punch and die [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Overview of the two-stage inference workflow. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Finite element model of the SPT specimen: (a) Boundary conditions. The punch reference point (RP) is loaded [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Truncation of the F–D curve for the estimation of E and σy. The force F (black, left axis) and the stiffness dF/dD (red, right axis) are plotted against punch displacement. The displacement Dpeak corresponding to the maximum in the dF/dD was used to truncate the F–D cu…
Figure 6
Figure 6. Figure 6: Parity plots of GP-predicted versus FE-extracted values for all five retained features on the 30% test set. The [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: GP surrogate F–D reconstruction quality on the 30% test set. (a) Best-case (left) and worst-case (right) reconstructions: the predicted curve (dashed) agrees closely with the FE output (solid) in the best case; the worst-case NMAE reaches 1.41% with the largest discrep…
Figure 8
Figure 8. Figure 8: Worst-case GP surrogate reconstruction of the DIC displacement field [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Joint posterior distributions for Young’s modulus [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Marginal posterior distributions for E (left) and σy (right) using F–D features only and combined F–D plus DIC features. Reference values from independent tensile tests are indicated by dashed vertical lines. The DIC features contract the marginal posteriors for both …
Figure 11
Figure 11. Figure 11: Posterior calibration diagnostics for the [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]
Figure 12
Figure 12. Figure 12: Posterior predictive checks for the F–D curve under the two inference scenarios. The 95% credible band (shaded) and median curve (solid blue) are obtained by propagating 20 000 posterior samples through the GP surrogate; the measured curve is shown in black. Red and b…
Figure 13
Figure 13. Figure 13: Comparison of the experimentally measured DIC out-of-plane displacement field [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]

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Works this paper leans on

43 extracted references · 30 canonical work pages

  1. [1]

    M. P. Manahan, A. S. Argon, O. K. Harling, The development of a miniaturized disk bend test for the determination of postirradiation mechanical properties, Journal of Nuclear Ma- terials 103–104 (1981) 1545–1550.doi:10.1016/0022-3115(82)90820-0

    work page doi:10.1016/0022-3115(82)90820-0 1981

  2. [2]

    T. E. García, C. Rodríguez, F. J. Belzunce, C. Suárez, Estimation of the mechanical prop- erties of metallic materials by means of the small punch test, Journal of Alloys and Com- pounds 582 (2014) 708–717.doi:10.1016/j.jallcom.2013.08.009

    work page doi:10.1016/j.jallcom.2013.08.009 2014

  3. [3]

    Seyed Mahmoud, G

    S. Seyed Mahmoud, G. Faraji, M. Baghani, M. Hashemi, A. Sheidaei, M. Baniassadi, Design of refractory alloys for desired thermal conductivity via ai-assisted in-silico mi- crostructure realization, Materials 16 (2023) 1088

    work page 2023

  4. [4]

    ASTM International, ASTM E3205-20: Standard test method for small punch testing of metallic materials, Tech. Rep. ASTM E3205-20, ASTM International, West Conshohocken, PA (2020)

    work page 2020

  5. [5]

    M. A. Sutton, J.-J. Orteu, H. W. Schreier, Image Correlation for Shape, Motion and Defor- mation Measurements, Springer, New York, 2009.doi:10.1007/978-0-387-78747-3

    doi:10.1007/978-0-387-78747-3 2009

  6. [6]

    B. Pan, K. Qian, H. Xie, A. Asundi, Two-dimensional digital image correlation for in-plane displacement and strain measurement: A review, Measurement Science and Technology 20 (6) (2009) 062001.doi:10.1088/0957-0233/20/6/062001

    doi:10.1088/0957-0233/20/6/062001 2009

  7. [7]

    Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation, Soci- ety for Industrial and Applied Mathematics, Philadelphia, PA, 2005

    A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation, Soci- ety for Industrial and Applied Mathematics, Philadelphia, PA, 2005

    work page 2005

  8. [8]

    Kaipio, E

    J. Kaipio, E. Somersalo, Statistical and Computational Inverse Problems, Springer, New York, 2005

    work page 2005

Show all 43 references
  1. [9]

    Leclerc, A

    N. Leclerc, A. Khosravani, S. Hashemi, D. B. Miracle, S. R. Kalidindi, Correlation of measured load-displacement curves in small punch tests with tensile stress-strain curves, Acta Materialia 204 (2021) 116501.doi:10.1016/j.actamat.2020.116501

    2021

  2. [10]

    E. N. Campitelli, P. Spätig, R. Bonadé, W. Hoffelner, M. Victoria, Assessment of the consti- tutive properties from small ball punch test: Experiment and modeling, Journal of Nuclear Materials 335 (3) (2004) 366–378.doi:10.1016/j.jnucmat.2004.07.052. 29

