{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,18]],"date-time":"2026-09-18T05:04:29Z","timestamp":1789707869491,"version":"4.0.1"},"reference-count":30,"publisher":"MDPI AG","issue":"18","license":[{"start":{"date-parts":[[2026,9,16]],"date-time":"2026-09-16T00:00:00Z","timestamp":1789516800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Key R&D Program Project of Xinjiang Uygur Autonomous Region","award":["2024B01014-1"],"award-info":[{"award-number":["2024B01014-1"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["52271299"],"award-info":[{"award-number":["52271299"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"award":["52271299"],"award-info":[{"award-number":["52271299"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]},{"name":"Ministry of Science and Technology of the People\u2019s Republic of China","award":["2024ZD1000800"],"award-info":[{"award-number":["2024ZD1000800"]}],"id":[{"id":"https:\/\/ror.org\/027s68j25","id-type":"ROR","asserted-by":"crossref"}]},{"name":"Ministry of Science and Technology of the People\u2019s Republic of China","award":["2024ZD1000806"],"award-info":[{"award-number":["2024ZD1000806"]}],"id":[{"id":"https:\/\/ror.org\/027s68j25","id-type":"ROR","asserted-by":"crossref"}]},{"award":["2024ZD1000800"],"award-info":[{"award-number":["2024ZD1000800"]}],"id":[{"id":"https:\/\/ror.org\/027s68j25","id-type":"ROR","asserted-by":"publisher"}]},{"award":["2024ZD1000806"],"award-info":[{"award-number":["2024ZD1000806"]}],"id":[{"id":"https:\/\/ror.org\/027s68j25","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>A surrogate-assisted inverse design framework is presented for selecting the power-law index of clamped\u2013clamped axially functionally graded (AFG) Timoshenko pipes conveying fluid. The GITT model, 336-sample database, and trained MLP forward surrogate were developed in our previous study; the present contribution begins with the formulation and solution of the inverse problem. Millisecond-speed MLP inference is embedded in grid, particle swarm optimization (PSO), and genetic algorithm (GA) searches for prescribed modal-frequency and deflection targets. Single-variable, two-variable, weighted-sum, constrained, and Pareto formulations are examined. Continuous candidates from the single-variable cases are subjected to GITT-database interpolation verification, which is explicitly distinguished from a new independent GITT calculation. The feasible single- and dual-modal examples produce small database-interpolated target residuals, whereas an intentionally unattainable triplet case retains an approximately 16% fundamental-frequency residual and demonstrates the need for feasibility screening. The two-variable maps reveal non-unique parameter couplings and are treated as exploratory surrogate results unless the complete candidate coincides with, or is independently assessed against, the available database. A five-network ensemble assesses repeatability with respect to training-data partitioning only, while a sensitivity analysis connects the selected indices to the high-sensitivity gradation range. In the synchronized Case A benchmark, the online MLP grid and PSO searches require approximately 0.044 and 0.210 s, respectively, excluding the inherited offline GITT-database construction cost.<\/jats:p>","DOI":"10.3390\/ma19183936","type":"journal-article","created":{"date-parts":[[2026,9,16]],"date-time":"2026-09-16T13:50:27Z","timestamp":1789566627000},"page":"3936","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Surrogate-Assisted Inverse Design of the Power-Law Index in Axially Functionally Graded Fluid-Conveying Pipes for Target Modal and Deflection Performance"],"prefix":"10.3390","volume":"19","author":[{"given":"Lun","family":"Gao","sequence":"first","affiliation":[{"name":"School of Safety Science and Engineering, Xinjiang Institute of Engineering, Urumqi 830023, China"},{"name":"Xinjiang Key Laboratory of Coal Mine Disaster Intelligent Prevention and Emergency Response, Xinjiang Institute of Engineering, Urumqi 830023, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"methodology","vocabulary":"credit"},{"role":"validation","vocabulary":"credit"},{"role":"formal-analysis","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"data-curation","vocabulary":"credit"},{"role":"writing-original-draft","vocabulary":"credit"},{"role":"visualization","vocabulary":"credit"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1098-988X","authenticated-orcid":false,"given":"Jijun","family":"Gu","sequence":"additional","affiliation":[{"name":"Xinjiang Key Laboratory of Coal Mine Disaster Intelligent Prevention and Emergency Response, Xinjiang Institute of Engineering, Urumqi 830023, China"},{"name":"College of Mechanical and Transportation Engineering, China University of Petroleum\u2013Beijing, Beijing 102249, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"methodology","vocabulary":"credit"},{"role":"supervision","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"},{"role":"project-administration","vocabulary":"credit"},{"role":"funding-acquisition","vocabulary":"credit"},{"role":"validation","vocabulary":"credit"}]},{"given":"Tianjin","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Safety Science and Engineering, Xinjiang Institute of Engineering, Urumqi 830023, China"},{"name":"Xinjiang Key Laboratory of Coal Mine Disaster Intelligent Prevention and Emergency Response, Xinjiang Institute of Engineering, Urumqi 830023, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"validation","vocabulary":"credit"},{"role":"data-curation","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"formal-analysis","vocabulary":"credit"},{"role":"visualization","vocabulary":"credit"}]},{"given":"Shanshan","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Safety Science and Engineering, Xinjiang Institute of Engineering, Urumqi 830023, China"},{"name":"Xinjiang Key Laboratory of Coal Mine Disaster Intelligent Prevention and Emergency Response, Xinjiang Institute of Engineering, Urumqi 830023, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"software","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"resources","vocabulary":"credit"},{"role":"visualization","vocabulary":"credit"},{"role":"data-curation","vocabulary":"credit"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-9025-3922","authenticated-orcid":false,"given":"Junjie","family":"Li","sequence":"additional","affiliation":[{"name":"School of Safety Science and Engineering, Xinjiang Institute of Engineering, Urumqi 830023, China"},{"name":"Xinjiang Key Laboratory of Coal Mine Disaster Intelligent Prevention and Emergency Response, Xinjiang Institute of Engineering, Urumqi 830023, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"investigation","vocabulary":"credit"},{"role":"data-curation","vocabulary":"credit"},{"role":"visualization","vocabulary":"credit"},{"role":"validation","vocabulary":"credit"}]}],"member":"1968","published-online":{"date-parts":[[2026,9,16]]},"reference":[{"key":"ref_1","unstructured":"Zhong, Z., Wu, L., and Chen, W. 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