{"id":"https://openalex.org/W2326558385","doi":"https://doi.org/10.1142/s0219876216500304","title":"A Multiscale Computational Formulation for Gradient Elasticity Problems of Heterogeneous Structures","display_name":"A Multiscale Computational Formulation for Gradient Elasticity Problems of Heterogeneous Structures","publication_year":2016,"publication_date":"2016-04-11","ids":{"openalex":"https://openalex.org/W2326558385","doi":"https://doi.org/10.1142/s0219876216500304","mag":"2326558385"},"language":"en","primary_location":{"id":"doi:10.1142/s0219876216500304","doi":"https://doi.org/10.1142/s0219876216500304","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0219876216500304","pdf_url":null,"source":{"id":"https://openalex.org/S42084661","display_name":"International Journal of Computational Methods","issn_l":"0219-8762","issn":["0219-8762","1793-6969"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","jufo-1","norway-1"],"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computational Methods","raw_type":"journal-article"},"volume":null,"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/A5024210657","display_name":"Ping Fu","orcid":"https://orcid.org/0000-0002-1003-2909","observed_orcids":["https://orcid.org/0000-0002-1003-2909"]},"institutions":[{"id":"https://openalex.org/I37461747","display_name":"Wuhan University","ror":"https://ror.org/033vjfk17","country_code":"CN","type":"education","lineage":["https://openalex.org/I37461747"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Ping Fu","raw_affiliation_strings":["Department of Engineering Mechanics, School of Civil Engineering, Wuhan University, Wuhan 430072, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Engineering Mechanics, School of Civil Engineering, Wuhan University, Wuhan 430072, P. R. China","institution_ids":["https://openalex.org/I37461747"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100658253","display_name":"Hui Liu","orcid":"https://orcid.org/0000-0002-8051-2932","observed_orcids":["https://orcid.org/0000-0002-8051-2932"]},"institutions":[{"id":"https://openalex.org/I37461747","display_name":"Wuhan University","ror":"https://ror.org/033vjfk17","country_code":"CN","type":"education","lineage":["https://openalex.org/I37461747"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Hui Liu","raw_affiliation_strings":["Department of Engineering Mechanics, School of Civil Engineering, Wuhan University, Wuhan 430072, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Engineering Mechanics, School of Civil Engineering, Wuhan University, Wuhan 430072, P. R. China","institution_ids":["https://openalex.org/I37461747"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5080316278","display_name":"Xihua Chu","orcid":"https://orcid.org/0000-0002-1663-2315","observed_orcids":["https://orcid.org/0000-0002-1663-2315","https://orcid.org/0000-0003-4837-4266"]},"institutions":[{"id":"https://openalex.org/I37461747","display_name":"Wuhan University","ror":"https://ror.org/033vjfk17","country_code":"CN","type":"education","lineage":["https://openalex.org/I37461747"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xihua Chu","raw_affiliation_strings":["Department of Engineering Mechanics, School of Civil Engineering, Wuhan University, Wuhan 430072, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Engineering Mechanics, School of Civil Engineering, Wuhan University, Wuhan 430072, P. R. China","institution_ids":["https://openalex.org/I37461747"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5001550510","display_name":"Yuanjie Xu","orcid":"https://orcid.org/0000-0002-4719-483X","observed_orcids":["https://orcid.org/0000-0002-4719-483X"]},"institutions":[{"id":"https://openalex.org/I37461747","display_name":"Wuhan University","ror":"https://ror.org/033vjfk17","country_code":"CN","type":"education","lineage":["https://openalex.org/I37461747"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yuanjie Xu","raw_affiliation_strings":["Department of Engineering Mechanics, School of Civil Engineering, Wuhan University, Wuhan 430072, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Engineering Mechanics, School of Civil Engineering, Wuhan University, Wuhan 430072, P. R. China","institution_ids":["https://openalex.org/I37461747"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I37461747"],"apc_list":null,"apc_paid":null,"fwci":0.2917,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.55066785,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":96},"biblio":{"volume":"13","issue":"05","first_page":"1650030","last_page":"1650030"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11646","display_name":"Nonlocal