{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T21:17:56Z","timestamp":1778534276326,"version":"3.51.4"},"reference-count":64,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,5,10]],"date-time":"2023-05-10T00:00:00Z","timestamp":1683676800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,5,10]],"date-time":"2023-05-10T00:00:00Z","timestamp":1683676800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100011033","name":"Agencia Estatal de Investigaci\u00f3n","doi-asserted-by":"publisher","award":["DPI2016-77538-R"],"award-info":[{"award-number":["DPI2016-77538-R"]}],"id":[{"id":"10.13039\/501100011033","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005741","name":"Universidad Polit\u00e9cnica de Cartagena","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100005741","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":[[2024,2]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This work presents an efficient parallel implementation of density-based robust topology optimization (RTO) using adaptive mesh refinement (AMR) schemes permitting us to address the problem with modest computational resources. We use sparse grid stochastic collocation methods (SCMs) for transforming the RTO problem into a weighted multiple-loading deterministic problem at the collocation points. The calculation of these deterministic problems and the functional sensitivity is computationally expensive. We combine distributed-memory parallel computing and AMR techniques to address the problem efficiently. The former allows us to exploit the computational resources available, whereas the latter permits us to increase performance significantly. We propose the parallel incremental calculation of the deterministic problems and the contribution to the functional sensitivity maintaining a similar memory allocation to the one used in the deterministic counterpart. The cumulative computing uses buffers to adapt the evaluation at the collocation points to the parallel computing resources permitting the exploitation of the embarrassing parallelism of SCMs. We evaluate the deterministic problems in a coarse mesh generated for each topology optimization iteration to increase the performance. We perform the regularization and design variable update in a fine mesh to obtain an equivalent design to the one generated in such a mesh. We evaluate the proposal in two- and three-dimensional problems to test its feasibility and scalability. We also check the performance improvement using computational buffers in parallel computing nodes. Finally, we compare the proposal to the same approach using different preconditioners without AMR schemes showing significant performance improvements.<\/jats:p>","DOI":"10.1007\/s00366-023-01823-w","type":"journal-article","created":{"date-parts":[[2023,5,11]],"date-time":"2023-05-11T09:32:36Z","timestamp":1683797556000},"page":"21-43","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Adaptive density-based robust topology optimization under uncertain loads using parallel computing"],"prefix":"10.1007","volume":"40","author":[{"given":"David","family":"Herrero-P\u00e9rez","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sebasti\u00e1n Gin\u00e9s","family":"Pic\u00f3-Vicente","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Humberto","family":"Mart\u00ednez-Barber\u00e1","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,5,10]]},"reference":[{"issue":"4","key":"1823_CR1","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1007\/s00158-012-0869-2","volume":"47","author":"N Aage","year":"2013","unstructured":"Aage N, Lazarov BS (2013) Parallel framework for topology optimization using the method of moving