{"id":"https://openalex.org/W3104284044","doi":"https://doi.org/10.1137/20m1321176","title":"Efficient Numerical Methods for Computing the Stationary States of Phase Field Crystal Models","display_name":"Efficient Numerical Methods for Computing the Stationary States of Phase Field Crystal Models","publication_year":2020,"publication_date":"2020-01-01","ids":{"openalex":"https://openalex.org/W3104284044","doi":"https://doi.org/10.1137/20m1321176","mag":"3104284044"},"language":"en","primary_location":{"id":"doi:10.1137/20m1321176","is_oa":false,"landing_page_url":"https://doi.org/10.1137/20m1321176","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Scientific Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101621262","display_name":"Kai Jiang","orcid":"https://orcid.org/0000-0001-9921-2043"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kai Jiang","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5082778273","display_name":"Wei Si","orcid":"https://orcid.org/0000-0001-7930-0126"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wei Si","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101806205","display_name":"Chen Chang","orcid":"https://orcid.org/0000-0002-8553-8736"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Chang Chen","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5054041500","display_name":"Chenglong Bao","orcid":"https://orcid.org/0000-0002-1201-1212"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Chenglong Bao","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.5564,"has_fulltext":false,"cited_by_count":17,"citation_normalized_percentile":{"value":0.60397671,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"42","issue":"6","first_page":"B1350","last_page":"B1377"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11087","display_name":"Solidification and crystal growth phenomena","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11087","display_name":"Solidification and crystal growth phenomena","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9887999892234802,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10670","display_name":"Aluminum Alloy Microstructure Properties","score":0.9775999784469604,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.6967483758926392},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5876045823097229},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.5829809904098511},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5621523261070251},{"id":"https://openalex.org/keywords/acceleration","display_name":"Acceleration","score":0.5330644249916077},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5295432806015015},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.49501579999923706},{"id":"https://openalex.org/keywords/dissipation","display_name":"Dissipation","score":0.4662092626094818},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.4517410099506378},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.4206704795360565},{"id":"https://openalex.org/keywords/kuramoto-model","display_name":"Kuramoto model","score":0.41173025965690613},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.34534478187561035},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3388552665710449},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.2057127058506012},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1696455180644989},{"id":"https://openalex.org/keywords/synchronization","display_name":"Synchronization (alternating current)","score":0.11142632365226746},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.09487178921699524},{"id":"https://openalex.org/keywords/classical-mechanics","display_name":"Classical mechanics","score":0.07983776926994324}],"concepts":[{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.6967483758926392},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5876045823097229},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.5829809904098511},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5621523261070251},{"id":"https://openalex.org/C117896860","wikidata":"https://www.wikidata.org/wiki/Q11376","display_name":"Acceleration","level":2,"score":0.5330644249916077},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5295432806015015},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.49501579999923706},{"id":"https://openalex.org/C135402231","wikidata":"https://www.wikidata.org/wiki/Q898440","display_name":"Dissipation","level":2,"score":0.4662092626094818},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.4517410099506378},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.4206704795360565},{"id":"https://openalex.org/C2780042749","wikidata":"https://www.wikidata.org/wiki/Q604374","display_name":"Kuramoto model","level":4,"score":0.41173025965690613},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.34534478187561035},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3388552665710449},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.2057127058506012},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1696455180644989},{"id":"https://openalex.org/C2778562939","wikidata":"https://www.wikidata.org/wiki/Q1298791","display_name":"Synchronization (alternating current)","level":3,"score":0.11142632365226746},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.09487178921699524},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.07983776926994324},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/20m1321176","is_oa":false,"landing_page_url":"https://doi.org/10.1137/20m1321176","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Scientific