{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T13:14:14Z","timestamp":1783602854659,"version":"3.55.0"},"reference-count":47,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12201021"],"award-info":[{"award-number":["12201021"]}],"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":["12471363"],"award-info":[{"award-number":["12471363"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Math."],"published-print":{"date-parts":[[2026,8,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>We investigate the self-assembly of triblock copolymers under soft confinement with a multicomponent triblock homopolymer-copolymer (THC) model. To address computational challenges associated with multiple components and mass conservation, we develop a novel accelerated projection saddle dynamics (A-PSD) method and prove the linear stability theory of the continuous A-PSD rigorously. With this algorithm, we numerically construct solution landscapes, a bifurcation diagram, and heatmaps to systematically study the THC model. First, we derive a multitude of extreme points, which can be categorized into several classes, including Layer class, Flower class, Circle class and Core-Shell class. The connections between these classes are established and some novel structures are observed. Next, we illustrate the emergence of critical points and the impact of domain size with a bifurcation diagram. Finally, we employ heatmaps to visualize the impact of volume fractions on the morphological behavior of the system.<\/jats:p>","DOI":"10.1137\/25m1753152","type":"journal-article","created":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T12:22:14Z","timestamp":1783599734000},"page":"1609-1631","source":"Crossref","is-referenced-by-count":0,"title":["Self-Assembly of Triblock Copolymers Under Soft Confinement: Solution Landscape, Bifurcation Diagram,\u00a0and Heatmap"],"prefix":"10.1137","volume":"86","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-0869-5511","authenticated-orcid":true,"given":"Longjing","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People\u2019s Republic of China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9157-025X","authenticated-orcid":true,"given":"Guanghua","family":"Ji","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People\u2019s Republic of China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9357-5409","authenticated-orcid":true,"given":"Zhen","family":"Xu","sequence":"additional","affiliation":[{"name":"Corresponding author. School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing, 100124, China."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2026,7,9]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.3390\/polym15040848"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1146\/annurev.pc.41.100190.002521"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1016\/j.jtice.2015.05.008"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2020.125463"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1016\/0003-9861(71)90349-3"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2009.v7.n4.a11"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.66.052301"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/24\/6\/008"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1088\/0370-1328\/85\/4\/301"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/3096"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198502180.001.0001"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6544\/abc5d4"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1063\/1.480097"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1021\/acsomega.9b00991"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827500366124"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.3390\/polym14224900"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827599365641"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-024-02760-6"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1137\/22M1487965"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1515\/9781400881802"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1021\/ma00072a031"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1016\/j.bulsci.2011.12.004"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(95)00005-O"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.1021\/ma00164a028"},{"key":"ref25","doi-asserted-by":"publisher","DOI":"10.1016\/0041-5553(64)90137-5"},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1002\/smm2.1232"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1016\/j.polymer.2012.05.026"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(02)00808-4"},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1039\/C2SM27031E"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1137\/23M1604606"},{"key":"ref31","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6544\/acc62d"},{"key":"ref32","first-page":"1","volume":"17","author":"Su W.","year":"2016","journal-title":"J. Mach. Learn. Res."},{"key":"ref33","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.77.016701"},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1007\/s13160-010-0014-9"},{"key":"ref35","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-7400-6"},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-17798-9"},{"key":"ref37","doi-asserted-by":"publisher","DOI":"10.1021\/acs.macromol.5b00335"},{"key":"ref38","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.104.014505"},{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.4208\/csiam-am.SO-2022-0036"},{"key":"ref40","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-022-01838-3"},{"key":"ref41","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.124.090601"},{"key":"ref42","doi-asserted-by":"publisher","DOI":"10.1007\/s11425-020-1737-1"},{"key":"ref43","doi-asserted-by":"publisher","DOI":"10.1137\/19M1253356"},{"key":"ref44","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2021.133081"},{"key":"ref45","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.96.138306"},{"key":"ref46","doi-asserted-by":"publisher","DOI":"10.1137\/110843149"},{"key":"ref47","doi-asserted-by":"publisher","DOI":"10.1137\/140972676"}],"container-title":["SIAM Journal on Applied Mathematics"],"original-title":[],"language":"en","deposited":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T12:22:15Z","timestamp":1783599735000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/25M1753152"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7,9]]},"references-count":47,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2026,8,31]]}},"alternative-id":["10.1137\/25M1753152"],"URL":"https:\/\/doi.org\/10.1137\/25m1753152","relation":{},"ISSN":["0036-1399","1095-712X"],"issn-type":[{"value":"0036-1399","type":"print"},{"value":"1095-712X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7,9]]}}}