{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,18]],"date-time":"2026-05-18T10:20:32Z","timestamp":1779099632409,"version":"3.51.4"},"reference-count":58,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2023,10,11]],"date-time":"2023-10-11T00:00:00Z","timestamp":1696982400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,10,11]],"date-time":"2023-10-11T00:00:00Z","timestamp":1696982400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100005714","name":"Technische Universit\u00e4t Darmstadt","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100005714","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Engineering with Computers"],"published-print":{"date-parts":[[2024,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this work, an experimentally validated multiscale modeling framework for additively manufactured shell lattice structures with graded parameters is introduced. It is exemplified in application to the Schwarz primitive triply periodic minimal surface microstructure and 3D printing using masked stereolithography of a photopolymer material. The systematic procedure starts with the characterization of a hyperelastic material model for the 3D printed material. This constitutive model is then employed in the finite element simulation of shell lattices at finite deformations. The computational model is validated with experimental compression tests of printed lattice structures. In this way, the numerical convergence behavior and size dependence of the model are assessed, and the range in which it is reasonable to assume linear elastic behavior is determined. Then, representative volume elements subject to periodic boundary conditions are simulated to homogenize the mechanical behavior of Schwarz primitives with varying aspect ratios and shell thicknesses. Subsequently, the parameterized effective linear elasticity tensor of the metamaterial is represented by a physics-augmented neural network model. With this constitutive model, functionally graded shell lattice structures with varying microstructural parameters are simulated as macroscale continua using finite element and differential quadrature methods. The accuracy, reliability and effectiveness of this multiscale simulation approach are investigated and discussed. Overall, it is shown that this experimentally validated multiscale simulation framework, which is likewise applicable to other shell-like metamaterials, facilitates the design of functionally graded structures through additive manufacturing.<\/jats:p>\n                <jats:p><jats:bold>Graphical Abstract<\/jats:bold><\/jats:p>","DOI":"10.1007\/s00366-023-01906-8","type":"journal-article","created":{"date-parts":[[2023,10,11]],"date-time":"2023-10-11T20:13:41Z","timestamp":1697055221000},"page":"2019-2036","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":27,"title":["Multiscale modeling of functionally graded shell lattice metamaterials for additive manufacturing"],"prefix":"10.1007","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5601-6867","authenticated-orcid":false,"given":"M.","family":"Shojaee","sequence":"first","affiliation":[]},{"given":"I.","family":"Valizadeh","sequence":"additional","affiliation":[]},{"given":"D. K.","family":"Klein","sequence":"additional","affiliation":[]},{"given":"P.","family":"Sharifi","sequence":"additional","affiliation":[]},{"given":"O.","family":"Weeger","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,10,11]]},"reference":[{"issue":"10","key":"1906_CR1","doi-asserted-by":"publisher","first-page":"750","DOI":"10.1557\/mrs.2019.232","volume":"44","author":"JR Greer","year":"2019","unstructured":"Greer JR, Deshpande VS (2019) Three-dimensional architected materials and structures: Design, fabrication, and mechanical behavior. MRS Bull 44(10):750\u2013757. https:\/\/doi.org\/10.1557\/mrs.2019.232","journal-title":"MRS Bull"},{"key":"1906_CR2","doi-asserted-by":"publisher","unstructured":"A.