{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T01:52:20Z","timestamp":1772848340467,"version":"3.50.1"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"15","license":[{"start":{"date-parts":[[2025,2,28]],"date-time":"2025-02-28T00:00:00Z","timestamp":1740700800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,2,28]],"date-time":"2025-02-28T00:00:00Z","timestamp":1740700800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100006199","name":"Langley Research Center","doi-asserted-by":"publisher","award":["011042"],"award-info":[{"award-number":["011042"]}],"id":[{"id":"10.13039\/100006199","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006199","name":"Langley Research Center","doi-asserted-by":"publisher","award":["012053"],"award-info":[{"award-number":["012053"]}],"id":[{"id":"10.13039\/100006199","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006199","name":"Langley Research Center","doi-asserted-by":"publisher","award":["012951"],"award-info":[{"award-number":["012951"]}],"id":[{"id":"10.13039\/100006199","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006199","name":"Langley Research Center","doi-asserted-by":"publisher","award":["014070"],"award-info":[{"award-number":["014070"]}],"id":[{"id":"10.13039\/100006199","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006199","name":"Langley Research Center","doi-asserted-by":"publisher","award":["015326"],"award-info":[{"award-number":["015326"]}],"id":[{"id":"10.13039\/100006199","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006199","name":"Langley Research Center","doi-asserted-by":"publisher","award":["016042"],"award-info":[{"award-number":["016042"]}],"id":[{"id":"10.13039\/100006199","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Neural Comput &amp; Applic"],"published-print":{"date-parts":[[2025,5]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>This paper proposes a general method of creating an accurate neural network-based surrogate model for postprocessing a topologically optimized structure. When topology optimization results are converted into computer-aided design (CAD) files with smooth boundaries for manufacturability, finite element method (FEM) based stresses often do not agree with the topology optimized results due to changes of surface and mesh density. The conversion between topology optimization derived results and CAD files often requires postprocessing, an additional fine tuning of the geometry parameters to reconcile the change of the stress values. In this work, a feedforward, deep artificial neural network (DANN) is presented with varying architecture parameters that are found for each stress output of interest. This network is trained with the data based on a combination of Design of Experiments (DoE) models that have the geometry dimensions as inputs and stress readings under various loads as the outputs. A DANN-based surrogate model is constructed to enable fine tuning of all relevant stress performance metrics. This method of constructing an artificial network-based surrogate model minimizes the number of FEM computations required to generate an optimized, post-processed design. We present a case study of postprocessing a wind tunnel balance, a measurement device that yields the six force and moment components of a test aircraft. It needs to be designed considering multiple stress measures under combinations of the six loading conditions. Excellent performance of a neural network is presented in this paper in terms of accurate prediction of the highly nonlinear stresses under combinations of the six loads. Von Mises stress predictions are within 10% and axial force sensor stress predictions are within 2% for the final post-processed topology. The results support its usefulness for postprocessing of topology optimized structures.