    2004 doi

  3. [11]

    Abendroth, M

    M. Abendroth, M. Kuna, Identification of ductile damage and fracture parameters from the small punch test using neural networks, Engineering Fracture Mechanics 73 (2006) 710– 725.doi:10.1016/j.engfracmech.2005.10.007

    2006 doi

  4. [12]

    Calaf-Chica, P

    J. Calaf-Chica, P. M. Bravo Díez, M. Preciado Calzada, Improved correlation for elastic modulus prediction of metallic materials in the small punch test, International Journal of Mechanical Sciences 134 (2017) 112–122.doi:10.1016/j.ijmecsci.2017.10.006

    2017 doi

  5. [13]

    V . D. Vijayanand, M. Mokhtarishirazabad, J. Peng, Y . Wang, M. Gorley, D. M. Knowles, M. Mostafavi, A novel methodology for estimating tensile properties in a small punch test employing in-situ DIC based deflection mapping, Journal of Nuclear Materials 538 (2020) 152260.doi:1...

    2020

  6. [14]

    Venkatraman, D

    A. Venkatraman, D. L. McDowell, S. R. Kalidindi, Bayesian analysis of parametric uncer- tainties and model form probabilities for crystal plasticity models of lamellar alpha+beta titanium alloys, International Journal of Plasticity 154 (2022) 103289.doi:10.1016/j. ijplas.2022.103289

    2022

  7. [15]

    Girolami, B

    M. Girolami, B. Calderhead, Riemann manifold langevin and hamiltonian monte carlo methods, Journal of the Royal Statistical Society: Series B (Statistical Methodology) 73 (2) (2011) 123–214.doi:10.1111/j.1467-9868.2010.00765.x

    2011

  8. [16]

    Beskos, M

    A. Beskos, M. Girolami, S. Lan, P. E. Farrell, A. M. Stuart, Geometric MCMC for infinite- dimensional inverse problems, Journal of Computational Physics 335 (2017) 327–351. doi:10.1016/j.jcp.2016.12.041

    2017 doi

  9. [17]

    R. M. Neal, MCMC using hamiltonian dynamics, in: S. Brooks, A. Gelman, G. L. Jones, X.-L. Meng (Eds.), Handbook of Markov Chain Monte Carlo, CRC Press, Boca Raton, FL, 2011, pp. 113–162

    2011

  10. [18]

    Venkatraman, C

    A. Venkatraman, C. E. Johnson, D. L. McDowell, S. R. Kalidindi, Bayesian protocols for high-throughput identification of kinematic hardening model forms, International Journal of Mechanical Sciences 285 (2025) 109791.doi:10.1016/j.ijmecsci.2024.109791

    2025

  11. [19]

    D. E. Ricciardi, D. T. Seidl, B. T. Lester, A. R. Jones, E. M. C. Jones, Advancements in constitutive model calibration: Leveraging the power of full-field DIC measurements and in situ load path selection for reliable parameter inference, Strain 61 (4) (2025) e70008. doi:10.11...

    2025 doi

  12. [20]

    M. A. Seyed Mahmoud, D. Renner, A. Khosravani, S. R. Kalidindi, Sequential bayesian inference of the GTN damage model using multimodal experimental data, Acta Materialia 306 (2026) 121902.doi:10.1016/j.actamat.2026.121902

    2026

  13. [21]

    C. E. Rasmussen, C. K. I. Williams, Gaussian Processes for Machine Learning, MIT Press, Cambridge, MA, 2006

    2006

  14. [22]

    A. R. Castillo, S. R. Kalidindi, A bayesian framework for the estimation of the single crystal elastic parameters from spherical indentation stress–strain measurements, Frontiers in Materials 6 (2019) 136.doi:10.3389/fmats.2019.00136. 30

    2019

  15. [23]

    Lipman, R

    Y . Lipman, R. T. Q. Chen, H. Ben-Hamu, M. Nickel, M. Le, Flow matching for generative modeling, International Conference on Learning Representations (2023)

    2023

  16. [24]

    M. S. Albergo, E. Vanden-Eijnden, Building normalizing flows with stochastic interpolants, International Conference on Learning Representations (2023)

    2023

  17. [25]

    Cranmer, J

    K. Cranmer, J. Brehmer, G. Louppe, The frontier of simulation-based inference, Pro- ceedings of the National Academy of Sciences 117 (48) (2020) 30055–30062.doi: 10.1073/pnas.1912789117

    2020 doi

  18. [26]

    Papamakarios, E

    G. Papamakarios, E. Nalisnick, D. J. Rezende, S. Mohamed, B. Lakshminarayanan, Nor- malizing flows for probabilistic modeling and inference, Journal of Machine Learning Re- search 22 (57) (2021) 1–64

