{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T17:41:30Z","timestamp":1769622090830,"version":"3.49.0"},"reference-count":78,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T00:00:00Z","timestamp":1769558400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T00:00:00Z","timestamp":1769558400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Royal Melbourne Institute of Technology"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Engineering with Computers"],"published-print":{"date-parts":[[2026,2]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>This work designs fluid structures that interact with steady Navier\u2013Stokes flows under a Brinkman body force by setting the density at the control points of an NURBS mesh. It approximates the velocity using standard NURBS basis functions and the pressure with one order of accuracy lower. Such a so-called Taylor\u2013Hood pair selection ensures inf-sup stability while providing a smooth, CAD-consistent geometry that enables seamless integration between design and fabrication models. The proposed method employs an optimality-criteria algorithm that utilizes adjoint-based sensitivities to update design variables, along with a bisection method to meet the volume constraint. Numerical benchmarks, including diffuser, dual-inlet\/dual-outlet, flow-reversal, and a 3D bend-pipe, confirm that the proposed method exhibits robust convergence over a range of Reynolds numbers and consistently generates smooth, near-binary designs with sharp interfaces and reduced numerical artefacts, achieving lower objective values without artificially altering sensitivities through filtering. These findings demonstrate that NURBS-based isogeometric analysis is an effective method for fluid topology optimization and shows strong correlation with traditional finite-element techniques.<\/jats:p>","DOI":"10.1007\/s00366-026-02276-7","type":"journal-article","created":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T04:00:38Z","timestamp":1769572838000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["NURBS-based isogeometric topology optimization for steady Navier\u2013Stokes flow"],"prefix":"10.1007","volume":"42","author":[{"given":"He","family":"Li","sequence":"first","affiliation":[]},{"given":"Xuyu","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Jianhu","family":"Shen","sequence":"additional","affiliation":[]},{"given":"Sihan","family":"Ruan","sequence":"additional","affiliation":[]},{"given":"Shiwei","family":"Zhou","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,1,28]]},"reference":[{"key":"2276_CR1","doi-asserted-by":"publisher","first-page":"635","DOI":"10.1007\/s004190050248","volume":"69","author":"MP Bends\u00f8e","year":"1999","unstructured":"Bends\u00f8e MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69:635\u2013654","journal-title":"Arch Appl Mech"},{"key":"2276_CR2","doi-asserted-by":"publisher","first-page":"120","DOI":"10.1007\/s001580050176","volume":"21","author":"O Sigmund","year":"2001","unstructured":"Sigmund O (2001) A 99 line topology optimization code written in matlab. Struct Multidiscip Optim 21:120\u2013127","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR3","unstructured":"Bendsoe MP, Sigmund O (2003) Topology optimization: theory, methods, and applications. Springer Science & Business Media"},{"key":"2276_CR4","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1016\/0045-7825(88)90086-2","volume":"71","author":"MP Bends\u00f8e","year":"1988","unstructured":"Bends\u00f8e MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197\u2013224","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2276_CR5","doi-asserted-by":"publisher","first-page":"262","DOI":"10.1007\/s00158-004-0436-6","volume":"28","author":"MY Wang","year":"2004","unstructured":"Wang MY, Zhou S, Ding H (2004) Nonlinear diffusions in topology optimization. Struct Multidiscip Optim 28:262\u2013276","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR6","doi-asserted-by":"publisher","first-page":"1911","DOI":"10.1002\/nme.1347","volume":"63","author":"MY Wang","year":"2005","unstructured":"Wang MY, Wang S (2005) Bilateral filtering for structural topology optimization. Int J Numer