Industrial-scale production applications of additive manufacturing (AM) are growing rapidly, and scalable AM production requires quality systems that monitor and control part geometric accuracy across multiple machines that operate within a factory. However, most published research on AM geometric accuracy focuses on a single part or a single set of hardware. This research develops a hybrid hierarchical modeling (HHM) approach to characterize the geometric accuracy of parts produced across multiple identical AM machines. Our approach organizes the geometric accuracy data into a hierarchy that represents data from individual parts, the positions of parts within the builds, and the machines that produced those parts. The part accuracy is modeled as the sum of a part-to-part trend and a small-scale variability at the feature level. A hierarchical Bayesian linear model accounts for this hierarchy and captures how the part-level accuracy depends upon each factor. The small-scale feature-level variability is modeled using Gaussian process (GP) regression. The effectiveness of the proposed method is demonstrated by case studies using experimental data collected from 70 polymer hexagonal lattice parts over seven builds produced by three identical Carbon M2 printers. Each part contains 237 walls, and the wall thickness is measured as the geometric feature. By leveraging data from different printers, the performance of the part-level geometric accuracy modeling is substantially improved compared with competing methods. The average root mean squared error (RMSE) of the part-level modeling is as small as 4.8 µm even when sampling only three of ten parts in one build, demonstrating excellent data efficiency. The part-level modeling method also demonstrates outstanding robustness against sampling randomness with the interquartile ranges of the prediction errors on most positions on a build smaller than 3 µm. The data efficiency is further improved by measuring selected geometric features from each sampled parts and utilizing feature-level modeling. HHM consistently achieves accurate feature-level prediction, with RMSE less than 10.1 µm when the feature measurement density is only 30%. On average, HHM permits a 0.62-µm reduction in RMSE compared to using part-level modeling only. HHM also provides fine-scale information about the statistical distribution of within-part geometric accuracy. We develop empirical rules for the part-level sampling design to improve the robustness of geometric accuracy modeling, which can reduce the RMSE by 78.8% compared with random sampling. The modeling approach is extensible to other types of AM and could be used as part of a quality system within AM factory.

Original languageEnglish (US)
Article number102621
JournalAdditive Manufacturing
StatePublished - Mar 2022


  • Additive manufacturing
  • Data-efficient learning
  • Gaussian process
  • Hierarchical Bayesian linear model
  • Hierarchical modeling
  • Metrology
  • Quality

ASJC Scopus subject areas

  • Biomedical Engineering
  • General Materials Science
  • Engineering (miscellaneous)
  • Industrial and Manufacturing Engineering


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