{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T01:23:41Z","timestamp":1784251421192,"version":"3.55.0"},"reference-count":79,"publisher":"Oxford University Press (OUP)","issue":"8","license":[{"start":{"date-parts":[[2025,7,16]],"date-time":"2025-07-16T00:00:00Z","timestamp":1752624000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["92371206"],"award-info":[{"award-number":["92371206"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12372123"],"award-info":[{"award-number":["12372123"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Postgraduate Scientific Research Innovation Project of Hunan Province","award":["XJCX2023185"],"award-info":[{"award-number":["XJCX2023185"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,8,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>In recent years, although deep learning (DL) has been a powerful tool for accelerating the design of multiscale structures, the existing research focuses on the single-material structural design, and it has not yet proposed an effective solution in the design of multi-material and multi-scale structures. To address this limitation, we introduce a novel multi-material multi-scale topology optimization framework based on deep learning, termed MMTO-DL, specifically tailored for designing multi-material multi-scale structures. The MMTO-DL framework leverages the level set method to implicitly represent heterogeneous microstructures. Each microstructure is governed by height and material variables, which respectively control the configuration and material selection. Furthermore, MMTO-DL incorporates two neural networks for neural reparametrization and surrogate models respectively. The input to the reparametrization network comprises spatial point coordinates within the design domain, while the outputs are sets of height and material variables. Innovatively, MMTO-DL introduces a penalty factor into the Softmax function to prevent the formation of mixed material units, facilitating the regional distribution of multi-material structures. Additionally, a separate neural network is used to build a high-precision surrogate model linking height variables and mechanical properties, substantially cutting computational needs for microstructure analysis. The MMTO-DL framework facilitates sensitivity analysis and gradient information acquisition. The framework\u2019s effectiveness and flexibility are demonstrated through numerical examples involving compliance minimization and shape matching problems.<\/jats:p>","DOI":"10.1093\/jcde\/qwaf064","type":"journal-article","created":{"date-parts":[[2025,7,16]],"date-time":"2025-07-16T13:25:28Z","timestamp":1752672328000},"page":"127-153","source":"Crossref","is-referenced-by-count":3,"title":["Multi-scale topology optimization of multi-material heterogeneous microstructures based on deep learning"],"prefix":"10.1093","volume":"12","author":[{"given":"Jiaxiang","family":"Luo","sequence":"first","affiliation":[{"name":"National University of Defense Technology College of Aerospace Science and Engineering, , Changsha 410073, Hunan ,","place":["China"]},{"name":"Chinese Academy of Military Science Defense Innovation Institute, , Beijing 100071 ,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yu","family":"Li","sequence":"additional","affiliation":[{"name":"Chinese Academy of Military Science Defense Innovation Institute, , Beijing 100071 ,","place":["China"]},{"name":"Intelligent Game and Decision Laboratory , Beijing 100071 ,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Weien","family":"Zhou","sequence":"additional","affiliation":[{"name":"Chinese Academy of Military Science Defense Innovation Institute, , Beijing 100071 ,","place":["China"]},{"name":"Intelligent Game and Decision Laboratory , Beijing 100071 ,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zeyu","family":"Zhang","sequence":"additional","affiliation":[{"name":"National University of Defense Technology College of Aerospace Science and Engineering, , Changsha 410073, Hunan ,","place":["China"]},{"name":"Chinese Academy of Military Science Defense Innovation Institute, , Beijing 100071 ,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5224-9834","authenticated-orcid":false,"given":"Wen","family":"Yao","sequence":"additional","affiliation":[{"name":"Chinese Academy of Military Science