\u00a0J.\u00a0D. Shaikeea, H.\u00a0Cui, M.\u00a0O\u2019Masta, X.\u00a0R. Zheng, V.\u00a0S. Deshpande, The toughness of mechanical metamaterials, Nature Materials https:\/\/doi.org\/10.1038\/s41563-021-01182-1","DOI":"10.1038\/s41563-021-01182-1"},{"issue":"2","key":"1906_CR3","doi-asserted-by":"publisher","DOI":"10.1063\/5.0144454","volume":"11","author":"AA Zadpoor","year":"2023","unstructured":"Zadpoor AA, Mirzaali MJ, Valdevit L, Hopkins JB (2023) Design, material, function, and fabrication of metamaterials. APL Mater 11(2):020401. https:\/\/doi.org\/10.1063\/5.0144454","journal-title":"APL Mater"},{"key":"1906_CR4","doi-asserted-by":"publisher","first-page":"114","DOI":"10.1016\/j.matdes.2018.01.011","volume":"142","author":"Y Wang","year":"2018","unstructured":"Wang Y, Zhang L, Daynes S, Zhang H, Feih S, Wang MY (2018) Design of graded lattice structure with optimized mesostructures for additive manufacturing. Mater Des 142:114\u2013123. https:\/\/doi.org\/10.1016\/j.matdes.2018.01.011","journal-title":"Mater Des"},{"issue":"11","key":"1906_CR5","doi-asserted-by":"publisher","first-page":"2200483","DOI":"10.1002\/adem.202200483","volume":"24","author":"F Veloso","year":"2022","unstructured":"Veloso F, Gomes-Fonseca J, Morais P, Correia-Pinto J, Pinho AC, Vila\u00e7a JL (2022) Overview of methods and software for the design of functionally graded lattice structures. Adv Eng Mater 24(11):2200483. https:\/\/doi.org\/10.1002\/adem.202200483","journal-title":"Adv Eng Mater"},{"key":"1906_CR6","doi-asserted-by":"publisher","first-page":"112","DOI":"10.1016\/j.matdes.2017.06.006","volume":"131","author":"SY Choy","year":"2017","unstructured":"Choy SY, Sun C-N, Leong KF, Wei J (2017) Compressive properties of functionally graded lattice structures manufactured by selective laser melting. Mater Des 131:112\u2013120. https:\/\/doi.org\/10.1016\/j.matdes.2017.06.006","journal-title":"Mater Des"},{"key":"1906_CR7","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijmecsci.2020.105735","volume":"182","author":"L Bai","year":"2020","unstructured":"Bai L, Gong C, Chen X, Sun Y, Xin L, Pu H, Peng Y, Luo J (2020) Mechanical properties and energy absorption capabilities of functionally graded lattice structures: Experiments and simulations. Int J Mech Sci 182:105735. https:\/\/doi.org\/10.1016\/j.ijmecsci.2020.105735","journal-title":"Int J Mech Sci"},{"issue":"4","key":"1906_CR8","doi-asserted-by":"publisher","first-page":"204","DOI":"10.1089\/3dp.2016.0039","volume":"3","author":"O Weeger","year":"2016","unstructured":"Weeger O, Kang Y, Yeung S-K, Dunn M (2016) Optimal design and manufacture of active rod structures with spatially variable materials, 3D Printing and Additive Manufacturing 3(4):204\u2013215. https:\/\/doi.org\/10.1089\/3dp.2016.0039","journal-title":"Optimal design and manufacture of active rod structures with spatially variable materials, 3D Printing and Additive Manufacturing"},{"key":"1906_CR9","doi-asserted-by":"publisher","DOI":"10.1016\/j.addma.2021.102108","volume":"47","author":"I Valizadeh","year":"2021","unstructured":"Valizadeh I, Al boud A, D\u00f6rsam E, Weeger O (2021) Tailoring of functionally graded hyperelastic materials via grayscale mask stereolithography 3d printing. Addit Manuf 47:102108. https:\/\/doi.org\/10.1016\/j.addma.2021.102108","journal-title":"Addit Manuf"},{"key":"1906_CR10","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijmecsci.2022.107335","volume":"226","author":"I Valizadeh","year":"2022","unstructured":"Valizadeh I, Weeger O (2022) Parametric visco-hyperelastic constitutive modeling of functionally graded 3d printed polymers. Int J Mech Sci 226:107335. https:\/\/doi.org\/10.1016\/j.ijmecsci.2022.107335","journal-title":"Int J Mech