and gradient elasticity in micro/nano structures","score":0.8988999724388123,"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/T11646","display_name":"Nonlocal and gradient elasticity in micro/nano structures","score":0.8988999724388123,"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":0.02979999966919422,"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/T10514","display_name":"Numerical methods in engineering","score":0.015399999916553497,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"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/gradient-elasticity","display_name":"gradient elasticity","score":0.9319999814033508},{"id":"https://openalex.org/keywords/multiscale-modeling","display_name":"multiscale modeling","score":0.8040000200271606},{"id":"https://openalex.org/keywords/heterogeneous-structures","display_name":"heterogeneous structures","score":0.9620000123977661},{"id":"https://openalex.org/keywords/downscaling","display_name":"downscaling","score":0.5509999990463257},{"id":"https://openalex.org/keywords/equivalent-material-properties","display_name":"equivalent material properties","score":0.4819999933242798},{"id":"https://openalex.org/keywords/computational-cost-reduction","display_name":"computational complexity reduction","score":0.828000009059906}],"concepts":[{"id":"https://openalex.org/C121854251","wikidata":"https://www.wikidata.org/wiki/Q62932","display_name":"Elasticity (physics)","level":2,"score":0.6152515411376953},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.5709066390991211},{"id":"https://openalex.org/C137800194","wikidata":"https://www.wikidata.org/wiki/Q11713455","display_name":"Interpolation (computer graphics)","level":3,"score":0.5575277209281921},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.5455020666122437},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.4869130849838257},{"id":"https://openalex.org/C63632240","wikidata":"https://www.wikidata.org/wiki/Q1935004","display_name":"Boundary element method","level":3,"score":0.428339421749115},{"id":"https://openalex.org/C29660869","wikidata":"https://www.wikidata.org/wiki/Q5282615","display_name":"Displacement field","level":3,"score":0.4114692509174347},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3867137134075165},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3749207854270935},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.35043853521347046},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3432236909866333},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3181997835636139},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2442149519920349},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.21357977390289307},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.15027514100074768},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C121684516","wikidata":"https://www.wikidata.org/wiki/Q7600677","display_name":"Computer graphics (images)","level":1,"score":0.0},{"id":"https://openalex.org/C502989409","wikidata":"https://www.wikidata.org/wiki/Q11425","display_name":"Animation","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0219876216500304","doi":"https://doi.org/10.1142/s0219876216500304","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0219876216500304","pdf_url":null,"source":{"id":"https://openalex.org/S42084661","display_name":"International Journal of Computational Methods","issn_l":"0219-8762","issn":["0219-8762","1793-6969"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","jufo-1","norway-1"],"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computational Methods","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G343292639","display_name":null,"funder_award_id":"2014M552078","funder_id":"https://openalex.org/F4320321543","funder_display_name":"China Postdoctoral Science Foundation"},{"id":"https://openalex.org/G5337257288","display_name":"\u975e\u5747\u5300\u9897\u7c92\u6750\u6599\u7684\u7279\u5f81\u957f\u5ea6\u4e0e\u76f8\u5173\u529b\u5b66\u884c\u4e3a\u7684\u591a\u5c3a\u5ea6\u7814\u7a76","funder_award_id":"11472196","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5642405869","display_name":null,"funder_award_id":"2015T80831","funder_id":"https://openalex.org/F4320321543","funder_display_name":"China Postdoctoral Science