Defense Innovation Institute, , Beijing 100071 ,","place":["China"]},{"name":"Intelligent Game and Decision Laboratory , Beijing 100071 ,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2025,7,16]]},"reference":[{"key":"2025102210123189300_bib1","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1016\/j.jcp.2003.09.032","article-title":"Structural optimization using sensitivity analysis and a level-set method","volume":"194","author":"Allaire","year":"2004","journal-title":"Journal of Computational Physics"},{"key":"2025102210123189300_bib2","doi-asserted-by":"publisher","first-page":"488","DOI":"10.1016\/j.commatsci.2013.09.006","article-title":"How to determine composite material properties using numerical homogenization","volume":"83","author":"Andreassen","year":"2014","journal-title":"Computational Materials Science"},{"key":"2025102210123189300_bib3","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1016\/0045-7825(88)90086-2","article-title":"Generating optimal topologies in structural design using a homogenization method","volume":"71","author":"Bends\u00f8e","year":"1988","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"2025102210123189300_bib4","doi-asserted-by":"publisher","first-page":"635","DOI":"10.1007\/s004190050248","article-title":"Material interpolation schemes in topology optimization","volume":"69","author":"Bends\u00f8e","year":"1999","journal-title":"Archive of Applied Mechanics (Ingenieur Archiv)"},{"key":"2025102210123189300_bib5","volume-title":"Topology Optimization: Theory, Methods, and Applications","author":"Bends\u00f8e","year":"2011"},{"key":"2025102210123189300_bib6","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1007\/s00158-022-03471-y","article-title":"Deep neural networks for parameterized homogenization in concurrent multiscale structural optimization","volume":"66","author":"Black","year":"2023","journal-title":"Structural and Multidisciplinary Optimization"},{"key":"2025102210123189300_bib7","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1007\/s11465-019-0531-4","article-title":"Robust topology optimization of multi-material lattice structures under material and load uncertainties","volume":"14","author":"Chan","year":"2019","journal-title":"Frontiers of Mechanical Engineering"},{"key":"2025102210123189300_bib8","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1007\/s00158-022-03224-x","article-title":"Remixing functionally graded structures: data-driven topology optimization with multiclass shape blending","volume":"65","author":"Chan","year":"2022","journal-title":"Structural and Multidisciplinary Optimization"},{"key":"2025102210123189300_bib9","doi-asserted-by":"publisher","first-page":"103017","DOI":"10.1016\/j.cad.2021.103017","article-title":"Multi-material topology optimization using neural networks","volume":"136","author":"Chandrasekhar","year":"2021","journal-title":"Computer-Aided Design"},{"key":"2025102210123189300_bib10","doi-asserted-by":"publisher","first-page":"1135","DOI":"10.1007\/s00158-020-02748-4","article-title":"Tounn: Topology optimization using neural networks","volume":"63","author":"Chandrasekhar","year":"2021","journal-title":"Structural and Multidisciplinary Optimization"},{"key":"2025102210123189300_bib11","doi-asserted-by":"publisher","first-page":"103359","DOI":"10.1016\/j.advengsoft.2022.103359","article-title":"Graded multiscale topology optimization using neural networks","volume":"175","author":"Chandrasekhar","year":"2023","journal-title":"Advances in Engineering Software"},{"key":"2025102210123189300_bib12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41598-018-26980-7","article-title":"Multi-material additive manufacturing of metamaterials with giant, tailorable negative poisson\u2019s ratios","volume":"8","author":"Chen","year":"2018","journal-title":"Scientific Reports"},{"key":"2025102210123189300_bib13","doi-asserted-by":"publisher","first-page":"528863","DOI":"10.7527\/S1000-6893.2023.28863","article-title":"A review of topology optimization design methods for multi-scale structures","volume":"44","author":"Chen","year":"2023","journal-title":"Acta Aeronautica et Astronautica Sinica"},{"key":"2025102210123189300_bib14","doi-asserted-by":"publisher","first-page":"024012","DOI":"10.1103\/PhysRevApplied.7.024012","article-title":"Lattice metamaterials with mechanically