Sci"},{"key":"1906_CR11","doi-asserted-by":"publisher","DOI":"10.1016\/j.addma.2021.102254","volume":"47","author":"TY Kim","year":"2021","unstructured":"Kim TY, Park S-H, Park K (2021) Development of functionally graded metamaterial using selective polymerization via digital light processing additive manufacturing. Addit Manuf 47:102254. https:\/\/doi.org\/10.1016\/j.addma.2021.102254","journal-title":"Addit Manuf"},{"issue":"1","key":"1906_CR12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41524-020-0341-6","volume":"6","author":"S Kumar","year":"2020","unstructured":"Kumar S, Tan S, Zheng L, Kochmann DM (2020) Inverse-designed spinodoid metamaterials. npj Computational Materials 6(1):1\u201310. https:\/\/doi.org\/10.1038\/s41524-020-0341-6","journal-title":"npj Computational Materials"},{"issue":"17","key":"1906_CR13","doi-asserted-by":"publisher","first-page":"1705708","DOI":"10.1002\/adma.201705708","volume":"30","author":"L Han","year":"2018","unstructured":"Han L, Che S (2018) An overview of materials with triply periodic minimal surfaces and related geometry: From biological structures to self-assembled systems. Adv Mater 30(17):1705708. https:\/\/doi.org\/10.1002\/adma.201705708","journal-title":"Adv Mater"},{"issue":"6012","key":"1906_CR14","doi-asserted-by":"publisher","first-page":"604","DOI":"10.1038\/314604a0","volume":"314","author":"AL Mackay","year":"1985","unstructured":"Mackay AL (1985) Periodic minimal surfaces. Nature 314(6012):604\u2013606. https:\/\/doi.org\/10.1038\/314604a0","journal-title":"Nature"},{"key":"1906_CR15","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1016\/j.addma.2018.08.007","volume":"23","author":"L Zhang","year":"2018","unstructured":"Zhang L, Feih S, Daynes S, Chang S, Wang MY, Wei J, Lu WF (2018) Energy absorption characteristics of metallic triply periodic minimal surface sheet structures under compressive loading. Addit Manuf 23:505\u2013515. https:\/\/doi.org\/10.1016\/j.addma.2018.08.007","journal-title":"Addit Manuf"},{"key":"1906_CR16","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jmps.2018.08.022","volume":"122","author":"C Bonatti","year":"2019","unstructured":"Bonatti C, Mohr D (2019) Mechanical performance of additively-manufactured anisotropic and isotropic smooth shell-lattice materials: Simulations and experiments. J Mech Phys Solids 122:1\u201326. https:\/\/doi.org\/10.1016\/j.jmps.2018.08.022","journal-title":"J Mech Phys Solids"},{"key":"1906_CR17","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijmecsci.2021.106977","volume":"216","author":"X Guo","year":"2022","unstructured":"Guo X, Ding J, Li X, Qu S, Song X, Fuh JYH, Lu WF, Zhai W (2022) Enhancement in the mechanical behaviour of a Schwarz primitive periodic minimal surface lattice structure design. Int J Mech Sci 216:106977. https:\/\/doi.org\/10.1016\/j.ijmecsci.2021.106977","journal-title":"Int J Mech Sci"},{"issue":"2","key":"1906_CR18","doi-asserted-by":"publisher","DOI":"10.1088\/2631-7990\/ac5be6","volume":"4","author":"J Feng","year":"2022","unstructured":"Feng J, Fu J, Yao X, He Y (2022) Triply periodic minimal surface (TPMS) porous structures: From multi-scale design, precise additive manufacturing to multidisciplinary applications. International Journal of Extreme Manufacturing 4(2):022001. https:\/\/doi.org\/10.1088\/2631-7990\/ac5be6","journal-title":"International Journal of Extreme Manufacturing"},{"key":"1906_CR19","doi-asserted-by":"publisher","DOI":"10.1016\/j.addma.2020.101171","volume":"33","author":"J Plocher","year":"2020","unstructured":"Plocher J, Panesar A (2020) Effect of density and unit cell size grading on the stiffness and energy absorption of short fibre-reinforced functionally graded lattice structures. Addit Manuf 33:101171. https:\/\/doi.org\/10.1016\/j.addma.2020.101171","journal-title":"Addit Manuf"},{"key":"1906_CR20","doi-asserted-by":"publisher","unstructured":"N.