Methods Eng 63:1911\u20131938","journal-title":"Int J Numer Methods Eng"},{"key":"2276_CR7","doi-asserted-by":"publisher","first-page":"885","DOI":"10.1016\/0045-7949(93)90035-C","volume":"49","author":"YM Xie","year":"1993","unstructured":"Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49:885\u2013896","journal-title":"Comput Struct"},{"key":"2276_CR8","doi-asserted-by":"crossref","unstructured":"Xie YM, Steven GP, Xie Y, Steven G (1997) Basic evolutionary structural optimization. Springer","DOI":"10.1007\/978-1-4471-0985-3"},{"key":"2276_CR9","doi-asserted-by":"publisher","first-page":"1031","DOI":"10.1108\/02644409810244129","volume":"15","author":"OM Querin","year":"1998","unstructured":"Querin OM, Steven GP, Xie YM (1998) Evolutionary structural optimisation (ESO) using a bidirectional algorithm. Eng Comput 15:1031\u20131048","journal-title":"Eng Comput"},{"key":"2276_CR10","doi-asserted-by":"publisher","first-page":"1483","DOI":"10.2514\/2.626","volume":"37","author":"XY Yang","year":"1999","unstructured":"Yang XY, Xie YM, Steven GP, Querin O (1999) Bidirectional evolutionary method for stiffness optimization. AIAA J 37:1483\u20131488","journal-title":"AIAA J"},{"key":"2276_CR11","doi-asserted-by":"crossref","unstructured":"Huang X, Xie M (2010) Evolutionary topology optimization of continuum structures: methods and applications. Wiley","DOI":"10.1002\/9780470689486"},{"key":"2276_CR12","doi-asserted-by":"publisher","first-page":"489","DOI":"10.1006\/jcph.2000.6581","volume":"163","author":"JA Sethian","year":"2000","unstructured":"Sethian JA, Wiegmann A (2000) Structural boundary design via level set and immersed interface methods. J Comput Phys 163:489\u2013528","journal-title":"J Comput Phys"},{"key":"2276_CR13","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1016\/S0045-7825(02)00559-5","volume":"192","author":"MY Wang","year":"2003","unstructured":"Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192:227\u2013246","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2276_CR14","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1016\/j.jcp.2003.09.032","volume":"194","author":"G Allaire","year":"2004","unstructured":"Allaire G, Jouve F, Toader A-M (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194:363\u2013393","journal-title":"J Comput Phys"},{"key":"2276_CR15","doi-asserted-by":"publisher","DOI":"10.1115\/1.4027609","volume":"81","author":"X Guo","year":"2014","unstructured":"Guo X, Zhang W, Zhong W (2014) Doing topology optimization explicitly and geometrically\u2014a new moving morphable components based framework. J Appl Mech 81:081009","journal-title":"J Appl Mech"},{"key":"2276_CR16","doi-asserted-by":"publisher","first-page":"1243","DOI":"10.1007\/s00158-015-1372-3","volume":"53","author":"W Zhang","year":"2016","unstructured":"Zhang W, Yuan J, Zhang J, Guo X (2016) A new topology optimization approach based on moving morphable components (MMC) and the ersatz material model. Struct Multidiscip Optim 53:1243\u20131260","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR17","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1002\/fld.426","volume":"41","author":"T Borrvall","year":"2003","unstructured":"Borrvall T, Petersson J (2003) Topology optimization of fluids in Stokes flow. Int J Numer Methods Fluids 41:77\u2013107","journal-title":"Int J Numer Methods Fluids"},{"key":"2276_CR18","doi-asserted-by":"publisher","first-page":"461","DOI":"10.1002\/nme.1560","volume":"66","author":"JK Guest","year":"2006","unstructured":"Guest JK, Pr\u00e9vost JH (2006) Topology optimization of creeping fluid flows using a Darcy\u2013Stokes finite element. Int J Numer Methods Eng 66:461\u2013484","journal-title":"Int J Numer Methods Eng"},{"key":"2276_CR19","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1007\/s00158-004-0508-7","volume":"30","author":"A Gersborg-Hansen","year":"2005","unstructured":"Gersborg-Hansen A, Sigmund O, Haber RB (2005) Topology optimization of channel flow problems. Struct Multidiscip Optim 30:181\u2013192","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR20","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1016\/j.finel.2017.10.006","volume":"139","author":"R