asymptotes. Struct Multidisc Opt 47(4):493\u2013505. https:\/\/doi.org\/10.1007\/s00158-012-0869-2","journal-title":"Struct Multidisc Opt"},{"issue":"3","key":"1823_CR2","doi-asserted-by":"publisher","first-page":"565","DOI":"10.1007\/s00158-014-1157-0","volume":"51","author":"N Aage","year":"2015","unstructured":"Aage N, Andreassen E, Lazarov BS (2015) Topology optimization using PETSc: An easy-to-use, fully parallel, open source topology optimization framework. Struct Multidiscip Opt 51(3):565\u2013572. https:\/\/doi.org\/10.1007\/s00158-014-1157-0","journal-title":"Struct Multidiscip Opt"},{"issue":"12","key":"1823_CR3","doi-asserted-by":"publisher","first-page":"1474","DOI":"10.1002\/nme.2536","volume":"78","author":"O Amir","year":"2009","unstructured":"Amir O, Bends\u00f8e MP, Sigmund O (2009) Approximate reanalysis in topology optimization. Int J Numer Methods Eng 78(12):1474\u20131491. https:\/\/doi.org\/10.1002\/nme.2536","journal-title":"Int J Numer Methods Eng"},{"issue":"1","key":"1823_CR4","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1007\/s00158-009-0463-4","volume":"42","author":"O Amir","year":"2010","unstructured":"Amir O, Stolpe M, Sigmund O (2010) Efficient use of iterative solvers in nested topology optimization. Struct Multidiscip Opt 42(1):55\u201372. https:\/\/doi.org\/10.1007\/s00158-009-0463-4","journal-title":"Struct Multidiscip Opt"},{"issue":"1","key":"1823_CR5","doi-asserted-by":"publisher","first-page":"815","DOI":"10.1007\/s00158-013-1015-5","volume":"49","author":"O Amir","year":"2014","unstructured":"Amir O, Aage N, Lazarov BS (2014) On multigrid-CG for efficient topology optimization. Struct Multidiscip Opt 49(1):815\u2013829. https:\/\/doi.org\/10.1007\/s00158-013-1015-5","journal-title":"Struct Multidiscip Opt"},{"key":"1823_CR6","doi-asserted-by":"publisher","first-page":"1131","DOI":"10.1016\/j.compstruc.2010.11.004","volume":"89","author":"A Asadpoure","year":"2011","unstructured":"Asadpoure A, Tootkaboni M, Guest JK (2011) Robust topology optimization of structures with uncertainties in stiffness - Application to truss structures. Comput Struct 89:1131\u20131141. https:\/\/doi.org\/10.1016\/j.compstruc.2010.11.004","journal-title":"Comput Struct"},{"key":"1823_CR7","doi-asserted-by":"publisher","first-page":"186","DOI":"10.1016\/j.cma.2018.08.028","volume":"343","author":"J Baiges","year":"2019","unstructured":"Baiges J, Mart\u00ednez-Frutos J, Herrero-P\u00e9rez D et al (2019) Large-scale stochastic topology optimization using adaptive mesh refinement and coarsening through a two-level parallelization scheme. Comput Methods Appl Mech Engrg 343:186\u2013206. https:\/\/doi.org\/10.1016\/j.cma.2018.08.028","journal-title":"Comput Methods Appl Mech Engrg"},{"issue":"9\u201310","key":"1823_CR8","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(9\u201310):635\u2013654. https:\/\/doi.org\/10.1007\/s004190050248","journal-title":"Arch Appl Mech"},{"issue":"46\u201347","key":"1823_CR9","doi-asserted-by":"publisher","first-page":"6201","DOI":"10.1016\/S0045-7825(01)00216-X","volume":"190","author":"T Borrvall","year":"2001","unstructured":"Borrvall T, Petersson J (2001) Large-scale topology optimization in 3D using parallel computing. Comput Methods Appl Mech Eng 190(46\u201347):6201\u20136229. https:\/\/doi.org\/10.1016\/S0045-7825(01)00216-X","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"9","key":"1823_CR10","doi-asserted-by":"publisher","first-page":"2143","DOI":"10.1002\/nme.116","volume":"50","author":"B Bourdin","year":"2001","unstructured":"Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143\u20132158. https:\/\/doi.org\/10.1002\/nme.116","journal-title":"Int J Numer Methods