\u00a0Sathishkumar, N.\u00a0Arunkumar, S.\u00a0V. Rohith, R.\u00a0R. Hariharan, Effect of varying unit cell size on energy absorption behaviour of additive manufactured TPMS PETG lattice structure, Progress in Additive Manufacturing https:\/\/doi.org\/10.1007\/s40964-023-00407-w","DOI":"10.1007\/s40964-023-00407-w"},{"key":"1906_CR21","doi-asserted-by":"publisher","DOI":"10.1016\/j.euromechsol.2022.104676","volume":"96","author":"N Roudbarian","year":"2022","unstructured":"Roudbarian N, Jebellat E, Famouri S, Baniasadi M, Hedayati R, Baghani M (2022) Shape-memory polymer metamaterials based on triply periodic minimal surfaces. Eur J Mech A Solids 96:104676. https:\/\/doi.org\/10.1016\/j.euromechsol.2022.104676","journal-title":"Eur J Mech A Solids"},{"issue":"10","key":"1906_CR22","doi-asserted-by":"publisher","first-page":"773","DOI":"10.1557\/mrs.2019.228","volume":"44","author":"DM Kochmann","year":"2019","unstructured":"Kochmann DM, Hopkins JB, Valdevit L (2019) Multiscale modeling and optimization of the mechanics of hierarchical metamaterials. MRS Bull 44(10):773\u2013781. https:\/\/doi.org\/10.1557\/mrs.2019.228","journal-title":"MRS Bull"},{"issue":"Supplement C","key":"1906_CR23","doi-asserted-by":"publisher","first-page":"102","DOI":"10.1016\/j.mechmat.2016.01.004","volume":"95","author":"DW Abueidda","year":"2016","unstructured":"Abueidda DW, Abu Al-Rub RK, Dalaq AS, Lee D-W, Khan KA, Jasiuk I (2016) Effective conductivities and elastic moduli of novel foams with triply periodic minimal surfaces. Mech Mater 95(Supplement C):102\u2013115. https:\/\/doi.org\/10.1016\/j.mechmat.2016.01.004","journal-title":"Mech Mater"},{"key":"1906_CR24","doi-asserted-by":"publisher","DOI":"10.1016\/j.matdes.2021.110050","volume":"210","author":"J Feng","year":"2021","unstructured":"Feng J, Liu B, Lin Z, Fu J (2021) Isotropic porous structure design methods based on triply periodic minimal surfaces. Materials & Design 210:110050. https:\/\/doi.org\/10.1016\/j.matdes.2021.110050","journal-title":"Materials & Design"},{"issue":"11","key":"1906_CR25","doi-asserted-by":"publisher","first-page":"2397","DOI":"10.1007\/s00419-020-01728-w","volume":"90","author":"I Valizadeh","year":"2020","unstructured":"Valizadeh I, Weeger O (2020) Nonlinear multiscale simulation of instabilities due to growth of an elastic film on a microstructured substrate. Arch Appl Mech 90(11):2397\u20132412. https:\/\/doi.org\/10.1007\/s00419-020-01728-w","journal-title":"Arch Appl Mech"},{"key":"1906_CR26","doi-asserted-by":"publisher","unstructured":"N.\u00a0Kladovasilakis, K.\u00a0Tsongas, I.\u00a0Kostavelis, D.\u00a0Tzovaras, D.\u00a0Tzetzis, Effective mechanical properties of additive manufactured triply periodic minimal surfaces: Experimental and finite element study, The International Journal of Advanced Manufacturing Technology https:\/\/doi.org\/10.1007\/s00170-022-09651-w","DOI":"10.1007\/s00170-022-09651-w"},{"key":"1906_CR27","doi-asserted-by":"publisher","unstructured":"A.\u00a0M. Abou-Ali, O.\u00a0Al-Ketan, D.-W. Lee, R.\u00a0Rowshan, R.\u00a0K. Abu Al-Rub, Mechanical behavior of polymeric selective laser sintered ligament and sheet based lattices of triply periodic minimal surface architectures, Materials and Design 196 (2020) 109100. https:\/\/doi.org\/10.1016\/j.matdes.2020.109100","DOI":"10.1016\/j.matdes.2020.109100"},{"key":"1906_CR28","doi-asserted-by":"publisher","DOI":"10.1016\/j.matdes.2022.111036","volume":"222","author":"F G\u00fcnther","year":"2022","unstructured":"G\u00fcnther F, Hirsch F, Pilz S, Wagner M, Gebert A, K\u00e4stner M, Zimmermann M (2022) Structure-property relationships of imperfect additively manufactured lattices based on triply periodic minimal surfaces. Mater Des 222:111036. https:\/\/doi.org\/10.1016\/j.matdes.2022.111036","journal-title":"Mater Des"},{"key":"1906_CR29","doi-asserted-by":"publisher","DOI":"10.1016\/j.matdes.2021.109655","volume":"204","author":"W Jiang","year":"2021","unstructured":"Jiang W, Liao W, Liu T, Shi X, Wang C, Qi J, Chen Y, Wang Z, Zhang C (2021) A voxel-based method of multiscale mechanical property optimization for the design of graded TPMS structures. Mater Des 204:109655. https:\/\/doi.org\/10.1016\/j.matdes.2021.109655","journal-title":"Mater Des"},{"key":"1906_CR30","doi-asserted-by":"publisher","unstructured":"G.