Sivapuram","year":"2018","unstructured":"Sivapuram R, Picelli R (2018) Topology optimization of binary structures using integer linear programming. Finite Elem Anal Des 139:49\u201361","journal-title":"Finite Elem Anal Des"},{"key":"2276_CR21","doi-asserted-by":"publisher","first-page":"1221","DOI":"10.1007\/s00158-021-02910-6","volume":"64","author":"B Souza","year":"2021","unstructured":"Souza B, Yamabe P, S\u00e1 L et al (2021) Topology optimization of fluid flow by using integer linear programming. Struct Multidiscip Optim 64:1221\u20131240","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR22","doi-asserted-by":"publisher","DOI":"10.1002\/nme.7480","volume":"125","author":"A da Soares Costa Azev\u00eado","year":"2024","unstructured":"da Soares Costa Azev\u00eado A, Li H, Ishida N et al (2024) Body-fitted topology optimization via integer linear programming using surface capturing techniques. Int J Numer Methods Eng 125:e7480","journal-title":"Int J Numer Methods Eng"},{"key":"2276_CR23","doi-asserted-by":"publisher","first-page":"2101","DOI":"10.1007\/s00158-020-02598-0","volume":"62","author":"R Picelli","year":"2020","unstructured":"Picelli R, Ranjbarzadeh S, Sivapuram R et al (2020) Topology optimization of binary structures under design-dependent fluid-structure interaction loads. Struct Multidiscip Optim 62:2101\u20132116","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/s00158-021-03118-4","volume":"65","author":"R Picelli","year":"2022","unstructured":"Picelli R, Moscatelli E, Yamabe PVM et al (2022) Topology optimization of turbulent fluid flow via the tobs method and a geometry trimming procedure. Struct Multidiscip Optim 65:34","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR25","doi-asserted-by":"publisher","DOI":"10.1007\/s00158-024-03931-7","volume":"68","author":"H Li","year":"2025","unstructured":"Li H, Jolivet P, Alexandersen J (2025) Multi-scale topology optimisation of microchannel cooling using a homogenisation-based method. Struct Multidiscip Optim 68:8","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR26","doi-asserted-by":"publisher","first-page":"2045","DOI":"10.1007\/s00158-018-1966-7","volume":"57","author":"L S\u00e1","year":"2018","unstructured":"S\u00e1 L, Romero J, Horikawa O, Silva ECN (2018) Topology optimization applied to the development of small scale pump. Struct Multidiscip Optim 57:2045\u20132059","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR27","doi-asserted-by":"publisher","DOI":"10.1115\/1.4030297","volume":"137","author":"S Lin","year":"2015","unstructured":"Lin S, Zhao L, Guest JK et al (2015) Topology optimization of fixed-geometry fluid diodes. J Mech Des 137:081402","journal-title":"J Mech Des"},{"key":"2276_CR28","doi-asserted-by":"publisher","DOI":"10.1007\/s00158-024-03920-w","volume":"67","author":"T Sasaki","year":"2024","unstructured":"Sasaki T, Furuta K, Ishida N et al (2024) Topology optimization for 3D fluid diode design considering wall-connected structures. Struct Multidiscip Optim 67:209","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR29","doi-asserted-by":"publisher","first-page":"975","DOI":"10.1002\/nme.1468","volume":"65","author":"LH Olesen","year":"2006","unstructured":"Olesen LH, Okkels F, Bruus H (2006) A high-level programming-language implementation of topology optimization applied to steady-state Navier\u2013Stokes flow. Int J Numer Methods Eng 65:975\u20131001","journal-title":"Int J Numer Methods Eng"},{"key":"2276_CR30","doi-asserted-by":"publisher","first-page":"1345","DOI":"10.1007\/s00158-014-1182-z","volume":"54","author":"A Pereira","year":"2016","unstructured":"Pereira A, Talischi C, Paulino GH et al (2016) Fluid flow topology optimization in PolyTop: stability and computational implementation. Struct Multidisciplinary Optim 54:1345\u20131364","journal-title":"Struct Multidisciplinary Optim"},{"key":"2276_CR31","doi-asserted-by":"publisher","DOI":"10.1007\/s00158-022-03420-9","volume":"66","author":"J Alexandersen","year":"2023","unstructured":"Alexandersen J (2023) A detailed introduction to density-based topology optimisation of fluid flow problems with implementation in MATLAB. Struct Multidiscip Optim 