Eng"},{"key":"1823_CR11","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1016\/j.cma.2015.02.003","volume":"289","author":"M Carrasco","year":"2015","unstructured":"Carrasco M, Ivorra B, Ramos AM (2015) Stochastic topology design optimization for continuous elastic materials. Comput Meth Appl Mech Eng 289:131\u2013154. https:\/\/doi.org\/10.1016\/j.cma.2015.02.003","journal-title":"Comput Meth Appl Mech Eng"},{"issue":"1\u20133","key":"1823_CR12","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1023\/B:ELAS.0000018772.09023.6c","volume":"72","author":"E Cherkaev","year":"2003","unstructured":"Cherkaev E, Cherkaev A (2003) Principal Compliance and Robust Optimal Design. J Elast 72(1\u20133):71\u201398. https:\/\/doi.org\/10.1023\/B:ELAS.0000018772.09023.6c","journal-title":"J Elast"},{"issue":"5","key":"1823_CR13","doi-asserted-by":"publisher","first-page":"1804","DOI":"10.1137\/S106482759833913X","volume":"21","author":"E Chow","year":"2000","unstructured":"Chow E (2000) A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM J Sci Comput 21(5):1804\u20131822. https:\/\/doi.org\/10.1137\/S106482759833913X","journal-title":"SIAM J Sci Comput"},{"issue":"1","key":"1823_CR14","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1002\/nme.759","volume":"58","author":"JCA Costa","year":"2003","unstructured":"Costa JCA, Alves MC (2003) Layout optimization with h-adaptivity of structures. Int J Numer Meth Eng 58(1):83\u2013102. https:\/\/doi.org\/10.1002\/nme.759","journal-title":"Int J Numer Meth Eng"},{"key":"1823_CR15","first-page":"1","volume-title":"Int","author":"E De Sturler","year":"2008","unstructured":"De Sturler E, Paulino G, Wang S (2008) Topology optimization with adaptive mesh refinement. Int. Conf. on Computation of Shell and Spatial Structures, Spanning Nano to Mega, Ithaca, NY, USA, pp 1\u20134"},{"issue":"1","key":"1823_CR16","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00158-013-0956-z","volume":"49","author":"JD Deaton","year":"2014","unstructured":"Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Opt 49(1):1\u201338. https:\/\/doi.org\/10.1007\/s00158-013-0956-z","journal-title":"Struct Multidiscip Opt"},{"key":"1823_CR17","doi-asserted-by":"publisher","unstructured":"Falgout RD, Jones JE, Yang UM (2006) The Design and Implementation of hypre, a Library of Parallel High Performance Preconditioners. In: Bruaset AM, Tveito A (eds) Numerical Solution of Partial Differential Equations on Parallel Computers, Lecture Notes in Computational Science and Engineering, vol\u00a051. Springer, Berlin, Heidelberg, p 267\u2013294, https:\/\/doi.org\/10.1007\/3-540-31619-1_8","DOI":"10.1007\/3-540-31619-1_8"},{"issue":"3\u20134","key":"1823_CR18","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1023\/A:1019129717644","volume":"18","author":"T Gerstner","year":"1998","unstructured":"Gerstner T, Griebel M (1998) Numerical integration using sparse grids. Numer Algorithms 18(3\u20134):209\u2013232. https:\/\/doi.org\/10.1023\/A:1019129717644","journal-title":"Numer Algorithms"},{"issue":"1","key":"1823_CR19","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1051\/cocv:2007048","volume":"14","author":"FD Gournay","year":"2008","unstructured":"Gournay FD, Allaire G, Jouve F (2008) Shape and topology optimization of the robust compliance via the level set method. ESAIM\u2014Control Opt Calc 14(1):43\u201370. https:\/\/doi.org\/10.1051\/cocv:2007048","journal-title":"ESAIM\u2014Control Opt Calc"},{"issue":"2","key":"1823_CR20","doi-asserted-by":"publisher","first-page":"238","DOI":"10.1002\/nme.1064","volume":"61","author":"J Guest","year":"2004","unstructured":"Guest J, Pr\u00e9vost J, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Meth