\u00a0C.\u00a0Y. Peng, M.\u00a0Alber, A.\u00a0Buganza\u00a0Tepole, W.\u00a0R. Cannon, S.\u00a0De, S.\u00a0Dura-Bernal, K.\u00a0Garikipati, G.\u00a0Karniadakis, W.\u00a0W. Lytton, P.\u00a0Perdikaris, L.\u00a0Petzold, E.\u00a0Kuhl, Multiscale modeling meets machine learning: What can we learn?, Archives of Computational Methods in Engineering https:\/\/doi.org\/10.1007\/s11831-020-09405-5","DOI":"10.1007\/s11831-020-09405-5"},{"key":"1906_CR31","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-94463-0","volume-title":"Neural Networks and Deep Learning","author":"CC Aggarwal","year":"2018","unstructured":"Aggarwal CC (2018) Neural Networks and Deep Learning, 1st edn. Springer, New York","edition":"1"},{"key":"1906_CR32","doi-asserted-by":"publisher","unstructured":"S.\u00a0Kollmannsberger, D.\u00a0D\u2019Angella, M.\u00a0Jokeit, L.\u00a0Herrmann, Deep Learning in Computational Mechanics, Vol. 977 of Studies in Computational Intelligence, Springer, 2021. https:\/\/doi.org\/10.1007\/978-3-030-76587-3","DOI":"10.1007\/978-3-030-76587-3"},{"issue":"2","key":"1906_CR33","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1016\/0893-6080(91)90009-T","volume":"4","author":"K Hornik","year":"1991","unstructured":"Hornik K (1991) Approximation capabilities of multilayer feedforward networks. Neural Netw 4(2):251\u2013257. https:\/\/doi.org\/10.1016\/0893-6080(91)90009-T","journal-title":"Neural Netw"},{"key":"1906_CR34","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.110072","volume":"428","author":"K Xu","year":"2021","unstructured":"Xu K, Huang DZ, Darve E (2021) Learning constitutive relations using symmetric positive definite neural networks. J Comput Phys 428:110072. https:\/\/doi.org\/10.1016\/j.jcp.2020.110072","journal-title":"J Comput Phys"},{"key":"1906_CR35","unstructured":"C.\u00a0F. Jekel, K.\u00a0E. Swartz, D.\u00a0A. White, D.\u00a0A. Tortorelli, S.\u00a0E. Watts, Neural network layers for prediction of positive definite elastic stiffness tensors, Pre-print under review arXiv:2203.13938"},{"key":"1906_CR36","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmps.2021.104703","volume":"159","author":"DK Klein","year":"2022","unstructured":"Klein DK, Fern\u00e1ndez M, Martin RJ, Neff P, Weeger O (2022) Polyconvex anisotropic hyperelasticity with neural networks. J Mech Phys Solids 159:104703. https:\/\/doi.org\/10.1016\/j.jmps.2021.104703","journal-title":"J Mech Phys Solids"},{"key":"1906_CR37","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115501","volume":"400","author":"DK Klein","year":"2022","unstructured":"Klein DK, Ortigosa R, Mart\u00ednez-Frutos J, Weeger O (2022) Finite electro-elasticity with physics-augmented neural networks. Comput Methods Appl Mech Eng 400:115501. https:\/\/doi.org\/10.1016\/j.cma.2022.115501","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1906_CR38","unstructured":"L.\u00a0Linden, D.\u00a0K. Klein, K.\u00a0A. Kalina, J.\u00a0Brummund, O.\u00a0Weeger, M.