tunable Poisson\u2019s ratio for vibration control","volume":"7","author":"Chen","year":"2017","journal-title":"Physical Review Applied"},{"key":"2025102210123189300_bib15","doi-asserted-by":"publisher","first-page":"408","DOI":"10.1016\/j.cma.2017.12.024","article-title":"Coupling lattice structure topology optimization with design-dependent feature evolution for additive manufactured heat conduction design","volume":"332","author":"Cheng","year":"2018","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"2025102210123189300_bib16","doi-asserted-by":"publisher","first-page":"115402","DOI":"10.1016\/j.compstruct.2022.115402","article-title":"Multi-scale and multi-material topology optimization of gradient lattice structures using surrogate models","volume":"289","author":"Costa","year":"2022","journal-title":"Composite Structures"},{"key":"2025102210123189300_bib17","doi-asserted-by":"publisher","first-page":"126","DOI":"10.1108\/EC-01-2018-0007","article-title":"Multiscale concurrent topology optimization of structures and microscopic multi-phase materials for thermal conductivity","volume":"36","author":"Da","year":"2019","journal-title":"Engineering Computations"},{"key":"2025102210123189300_bib18","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1007\/s11465-019-0532-3","article-title":"Connected morphable components-based multiscale topology optimization","volume":"14","author":"Deng","year":"2019","journal-title":"Frontiers of Mechanical Engineering"},{"key":"2025102210123189300_bib19","doi-asserted-by":"publisher","first-page":"011005","DOI":"10.1115\/1.4040555","article-title":"A 149 line homogenization code for three-dimensional cellular materials written in matlab","volume":"141","author":"Dong","year":"2019","journal-title":"Journal of Engineering Materials and Technology"},{"key":"2025102210123189300_bib20","doi-asserted-by":"publisher","first-page":"2652","DOI":"10.1016\/j.jmps.2007.04.005","article-title":"Shock enhancement of cellular structures under impact loading: Part i experiments","volume":"55","author":"Elnasri","year":"2007","journal-title":"Journal of the Mechanics and Physics of Solids"},{"key":"2025102210123189300_bib21","doi-asserted-by":"publisher","first-page":"2621","DOI":"10.1007\/s00158-019-02323-6","article-title":"Concurrent topology optimization of multiscale composite structures in Matlab","volume":"60","author":"Gao","year":"2019","journal-title":"Structural and Multidisciplinary Optimization"},{"key":"2025102210123189300_bib22","doi-asserted-by":"publisher","first-page":"100755","DOI":"10.1016\/j.mser.2023.100755","article-title":"Rational designs of mechanical metamaterials: Formulations, architectures, tessellations and prospects","volume":"156","author":"Gao","year":"2023","journal-title":"Materials Science and Engineering: R: Reports"},{"key":"2025102210123189300_bib23","doi-asserted-by":"publisher","first-page":"116036","DOI":"10.1016\/j.cma.2023.116036","article-title":"Multi-patch isogeometric topology optimization for cellular structures with flexible designs using nitsche\u2019s method","volume":"410","author":"Gao","year":"2023","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"2025102210123189300_bib24","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1016\/0045-7825(90)90148-F","article-title":"Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods","volume":"83","author":"Guedes","year":"1990","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"2025102210123189300_bib25","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1016\/0893-6080(89)90020-8","article-title":"Multilayer feedforward networks are universal approximators","volume":"2","author":"Hornik","year":"1989","journal-title":"Neural Networks"},{"key":"2025102210123189300_bib26","article-title":"Neural reparameterization improves structural optimization","author":"Hoyer","year":"2019"},{"key":"2025102210123189300_bib27","doi-asserted-by":"publisher","first-page":"116922","DOI":"10.1016\/j.compstruct.2023.116922","article-title":"Thermal design of functionally graded cellular structures with multiple microstructure configurations through topology optimization","volume":"313","author":"Huo","year":"2023","journal-title":"Composite Structures"},{"key":"2025102210123189300_bib28","doi-asserted-by":"publisher","first-page":"116864","DOI":"10.1016\/j.cma.2024.116864","article-title":"Topology