Eng 61(2):238\u2013254. https:\/\/doi.org\/10.1002\/nme.1064","journal-title":"Int J Numer Meth Eng"},{"issue":"1","key":"1823_CR21","doi-asserted-by":"publisher","first-page":"116","DOI":"10.1016\/j.cma.2008.04.009","volume":"198","author":"JK Guest","year":"2008","unstructured":"Guest JK, Igusa T (2008) Structural optimization under uncertain loads and nodal locations. Comput Meth Appl Mech Eng 198(1):116\u2013124. https:\/\/doi.org\/10.1016\/j.cma.2008.04.009","journal-title":"Comput Meth Appl Mech Eng"},{"issue":"3","key":"1823_CR22","doi-asserted-by":"publisher","first-page":"450","DOI":"10.1002\/nme.6217","volume":"121","author":"DK Gupta","year":"2020","unstructured":"Gupta DK, van Keulen F, Langelaar M (2020) Design and analysis adaptivity in multiresolution topology optimization. Int J Numer Methods Eng 121(3):450\u2013476. https:\/\/doi.org\/10.1002\/nme.6217","journal-title":"Int J Numer Methods Eng"},{"issue":"1","key":"1823_CR23","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1016\/S0168-9274(01)00115-5","volume":"41","author":"VE Henson","year":"2002","unstructured":"Henson VE, Yang UM (2002) BoomerAMG: A parallel algebraic multigrid solver and preconditioner. Appl Numer Math 41(1):155\u2013177. https:\/\/doi.org\/10.1016\/S0168-9274(01)00115-5","journal-title":"Appl Numer Math"},{"issue":"103","key":"1823_CR24","doi-asserted-by":"publisher","first-page":"006","DOI":"10.1016\/j.advengsoft.2021.103006","volume":"157\u2013158","author":"D Herrero-P\u00e9rez","year":"2021","unstructured":"Herrero-P\u00e9rez D, Mart\u00ednez-Castej\u00f3n P (2021) Multi-GPU acceleration of large-scale density-based topology optimization. Adv Eng Softw 157\u2013158(103):006. https:\/\/doi.org\/10.1016\/j.advengsoft.2021.103006","journal-title":"Adv Eng Softw"},{"key":"1823_CR25","doi-asserted-by":"publisher","unstructured":"Herrero-P\u00e9rez D, Mart\u00ednez-Frutos J (2017) A Multi-GPU Framework for Structural Optimization under Uncertainty. In: Iv\u00e1nyi P, Topping B, V\u00e1rady G (eds) Advances in Parallel, Distributed, Grid and Cloud Computing for Engineering. Saxe-Coburg Publications, Stirlingshire, UK, chap\u00a02, p 9\u201327, https:\/\/doi.org\/10.4203\/csets.40.2","DOI":"10.4203\/csets.40.2"},{"issue":"106","key":"1823_CR26","doi-asserted-by":"publisher","first-page":"770","DOI":"10.1016\/j.compstruc.2022.106770","volume":"265","author":"D Herrero-P\u00e9rez","year":"2022","unstructured":"Herrero-P\u00e9rez D, Pic\u00f3-Vicente SG, Mart\u00ednez-Barber\u00e1 H (2022) Efficient distributed approach for density-based topology optimization using coarsening and h-refinement. Comput Struct 265(106):770. https:\/\/doi.org\/10.1016\/j.compstruc.2022.106770","journal-title":"Comput Struct"},{"key":"1823_CR27","unstructured":"Hypre (2021) A library of high performance preconditioners. http:\/\/www.llnl.gov\/CASC\/hypre\/"},{"key":"1823_CR28","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1007\/s00158-003-0322-7","volume":"26","author":"G Kharmanda","year":"2004","unstructured":"Kharmanda G, Olhoff N, Mohamed A et al (2004) Reliability-based topology optimization. Struct Multidiscip Opt 26:295\u2013307. https:\/\/doi.org\/10.1007\/s00158-003-0322-7","journal-title":"Struct Multidiscip Opt"},{"issue":"1","key":"1823_CR29","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1016\/0045-7825(86)90071-X","volume":"57","author":"N Kikuchi","year":"1986","unstructured":"Kikuchi N, Chung KY, Torigaki T et al (1986) Adaptive finite element methods for shape optimization of linearly elastic structures. Comp Meth Appl Mech Eng 57(1):67\u201389. https:\/\/doi.org\/10.1016\/0045-7825(86)90071-X","journal-title":"Comp