{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T03:01:08Z","timestamp":1760151668397,"version":"build-2065373602"},"reference-count":57,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,4,19]],"date-time":"2022-04-19T00:00:00Z","timestamp":1650326400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["52105255"],"award-info":[{"award-number":["52105255"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key R&amp;D Program of China","doi-asserted-by":"publisher","award":["2020YFB1708300"],"award-info":[{"award-number":["2020YFB1708300"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Program for HUST Academic Frontier Youth Team","award":["2017QYTD04"],"award-info":[{"award-number":["2017QYTD04"]}]},{"name":"Fundamental Research Funds for the Central Universities of the Huazhong University of Science and Technology (HUST)","award":["5003123025"],"award-info":[{"award-number":["5003123025"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Recently, isogeometric analysis (IGA), which unifies the computer-aided design (CAD) model and the computer-aided engineering (CAE) model, has been adopted to develop the isogeometric topology optimization (ITO) framework. However, a critical study on the indispensability of IGA in topology optimization to take the place of the conventional finite element method (FEM) is still lacking. In the current work, two important problems are extensively discussed: (1) The lower numerical precision of the FEM resulting from the disunification between the CAD and CAE models damages the effectiveness of the topology optimization, which suggests the indispensability of IGA in the replacement of the FEM in optimization; (2) a material penalization model is required to ensure the generation of a full loading-transmission path during optimization in classic density-based methods, which causes a greater overestimation of structural stiffness and also suggests the necessity of an ersatz material model. The current paper describes a promising ITO method with point-wise design to gain smooth or binary symmetrical topologies, for which an extended density distribution function (DDF) was constructed to describe the structural topology. Two benchmarks of the stiffness-maximization problem and compliant mechanism are studied in the context of the above issues. Finally, several topologically optimized designs with symmetry are obtained using the ITO method.<\/jats:p>","DOI":"10.3390\/sym14050845","type":"journal-article","created":{"date-parts":[[2022,4,20]],"date-time":"2022-04-20T00:22:43Z","timestamp":1650414163000},"page":"845","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On the Indispensability of Isogeometric Analysis in Topology Optimization for Smooth or Binary Designs"],"prefix":"10.3390","volume":"14","author":[{"given":"Xiaomeng","family":"Wu","sequence":"first","affiliation":[{"name":"Department of Engineering Mechanics, School of Aerospace Engineering, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan 430074, China"},{"name":"Hubei Key Laboratory for Engineering Structural Analysis and Safety Assessment, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan 430074, China"}]},{"given":"Yan","family":"Zhang","sequence":"additional","affiliation":[{"name":"Key Laboratory of Metallurgical Equipment and Control Technology, Ministry of Education, Wuhan University of Science and Technology, Wuhan 430081, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1485-0722","authenticated-orcid":false,"given":"Liang","family":"Gao","sequence":"additional","affiliation":[{"name":"State Key Lab of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan 430074, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4760-2768","authenticated-orcid":false,"given":"Jie","family":"Gao","sequence":"additional","affiliation":[{"name":"Department of Engineering Mechanics, School of Aerospace Engineering, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan 430074, China"},{"name":"Hubei Key Laboratory for Engineering Structural Analysis and Safety Assessment, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan 430074, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bends\u00f8e, M.P., and Sigmund, O. (2003). Topology Optimization: Theory, Methods and Applications, Springer Science & Business Media.","DOI":"10.1007\/978-3-662-05086-6_2"},{"key":"ref_2","doi-asserted-by":"crossref","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":"Kikuchi","year":"1988","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/0045-7825(91)90046-9","article-title":"The COC Algorithm, Part II: Topological, Geometrical and Generalized Shape Optimization","volume":"89","author":"Zhou","year":"1991","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1007\/s004190050248","article-title":"Material Interpolation Schemes in Topology Optimization","volume":"69","author":"Sigmund","year":"1999","journal-title":"Arch. Appl. Mech."},{"key":"ref_5","doi-asserted-by":"crossref","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":"Comput. Struct."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1007\/s00466-008-0312-0","article-title":"Bi-Directional Evolutionary Topology Optimization of Continuum Structures with One or Multiple Materials","volume":"43","author":"Huang","year":"2009","journal-title":"Comput. Mech."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1006\/jcph.2000.6581","article-title":"Structural Boundary Design via Level Set and Immersed Interface Methods","volume":"163","author":"Sethian","year":"2000","journal-title":"J. Comput. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","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":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","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":"J. Comput. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"081009","DOI":"10.1115\/1.4027609","article-title":"Doing Topology Optimization Explicitly and Geometrically\u2014A Mew Moving Morphable Components Based Framework","volume":"81","author":"Guo","year":"2014","journal-title":"J. Appl. Mech."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"011011","DOI":"10.1115\/1.4034972","article-title":"Structural Topology Optimization through Explicit Boundary Evolution","volume":"84","author":"Zhang","year":"2017","journal-title":"J. Appl. Mech."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/j.cma.2019.04.021","article-title":"Topology Optimization for Auxetic Metamaterials Based on Isogeometric Analysis","volume":"352","author":"Gao","year":"2019","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1016\/j.cma.2018.10.017","article-title":"Topology Optimization for Multiscale Design of Porous Composites with Multi-Domain Microstructures","volume":"344","author":"Gao","year":"2019","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"106369","DOI":"10.1016\/j.ymssp.2019.106369","article-title":"Multiscale Topology Optimization for Minimizing Frequency Responses of Cellular Composites with Connectable Graded Microstructures","volume":"135","author":"Zhang","year":"2020","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"114149","DOI":"10.1016\/j.cma.2021.114149","article-title":"Reduced-Order Methods for Dynamic Problems in Topology Optimization: A Comparative Study","volume":"387","author":"Li","year":"2021","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2768","DOI":"10.2514\/1.J060015","article-title":"Simultaneous Layout and Topology Optimization of Curved Stiffened Panels","volume":"59","author":"Chu","year":"2021","journal-title":"AIAA J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3123","DOI":"10.1007\/s00158-021-03062-3","article-title":"Design of Stiffened Panels for Stress and Buckling via Topology Optimization","volume":"64","author":"Chu","year":"2021","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"113949","DOI":"10.1016\/j.cma.2021.113949","article-title":"Design of Graded Lattice Sandwich Structures by Multiscale Topology Optimization","volume":"384","author":"Xiao","year":"2021","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"113720","DOI":"10.1016\/j.cma.2021.113720","article-title":"Topology Optimization Design of Quasi-Periodic Cellular Structures Based on Erode\u2013Dilate Operators","volume":"377","author":"Li","year":"2021","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"110404","DOI":"10.1016\/j.matdes.2022.110404","article-title":"A Multiscale Topological Design Method of Geometrically Asymmetric Porous Sandwich Structures for Minimizing Dynamic Compliance","volume":"214","author":"Zhang","year":"2022","journal-title":"Mater. Des."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1925","DOI":"10.1002\/nme.945","article-title":"Continuous Approximation of Material Distribution for Topology Optimization","volume":"59","author":"Matsui","year":"2004","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1002\/nme.1064","article-title":"Achieving Minimum Length Scale in Topology Optimization Using Nodal Design Variables and Projection Functions","volume":"61","author":"Guest","year":"2004","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1007\/s00158-003-0365-9","article-title":"A Q4\/Q4 Continuum Structural Topology Optimization Implementation","volume":"27","author":"Rahmatalla","year":"2004","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/s00158-008-0228-5","article-title":"A Modified Q4\/Q4 Element for Topology Optimization","volume":"37","author":"Paulino","year":"2009","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3515","DOI":"10.1016\/j.cma.2011.09.001","article-title":"Structural Topology Optimization Based on Non-Local Shepard Interpolation of Density Field","volume":"200","author":"Kang","year":"2011","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1002\/nme.3321","article-title":"A Nodal Variable Method of Structural Topology Optimization Based on Shepard Interpolant","volume":"90","author":"Kang","year":"2012","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2143","DOI":"10.1007\/s00158-017-1846-6","article-title":"Evolutionary Topology Optimization of Continuum Structures with Smooth Boundary Representation","volume":"57","author":"Da","year":"2018","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"685","DOI":"10.1007\/s00158-020-02527-1","article-title":"Level Set Topology and Shape Optimization by Density Methods Using Cut Elements with Length Scale Control","volume":"20","author":"Andreasen","year":"2020","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"110330","DOI":"10.1016\/j.engstruct.2020.110330","article-title":"Smooth Topological Design of Structures Using the Floating Projection","volume":"208","author":"Huang","year":"2020","journal-title":"Eng. Struct."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"102942","DOI":"10.1016\/j.advengsoft.2020.102942","article-title":"On Smooth or 0\/1 Designs of the Fixed-Mesh Element-Based Topology Optimization","volume":"151","author":"Huang","year":"2021","journal-title":"Adv. Eng. Softw."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"102921","DOI":"10.1016\/j.advengsoft.2020.102921","article-title":"SEMDOT: Smooth-Edged Material Distribution for Optimizing Topology Algorithm","volume":"150","author":"Fu","year":"2020","journal-title":"Adv. Eng. Softw."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1137\/S0036144502417715","article-title":"Barycentric Lagrange Interpolation","volume":"46","author":"Berrut","year":"2004","journal-title":"SIAM Rev."},{"key":"ref_33","first-page":"325","article-title":"A Practical Guide to Splines","volume":"27","year":"1978","journal-title":"Math. Comput."},{"key":"ref_34","unstructured":"Piegl, L., and Tiller, W. (2012). The NURBS Book, Springer Science & Business Media."