66:12","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR32","doi-asserted-by":"publisher","first-page":"10178","DOI":"10.1016\/j.jcp.2008.08.022","volume":"227","author":"S Zhou","year":"2008","unstructured":"Zhou S, Li Q (2008) A variational level set method for the topology optimization of steady-state Navier\u2013Stokes flow. J Comput Phys 227:10178\u201310195","journal-title":"J Comput Phys"},{"key":"2276_CR33","doi-asserted-by":"publisher","first-page":"1284","DOI":"10.1002\/nme.2616","volume":"79","author":"VJ Challis","year":"2009","unstructured":"Challis VJ, Guest JK (2009) Level set topology optimization of fluids in Stokes flow. Int J Numer Methods Eng 79:1284\u20131308","journal-title":"Int J Numer Methods Eng"},{"key":"2276_CR34","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1007\/s00158-009-0405-1","volume":"41","author":"G Pingen","year":"2010","unstructured":"Pingen G, Waidmann M, Evgrafov A, Maute K (2010) A parametric level-set approach for topology optimization of flow domains. Struct Multidiscip Optim 41:117\u2013131","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR35","doi-asserted-by":"publisher","first-page":"306","DOI":"10.1016\/j.cma.2012.11.015","volume":"255","author":"Y Deng","year":"2013","unstructured":"Deng Y, Liu Z, Wu J, Wu Y (2013) Topology optimization of steady Navier\u2013Stokes flow with body force. Comput Methods Appl Mech Eng 255:306\u2013321","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2276_CR36","doi-asserted-by":"publisher","first-page":"276","DOI":"10.1016\/j.apm.2021.08.021","volume":"101","author":"H Li","year":"2022","unstructured":"Li H, Kondoh T, Jolivet P et al (2022) Three-dimensional topology optimization of a fluid\u2013structure system using body-fitted mesh adaption based on the level-set method. Appl Math Model 101:276\u2013308","journal-title":"Appl Math Model"},{"key":"2276_CR37","doi-asserted-by":"publisher","DOI":"10.1007\/s00158-022-03314-w","volume":"65","author":"H Li","year":"2022","unstructured":"Li H, Kondoh T, Jolivet P et al (2022) Topology optimization for lift\u2013drag problems incorporated with distributed unstructured mesh adaptation. Struct Multidiscip Optim 65:222","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR38","doi-asserted-by":"publisher","first-page":"198","DOI":"10.3901\/JME.2019.10.198","volume":"55","author":"L Hao","year":"2019","unstructured":"Hao L, Xiaohong D, Dalei J (2019) Experimental and numerical investigation of fluid cooling channel layout designed by topology optimization. J Mech Eng 55:198\u2013206","journal-title":"J Mech Eng"},{"key":"2276_CR39","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1016\/j.jcp.2015.12.023","volume":"307","author":"S N\u00f8rgaard","year":"2016","unstructured":"N\u00f8rgaard S, Sigmund O, Lazarov B (2016) Topology optimization of unsteady flow problems using the lattice Boltzmann method. J Comput Phys 307:291\u2013307","journal-title":"J Comput Phys"},{"key":"2276_CR40","doi-asserted-by":"publisher","first-page":"86","DOI":"10.1016\/j.compfluid.2019.05.010","volume":"188","author":"Y Sasaki","year":"2019","unstructured":"Sasaki Y, Sato Y, Yamada T et al (2019) Topology optimization for fluid flows using the MPS method incorporating the level set method. Comput Fluids 188:86\u2013101","journal-title":"Comput Fluids"},{"key":"2276_CR41","doi-asserted-by":"publisher","first-page":"288","DOI":"10.1016\/j.cma.2016.01.014","volume":"303","author":"GH Yoon","year":"2016","unstructured":"Yoon GH (2016) Topology optimization for turbulent flow with Spalart\u2013Allmaras model. Comput Methods Appl Mech Eng 303:288\u2013311","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2276_CR42","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1016\/j.cma.2017.11.029","volume":"331","author":"CB Dilgen","year":"2018","unstructured":"Dilgen CB, Dilgen SB, Fuhrman DR et al (2018) Topology optimization of turbulent flows. Comput Methods Appl Mech Eng 331:363\u2013393","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2276_CR43","unstructured":"Wu C, Zhang Y (2022) Flow topology optimization at high Reynolds numbers. Based on Modified Turbulence Models"},{"key":"2276_CR44","doi-asserted-by":"publisher","first-page":"2340","DOI":"10.1016\/j.camwa.2009.08.044","volume":"59","author":"G