optimization of irregular multiscale structures with tunable responses using a virtual growth rule","volume":"425","author":"Jia","year":"2024","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"2025102210123189300_bib29","first-page":"935","article-title":"Multiscale design and multiobjective optimization of orthopaedic cellular hip implants","volume-title":"International Design Engineering Technical Conferences and Computers and Information in Engineering Conference","author":"Khanoki","year":"2011"},{"key":"2025102210123189300_bib30","doi-asserted-by":"publisher","first-page":"114158","DOI":"10.1016\/j.cma.2021.114158","article-title":"Machine learning-combined topology optimization for functionary graded composite structure design","volume":"387","author":"Kim","year":"2021","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"2025102210123189300_bib31","doi-asserted-by":"publisher","first-page":"1602","DOI":"10.1093\/jcde\/qwac078","article-title":"Computational homogenization of additively manufactured lightweight structures with multiscale topology optimization","volume":"9","author":"Kim","year":"2022","journal-title":"Journal of Computational Design and Engineering"},{"key":"2025102210123189300_bib32","doi-asserted-by":"publisher","first-page":"1129","DOI":"10.1016\/j.ijheatmasstransfer.2003.10.012","article-title":"Fluid-flow and endwall heat-transfer characteristics of an ultralight lattice-frame material","volume":"47","author":"Kim","year":"2004","journal-title":"International Journal of Heat and Mass Transfer"},{"key":"2025102210123189300_bib33","article-title":"Adam: a method for stochastic optimization","author":"Kingma","year":"2017"},{"key":"2025102210123189300_bib34","doi-asserted-by":"publisher","first-page":"1653","DOI":"10.1007\/s40964-023-00529-1","article-title":"Energy absorption of 3d printed multi-material elastic lattice structures","volume":"9","author":"Kreide","year":"2023","journal-title":"Progress in Additive Manufacturing"},{"key":"2025102210123189300_bib35","doi-asserted-by":"publisher","first-page":"116969","DOI":"10.1016\/j.cma.2024.116969","article-title":"A multi-material topology optimization approach to hybrid material structures with gradient lattices","volume":"425","author":"Li","year":"2024","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"2025102210123189300_bib36","doi-asserted-by":"publisher","first-page":"081008","DOI":"10.1115\/1.4036941","article-title":"Additive manufacturing-oriented design of graded lattice structures through explicit topology optimization","volume":"84","author":"Liu","year":"2017","journal-title":"Journal of Applied Mechanics"},{"key":"2025102210123189300_bib37","doi-asserted-by":"publisher","first-page":"113154","DOI":"10.1016\/j.cma.2020.113154","article-title":"M-vcut level set method for optimizing cellular structures","volume":"367","author":"Liu","year":"2020","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"2025102210123189300_bib38","doi-asserted-by":"publisher","first-page":"116485","DOI":"10.1016\/j.compstruct.2022.116485","article-title":"Data-driven design of graded composite lattice structures with multiple microstructure prototypes and materials","volume":"305","author":"Liu","year":"2023","journal-title":"Composite Structures"},{"key":"2025102210123189300_bib39","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1093\/jcde\/qwab069","article-title":"A heterogeneous lattice structure modeling technique supported by multiquadric radial basis function networks","volume":"9","author":"Liu","year":"2022","journal-title":"Journal of Computational Design and Engineering"},{"key":"2025102210123189300_bib40","article-title":"Dying relu and initialization: Theory and numerical examples","author":"Lu","year":"2019"},{"key":"2025102210123189300_bib41","doi-asserted-by":"publisher","first-page":"1729","DOI":"10.1021\/acsphotonics.0c00327","article-title":"Inverse design of photonic crystals through automatic differentiation","volume":"7","author":"Minkov","year":"2020","journal-title":"ACS Photonics"},{"key":"2025102210123189300_bib42","doi-asserted-by":"publisher","first-page":"813","DOI":"10.1093\/jcde\/qwab016","article-title":"Implicit-based computer-aided design for additively manufactured functionally graded cellular