{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,2]],"date-time":"2026-09-02T05:19:31Z","timestamp":1788326371187,"version":"build-2803163510"},"reference-count":41,"publisher":"MDPI AG","issue":"17","license":[{"start":{"date-parts":[[2026,8,31]],"date-time":"2026-08-31T00:00:00Z","timestamp":1788134400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100018617","name":"LiaoNing Revitalization Talents Program","doi-asserted-by":"crossref","award":["YS2023004"],"award-info":[{"award-number":["YS2023004"]}],"id":[{"id":"10.13039\/501100018617","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Sequential tightening redistributes load among bolts and produces a residual preload field that depends on the complete operation history. This study presents a physics-informed graph-learning framework for a 60-bolt circular flange. The model receives initial preloads and tightening order and predicts the residual preload field on a complete causal directed graph. All 1770 admissible later-to-earlier edges are retained. Among them, 240 local edges carry non-zero Elastic Interaction Coefficient Method (EICM) attributes, while 1530 distant edges remain available to learned message passing. The multi-fidelity dataset combines online EICM responses, prescribed nonlinear-discrepancy labels, and repeated ultrasonic preload measurements. On the augmented development split, TransformerConv-L3 achieved an MAE of 138.3 N, an RMSE of 170.9 N, an NLL of 8.839, and 93.9% coverage for the nominal 95% prediction interval. Its RMSE was 56.9% lower than that of the strongest recurrent baseline evaluated under the same protocol. These results demonstrate the value of combining causal topology with mechanics-informed edge attributes for residual preload prediction.<\/jats:p>","DOI":"10.3390\/s26175534","type":"journal-article","created":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T03:19:26Z","timestamp":1788232766000},"page":"5534","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Physics-Informed Multi-Fidelity Graph Learning for Sequence-Aware Residual Bolt Preload Prediction in Hyperelastic-Sealed Flanges"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-9339-030X","authenticated-orcid":false,"given":"Junyi","family":"Wang","sequence":"first","affiliation":[{"name":"School of Instrumentation Science and Engineering, Harbin Institute of Technology, Harbin 150001, China"},{"name":"Institute of AI for Industries, Chinese Academy of Sciences, Nanjing 211135, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiubin","family":"Tan","sequence":"additional","affiliation":[{"name":"School of Instrumentation Science and Engineering, Harbin Institute of Technology, Harbin 150001, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yongmeng","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Instrumentation Science and Engineering, Harbin Institute of Technology, Harbin 150001, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiacheng","family":"Zhong","sequence":"additional","affiliation":[{"name":"Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110169, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"021004","DOI":"10.1115\/1.3042149","article-title":"Formulation of Elastic Interaction Between Bolts During the Tightening of Flat-Face Gasketed Joints","volume":"131","author":"Alkelani","year":"2009","journal-title":"J. Mech. Des."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/s40544-021-0497-1","article-title":"Review of Research on Loosening of Threaded Fasteners","volume":"10","author":"Gong","year":"2022","journal-title":"Friction"},{"key":"ref_3","first-page":"283","article-title":"Research on the Tightening Force Consistency of Aero-Engine Rotor Flange Bolt Connection","volume":"45","author":"Zhao","year":"2025","journal-title":"Robot. Intell. Autom."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Zhang, J., Wang, Y., Wang, J., Cao, R., and Xu, Z. (2023). Online O-ring Stress Prediction and Bolt Tightening Sequence Optimization Method for Solid Rocket Motor Assembly. Machines, 11.","DOI":"10.3390\/machines11030387"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"123","DOI":"10.5194\/ms-17-123-2026","article-title":"Sealing Performance of o-Ring Under Coupled Thermo-Mechanical Conditions","volume":"17","author":"Jing","year":"2026","journal-title":"Mech. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.jcsr.2019.01.016","article-title":"Research on Preload of Bolted Joints Tightening Sequence-Related Relaxation Considering Elastic Interaction Between Bolts","volume":"160","author":"Li","year":"2019","journal-title":"J. Constr. Steel Res."