{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:02:46Z","timestamp":1760144566818,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,5,4]],"date-time":"2024-05-04T00:00:00Z","timestamp":1714780800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior (CAPES)"},{"name":"Petr\u00f3leo Brasileiro S.A. (PETROBRAS)"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>Thermally characterizing high-thermal conductivity materials is challenging, especially considering high temperatures. However, the modeling of heat transfer processes requires specific material information. The present study addresses an inverse approach to estimate the thermal conductivity of SAE 1020 relative to temperature during an autogenous LASER Beam Welding (LBW) experiment. The temperature profile during LBW is computed with the aid of an in-house CUDA-C algorithm. Here, the governing three-dimensional heat diffusion equation is discretized through the Finite Volume Method (FVM) and solved using the Successive Over-Relaxation (SOR) parallelized iterative solver. With temperature information, one may employ a minimization procedure to assess thermal properties or process parameters. In this work, the Quadrilateral Optimization Method (QOM) is applied to perform estimations because it allows for the simultaneous optimization of variables with no quantity restriction and renders the assessment of parameters in unsteady states valid, thereby preventing the requirement for steady-state experiments. We extended QOM\u2019s prior applicability to account for more parameters concurrently. In Case I, the optimization of the three parameters that compose the second-degree polynomial function model of thermal conductivity is performed. In Case II, the heat distribution model\u2019s gross heat rate (\u03a9) is also estimated in addition to the previous parameters. \u03a9 [W] quantifies the power the sample receives and is related to the process\u2019s efficiency. The method\u2019s suitability for estimating the parameters was confirmed by investigating the reduced sensitivity coefficients, while the method\u2019s stability was corroborated by performing the estimates with noisy data. There is a good agreement between the reference and estimated values. Hence, this study introduces a proper methodology for estimating a temperature-dependent thermal property and an LBW parameter. As the performance of the present algorithm is increased using parallel computation, a pondered solution between estimation reliability and computational cost is achieved.<\/jats:p>","DOI":"10.3390\/computation12050092","type":"journal-article","created":{"date-parts":[[2024,5,6]],"date-time":"2024-05-06T15:09:36Z","timestamp":1715008176000},"page":"92","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Numerical Estimation of Nonlinear Thermal Conductivity of SAE 1020 Steel"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-8339-2168","authenticated-orcid":false,"given":"Ariel Flores Monteiro","family":"de Oliveira","sequence":"first","affiliation":[{"name":"Instituto Tecnol\u00f3gico de Aeron\u00e1utica\u2014ITA, S\u00e3o Jos\u00e9 dos Campos 12228-900, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0572-4943","authenticated-orcid":false,"given":"Elisan dos Santos","family":"Magalh\u00e3es","sequence":"additional","affiliation":[{"name":"Instituto Tecnol\u00f3gico de Aeron\u00e1utica\u2014ITA, S\u00e3o Jos\u00e9 dos Campos 12228-900, Brazil"}]},{"given":"Kahl Dick","family":"Zilnyk","sequence":"additional","affiliation":[{"name":"Instituto Tecnol\u00f3gico de Aeron\u00e1utica\u2014ITA, S\u00e3o Jos\u00e9 dos Campos 12228-900, Brazil"}]},{"given":"Philippe","family":"Le Masson","sequence":"additional","affiliation":[{"name":"IRDL, Universit\u00e9 de Bretagne-Sud\u2014UBS, CNRS 6027, Rue de Saint Maud\u00e9, 56100 Lorient, France"}]},{"given":"Ernandes Jos\u00e9 Gon\u00e7alves do","family":"Nascimento","sequence":"additional","affiliation":[{"name":"Instituto Tecnol\u00f3gico de Aeron\u00e1utica\u2014ITA, S\u00e3o Jos\u00e9 dos Campos 12228-900, Brazil"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"107493","DOI":"10.1016\/j.optlastec.2021.107493","article-title":"Prediction of Temperature Field and Residual Stress of Oscillation Laser Welding of 316LN Stainless Steel","volume":"145","author":"Yan","year":"2022","journal-title":"Opt. Laser Technol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1016\/j.applthermaleng.2018.10.021","article-title":"Simultaneous Retrieval of High Temperature Thermal Conductivities, Anisotropic Radiative Properties, and Thermal Contact Resistance for Ceramic Foams","volume":"146","author":"Zhao","year":"2019","journal-title":"Appl. Therm. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"100724","DOI":"10.1016\/j.rineng.2022.100724","article-title":"Development of New Composite Materials Based on TiN\u2013Ni Cermets during Thermochemical Pressing","volume":"16","author":"Belokon","year":"2022","journal-title":"Results Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"107849","DOI":"10.1016\/j.polymertesting.2022.107849","article-title":"Development of a New Birthing Model Material Based on Silicone Rubber\/Natural Rubber Blend","volume":"117","author":"Panmanee","year":"2023","journal-title":"Polym. Test."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"101488","DOI":"10.1016\/j.aei.2021.101488","article-title":"A Collaborative Design Platform for New Alloy Material Development","volume":"51","author":"Peng","year":"2022","journal-title":"Adv. Eng. Inform."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"17041","DOI":"10.1063\/5.0074613","article-title":"Charge-Density-Wave Quantum Materials and Devices-New Developments and Future Prospects","volume":"119","author":"Balandin","year":"2021","journal-title":"Appl. Phys. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Wardak, C., Morawska, K., and Pietrzak, K. (2023). New Materials Used for the Development of Anion-Selective Electrodes\u2014A Review. Materials, 16.","DOI":"10.3390\/ma16175779"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"106338","DOI":"10.1016\/j.ijthermalsci.2020.106338","article-title":"Thermal Conductivity of Evacuated Expanded Perlite Measured with Guarded-Hot-Plate and Transient-Hot-Wire Method at Temperatures between 295 K and 1073 K","volume":"152","author":"Rottmann","year":"2020","journal-title":"Int. J. Therm. Sci."},{"key":"ref_9","first-page":"11","article-title":"Measurement of Thermal Conductivity and Thermal Resistance with a Tiny Hot Plate","volume":"39","author":"Jannot","year":"2010","journal-title":"High Temp. -High Press."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1051\/matecconf\/201711001037","article-title":"Evaluation of the Laser-Flash Method and Its Errors for Determining of Materials Thermal Diffusivity at High Temperatures","volume":"110","author":"Katz","year":"2017","journal-title":"MATEC Web Conf."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2801","DOI":"10.1007\/s11661-003-0181-2","article-title":"Thermal Conductivities of Molten Iron, Cobalt, and Nickel by Laser Flash Method","volume":"34","author":"Nishi","year":"2003","journal-title":"Metall. Mater. Trans. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1007\/s10765-020-02676-x","article-title":"A Comparative Fluxmetric (CFM) Method for Apparent Thermal Conductivity Measurement of Insulating Materials at High Temperature","volume":"41","author":"Jannot","year":"2020","journal-title":"Int. J. Thermophys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/j.ijheatmasstransfer.2019.06.054","article-title":"A Bayesian Approach for the Estimation of the Thermal Diffusivity of Aerodynamically Levitated Solid Metals at High Temperatures","volume":"141","author":"Lamien","year":"2019","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/j.ijthermalsci.2017.06.001","article-title":"Investigations on Design and Construction of a Square Guarded Hot Plate (SGHP) Apparatus for Thermal Conductivity Measurement of Insulation Materials","volume":"120","author":"Reddy","year":"2017","journal-title":"Int. J. Therm. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/s10765-019-2594-1","article-title":"Thermal Properties Investigation of Inhomogeneous Ternary Alloy (Se\u2013Te\u2013S) Using Flash Method Technique","volume":"41","author":"Dessouky","year":"2020","journal-title":"Int. J. Thermophys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"104736","DOI":"10.1016\/j.icheatmasstransfer.2020.104736","article-title":"Experimental Estimation of Temperature-Dependent Thermal Conductivity Coefficient by Using Inverse Method and Remote Boundary Condition","volume":"117","author":"Farahani","year":"2020","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Mohebbi, F., and Sellier, M. (2021). Estimation of Functional Form of Time-Dependent Heat Transfer Coefficient Using an Accurate and Robust Parameter Estimation Approach: An Inverse Analysis. Energies, 14.","DOI":"10.3390\/en14165073"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1470","DOI":"10.1007\/s10765-007-0165-3","article-title":"Simultaneous Estimation of Thermal Properties of Living Tissue Using Noninvasive Method","volume":"28","author":"Yue","year":"2007","journal-title":"Int. J. Thermophys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"13297","DOI":"10.1007\/s10973-023-12626-y","article-title":"Estimation of the Energy Requirement of Bread during Baking by Inverse Heat Transfer Method","volume":"148","author":"Ravula","year":"2023","journal-title":"J. Therm. Anal. Calorim."