Foundation"},{"id":"https://openalex.org/G6021287858","display_name":null,"funder_award_id":"11402178","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8525770625","display_name":null,"funder_award_id":"2014CFB336","funder_id":"https://openalex.org/F4320322186","funder_display_name":"Natural Science Foundation of Hubei Province"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320321543","display_name":"China Postdoctoral Science Foundation","ror":"https://ror.org/0426zh255"},{"id":"https://openalex.org/F4320322186","display_name":"Natural Science Foundation of Hubei Province","ror":null}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":45,"referenced_works":["https://openalex.org/W1608481957","https://openalex.org/W1966480196","https://openalex.org/W1968164510","https://openalex.org/W1970614773","https://openalex.org/W1971171468","https://openalex.org/W1975997259","https://openalex.org/W1979184230","https://openalex.org/W1980606311","https://openalex.org/W1983562272","https://openalex.org/W1984980836","https://openalex.org/W1989581196","https://openalex.org/W2003637147","https://openalex.org/W2007901165","https://openalex.org/W2008682464","https://openalex.org/W2014293274","https://openalex.org/W2014826436","https://openalex.org/W2017666205","https://openalex.org/W2018142698","https://openalex.org/W2018204879","https://openalex.org/W2022861432","https://openalex.org/W2027911250","https://openalex.org/W2029918260","https://openalex.org/W2030892014","https://openalex.org/W2061176151","https://openalex.org/W2063882799","https://openalex.org/W2065696053","https://openalex.org/W2076380814","https://openalex.org/W2077679705","https://openalex.org/W2078233671","https://openalex.org/W2082042462","https://openalex.org/W2084113519","https://openalex.org/W2085212323","https://openalex.org/W2092726980","https://openalex.org/W2101241849","https://openalex.org/W2106974008","https://openalex.org/W2111513248","https://openalex.org/W2129408955","https://openalex.org/W2141879489","https://openalex.org/W2143218676","https://openalex.org/W2145452318","https://openalex.org/W2148589186","https://openalex.org/W2151194631","https://openalex.org/W2157721313","https://openalex.org/W2334316871","https://openalex.org/W2492156635"],"related_works":["https://openalex.org/W2506264689","https://openalex.org/W3016108315","https://openalex.org/W1979221084","https://openalex.org/W2080161047","https://openalex.org/W4285737854","https://openalex.org/W2008452721","https://openalex.org/W2144133766","https://openalex.org/W2385060466","https://openalex.org/W599820626","https://openalex.org/W2029680307"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"a":[3,30],"multiscale":[4,32,145],"computational":[5,126],"formulation":[6],"is":[7,38],"developed":[8],"for":[9],"modeling":[10],"two-":[11],"and":[12,99,140],"three-dimensional":[13],"gradient":[14,50],"elasticity":[15,51],"behaviors":[16],"of":[17,35,55,61,86,117,142],"heterogeneous":[18],"structures.":[19],"To":[20],"capture":[21],"the":[22,26,42,48,58,62,72,79,83,87,96,100,111,115,125,138,143],"microscopic":[23,74,102],"properties":[24],"at":[25,95],"macroscopic":[27,80,97],"level":[28],"effectively,":[29],"numerical":[31,132],"interpolation":[33],"function":[34],"coarse":[36,63,119],"element":[37,44,64,120],"constructed":[39],"by":[40,109],"employing":[41],"oversampling":[43],"technique":[45],"based":[46],"on":[47,114],"staggered":[49],"scheme.":[52],"By":[53],"virtue":[54],"these":[56],"functions,":[57],"equivalent":[59],"quantities":[60],"could":[65,92,105],"be":[66,93,107],"obtained":[67],"easily,":[68],"resulting":[69],"in":[70],"that":[71],"material":[73],"characteristics":[75],"are":[76,134],"reflected":[77],"to":[78,136],"scale.":[81],"Consequently,":[82],"displacement":[84],"field":[85],"original":[88],"boundary":[89],"value":[90],"problem":[91],"calculated":[94],"level,":[98],"corresponding":[101],"gradient-enriched":[103],"solutions":[104],"also":[106],"evaluated":[108],"adopting":[110],"downscaling":[112],"computation":[113],"sub-grids":[116],"each":[118],"domain,":[121],"which":[122],"will":[123],"reduce":[124],"cost":[127],"significantly.":[128],"Furthermore,":[129],"several":[130],"representative":[131],"experiments":[133],"performed":[135],"demonstrate":[137],"validity":[139],"efficiency":[141],"proposed":[144],"formulation.":[146]},"counts_by_year":[{"year":2018,"cited_by_count":2}],"updated_date":"2026-09-24T07:57:37.273135","created_date":"2025-10-10T00:00:00"}