structures","volume":"8","author":"Nguyen","year":"2021","journal-title":"Journal of Computational Design and Engineering"},{"key":"2025102210123189300_bib43","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1093\/jcde\/qwab078","article-title":"Lightweight design with metallic additively manufactured cellular structures","volume":"9","author":"Nguyen","year":"2022","journal-title":"Journal of Computational Design and Engineering"},{"key":"2025102210123189300_bib44","doi-asserted-by":"publisher","first-page":"117213","DOI":"10.1016\/j.cma.2024.117213","article-title":"Topology optimization of two-scale hierarchical structures with high-cycle fatigue resistance","volume":"430","author":"Ni","year":"2024","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"2025102210123189300_bib45","doi-asserted-by":"publisher","first-page":"8024","DOI":"10.48550\/arXiv.1912.01703","article-title":"Pytorch: An imperative style, high-performance deep learning library","volume":"32","author":"Paszke","year":"2019","journal-title":"Advances in Neural Information Processing Systems"},{"key":"2025102210123189300_bib46","doi-asserted-by":"publisher","first-page":"266","DOI":"10.1007\/s00158-022-03369-9","article-title":"A survey of machine learning techniques in structural and multidisciplinary optimization","volume":"65","author":"Ramu","year":"2022","journal-title":"Structural and Multidisciplinary Optimization"},{"key":"2025102210123189300_bib47","doi-asserted-by":"publisher","first-page":"011701","DOI":"10.1115\/1.4055505","article-title":"Neural network-assisted design: A study of multiscale topology optimization with smoothly graded cellular structures","volume":"145","author":"Rastegarzadeh","year":"2023","journal-title":"Journal of Mechanical Design"},{"key":"2025102210123189300_bib48","doi-asserted-by":"publisher","first-page":"4296","DOI":"10.1002\/adma.201501708","article-title":"Multistable architected materials for trapping elastic strain energy","volume":"27","author":"Shan","year":"2015","journal-title":"Advanced Materials"},{"key":"2025102210123189300_bib49","doi-asserted-by":"publisher","first-page":"1640006","DOI":"10.1142\/S1758825116400068","article-title":"Energy absorption of thermoplastic polyurethane lattice structures via 3d printing: modeling and prediction","volume":"8","author":"Shen","year":"2016","journal-title":"International Journal of Applied Mechanics"},{"key":"2025102210123189300_bib50","doi-asserted-by":"publisher","first-page":"1736","DOI":"10.1093\/jcde\/qwad072","article-title":"Topology optimization via machine learning and deep learning: A review","volume":"10","author":"Shin","year":"2023","journal-title":"Journal of Computational Design and Engineering"},{"key":"2025102210123189300_bib51","doi-asserted-by":"publisher","first-page":"1267","DOI":"10.1007\/s00158-016-1519-x","article-title":"Simultaneous material and structural optimization by multiscale topology optimization","volume":"54","author":"Sivapuram","year":"2016","journal-title":"Structural and multidisciplinary optimization"},{"key":"2025102210123189300_bib52","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1002\/nme.1620240207","article-title":"The method of moving asymptotes\u2014a new method for structural optimization","volume":"24","author":"Svanberg","year":"1987","journal-title":"International Journal for Numerical Methods in Engineering"},{"key":"2025102210123189300_bib53","doi-asserted-by":"publisher","first-page":"621","DOI":"10.1007\/s00158-013-0999-1","article-title":"Alternating active-phase algorithm for multimaterial topology optimization problems: A 115-line MATLAB implementation","volume":"49","author":"Tavakoli","year":"2014","journal-title":"Structural and Multidisciplinary Optimization"},{"key":"2025102210123189300_bib54","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1016\/j.promfg.2021.06.089","article-title":"Multi-material topology optimization using variable density lattice structures for additive manufacturing","volume":"53","author":"Venugopal","year":"2021","journal-title":"Procedia Manufacturing"},{"key":"2025102210123189300_bib55","doi-asserted-by":"publisher","first-page":"113377","DOI":"10.1016\/j.cma.2020.113377","article-title":"Deep generative modeling for mechanistic-based learning and design of metamaterial