Meth Appl Mech Eng"},{"issue":"6","key":"1823_CR30","doi-asserted-by":"publisher","first-page":"765","DOI":"10.1002\/nme.3072","volume":"86","author":"BS Lazarov","year":"2011","unstructured":"Lazarov BS, Sigmund O (2011) Filters in topology optimization based on Helmholtz-type differential equations. Int J Numer Methods Eng 86(6):765\u2013781. https:\/\/doi.org\/10.1002\/nme.3072","journal-title":"Int J Numer Methods Eng"},{"issue":"103","key":"1823_CR31","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1016\/j.finel.2021.103561","volume":"194","author":"H Li","year":"2021","unstructured":"Li H, Yamada T, Jolivet P et al (2021) Full-scale 3D structural topology optimization using adaptive mesh refinement based on the level-set method. Finite Elem Anal Des 194(103):561. https:\/\/doi.org\/10.1016\/j.finel.2021.103561","journal-title":"Finite Elem Anal Des"},{"issue":"6","key":"1823_CR32","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/3272127.3275012","volume":"37","author":"H Liu","year":"2018","unstructured":"Liu H, Hu Y, Zhu B et al (2018) Narrow-band topology optimization on a sparsely populated grid. ACM Trans Graph 37(6):1\u201314. https:\/\/doi.org\/10.1145\/3272127.3275012","journal-title":"ACM Trans Graph"},{"key":"1823_CR33","doi-asserted-by":"publisher","first-page":"1411","DOI":"10.1007\/s00158-018-1972-9","volume":"58","author":"H Liu","year":"2018","unstructured":"Liu H, Wang Y, Zong H et al (2018) Efficient structure topology optimization by using the multiscale finite element method. Struct Multidiscip Opt 58:1411\u20131430. https:\/\/doi.org\/10.1007\/s00158-018-1972-9","journal-title":"Struct Multidiscip Opt"},{"key":"1823_CR34","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1016\/j.compstruc.2019.05.010","volume":"221","author":"H Liu","year":"2019","unstructured":"Liu H, Tian Y, Zong H et al (2019) Fully parallel level set method for large-scale structural topology optimization. Comput Struct 221:13\u201327. https:\/\/doi.org\/10.1016\/j.compstruc.2019.05.010","journal-title":"Comput Struct"},{"issue":"3\u20134","key":"1823_CR35","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/j.compstruc.2005.10.001","volume":"84","author":"Z Luo","year":"2006","unstructured":"Luo Z, Chen LP, Yang J et al (2006) Fuzzy tolerance multilevel approach for structural topology optimization. Comput Struct 84(3\u20134):127\u2013140. https:\/\/doi.org\/10.1016\/j.compstruc.2005.10.001","journal-title":"Comput Struct"},{"key":"1823_CR36","doi-asserted-by":"publisher","first-page":"393","DOI":"10.1016\/j.cma.2016.08.016","volume":"311","author":"J Mart\u00ednez-Frutos","year":"2016","unstructured":"Mart\u00ednez-Frutos J, Herrero-P\u00e9rez D (2016) Large-scale robust topology optimization using multi-GPU systems. Comp Meth Appl Mech Eng 311:393\u2013414. https:\/\/doi.org\/10.1016\/j.cma.2016.08.016","journal-title":"Comp Meth Appl Mech Eng"},{"key":"1823_CR37","doi-asserted-by":"publisher","first-page":"271","DOI":"10.1016\/j.cma.2016.03.003","volume":"305","author":"J Mart\u00ednez-Frutos","year":"2016","unstructured":"Mart\u00ednez-Frutos J, Herrero-P\u00e9rez D, Kessler M et al (2016) Robust shape optimization of continuous structures via the level set method. Comput Meth Appl Mech Eng 305:271\u2013291. https:\/\/doi.org\/10.1016\/j.cma.2016.03.003","journal-title":"Comput Meth Appl Mech Eng"},{"key":"1823_CR38","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1016\/j.advengsoft.2017.01.009","volume":"106","author":"J Mart\u00ednez-Frutos","year":"2017","unstructured":"Mart\u00ednez-Frutos J, Mart\u00ednez-Castej\u00f3n P, Herrero-P\u00e9rez D (2017) Efficient topology optimization using GPU computing with