<\/jats:p>","DOI":"10.1007\/s00521-025-11039-2","type":"journal-article","created":{"date-parts":[[2025,2,28]],"date-time":"2025-02-28T06:53:34Z","timestamp":1740725614000},"page":"8845-8867","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Neural network-based surrogate model in postprocessing of topology optimized structures"],"prefix":"10.1007","volume":"37","author":[{"given":"Jude Thaddeus","family":"Persia","sequence":"first","affiliation":[]},{"given":"Myung Kyun","family":"Sung","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6418-7527","authenticated-orcid":false,"given":"Soobum","family":"Lee","sequence":"additional","affiliation":[]},{"given":"Devin E.","family":"Burns","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,2,28]]},"reference":[{"issue":"1","key":"11039_CR1","doi-asserted-by":"publisher","first-page":"6","DOI":"10.1007\/s00158-021-03088-7","volume":"65","author":"MK Sung","year":"2022","unstructured":"Sung MK, Lee S, Burns DE (2022) Robust topology optimization of a flexural structure considering multi-stress performance for force sensing and structural safety. Struct Multidiscip Optim 65(1):6. https:\/\/doi.org\/10.1007\/s00158-021-03088-7","journal-title":"Struct Multidiscip Optim"},{"key":"11039_CR2","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/j.compstruc.2015.03.011","volume":"154","author":"JN Richardson","year":"2015","unstructured":"Richardson JN, Filomeno Coelho R, Adriaenssens S (2015) Robust topology optimization of truss structures with random loading and material properties: a multiobjective perspective. Comput Struct 154:41\u201347. https:\/\/doi.org\/10.1016\/j.compstruc.2015.03.011","journal-title":"Comput Struct"},{"key":"11039_CR3","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1016\/j.cma.2014.08.027","volume":"282","author":"W Zhang","year":"2014","unstructured":"Zhang W, Zhong W, Guo X (2014) An explicit length scale control approach in SIMP-based topology optimization. Comput Methods Appl Mech Eng 282:71\u201386. https:\/\/doi.org\/10.1016\/j.cma.2014.08.027","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"2","key":"11039_CR4","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1007\/s00158-013-0971-0","volume":"49","author":"E Lee","year":"2014","unstructured":"Lee E, Gea HC (2014) A strain based topology optimization method for compliant mechanism design. Struct Multidiscip Optim 49(2):199\u2013207. https:\/\/doi.org\/10.1007\/s00158-013-0971-0","journal-title":"Struct Multidiscip Optim"},{"key":"11039_CR5","doi-asserted-by":"publisher","first-page":"V03BT03A013","DOI":"10.1115\/DETC2023-114920","volume-title":"International design engineering technical conferences and computers and information in engineering conference","author":"MK Sung","year":"2023","unstructured":"Sung MK, Lee S, Burns D, Persia JT (2023) Selective amplification and suppression of strain in a multi-axis force sensor using topology optimization. International design engineering technical conferences and computers and information in engineering conference, vol 87318. American Society of Mechanical Engineers, New York, p V03BT03A013. https:\/\/doi.org\/10.1115\/DETC2023-114920"},{"issue":"1\u20134","key":"11039_CR6","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1007\/s00170-015-7801-z","volume":"93","author":"R Bharanidaran","year":"2017","unstructured":"Bharanidaran R, Ramesh T (2017) A modified post-processing technique to design a compliant based microgripper with a plunger using topological optimization. Int J Adv Manuf Technol 93(1\u20134):103\u2013112. https:\/\/doi.org\/10.1007\/s00170-015-7801-z","journal-title":"Int J Adv Manuf Technol"},{"key":"11039_CR7","doi-asserted-by":"publisher","first-page":"112972","DOI":"10.1016\/j.cma.2020.112972","volume":"365","author":"GA da Silva","year":"2020","unstructured":"da Silva GA, Beck AT, Sigmund O (2020) Topology