Pingen","year":"2010","unstructured":"Pingen G, Maute K (2010) Optimal design for non-Newtonian flows using a topology optimization approach. Comput Math Appl 59:2340\u20132350","journal-title":"Comput Math Appl"},{"key":"2276_CR45","doi-asserted-by":"publisher","DOI":"10.1007\/s00366-022-01637-2","author":"MA Su\u00e1rez","year":"2022","unstructured":"Su\u00e1rez MA, Romero JS, Pereira A, Menezes IF (2022) On the virtual element method for topology optimization of non-Newtonian fluid-flow problems. Eng Comput. https:\/\/doi.org\/10.1007\/s00366-022-01637-2","journal-title":"Eng Comput"},{"key":"2276_CR46","doi-asserted-by":"publisher","first-page":"861","DOI":"10.1002\/fld.1770","volume":"58","author":"C Othmer","year":"2008","unstructured":"Othmer C (2008) A continuous adjoint formulation for the computation of topological and surface sensitivities of ducted flows. Int J Numer Methods Fluids 58:861\u2013877","journal-title":"Int J Numer Methods Fluids"},{"key":"2276_CR47","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1007\/s00158-007-0105-7","volume":"34","author":"G Pingen","year":"2007","unstructured":"Pingen G, Evgrafov A, Maute K (2007) Topology optimization of flow domains using the lattice Boltzmann method. Struct Multidiscip Optim 34:507\u2013524","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR48","doi-asserted-by":"publisher","first-page":"457","DOI":"10.1080\/10618560802238267","volume":"22","author":"G Pingen","year":"2008","unstructured":"Pingen G, Evgrafov A, Maute K (2008) A parallel schur complement solver for the solution of the adjoint steady-state lattice Boltzmann equations: application to design optimisation. Int J Comput Fluid Dyn 22:457\u2013464","journal-title":"Int J Comput Fluid Dyn"},{"key":"2276_CR49","doi-asserted-by":"publisher","first-page":"158","DOI":"10.1016\/j.jcp.2014.06.004","volume":"274","author":"K Yaji","year":"2014","unstructured":"Yaji K, Yamada T, Yoshino M et al (2014) Topology optimization using the lattice Boltzmann method incorporating level set boundary expressions. J Comput Phys 274:158\u2013181","journal-title":"J Comput Phys"},{"key":"2276_CR50","doi-asserted-by":"publisher","first-page":"1031","DOI":"10.1142\/S0218202506001455","volume":"16","author":"Y Bazilevs","year":"2006","unstructured":"Bazilevs Y, Beirao da Veiga L, Cottrell JA et al (2006) Isogeometric analysis: approximation, stability and error estimates for h-refined meshes. Math Models Methods Appl Sci 16:1031\u20131090","journal-title":"Math Models Methods Appl Sci"},{"key":"2276_CR51","doi-asserted-by":"crossref","unstructured":"Cottrell JA, Hughes TJ, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley","DOI":"10.1002\/9780470749081"},{"key":"2276_CR52","doi-asserted-by":"publisher","first-page":"4135","DOI":"10.1016\/j.cma.2004.10.008","volume":"194","author":"TJ Hughes","year":"2005","unstructured":"Hughes TJ, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135\u20134195","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2276_CR53","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1007\/s00158-011-0650-y","volume":"44","author":"AV Kumar","year":"2011","unstructured":"Kumar AV, Parthasarathy A (2011) Topology optimization using B-spline finite elements. Struct Multidiscip Optim 44:471\u2013481","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR54","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1007\/s00158-011-0680-5","volume":"45","author":"B Hassani","year":"2012","unstructured":"Hassani B, Khanzadi M, Tavakkoli SM (2012) An isogeometrical approach to structural topology optimization by optimality criteria. Struct Multidiscip Optim 45:223\u2013233","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR55","doi-asserted-by":"publisher","first-page":"991","DOI":"10.1002\/nme.6081","volume":"119","author":"J Gao","year":"2019","unstructured":"Gao J, Gao L, Luo Z, Li P (2019) Isogeometric topology optimization for continuum structures using density distribution function. Int J Numer Methods Eng 119:991\u20131017","journal-title":"Int J Numer Methods Eng"},{"key":"2276_CR56","doi-asserted-by":"publisher","first-page":"1669","DOI":"10.1007\/s00158-021-02858-7","volume":"64","author":"J