multilevel granularity. Adv Eng Softw 106:47\u201362. https:\/\/doi.org\/10.1016\/j.advengsoft.2017.01.009","journal-title":"Adv Eng Softw"},{"key":"1823_CR39","unstructured":"MFEM (2021) Modular finite element methods. http:\/\/mfem.org"},{"issue":"12\u201316","key":"1823_CR40","doi-asserted-by":"publisher","first-page":"1527","DOI":"10.1016\/j.cma.2004.03.019","volume":"194","author":"D Moens","year":"2005","unstructured":"Moens D, Vandepitte D (2005) A survey of non-probabilistic uncertainty treatment in finite element analysis. Comput Meth Appl Mech Eng 194(12\u201316):1527\u20131555. https:\/\/doi.org\/10.1016\/j.cma.2004.03.019","journal-title":"Comput Meth Appl Mech Eng"},{"key":"1823_CR41","doi-asserted-by":"publisher","first-page":"290","DOI":"10.1016\/j.advengsoft.2016.08.005","volume":"100","author":"A Nana","year":"2016","unstructured":"Nana A, Cuilli\u00e8re J, Francois V (2016) Towards adaptive topology optimization. Adv Eng Soft 100:290\u2013307. https:\/\/doi.org\/10.1016\/j.advengsoft.2016.08.005","journal-title":"Adv Eng Soft"},{"issue":"4","key":"1823_CR42","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1007\/s00158-009-0443-8","volume":"41","author":"TH Nguyen","year":"2010","unstructured":"Nguyen TH, Paulino GH, Song J et al (2010) A computational paradigm for multiresolution topology optimization (MTOP). Struct Multidiscip Opt 41(4):525\u2013539. https:\/\/doi.org\/10.1007\/s00158-009-0443-8","journal-title":"Struct Multidiscip Opt"},{"issue":"5","key":"1823_CR43","doi-asserted-by":"publisher","first-page":"2309","DOI":"10.1137\/060663660","volume":"46","author":"F Nobile","year":"2008","unstructured":"Nobile F, Tempone R, Webster CG (2008) A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data. SIAM J Numer Anal 46(5):2309\u20132345. https:\/\/doi.org\/10.1137\/060663660","journal-title":"SIAM J Numer Anal"},{"key":"1823_CR44","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1016\/j.cma.2015.12.021","volume":"302","author":"FJ Ram\u00edrez-Gil","year":"2016","unstructured":"Ram\u00edrez-Gil FJ, Nelli-Silva E, Montealegre-Rubio W (2016) Topology optimization design of 3D electrothermomechanical actuators by using GPU as a co-processor. Comp Meth Appl Mech Eng 302:44\u201369. https:\/\/doi.org\/10.1016\/j.cma.2015.12.021","journal-title":"Comp Meth Appl Mech Eng"},{"issue":"5","key":"1823_CR45","doi-asserted-by":"publisher","first-page":"2112","DOI":"10.3390\/app11052112","volume":"11","author":"TP Ribeiro","year":"2021","unstructured":"Ribeiro TP, Bernardo LFA, Andrade JMA (2021) Topology Optimisation in Structural Steel Design for Additive Manufacturing. Appl Sci 11(5):2112. https:\/\/doi.org\/10.3390\/app11052112","journal-title":"Appl Sci"},{"key":"1823_CR46","doi-asserted-by":"publisher","unstructured":"Ruge JW, St\u00fcben K (1987) Algebraic Multigrid. In: McCormick SF (ed) Multigrid Methods. Frontiers in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), chap\u00a04, p 73\u2013130, https:\/\/doi.org\/10.1137\/1.9781611971057.ch4","DOI":"10.1137\/1.9781611971057.ch4"},{"key":"1823_CR47","doi-asserted-by":"publisher","first-page":"2369","DOI":"10.1007\/s00158-018-2084-2","volume":"58","author":"M Salazar de Troya","year":"2018","unstructured":"Salazar de Troya M, Tortorelli D (2018) Adaptive mesh refinement in stress-constrained topology optimization. Struct Multidiscip Opt 58:2369\u20132386. https:\/\/doi.org\/10.1007\/s00158-018-2084-2","journal-title":"Struct Multidiscip Opt"},{"key":"1823_CR48","doi-asserted-by":"publisher","first-page":"2467","DOI":"10.1007\/s00158-020-02618-z","volume":"62","author":"M Salazar de Troya","year":"2020","unstructured":"Salazar de Troya M, Tortorelli D (2020) Three-dimensional adaptive mesh refinement in stress-constrained topology optimization. Struct Multidiscip Opt 62:2467\u20132479. https:\/\/doi.org\/10.1007\/s00158-020-02618-z","journal-title":"Struct Multidiscip Opt"},{"issue":"49\u201352","key":"1823_CR49","doi-asserted-by":"publisher","first-page":"3613","DOI":"10.1016\/j.cma.2011.08.006","volume":"200","author":"M Schevenels","year":"2011","unstructured":"Schevenels M, Lazarov BS, Sigmund O (2011) Robust topology optimization accounting for spatially varying manufacturing errors. Comput Meth Appl Mech Eng 200(49\u201352):3613\u20133627. https:\/\/doi.org\/10.1016\/j.cma.2011.08.006","journal-title":"Comput Meth Appl Mech Eng"},{"issue":"4\u20135","key":"1823_CR50","doi-asserted-by":"publisher","first-page":"401","DOI":"10.1007\/s00158-006-0087-x","volume":"33","author":"O Sigmund","year":"2007","unstructured":"Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Opt 33(4\u20135):401\u2013424. https:\/\/doi.org\/10.1007\/s00158-006-0087-x","journal-title":"Struct Multidiscip Opt"},{"issue":"1","key":"1823_CR51","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1007\/BF01214002","volume":"16","author":"O Sigmund","year":"1998","unstructured":"Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Opt 16(1):68\u201375. https:\/\/doi.org\/10.1007\/BF01214002","journal-title":"Struct Opt"},{"issue":"103","key":"1823_CR52","doi-asserted-by":"publisher","first-page":"039","DOI":"10.1016\/j.probengmech.2020.103039","volume":"59","author":"GA da Silva","year":"2020","unstructured":"da Silva GA, Cardoso EL, Beck AT (2020) Comparison of robust, reliability-based and non-probabilistic topology optimization under uncertain loads and stress constraints. Probabilistic Eng Mech 59(103):039. https:\/\/doi.org\/10.1016\/j.probengmech.2020.103039","journal-title":"Probabilistic Eng Mech"},{"issue":"5","key":"1823_CR53","first-page":"240","volume":"4","author":"SA Smolyak","year":"1963","unstructured":"Smolyak SA (1963) Quadrature and interpolation formulas for tensor products of certain classes of functions. Sov Math Doklady 4(5):240\u2013243","journal-title":"Sov Math Doklady"},{"issue":"9\u201312","key":"1823_CR54","doi-asserted-by":"publisher","first-page":"1031","DOI":"10.1016\/j.cma.2008.11.007","volume":"198","author":"G Stefanou","year":"2009","unstructured":"Stefanou G (2009) The stochastic finite element method: Past, present and future. Comput Meth Appl Mech Eng 198(9\u201312):1031\u20131051. https:\/\/doi.org\/10.1016\/j.cma.2008.11.007","journal-title":"Comput Meth Appl Mech Eng"},{"issue":"2\u20133","key":"1823_CR55","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1002\/nla.559","volume":"15","author":"HD Sterck","year":"2008","unstructured":"Sterck HD, Falgout R, Nolting J et al (2008) Distance-two interpolation for parallel algebraic multigrid. Numer Linear Algebra Appl 15(2\u20133):115\u2013139. https:\/\/doi.org\/10.1002\/nla.559","journal-title":"Numer Linear Algebra Appl"},{"issue":"1\u20132","key":"1823_CR56","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1016\/S0377-0427(00)00516-1","volume":"128","author":"K St\u00fcben","year":"2001","unstructured":"St\u00fcben K (2001) A review of algebraic multigrid. J Comput Appl Math 128(1\u20132):281\u2013309. https:\/\/doi.org\/10.1016\/S0377-0427(00)00516-1","journal-title":"J Comput Appl Math"},{"issue":"2","key":"1823_CR57","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1002\/nme.1620240207","volume":"24","author":"K Svanberg","year":"1987","unstructured":"Svanberg K (1987) The method of moving asymptotes-a new method for structural optimization. Int J Numer Methods Eng 24(2):359\u2013373. https:\/\/doi.org\/10.1002\/nme.1620240207","journal-title":"Int