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"109769","DOI":"10.1016\/j.jfoodeng.2019.109769","article-title":"Estimation of the Effective Moisture Diffusivity in Cake Baking by the Inversion of a Finite Element Model","volume":"270","author":"Cevoli","year":"2020","journal-title":"J. Food Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"106240","DOI":"10.1016\/j.ijthermalsci.2019.106240","article-title":"Development and Validation of an Inverse Method for Identification of Thermal Characteristics of a Short Laser Pulse","volume":"150","author":"Pietrak","year":"2020","journal-title":"Int. J. Therm. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"106395","DOI":"10.1016\/j.ijthermalsci.2020.106395","article-title":"Real-Time Estimation of Thermal Boundary of Unsteady Heat Conduction System Using PID Algorithm","volume":"153","author":"Wan","year":"2020","journal-title":"Int. J. Therm. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"122657","DOI":"10.1016\/j.ijheatmasstransfer.2022.122657","article-title":"Numerical and Experimental Verification of the Single Neural Adaptive PID Real-Time Inverse Method for Solving Inverse Heat Conduction Problems","volume":"189","author":"Wan","year":"2022","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.icheatmasstransfer.2015.05.023","article-title":"The Use of Non-Linear Inverse Problem and Enthalpy Method in GTAW Process of Aluminum","volume":"66","author":"Silva","year":"2015","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1016\/j.applthermaleng.2016.02.051","article-title":"Microstructural Analysis in GTA Aluminum Alloy Welding Using Inverse Problems","volume":"100","author":"Correa","year":"2016","journal-title":"Appl. Therm. Eng."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"121857","DOI":"10.1016\/j.ijheatmasstransfer.2021.121857","article-title":"dos S. A Quadrilateral Optimization Method for Non-Linear Thermal Properties Determination in Materials at High Temperatures","volume":"181","year":"2021","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"de Oliveira, A.F.M., Magalh\u00e3es, E.D.S., Paes, L.E.D.S., Pereira, M., and da Silva, L.R. (2023). A Thermal Analysis of LASER Beam Welding Using Statistical Approaches. Processes, 11.","DOI":"10.3390\/pr11072023"},{"key":"ref_28","unstructured":"\u00c7engel, Y.A., and Ghajar, A.J. (2010). Heat and Mass Transfer: Fundamentals and Applications, McGraw-Hill Science\/Engineering\/Math. [4th ed.]."},{"key":"ref_29","unstructured":"Bergman, T.L., Lavine, A.S., Incropera, F.P., and Dewitt, D.P. (2011). Fundamentals of Heat and Mass Transfer, John Wiley & Sons. [7th ed.]."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1590\/0104-9224\/si2201.10","article-title":"Two Heat Source Models to Simulate Welding Processes with Magnetic Deflection","volume":"22","author":"Clain","year":"2017","journal-title":"Soldag. E Insp."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Nascimento, E., Magalh\u00e3es, E., Azevedo, A., Paes, L.E.S., and Oliveira, A. (2024). An Implementation of LASER Beam Welding Simulation on Graphics Processing Unit Using CUDA. Computation, 12.","DOI":"10.3390\/computation12040083"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Magalh\u00e3es, E.D.S., Anselmo, B.D.C.S., Lima e Silva, A.L.F.D., and Lima e Silva, S.M.M. (2018). Time Traveling Regularization for Inverse Heat Transfer Problems. Energies, 11.","DOI":"10.3390\/en11030507"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3649","DOI":"10.1016\/0017-9310(96)00034-8","article-title":"Comparison of Some Inverse Heat Conduction Methods Using Experimental Data","volume":"39","author":"Beck","year":"1996","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2977","DOI":"10.1016\/0017-9310(93)90027-4","article-title":"Optimal Experimental Design for Estimating Thermal Properties of Composite Materials","volume":"36","author":"Taktak","year":"1993","journal-title":"Int. J. Heat Mass Transf."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/5\/92\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:39:34Z","timestamp":1760107174000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/5\/92"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,4]]},"references-count":34,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2024,5]]}},"alternative-id":["computation12050092"],"URL":"https:\/\/doi.org\/10.3390\/computation12050092","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2024,5,4]]}}}