Gao","year":"2021","unstructured":"Gao J, Wang L, Luo Z, Gao L (2021) IgaTop: an implementation of topology optimization for structures using IGA in MATLAB. Struct Multidiscip Optim 64:1669\u20131700","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR57","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2025.118365","volume":"447","author":"Z Yang","year":"2025","unstructured":"Yang Z, Gao L, Tang H, Gao J (2025) Full-scale topology optimization for dynamic responses of functionally graded porous infill designs using Nitsche-type multi-patch isogeometric analysis. Comput Methods Appl Mech Eng 447:118365","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2276_CR58","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2024.117095","volume":"428","author":"J Gao","year":"2024","unstructured":"Gao J, Chen C, Fang X et al (2024) Multi-objective topology optimization for solid-porous infill designs in regions-divided structures using multi-patch isogeometric analysis. Comput Methods Appl Mech Eng 428:117095","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2276_CR59","doi-asserted-by":"publisher","first-page":"e7391","DOI":"10.1002\/nme.7391","volume":"125","author":"D Wei","year":"2024","unstructured":"Wei D, Zhu G, Shi Z et al (2024) Isogeometric topology optimization for infill designs of porous structures with stress minimization in additive manufacturing. Int J Numer Methods Eng 125:e7391","journal-title":"Int J Numer Methods Eng"},{"key":"2276_CR60","doi-asserted-by":"publisher","first-page":"2025","DOI":"10.1002\/nme.5593","volume":"112","author":"QX Lieu","year":"2017","unstructured":"Lieu QX, Lee J (2017) Multiresolution topology optimization using isogeometric analysis. Int J Numer Methods Eng 112:2025\u20132047","journal-title":"Int J Numer Methods Eng"},{"key":"2276_CR61","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1007\/s00466-015-1219-1","volume":"57","author":"Y Wang","year":"2016","unstructured":"Wang Y, Benson DJ (2016) Isogeometric analysis for parameterized LSM-based structural topology optimization. Comput Mech 57:19\u201335","journal-title":"Comput Mech"},{"key":"2276_CR62","doi-asserted-by":"publisher","first-page":"240","DOI":"10.1016\/j.cma.2017.02.005","volume":"319","author":"HA Jahangiry","year":"2017","unstructured":"Jahangiry HA, Tavakkoli SM (2017) An isogeometrical approach to structural level set topology optimization. Comput Methods Appl Mech Eng 319:240\u2013257","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2276_CR63","doi-asserted-by":"publisher","DOI":"10.1016\/j.engstruct.2025.121025","volume":"343","author":"Z Yang","year":"2025","unstructured":"Yang Z, Gao L, Xiao M et al (2025) The ODE-driven isogeometric level set method with the parameterization for structural dynamic topology optimization problems. Eng Struct 343:121025. https:\/\/doi.org\/10.1016\/j.engstruct.2025.121025","journal-title":"Eng Struct"},{"key":"2276_CR64","doi-asserted-by":"publisher","first-page":"711","DOI":"10.1016\/j.cma.2016.07.018","volume":"310","author":"X Guo","year":"2016","unstructured":"Guo X, Zhang W, Zhang J, Yuan J (2016) Explicit structural topology optimization based on moving morphable components (MMC) with curved skeletons. Comput Methods Appl Mech Eng 310:711\u2013748","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2276_CR65","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1007\/s00158-021-02853-y","volume":"64","author":"B Zhu","year":"2021","unstructured":"Zhu B, Wang R, Wang N et al (2021) Explicit structural topology optimization using moving wide Bezier components with constrained ends. Struct Multidiscip Optim 64:53\u201370","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR66","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s00466-008-0315-x","volume":"43","author":"Y Bazilevs","year":"2008","unstructured":"Bazilevs Y, Calo VM, Hughes TJ, Zhang Y (2008) Isogeometric fluid-structure interaction: theory, algorithms, and computations. Comput Mech 43:3\u201337","journal-title":"Comput Mech"},{"key":"2276_CR67","doi-asserted-by":"publisher","first-page":"1407","DOI":"10.1002\/fld.2337","volume":"65","author":"A Buffa","year":"2011","unstructured":"Buffa A, De