\u00a0K\u00e4stner, Neural networks meet hyperelasticity: A guide to enforcing physics, Pre-print under review arXiv:2302.02403"},{"issue":"12","key":"1906_CR39","doi-asserted-by":"publisher","first-page":"2738","DOI":"10.1002\/nme.6957","volume":"123","author":"F As\u2019ad","year":"2022","unstructured":"As\u2019ad F, Avery P, Farhat C (2022) A mechanics-informed artificial neural network approach in data-driven constitutive modeling. Int J Numer Meth Eng 123(12):2738\u20132759. https:\/\/doi.org\/10.1002\/nme.6957","journal-title":"Int J Numer Meth Eng"},{"key":"1906_CR40","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.113695","volume":"377","author":"NN Vlassis","year":"2021","unstructured":"Vlassis NN, Sun W (2021) Sobolev training of thermodynamic-informed neural networks for interpretable elasto-plasticity models with level set hardening. Comput Methods Appl Mech Eng 377:113695. https:\/\/doi.org\/10.1016\/j.cma.2021.113695","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1906_CR41","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115190","volume":"398","author":"F Masi","year":"2022","unstructured":"Masi F, Stefanou I (2022) Multiscale modeling of inelastic materials with thermodynamics-based artificial neural networks (TANN). Comput Methods Appl Mech Eng 398:115190. https:\/\/doi.org\/10.1016\/j.cma.2022.115190","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1906_CR42","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmps.2022.105076","volume":"169","author":"P Thakolkaran","year":"2022","unstructured":"Thakolkaran P, Joshi A, Zheng Y, Flaschel M, De Lorenzis L, Kumar S (2022) NN-EUCLID: Deep-learning hyperelasticity without stress data. J Mech Phys Solids 169:105076. https:\/\/doi.org\/10.1016\/j.jmps.2022.105076","journal-title":"J Mech Phys Solids"},{"key":"1906_CR43","unstructured":"Z.\u00a0Liu, Y.\u00a0Du, Y.\u00a0Chen, M.\u00a0Tegmark, Physics-augmented learning: A new paradigm beyond physics-informed learning, 2021. arXiv:2109.13901"},{"key":"1906_CR44","doi-asserted-by":"publisher","unstructured":"J.\u00a0Mianroodi, S.\u00a0Rezaei, N.\u00a0Siboni, B.-X. Xu, D.\u00a0Raabe, Lossless multi-scale constitutive elastic relations with artificial intelligence, npj Computational Materials 8 (2022) 67. https:\/\/doi.org\/10.1038\/s41524-022-00753-3","DOI":"10.1038\/s41524-022-00753-3"},{"key":"1906_CR45","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115741","volume":"403","author":"B Eidel","year":"2023","unstructured":"Eidel B (2023) Deep CNNs as universal predictors of elasticity tensors in homogenization. Comput Methods Appl Mech Eng 403:115741. https:\/\/doi.org\/10.1016\/j.cma.2022.115741","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"2","key":"1906_CR46","doi-asserted-by":"publisher","first-page":"577","DOI":"10.1002\/nme.6869","volume":"123","author":"M Fern\u00e1ndez","year":"2022","unstructured":"Fern\u00e1ndez M, Fritzen F, Weeger O (2022) Material modeling for parametric, anisotropic finite strain hyperelasticity based on machine learning with application in optimization of metamaterials. Int J Numer Meth Eng 123(2):577\u2013609. https:\/\/doi.org\/10.1002\/nme.6869","journal-title":"Int J Numer Meth Eng"},{"key":"1906_CR47","doi-asserted-by":"publisher","unstructured":"T.\u00a0G\u00e4rtner, M.\u00a0Fern\u00e1ndez, O.\u00a0Weeger, Nonlinear multiscale simulation of elastic beam lattices with anisotropic homogenized constitutive models based on artificial neural networks, Computational Mechanics 68. https:\/\/doi.org\/10.1007\/s00466-021-02061-x","DOI":"10.1007\/s00466-021-02061-x"},{"key":"1906_CR48","unstructured":"Dassault Systemes SIMULIA Corp., ABAQUS\/CAE 2021: SIMULIA User Assistance (2021)"},{"key":"1906_CR49","volume-title":"Nonlinear Solid Mechanics: A Continuum Approach for Engineering","author":"GA Holzapfel","year":"2000","unstructured":"Holzapfel GA (2000) Nonlinear Solid Mechanics: A Continuum Approach for Engineering, 2nd edn. Wiley, Amsterdam","edition":"2"},{"key":"1906_CR50","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-32360-0","author":"TI Zohdi","year":"2005","unstructured":"Zohdi TI, Wriggers P (2005) An Introduction to Computational Micromechanics, Lecture Notes