    2021

  19. [27]

    Kukiełka, R

    L. Kukiełka, R. M. Patyk, Ł. Bohdal, A. M. Radchenko, S. Legutko, Modeling and exper- imental analysis of shear-slitting of AA6111-T4 aluminum alloy sheet, Materials 13 (14) (2020) 3175.doi:10.3390/ma13143175

    2020 doi

  20. [28]

    Moradi, W

    M. Moradi, W. N. Khan, E. Bitar-Nehme, Étienne Martin, Investigations on printability and sintering of additively manufactured binder jet co-cr-mo alloy parts for biomedical applications, Journal of Manufacturing Processes 170 (2026) 138–152.doi:https:// doi.org/10.1016/j.jmap...

    2026 doi

  21. [29]

    Sarkar, T

    J. Sarkar, T. R. G. Kutty, D. S. Wilkinson, J. D. Embury, D. J. Lloyd, Tensile properties and bendability of T4-treated AA6111 aluminum alloys, Materials Science and Engineering: A 369 (1–2) (2004) 258–266.doi:10.1016/j.msea.2003.11.022

    2004 doi

  22. [30]

    Dassault Systèmes Simulia Corp., Abaqus Analysis User’s Guide, Dassault Systèmes Simu- lia Corp., Vélizy-Villacoublay, France, 2022nd Edition (2022)

    2022

  23. [31]

    M. D. McKay, R. J. Beckman, W. J. Conover, A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics 21 (2) (1979) 239–245.doi:10.1080/00401706.1979.10489755

    1979

  24. [32]

    M. F. Ashby, Materials Selection in Mechanical Design, 4th Edition, Butterworth- Heinemann, Oxford, 2011

    2011

  25. [33]

    Calaf-Chica, P

    J. Calaf-Chica, P. M. Bravo Díez, M. Preciado Calzada, Development of an improved pre- diction method for the yield strength of steel alloys in the small punch test, Materials & Design 148 (2018) 153–166

    2018

  26. [34]

    Hähner, C

    P. Hähner, C. Soyarslan, B. Gülçimen Çakan, S. Bargmann, Determining tensile yield stresses from small punch tests: A numerical-based scheme, Materials & Design 182 (2019) 107974.doi:10.1016/j.matdes.2019.107974

    2019

  27. [35]

    Savitzky, M

    A. Savitzky, M. J. E. Golay, Smoothing and differentiation of data by simplified least squares procedures, Analytical Chemistry 36 (8) (1964) 1627–1639.doi:10.1021/ ac60214a047

    1964

  28. [36]

    S. Wold, K. Esbensen, P. Geladi, Principal component analysis, Chemometrics and Intelli- gent Laboratory Systems 2 (1–3) (1987) 37–52.doi:10.1016/0169-7439(87)80084-9. 31

    1987 doi

  29. [37]

    J. R. Gardner, G. Pleiss, D. Bindel, K. Q. Weinberger, A. G. Wilson, GPyTorch: Blackbox matrix-matrix gaussian process inference with GPU acceleration, in: Advances in Neural Information Processing Systems, V ol. 31, 2018, pp. 7576–7586

    2018

  30. [38]

    Sherki, I

    D. Sherki, I. Oseledets, E. Muravleva, Bayesian inverse problems meet flow matching: Efficient and flexible inference via transformers (2025).arXiv:2503.01375

    2025 arXiv

  31. [39]

    Dasgupta, A

    A. Dasgupta, A. Fardisi, M. Aminy, B. Binder, B. Shaddy, S. Moazami, A. A. Oberai, Solving physics-constrained inverse problems with conditional flow matching (2026). arXiv:2603.14135

    2026 arXiv

  32. [40]

    Gelman, X.-L

    A. Gelman, X.-L. Meng, H. Stern, Posterior predictive assessment of model fitness via realized discrepancies, Statistica Sinica 6 (4) (1996) 733–807

    1996

  33. [41]

    Talts, M

    S. Talts, M. Betancourt, D. Simpson, A. Vehtari, A. Gelman, Validating bayesian inference algorithms with simulation-based calibration (2018).arXiv:1804.06788

    2018 arXiv

  34. [42]

    Z. S. Courtright, A. Venkatraman, B. Yucel, V . S. K. Adapa, A. Diaz, S. R. Kalidindi, High-throughput experiments and machine learning strategies for efficient exploration of additively manufactured Inconel 625, Acta Materialia 288 (2025) 120875.doi:10.1016/ j.actamat.2025.120875

    2025

  35. [43]

    D. P. Kingma, J. Ba, Adam: A method for stochastic optimization, International Conference on Learning Representations (2015). 32

    2015

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.