Computing","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/7","score":0.8799999952316284,"display_name":"Affordable and clean energy"}],"awards":[{"id":"https://openalex.org/G1003609349","display_name":null,"funder_award_id":"18B057","funder_id":"https://openalex.org/F4320335817","funder_display_name":"National Social Science Fund Youth Project"},{"id":"https://openalex.org/G2596875127","display_name":"\u51c6\u6676\u7ed3\u6784\u7684\u6570\u5b66\u7406\u8bba\u7814\u7a76\u548c\u9ad8\u6548\u7b97\u6cd5\u8bbe\u8ba1","funder_award_id":"11771368","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3631543633","display_name":null,"funder_award_id":"19A500","funder_id":"https://openalex.org/F4320334986","funder_display_name":"Chinese Academy of Sciences Key Project"},{"id":"https://openalex.org/G5111578883","display_name":null,"funder_award_id":"11901338","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320311213","display_name":"Education Department of Hunan Province","ror":"https://ror.org/05ckg3w11"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320334986","display_name":"Chinese Academy of Sciences Key Project","ror":"https://ror.org/034t30j35"},{"id":"https://openalex.org/F4320335817","display_name":"National Social Science Fund Youth Project","ror":null}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":40,"referenced_works":["https://openalex.org/W1089797291","https://openalex.org/W1566292686","https://openalex.org/W1581623368","https://openalex.org/W1852068959","https://openalex.org/W1966366359","https://openalex.org/W1982014588","https://openalex.org/W1992244966","https://openalex.org/W1999107555","https://openalex.org/W2000883601","https://openalex.org/W2017855598","https://openalex.org/W2024261274","https://openalex.org/W2025944355","https://openalex.org/W2027982384","https://openalex.org/W2029449803","https://openalex.org/W2033468335","https://openalex.org/W2037227343","https://openalex.org/W2050969603","https://openalex.org/W2076605490","https://openalex.org/W2077380502","https://openalex.org/W2080928083","https://openalex.org/W2091320672","https://openalex.org/W2097973750","https://openalex.org/W2099799470","https://openalex.org/W2100556411","https://openalex.org/W2127447444","https://openalex.org/W2228017982","https://openalex.org/W2322734859","https://openalex.org/W2327012892","https://openalex.org/W2521393946","https://openalex.org/W2526274843","https://openalex.org/W2526298107","https://openalex.org/W2551153860","https://openalex.org/W2606274634","https://openalex.org/W2913992843","https://openalex.org/W2954429508","https://openalex.org/W2963255983","https://openalex.org/W2963967579","https://openalex.org/W2964580220","https://openalex.org/W3101256136","https://openalex.org/W3105240523"],"related_works":["https://openalex.org/W3163395947","https://openalex.org/W4299867368","https://openalex.org/W2006251942","https://openalex.org/W1989590366","https://openalex.org/W2883218885","https://openalex.org/W3185235544","https://openalex.org/W2364741597","https://openalex.org/W4297791327","https://openalex.org/W2347486132","https://openalex.org/W4388482258"],"abstract_inverted_index":{"Finding":[0],"the":[1,43,55,65,77,89,95,110,124,133,145,151,157,170,179],"stationary":[2],"states":[3],"of":[4,45,72,119,150],"a":[5,139,175,194],"free":[6],"energy":[7,29,56,125],"functional":[8,57],"is":[9,86,92,104,122],"an":[10],"important":[11],"problem":[12],"in":[13,169,178],"phase":[14],"field":[15],"crystal":[16],"(PFC)":[17],"models.":[18],"Many":[19],"efforts":[20],"have":[21,187],"been":[22],"devoted":[23],"to":[24,42,107,143,193],"designing":[25],"numerical":[26,61,162],"schemes":[27],"with":[28,113,156],"dissipation":[30,126],"and":[31,58,88,127,148,174],"mass":[32,128],"conservation":[33,129],"properties.":[34],"However,":[35],"most":[36,149],"existing":[37,152,199],"approaches":[38,153,186],"are":[39,76],"time-consuming":[40],"due":[41],"requirement":[44],"small":[46],"effective":[47],"step":[48,189],"sizes.":[49],"In":[50],"this":[51],"paper,":[52],"we":[53,137],"discretize":[54],"propose":[59],"efficient":[60],"algorithms":[62,121],"for":[63],"solving":[64],"constrained":[66],"nonconvex":[67],"minimization":[68],"problem.":[69],"A":[70,100],"class":[71],"gradient-based":[73],"approaches,":[74],"which":[75,191],"so-called":[78],"adaptive":[79,188],"accelerated":[80],"Bregman":[81],"proximal":[82],"gradient":[83],"(AA-BPG)":[84],"methods,":[85],"proposed,":[87],"convergence":[90,112,114],"property":[91],"established":[93],"without":[94],"global":[96],"Lipschitz":[97],"constant":[98],"requirements.":[99],"practical":[101,158],"Newton":[102,159],"method":[103],"also":[105],"designed":[106],"further":[108],"accelerate":[109,144],"local":[111],"guarantee.":[115],"One":[116],"key":[117],"feature":[118],"our":[120,185],"that":[123,184],"properties":[130],"hold":[131],"during":[132],"iteration":[134],"process.":[135],"Moreover,":[136],"develop":[138],"hybrid":[140],"acceleration":[141,196],"framework":[142],"AA-BPG":[146],"methods":[147,200],"through":[154],"coupling":[155],"method.":[160],"Extensive":[161],"experiments,":[163],"including":[164],"two":[165],"three-dimensional":[166],"periodic":[167],"crystals":[168],"Landau--Brazovskii":[171],"(LB)":[172],"model":[173],"two-dimensional":[176],"quasicrystal":[177],"Lifshitz--Petrich":[180],"(LP)":[181],"model,":[182],"demonstrate":[183],"sizes":[190],"lead":[192],"significant":[195],"over":[197],"many":[198],"when":[201],"computing":[202],"complex":[203],"structures.":[204]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}