in Applied and Computational Mechanics, Springer-Verlag. Berlin Heidelberg. https:\/\/doi.org\/10.1007\/978-3-540-32360-0","journal-title":"Berlin Heidelberg"},{"key":"1906_CR51","doi-asserted-by":"publisher","first-page":"322","DOI":"10.1016\/j.pmatsci.2018.02.003","volume":"96","author":"S Bargmann","year":"2018","unstructured":"Bargmann S, Klusemann B, Markmann J, Schnabel JE, Schneider K, Soyarslan C, Wilmers J (2018) Generation of 3d representative volume elements for heterogeneous materials: A review. Prog Mater Sci 96:322\u2013384. https:\/\/doi.org\/10.1016\/j.pmatsci.2018.02.003","journal-title":"Prog Mater Sci"},{"key":"1906_CR52","unstructured":"V.\u00a0Ebbing, Design of Polyconvex Energy Functions for All Anisotropy Classes, Ph.D. thesis, Universit\u00e4t Duisburg-Essen (2010)"},{"key":"1906_CR53","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-04775-0","volume-title":"Continuum Mechanics and Theory of Materials","author":"P Haupt","year":"2002","unstructured":"Haupt P (2002) Continuum Mechanics and Theory of Materials, 2nd edn. Springer, Berlin Heidelberg","edition":"2"},{"issue":"3","key":"1906_CR54","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1016\/j.compstruct.2008.08.007","volume":"89","author":"P Malekzadeh","year":"2009","unstructured":"Malekzadeh P (2009) Three-dimensional free vibration analysis of thick functionally graded plates on elastic foundations. Compos Struct 89(3):367\u2013373. https:\/\/doi.org\/10.1016\/j.compstruct.2008.08.007","journal-title":"Compos Struct"},{"key":"1906_CR55","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.tws.2016.07.019","volume":"108","author":"A Setoodeh","year":"2016","unstructured":"Setoodeh A, Shojaee M (2016) Application of TW-DQ method to nonlinear free vibration analysis of FG carbon nanotube-reinforced composite quadrilateral plates. Thin-Walled Structures 108:1\u201311. https:\/\/doi.org\/10.1016\/j.tws.2016.07.019","journal-title":"Thin-Walled Structures"},{"issue":"3","key":"1906_CR56","doi-asserted-by":"publisher","first-page":"833","DOI":"10.1007\/s00366-018-0633-3","volume":"35","author":"H Jrad","year":"2018","unstructured":"Jrad H, Mars J, Wali M, Dammak F (2018) Geometrically nonlinear analysis of elastoplastic behavior of functionally graded shells. Engineering with Computers 35(3):833\u2013847. https:\/\/doi.org\/10.1007\/s00366-018-0633-3","journal-title":"Engineering with Computers"},{"key":"1906_CR57","volume-title":"Matrix Analysis","author":"R Horn","year":"2013","unstructured":"Horn R, Johnson C (2013) Matrix Analysis, 2nd edn. Cambridge University Press, USA","edition":"2"},{"key":"1906_CR58","unstructured":"K.\u00a0A. Kalina, L.\u00a0Linden, J.\u00a0Brummund, M.\u00a0K\u00e4stner, FEANN \u2013 An efficient data-driven multiscale approach based on physics-constrained neural networks and automated data mining arXiv:1048550\/arXiv.2207.01045"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-023-01906-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-023-01906-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-023-01906-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,2]],"date-time":"2024-07-02T09:16:35Z","timestamp":1719911795000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-023-01906-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,11]]},"references-count":58,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2024,6]]}},"alternative-id":["1906"],"URL":"https:\/\/doi.org\/10.1007\/s00366-023-01906-8","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,11]]},"assertion":[{"value":"4 May 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 September 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 October 2023","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 have no competing interests to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}