systems","volume":"372","author":"Wang","year":"2020","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"2025102210123189300_bib56","doi-asserted-by":"publisher","first-page":"114967","DOI":"10.1016\/j.cma.2022.114967","article-title":"Generalized de-homogenization via sawtooth-function-based mapping and its demonstration on data-driven frequency response optimization","volume":"395","author":"Wang","year":"2022","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"2025102210123189300_bib57","doi-asserted-by":"publisher","first-page":"114949","DOI":"10.1016\/j.compstruct.2021.114949","article-title":"Data-driven multiscale design of cellular composites with multiclass microstructures for natural frequency maximization","volume":"280","author":"Wang","year":"2022","journal-title":"Composite Structures"},{"key":"2025102210123189300_bib58","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1016\/S0045-7825(02)00559-5","article-title":"A level set method for structural topology optimization","volume":"192","author":"Wang","year":"2003","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"2025102210123189300_bib59","doi-asserted-by":"publisher","first-page":"1145","DOI":"10.1007\/s00158-016-1430-5","article-title":"Structure-material integrated design by level sets","volume":"54","author":"Wang","year":"2016","journal-title":"Structural and Multidisciplinary Optimization"},{"key":"2025102210123189300_bib60","doi-asserted-by":"publisher","first-page":"84","DOI":"10.1016\/j.cma.2016.12.007","article-title":"Concurrent design with connectable graded microstructures","volume":"317","author":"Wang","year":"2017","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"2025102210123189300_bib61","doi-asserted-by":"publisher","first-page":"1118","DOI":"10.1016\/j.cma.2018.09.007","article-title":"Multiscale topology optimization using neural network surrogate models","volume":"346","author":"White","year":"2019","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"2025102210123189300_bib62","doi-asserted-by":"publisher","first-page":"294","DOI":"10.1007\/s00158-022-03347-1","article-title":"On the use of artificial neural networks in topology optimisation","volume":"65","author":"Woldseth","year":"2022","journal-title":"Structural and Multidisciplinary Optimization"},{"key":"2025102210123189300_bib64","doi-asserted-by":"publisher","first-page":"1127","DOI":"10.1109\/TVCG.2017.2655523","article-title":"Infill optimization for additive manufacturing-approaching bone-like porous structures","volume":"24","author":"Wu","year":"2017","journal-title":"IEEE Transactions on Visualization and Computer Graphics"},{"key":"2025102210123189300_bib65","doi-asserted-by":"publisher","first-page":"1455","DOI":"10.1007\/s00158-021-02881-8","article-title":"Topology optimization of multi-scale structures: A review","volume":"63","author":"Wu","year":"2021","journal-title":"Structural and Multidisciplinary Optimization"},{"key":"2025102210123189300_bib66","doi-asserted-by":"publisher","first-page":"514","DOI":"10.1093\/jcde\/qwaa001","article-title":"A local solution approach for level-set based structural topology optimization in isogeometric analysis","volume":"7","author":"Wu","year":"2020","journal-title":"Journal of Computational Design and Engineering"},{"key":"2025102210123189300_bib67","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1007\/s11831-016-9170-7","article-title":"Recent advances on topology optimization of multiscale nonlinear structures","volume":"24","author":"Xia","year":"2017","journal-title":"Archives of Computational Methods in Engineering"},{"key":"2025102210123189300_bib68","doi-asserted-by":"publisher","first-page":"885","DOI":"10.1016\/0045-7949(93)90035-C","article-title":"A simple evolutionary procedure for structural optimization","volume":"49","author":"Xie","year":"1993","journal-title":"Computers & Structures"},{"key":"2025102210123189300_bib69","doi-asserted-by":"publisher","first-page":"437","DOI":"10.1007\/s00466-015-1255-x","article-title":"Multi-scale concurrent material and structural design under mechanical and thermal loads","volume":"57","author":"Yan","year":"2016","journal-title":"Computational Mechanics"},{"key":"2025102210123189300_bib70","doi-asserted-by":"publisher","first-page":"967","DOI":"10.32604\/cmes.2022.017708","article-title":"Multi-material and multiscale topology design optimization of thermoelastic lattice