optimization of compliant mechanisms considering stress constraints, manufacturing uncertainty and geometric nonlinearity. Comput Methods Appl Mech Eng 365:112972. https:\/\/doi.org\/10.1016\/j.cma.2020.112972","journal-title":"Comput Methods Appl Mech Eng"},{"key":"11039_CR8","doi-asserted-by":"publisher","first-page":"104869","DOI":"10.1016\/j.euromechsol.2022.104869","volume":"98","author":"HT Mai","year":"2023","unstructured":"Mai HT, Lee S, Kim D, Lee J, Kang J, Lee J (2023) Optimum design of nonlinear structures via deep neural network-based parameterization framework. Eur J Mech, A\/Solids 98:104869. https:\/\/doi.org\/10.1016\/j.euromechsol.2022.104869","journal-title":"Eur J Mech, A\/Solids"},{"key":"11039_CR9","doi-asserted-by":"publisher","first-page":"795","DOI":"10.1016\/j.promfg.2020.10.111","volume":"51","author":"A Ktari","year":"2020","unstructured":"Ktari A, Elmansori M (2020) Bridging FEM and artificial neural network in gating system design for smart 3D sand casting. Procedia Manuf 51:795\u2013800. https:\/\/doi.org\/10.1016\/j.promfg.2020.10.111","journal-title":"Procedia Manuf"},{"key":"11039_CR10","doi-asserted-by":"publisher","first-page":"589","DOI":"10.1016\/j.compstruct.2016.10.007","volume":"159","author":"J Gajewski","year":"2017","unstructured":"Gajewski J, Golewski P, Sadowski T (2017) Geometry optimization of a thin-walled element for an air structure using hybrid system integrating artificial neural network and finite element method. Compos Struct 159:589\u2013599. https:\/\/doi.org\/10.1016\/j.compstruct.2016.10.007","journal-title":"Compos Struct"},{"key":"11039_CR11","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/j.sna.2014.01.001","volume":"209","author":"MK Kang","year":"2014","unstructured":"Kang MK, Lee S, Jim JH (2014) Shape optimization of a mechanically decoupled six-axis force\/torque sensor. Sens Actuators A Phys 209:41\u201351. https:\/\/doi.org\/10.1016\/j.sna.2014.01.001","journal-title":"Sens Actuators A Phys"},{"key":"11039_CR12","unstructured":"Burns DE, Parker PA, Phillips BD, Webb TL, D (2019) Wind tunnel balance design: a nasa langley perspective. [Online]. Available: http:\/\/www.sti.nasa.gov"},{"key":"11039_CR13","first-page":"115","volume-title":"Wind tunnels and experimental fluid dynamics research","author":"M Gonz\u00e1lez","year":"2011","unstructured":"Gonz\u00e1lez M, Ezquerro J, Lapuerta V, Laver\u00f3n A, Rodr\u00edguez J (2011) Wind tunnels and experimental fluid dynamics research. In: Boldes U, Lerner J (eds) Wind tunnels and experimental fluid dynamics research. InTech, Rijeka, pp 115\u2013135"},{"key":"11039_CR14","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/j.sna.2014.01.001","volume":"209","author":"M-K Kang","year":"2014","unstructured":"Kang M-K, Lee S, Kim J-H (2014) Shape optimization of a mechanically decoupled six-axis force\/torque sensor. Sens Actuators A Phys 209:41\u201351. https:\/\/doi.org\/10.1016\/j.sna.2014.01.001","journal-title":"Sens Actuators A Phys"},{"key":"11039_CR15","doi-asserted-by":"crossref","unstructured":"Cahill D, Steinle F, Richardson S (2004) Evaluation of wind tunnel internal force balances from seven vendors. AIAA Paper. 10438\u201310447","DOI":"10.2514\/6.2004-1292"},{"issue":"2","key":"11039_CR16","doi-asserted-by":"publisher","first-page":"238","DOI":"10.1002\/nme.1064","volume":"61","author":"JK Guest","year":"2004","unstructured":"Guest JK, Pr\u00e9vost JH, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61(2):238\u2013254. https:\/\/doi.org\/10.1002\/nme.1064","journal-title":"Int J Numer Methods Eng"},{"key":"11039_CR17","unstructured":"Sung MK (2022) Design methodology for multifunctional stress\/strain performances of multi-loading sensors using topology optimization. Dissertation, University of Maryland, Baltimore