Falco C, Sangalli G (2011) Isogeometric analysis: stable elements for the 2D Stokes equation. Int J Numer Methods Fluids 65:1407\u20131422","journal-title":"Int J Numer Methods Fluids"},{"key":"2276_CR68","doi-asserted-by":"publisher","first-page":"3242","DOI":"10.1016\/j.cma.2011.06.007","volume":"200","author":"PN Nielsen","year":"2011","unstructured":"Nielsen PN, Gersborg AR, Gravesen J, Pedersen NL (2011) Discretizations in isogeometric analysis of Navier\u2013Stokes flow. Comput Methods Appl Mech Eng 200:3242\u20133253","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2276_CR69","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1016\/j.matcom.2016.05.010","volume":"145","author":"B Bastl","year":"2018","unstructured":"Bastl B, Brandner M, Egermaier J et al (2018) Isogeometric analysis for turbulent flow. Math Comput Simul 145:3\u201317","journal-title":"Math Comput Simul"},{"key":"2276_CR70","doi-asserted-by":"publisher","first-page":"598","DOI":"10.1016\/j.cma.2019.06.011","volume":"356","author":"D Garcia","year":"2019","unstructured":"Garcia D, Pardo D, Calo VM (2019) Refined isogeometric analysis for fluid mechanics and electromagnetics. Comput Methods Appl Mech Eng 356:598\u2013628","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2276_CR71","doi-asserted-by":"publisher","first-page":"965","DOI":"10.1007\/s00158-013-0939-0","volume":"48","author":"B-U Park","year":"2013","unstructured":"Park B-U, Seo Y-D, Sigmund O, Youn S-K (2013) Shape optimization of the Stokes flow problem based on isogeometric analysis. Struct Multidiscip Optim 48:965\u2013977","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR72","doi-asserted-by":"publisher","first-page":"909","DOI":"10.1007\/s00158-013-0931-8","volume":"48","author":"P N\u00f8rtoft","year":"2013","unstructured":"N\u00f8rtoft P, Gravesen J (2013) Isogeometric shape optimization in fluid mechanics. Struct Multidiscip Optim 48:909\u2013925","journal-title":"Struct Multidiscip Optim"},{"key":"2276_CR73","doi-asserted-by":"publisher","first-page":"212","DOI":"10.1016\/j.camwa.2024.03.009","volume":"161","author":"C Wang","year":"2024","unstructured":"Wang C, Fang L, Wang X et al (2024) Topology optimization of steady Navier-Stokes flow using moving morphable void method. Comput Math Appl 161:212\u2013224","journal-title":"Comput Math Appl"},{"key":"2276_CR74","doi-asserted-by":"crossref","unstructured":"Piegl Les (1997) The NURBS Book, 2nd ed. 1997. Springer Berlin Heidelberg, Berlin, Heidelberg","DOI":"10.1007\/978-3-642-59223-2"},{"key":"2276_CR75","unstructured":"Maz\u2019ya V (2013) Sobolev spaces. Springer"},{"key":"2276_CR76","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1016\/j.matcom.2015.05.008","volume":"117","author":"VP Nguyen","year":"2015","unstructured":"Nguyen VP, Anitescu C, Bordas SP, Rabczuk T (2015) Isogeometric analysis: an overview and computer implementation aspects. Math Comput Simul 117:89\u2013116","journal-title":"Math Comput Simul"},{"key":"2276_CR77","doi-asserted-by":"crossref","unstructured":"Brezzi F, Franco) (1991) Mixed and hybrid finite element methods, 1st ed. 1991. Springer New York, New York","DOI":"10.1007\/978-1-4612-3172-1_1"},{"key":"2276_CR78","doi-asserted-by":"crossref","unstructured":"Bends\u00f8e MP (1995) Optimization of structural topology, shape, and material. Springer","DOI":"10.1007\/978-3-662-03115-5"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-026-02276-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-026-02276-7","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-026-02276-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T05:04:02Z","timestamp":1769576642000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-026-02276-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,28]]},"references-count":78,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,2]]}},"alternative-id":["2276"],"URL":"https:\/\/doi.org\/10.1007\/s00366-026-02276-7","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,28]]},"assertion":[{"value":"31 July 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 January 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 January 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"31"}}