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/s00158-011-0680-5","article-title":"An Isogeometrical Approach to Structural Topology Optimization by Optimality Criteria","volume":"45","author":"Hassani","year":"2012","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/j.cma.2013.06.001","article-title":"Topology Optimization in B-Spline Space","volume":"265","author":"Qian","year":"2013","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1016\/j.cma.2018.08.013","article-title":"Isogeometric Analysis Based Topology Optimization Design with Global Stress Constraint","volume":"342","author":"Liu","year":"2018","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"991","DOI":"10.1002\/nme.6081","article-title":"Isogeometric Topology Optimization for Continuum Structures Using Density Distribution Function","volume":"119","author":"Gao","year":"2019","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"818","DOI":"10.1016\/j.apm.2020.01.006","article-title":"A NURBS-Based Multi-Material Interpolation (N-MMI) for Isogeometric Topology Optimization of Structures","volume":"81","author":"Gao","year":"2020","journal-title":"Appl. Math. Model."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"4135","DOI":"10.1016\/j.cma.2004.10.008","article-title":"Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement","volume":"194","author":"Hughes","year":"2005","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Cottrell, J.A., Hughes, T.J.R., and Bazilevs, Y. (2009). Isogeometric Analysis: Toward Integration of CAD and FEA, John Wiley & Sons.","DOI":"10.1002\/9780470749081"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1016\/j.finel.2015.02.002","article-title":"Isogeometric Analysis of Topologically Complex Shell Structures","volume":"99","author":"Kang","year":"2015","journal-title":"Finite Elem. Anal. Des."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"704","DOI":"10.1016\/j.cma.2018.04.013","article-title":"Adaptive Multi-Patch Isogeometric Analysis Based on Locally Refined B-Splines","volume":"339","author":"Gu","year":"2018","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.finel.2018.05.003","article-title":"Multi-Inclusions Modeling by Adaptive XIGA Based on LR B-Splines and Multiple Level Sets","volume":"148","author":"Gu","year":"2018","journal-title":"Finite Elem. Anal. Des."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Huynh, G.D., Zhuang, X., Bui, H.G., Meschke, G., and Nguyen-Xuan, H. (2020). Elasto-Plastic Large Deformation Analysis of Multi-Patch Thin Shells by Isogeometric Approach. Finite Elem. Anal. Des., 173.","DOI":"10.1016\/j.finel.2020.103389"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1007\/s00466-015-1219-1","article-title":"Isogeometric Analysis for Parameterized LSM-Based Structural Topology Optimization","volume":"57","author":"Wang","year":"2016","journal-title":"Comput. Mech."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1016\/j.cma.2017.02.005","article-title":"An Isogeometrical Approach to Structural Level Set Topology Optimization","volume":"319","author":"Jahangiry","year":"2017","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"115263","DOI":"10.1016\/j.compstruct.2022.115263","article-title":"Isogeometric Topology and Shape Optimization for Composite Structures Using Level-Sets and Adaptive Gauss Quadrature","volume":"285","author":"Gao","year":"2022","journal-title":"Compos. Struct."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/j.cma.2018.04.048","article-title":"A New Isogeometric Topology Optimization Using Moving Morphable Components Based on R-Functions and Collocation Schemes","volume":"339","author":"Xie","year":"2018","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"112685","DOI":"10.1016\/j.cma.2019.112685","article-title":"Explicit Topology Optimization Using IGA-Based Moving Morphable Void (MMV) Approach","volume":"360","author":"Zhang","year":"2020","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"694","DOI":"10.1016\/j.cma.2017.08.021","article-title":"Explicit Isogeometric Topology Optimization Using Moving Morphable Components","volume":"326","author":"Hou","year":"2017","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1186\/s10033-020-00503-w","article-title":"A Comprehensive Review of Isogeometric Topology Optimization: Methods, Applications and Prospects","volume":"33","author":"Gao","year":"2020","journal-title":"Chinese J. Mech. Eng."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.1007\/s00158-013-0978-6","article-title":"Topology Optimization Approaches","volume":"48","author":"Sigmund","year":"2013","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00158-010-0594-7","article-title":"Efficient Topology Optimization in MATLAB Using 88 Lines of Code","volume":"43","author":"Andreassen","year":"2011","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1007\/s00158-006-0087-x","article-title":"Morphology-Based Black and White Filters for Topology Optimization","volume":"33","author":"Sigmund","year":"2007","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1007\/s00158-010-0602-y","article-title":"On Projection Methods, Convergence and Robust Formulations in Topology Optimization","volume":"43","author":"Wang","year":"2011","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1007\/s00158-009-0452-7","article-title":"Volume Preserving Nonlinear Density Filter Based on Heaviside Functions","volume":"41","author":"Xu","year":"2010","journal-title":"Struct. Multidiscip. Optim."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/845\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:56:37Z","timestamp":1760136997000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/845"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,19]]},"references-count":57,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["sym14050845"],"URL":"https:\/\/doi.org\/10.3390\/sym14050845","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,4,19]]}}}