County"},{"issue":"5","key":"11039_CR18","doi-asserted-by":"publisher","first-page":"929","DOI":"10.1007\/s00158-015-1279-z","volume":"52","author":"DM De Leon","year":"2015","unstructured":"De Leon DM, Alexandersen J, Fonseca JSO, Sigmund O (2015) Stress-constrained topology optimization for compliant mechanism design. Struct Multidiscip Optim 52(5):929\u2013943. https:\/\/doi.org\/10.1007\/s00158-015-1279-z","journal-title":"Struct Multidiscip Optim"},{"issue":"4","key":"11039_CR19","doi-asserted-by":"publisher","first-page":"605","DOI":"10.1007\/s00158-009-0440-y","volume":"41","author":"C Le","year":"2010","unstructured":"Le C, Norato J, Bruns T, Ha C, Tortorelli D (2010) Stress-based topology optimization for continua. Struct Multidiscip Optim 41(4):605\u2013620. https:\/\/doi.org\/10.1007\/s00158-009-0440-y","journal-title":"Struct Multidiscip Optim"},{"key":"11039_CR20","volume-title":"Design and analysis of experiments","author":"D Montgomery","year":"2013","unstructured":"Montgomery D (2013) Design and analysis of experiments. Wiley, Eighth"},{"issue":"8","key":"11039_CR21","doi-asserted-by":"publisher","first-page":"1553","DOI":"10.61356\/SMIJ.2024.8288","volume":"25","author":"M Bianchini","year":"2014","unstructured":"Bianchini M, Scarselli F (2014) On the complexity of neural network classifiers: a comparison between shallow and deep architectures. IEEE Trans Neural Netw Learn Syst 25(8):1553\u20131565. https:\/\/doi.org\/10.61356\/SMIJ.2024.8288","journal-title":"IEEE Trans Neural Netw Learn Syst"},{"key":"11039_CR22","doi-asserted-by":"publisher","first-page":"14","DOI":"10.61356\/SMIJ.2024.8288","volume":"8","author":"A Darwish","year":"2024","unstructured":"Darwish A (2024) A data-driven deep learning approach for remaining useful life in the ion mill etching process. Sustain Mach Intell J 8:14\u201334. https:\/\/doi.org\/10.61356\/SMIJ.2024.8288","journal-title":"Sustain Mach Intell J"},{"key":"11039_CR23","doi-asserted-by":"publisher","first-page":"14","DOI":"10.61356\/SMIJ.2024.9381","volume":"9","author":"A Darwish","year":"2024","unstructured":"Darwish A (2024) A novel deep learning model for tool wear estimation of cutting tools. Sustain Mach Intell J 9:14\u201333. https:\/\/doi.org\/10.61356\/SMIJ.2024.9381","journal-title":"Sustain Mach Intell J"},{"key":"11039_CR24","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1016\/j.neunet.2014.09.003","volume":"61","author":"J Schmidhuber","year":"2015","unstructured":"Schmidhuber J (2015) Deep Learning in neural networks: an overview. Neural Netw 61:85\u2013117. https:\/\/doi.org\/10.1016\/j.neunet.2014.09.003","journal-title":"Neural Netw"},{"key":"11039_CR25","unstructured":"Response Surface Designs (2021) E-Handbook., vol. Statistical Methods. NIST\/SEMATECH, 2021."},{"key":"11039_CR26","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1016\/j.cma.2017.09.010","volume":"328","author":"V Papadopoulos","year":"2018","unstructured":"Papadopoulos V, Soimiris G, Giovanis DG, Papadrakakis M (2018) A neural network-based surrogate model for carbon nanotubes with geometric nonlinearities. Comput Methods Appl Mech Eng 328:411\u2013430. https:\/\/doi.org\/10.1016\/j.cma.2017.09.010","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"3","key":"11039_CR27","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1007\/s00163-020-00336-7","volume":"31","author":"R Alizadeh","year":"2020","unstructured":"Alizadeh R, Allen JK, Mistree F (2020) Managing computational complexity using surrogate models: a critical review. Res Eng Des 31(3):275\u2013298. https:\/\/doi.org\/10.1007\/s00163-020-00336-7","journal-title":"Res Eng Des"},{"issue":"1","key":"11039_CR28","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1214\/aoms\/1177707047","volume":"28","author":"G Box","year":"1957","unstructured":"Box G, Hunter J (1957) Multi-factor experimental designs for exploring response surfaces. Ann