J Numer Methods Eng"},{"issue":"4","key":"1823_CR58","doi-asserted-by":"publisher","first-page":"C367","DOI":"10.1137\/18M1193992","volume":"41","author":"J \u010cerven\u00fd","year":"2019","unstructured":"\u010cerven\u00fd J, Dobrev V, Kolev T (2019) Nonconforming Mesh Refinement for High-Order Finite Elements. SIAM J Sci Comput 41(4):C367\u2013C392. https:\/\/doi.org\/10.1137\/18M1193992","journal-title":"SIAM J Sci Comput"},{"key":"1823_CR59","doi-asserted-by":"publisher","first-page":"767","DOI":"10.1007\/s00158-010-0602-y","volume":"43","author":"F Wang","year":"2011","unstructured":"Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Opt 43:767\u2013784. https:\/\/doi.org\/10.1007\/s00158-010-0602-y","journal-title":"Struct Multidiscip Opt"},{"issue":"12","key":"1823_CR60","doi-asserted-by":"publisher","first-page":"2441","DOI":"10.1002\/nme.1798","volume":"69","author":"S Wang","year":"2007","unstructured":"Wang S, De Sturler E, Paulino G (2007) Large-scale topology optimization using preconditioned Krylov subspace methods with recycling. Int J Numer Methods Eng 69(12):2441\u20132468. https:\/\/doi.org\/10.1002\/nme.1798","journal-title":"Int J Numer Methods Eng"},{"key":"1823_CR61","doi-asserted-by":"publisher","first-page":"50","DOI":"10.1016\/j.compstruc.2014.01.008","volume":"135","author":"Y Wang","year":"2014","unstructured":"Wang Y, Kang Z, He Q (2014) Adaptive topology optimization with independent error control for separated displacement and density fields. Comput Struct 135:50\u201361. https:\/\/doi.org\/10.1016\/j.compstruc.2014.01.008","journal-title":"Comput Struct"},{"key":"1823_CR62","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1007\/s00158-009-0452-7","volume":"41","author":"S Xu","year":"2010","unstructured":"Xu S, Cai Y, Cheng G (2010) Volume preserving nonlinear density filter based on heaviside functions. Struct Multidiscip Opt 41:495\u2013505. https:\/\/doi.org\/10.1007\/s00158-009-0452-7","journal-title":"Struct Multidiscip Opt"},{"key":"1823_CR63","doi-asserted-by":"publisher","unstructured":"Yang UM (2006) Parallel Algebraic Multigrid Methods \u2013 High Performance Preconditioners. In: Bruaset AM, Tveito A (eds) Numerical Solution of Partial Differential Equations on Parallel Computers, Lecture Notes in Computational Science and Engineering, vol\u00a051. Springer, Berlin, Heidelberg, p 209\u2013236, https:\/\/doi.org\/10.1007\/3-540-31619-1_6","DOI":"10.1007\/3-540-31619-1_6"},{"key":"1823_CR64","doi-asserted-by":"publisher","unstructured":"Zhao Q, Chen X, Ma ZD et al (2015) Robust Topology Optimization Based on Stochastic Collocation Methods under Loading Uncertainties. Math Probl Eng 580980. https:\/\/doi.org\/10.1155\/2015\/580980","DOI":"10.1155\/2015\/580980"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-023-01823-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-023-01823-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-023-01823-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,1]],"date-time":"2024-02-01T02:10:22Z","timestamp":1706753422000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-023-01823-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,10]]},"references-count":64,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,2]]}},"alternative-id":["1823"],"URL":"https:\/\/doi.org\/10.1007\/s00366-023-01823-w","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,10]]},"assertion":[{"value":"15 October 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 April 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 May 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 declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}