structures","volume":"130","author":"Yan","year":"2022","journal-title":"CMES \u2013 Computer Modeling in Engineering & Sciences"},{"key":"2025102210123189300_bib71","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1016\/j.compstruc.2013.12.001","article-title":"Concurrent topology optimization of structures and their composite microstructures","volume":"133","author":"Yan","year":"2014","journal-title":"Computers & Structures"},{"key":"2025102210123189300_bib72","doi-asserted-by":"publisher","first-page":"110613","DOI":"10.1016\/j.matdes.2022.110613","article-title":"Design and fabrication of architected multi-material lattices with tunable stiffness, strength, and energy absorption","volume":"217","author":"Yavas","year":"2022","journal-title":"Materials & Design"},{"key":"2025102210123189300_bib73","doi-asserted-by":"publisher","first-page":"1229","DOI":"10.1007\/s10999-024-09719-3","article-title":"Topology optimization of anisotropic multi-material structures considering negative poisson\u2019s ratio and high thermal conductivity based on iga approach","volume":"20","author":"Zhang","year":"2024","journal-title":"International Journal of Mechanics and Materials in Design"},{"key":"2025102210123189300_bib74","doi-asserted-by":"publisher","first-page":"4213","DOI":"10.2514\/1.J057241","article-title":"Vibration tests of 3d printed satellite structure made of lattice sandwich panels","volume":"56","author":"Zhang","year":"2018","journal-title":"AIAA Journal"},{"key":"2025102210123189300_bib75","doi-asserted-by":"publisher","first-page":"e2450276","DOI":"10.1080\/17452759.2025.2450276","article-title":"Microscopic stress-constrained two-scale topology optimisation for additive manufacturing","volume":"20","author":"Zhang","year":"2025","journal-title":"Virtual and Physical Prototyping"},{"key":"2025102210123189300_bib76","doi-asserted-by":"publisher","first-page":"116052","DOI":"10.1016\/j.cma.2023.116052","article-title":"Topology optimization via implicit neural representations","volume":"411","author":"Zhang","year":"2023","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"2025102210123189300_bib77","doi-asserted-by":"publisher","first-page":"113894","DOI":"10.1016\/j.cma.2021.113894","article-title":"Data-driven topology optimization of spinodoid metamaterials with seamlessly tunable anisotropy","volume":"383","author":"Zheng","year":"2021","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"2025102210123189300_bib78","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1007\/s00158-006-0035-9","article-title":"Multimaterial structural topology optimization with a generalized cahn-hilliard model of multiphase transition","volume":"33","author":"Zhou","year":"2007","journal-title":"Structural and Multidisciplinary Optimization"},{"key":"2025102210123189300_bib79","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1109\/JPROC.2020.3004555","article-title":"A Comprehensive Survey on Transfer Learning","volume":"109","author":"Zhuang","year":"2021","journal-title":"Proceedings of the IEEE"},{"key":"2025102210123189300_bib80","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1016\/j.cma.2019.05.029","article-title":"Vcut level set method for topology optimization of functionally graded cellular structures","volume":"354","author":"Zong","year":"2019","journal-title":"Computer Methods in Applied Mechanics and Engineering"}],"container-title":["Journal of Computational Design and Engineering"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/jcde\/advance-article-pdf\/doi\/10.1093\/jcde\/qwaf064\/63775581\/qwaf064.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jcde\/article-pdf\/12\/8\/127\/63775581\/qwaf064.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jcde\/article-pdf\/12\/8\/127\/63775581\/qwaf064.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,22]],"date-time":"2025-10-22T14:12:42Z","timestamp":1761142362000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jcde\/article\/12\/8\/127\/8203285"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,16]]},"references-count":79,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2025,8,4]]}},"URL":"https:\/\/doi.org\/10.1093\/jcde\/qwaf064","relation":{},"ISSN":["2288-5048"],"issn-type":[{"value":"2288-5048","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2025,8]]},"published":{"date-parts":[[2025,7,16]]}}}