Math Stat 28(1):195\u2013241","journal-title":"Ann Math Stat"},{"issue":"1","key":"11039_CR29","doi-asserted-by":"publisher","first-page":"75","DOI":"10.15282\/ijsecs.1.2015.6.0006","volume":"1","author":"AR Ajiboye","year":"2015","unstructured":"Ajiboye AR, Abdullah-Arshah R, Qin H, Isah-Kebbe H (2015) Evaluating the effect of dataset size on predictive model using supervised learning technique. Int J Comput Syst Softw Eng 1(1):75\u201384. https:\/\/doi.org\/10.15282\/ijsecs.1.2015.6.0006","journal-title":"Int J Comput Syst Softw Eng"},{"issue":"15","key":"11039_CR30","doi-asserted-by":"publisher","first-page":"9565","DOI":"10.1007\/s00521-021-05717-0","volume":"33","author":"N Le Chau","year":"2021","unstructured":"Le Chau N, Tran NT, Dao TP (2021) Design optimization for a compliant mechanism based on computational intelligence method. Neural Comput Appl 33(15):9565\u20139587. https:\/\/doi.org\/10.1007\/s00521-021-05717-0","journal-title":"Neural Comput Appl"},{"key":"11039_CR31","doi-asserted-by":"publisher","first-page":"109098","DOI":"10.1016\/j.matdes.2020.109098","volume":"196","author":"HT Kollmann","year":"2020","unstructured":"Kollmann HT, Abueidda DW, Koric S, Guleryuz E, Sobh NA (2020) Deep learning for topology optimization of 2D metamaterials. Mater Des 196:109098. https:\/\/doi.org\/10.1016\/j.matdes.2020.109098","journal-title":"Mater Des"},{"key":"11039_CR32","doi-asserted-by":"publisher","first-page":"103572","DOI":"10.1016\/j.finel.2021.103572","volume":"196","author":"HT Mai","year":"2021","unstructured":"Mai HT, Kang J, Lee J (2021) \u201cA machine learning-based surrogate model for optimization of truss structures with geometrically nonlinear behavior. Finite Elem Anal Des 196:103572. https:\/\/doi.org\/10.1016\/j.finel.2021.103572","journal-title":"Finite Elem Anal Des"},{"key":"11039_CR33","unstructured":"Tucci M, Barmada S, Sani L, Thomopulos D, Fontana N (2019) Deep neural networks based surrogate model for topology optimization of electromagnetic devices. In: 2019 International Applied Computational Electromagnetics Society Symposium (ACES), Miami, FL, USA, pp 1\u20132"},{"key":"11039_CR34","doi-asserted-by":"publisher","unstructured":"Reynaldi A, Lukas S, Margaretha H (2012) Backpropagation and Levenberg-Marquardt algorithm for training finite element neural network. In: Proceedings\u2014UKSim-AMSS 6th European Modelling Symposium, EMS 2012, pp 89\u201394. https:\/\/doi.org\/10.1109\/EMS.2012.56.","DOI":"10.1109\/EMS.2012.56"},{"key":"11039_CR35","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1137\/0111030","volume":"11","author":"D Marquardt","year":"1963","unstructured":"Marquardt D (1963) An algorithm for least-squares estimation of nonlinear parameters. SIAM J Appl Math 11:431\u2013441","journal-title":"SIAM J Appl Math"}],"container-title":["Neural Computing and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-025-11039-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00521-025-11039-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-025-11039-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,5]],"date-time":"2025-05-05T08:48:36Z","timestamp":1746434916000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00521-025-11039-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,28]]},"references-count":35,"journal-issue":{"issue":"15","published-print":{"date-parts":[[2025,5]]}},"alternative-id":["11039"],"URL":"https:\/\/doi.org\/10.1007\/s00521-025-11039-2","relation":{},"ISSN":["0941-0643","1433-3058"],"issn-type":[{"value":"0941-0643","type":"print"},{"value":"1433-3058","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,28]]},"assertion":[{"value":"10 February 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 January 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 February 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}