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Zhong, J., Bao, J., Wang, J., Xu, Z., Yang, M., and Shi, Y. (2025). A New Single-Step Bolt Tightening Method for Multi-Bolt Flange Structures. Buildings, 15.","DOI":"10.3390\/buildings15010002"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2009092","DOI":"10.1080\/23311916.2021.2009092","article-title":"Effect of Ultrasonic Coupling Media and Surface Roughness on Contact Transfer Loss","volume":"9","author":"Zuljan","year":"2022","journal-title":"Cogent Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1016\/j.ymssp.2016.05.016","article-title":"Ultrasound-Based Measurement of Liquid-Layer Thickness: A Novel Time-Domain Approach","volume":"82","author":"Praher","year":"2017","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"109029","DOI":"10.1016\/j.engappai.2024.109029","article-title":"Two-Stream Bolt Preload Prediction Network Using Hydraulic Pressure and Nut Angle Signals","volume":"136","author":"Xu","year":"2024","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1115\/1.2929252","article-title":"An Improved Flange Bolt-up Procedure Using Experimentally Determined Elastic Interaction Coefficients","volume":"114","author":"Bibel","year":"1992","journal-title":"J. Press. Vessel Technol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1115\/1.1613304","article-title":"Finite Element Simulation of Bolt-up Process of Pipe Flange Connections with Spiral Wound Gasket","volume":"125","author":"Fukuoka","year":"2003","journal-title":"J. Press. Vessel Technol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"012072","DOI":"10.1088\/1742-6596\/2174\/1\/012072","article-title":"Methodology for Optimization of Preload in a Bolted-Flange Connection Based on Markov Theory","volume":"2174","author":"Wang","year":"2022","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Yang, B., Zeng, Z., Ng, C.T., and Kotousov, A. (2026). Data-Driven Generalization for Guided Wave Damage Identification Across Beam-Like Structures. Struct. Health Monit., OnlineFirst.","DOI":"10.1177\/14759217251409192"},{"key":"ref_15","unstructured":"Battaglia, P.W., Hamrick, J.B., Bapst, V., Sanchez-Gonzalez, A., Zambaldi, V., Malinowski, M., Tacchetti, A., Raposo, D., Santoro, A., and Faulkner, R. (2018). Relational Inductive Biases, Deep Learning, and Graph Networks. arXiv."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1109\/MSP.2017.2693418","article-title":"Geometric Deep Learning: Going Beyond Euclidean Data","volume":"34","author":"Bronstein","year":"2017","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_17","unstructured":"Sanchez-Gonzalez, A., Godwin, J., Pfaff, T., Ying, R., Leskovec, J., and Battaglia, P. (2020). Learning to Simulate Complex Physics with Graph Networks. Proceedings of the 37th International Conference on Machine Learning, PMLR."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"114502","DOI":"10.1016\/j.cma.2021.114502","article-title":"Physics-Informed Graph Neural Galerkin Networks: A Unified Framework for Solving PDE-Governed Forward and Inverse Problems","volume":"390","author":"Gao","year":"2022","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_19","first-page":"8799","article-title":"Flexible Neural Representation for Physics Prediction","volume":"31","author":"Mrowca","year":"2018","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_20","unstructured":"Bapst, V., Sanchez-Gonzalez, A., Doersch, C., Stachenfeld, K., Kohli, P., Battaglia, P.W., and Hamrick, J.B. (2019). Structured Agents for Physical Construction. Proceedings of the International Conference on Machine Learning, PMLR."},{"key":"ref_21","unstructured":"Pfaff, T., Fortunato, M., Sanchez-Gonzalez, A., and Battaglia, P. (2020). Learning Mesh-Based Simulation with Graph Networks. arXiv."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"113299","DOI":"10.1016\/j.cma.2020.113299","article-title":"Geometric Deep Learning for Computational Mechanics Part I: Anisotropic Hyperelasticity","volume":"371","author":"Vlassis","year":"2020","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"66:1","DOI":"10.1145\/3514228","article-title":"Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems","volume":"55","author":"Willard","year":"2023","journal-title":"ACM Comput. Surv."},{"key":"ref_24","unstructured":"Thuerey, N., Holl, P., Mueller, M., Schnell, P., Trost, F., and Um, K. (2021). Physics-Based Deep Learning. arXiv."},{"key":"ref_25","first-page":"1","article-title":"Predictive Computational Science: Computer Predictions in the Presence of Uncertainty","volume":"Volume 1 (Fundamentals)","author":"Stein","year":"2017","journal-title":"Encyclopedia of Computational Mechanics"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","article-title":"Physics-Informed Neural Networks: A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations","volume":"378","author":"Raissi","year":"2019","journal-title":"J. Comput. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"550","DOI":"10.1137\/16M1082469","article-title":"Survey of Multifidelity Methods in Uncertainty Propagation, Inference, and Optimization","volume":"60","author":"Peherstorfer","year":"2018","journal-title":"SIAM Rev."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/biomet\/87.1.1","article-title":"Predicting the Output from a Complex Computer Code When Fast Approximations Are Available","volume":"87","author":"Kennedy","year":"2000","journal-title":"Biometrika"},{"key":"ref_29","first-page":"20160751","article-title":"Nonlinear Information Fusion Algorithms for Data-Efficient Multi-Fidelity Modelling","volume":"473","author":"Perdikaris","year":"2017","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"044001","DOI":"10.1088\/1681-7575\/acd6fd","article-title":"Virtual Experiments for the Assessment of Data Analysis and Uncertainty Quantification Methods in Scatterometry","volume":"60","author":"Kok","year":"2023","journal-title":"Metrologia"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1513","DOI":"10.1007\/s10462-023-10562-9","article-title":"A Survey of Uncertainty in Deep Neural Networks","volume":"56","author":"Gawlikowski","year":"2023","journal-title":"Artif. Intell. Rev."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Most, T., Zabel, V., Das, R.R., and Khadka, A. (2025). An Efficient Uncertainty Quantification Approach for Robust Design of Tuned Mass Dampers in Linear Structural Dynamics. Appl. Sci., 15.","DOI":"10.3390\/app15179329"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Le, B.T., and Siedziako, B. (2026). Finite Element-Based Evaluation of Minimum Detectable Damage in the Magerholm Linkspan Under Environmental, Noise, and Damage Asymmetry Effects. Struct. Health Monit., OnlineFirst.","DOI":"10.1177\/14759217261431577"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1109\/ICNN.1994.374138","article-title":"Estimating the Mean and Variance of the Target Probability Distribution","volume":"Volume 1","author":"Nix","year":"1994","journal-title":"Proceedings of the 1994 IEEE International Conference on Neural Networks (ICNN\u201994)"},{"key":"ref_35","first-page":"5574","article-title":"What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?","volume":"30","author":"Kendall","year":"2017","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1137\/22M148375X","article-title":"Discrepancy Modeling Framework: Learning Missing Physics, Modeling Systematic Residuals, and Disambiguating Between Deterministic and Random Effects","volume":"23","author":"Ebers","year":"2024","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"ref_37","first-page":"6402","article-title":"Simple and Scalable Predictive Uncertainty Estimation Using Deep Ensembles","volume":"30","author":"Lakshminarayanan","year":"2017","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1115\/1.2218343","article-title":"Clamp Load Loss Due to Elastic Interaction and Gasket Creep Relaxation in Bolted Joints","volume":"128","author":"Nassar","year":"2006","journal-title":"J. Press. Vessel Technol."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"041202","DOI":"10.1115\/1.4002541","article-title":"A Tetraparametric Metamodel for the Analysis and Design of Bolting Sequences for Wind Generator Flanges","volume":"133","author":"Abasolo","year":"2011","journal-title":"J. Press. Vessel Technol."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"111122","DOI":"10.1016\/j.ress.2025.111122","article-title":"The Bearing Multi-Sensor Fault Diagnosis Method Based on a Multi-Branch Parallel Perception Network and Feature Fusion Strategy","volume":"261","author":"Li","year":"2025","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Wu, Z., Nassar, S.A., and Yang, X. (2010, January 18\u201322). Nonlinear Deformation Behavior of Bolted Flanges under Tensile, Torsional and Bending Loads. Proceedings of the ASME 2010 Pressure Vessels and Piping Division\/K-PVP Conference, Bellevue, WA, USA. Paper No. PVP2010-25072.","DOI":"10.1115\/PVP2010-25072"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/26\/17\/5534\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,2]],"date-time":"2026-09-02T04:27:45Z","timestamp":1788323265000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/26\/17\/5534"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,31]]},"references-count":41,"journal-issue":{"issue":"17","published-online":{"date-parts":[[2026,9]]}},"alternative-id":["s26175534"],"URL":"https:\/\/doi.org\/10.3390\/s26175534","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,31]]}}}