{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,20]],"date-time":"2025-05-20T04:06:04Z","timestamp":1747713964664,"version":"3.40.5"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T00:00:00Z","timestamp":1743638400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T00:00:00Z","timestamp":1743638400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,7]]},"DOI":"10.1007\/s40314-025-03188-w","type":"journal-article","created":{"date-parts":[[2025,4,5]],"date-time":"2025-04-05T01:43:11Z","timestamp":1743817391000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Application of the fragile points method for two-dimensional generalized biharmonic equation on arbitrary domains"],"prefix":"10.1007","volume":"44","author":[{"given":"Donya","family":"Haghighi","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3385-4152","authenticated-orcid":false,"given":"Saeid","family":"Abbasbandy","sequence":"additional","affiliation":[]},{"given":"Yue","family":"Guan","sequence":"additional","affiliation":[]},{"given":"Elyas","family":"Shivanian","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,4,3]]},"reference":[{"issue":"1","key":"3188_CR1","first-page":"246","volume":"186","author":"H Adibi","year":"2007","unstructured":"Adibi H, Es\u2019haghi J (2007) Numerical solution for biharmonic equation using multilevel radial basis functions and domain decomposition methods. Appl Math Comput 186(1):246\u2013255","journal-title":"Appl Math Comput"},{"issue":"3","key":"3188_CR2","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1016\/j.icheatmasstransfer.2006.10.004","volume":"34","author":"S Chantasiriwan","year":"2007","unstructured":"Chantasiriwan S (2007) Solutions to harmonic and biharmonic problems with discontinuous boundary conditions by collocation methods using multiquadrics as basis functions. Int Commun Heat Mass Transf 34(3):313\u2013320","journal-title":"Int Commun Heat Mass Transf"},{"key":"3188_CR3","doi-asserted-by":"publisher","first-page":"124","DOI":"10.1016\/j.enganabound.2019.07.009","volume":"107","author":"L Dong","year":"2019","unstructured":"Dong L, Yang T, Wang K, Atluri SN (2019) A new Fragile Points Method (FPM) in computational mechanics, based on the concepts of point stiffnesses and numerical flux corrections. Eng Anal Bound Elem 107:124\u2013133","journal-title":"Eng Anal Bound Elem"},{"issue":"16","key":"3188_CR4","doi-asserted-by":"publisher","first-page":"4055","DOI":"10.1002\/nme.6692","volume":"122","author":"Y Guan","year":"2021","unstructured":"Guan Y, Atluri SN (2021) Meshless fragile points methods based on Petrov-Galerkin weak-forms for transient heat conduction problems in complex anisotropic nonhomogeneous media. Int J Numer Methods Eng 122(16):4055\u20134092","journal-title":"Int J Numer Methods Eng"},{"key":"3188_CR5","doi-asserted-by":"crossref","unstructured":"Guan Y, Dong L, Atluri SN (2020) A new meshless \u201cFragile Points Method (FPM)\u201d based on a Galerkin weak-form for 2D flexoelectric analysis. In: ASME international mechanical engineering congress and exposition, vol 84607. American Society of Mechanical Engineers","DOI":"10.1115\/IMECE2020-24527"},{"issue":"2","key":"3188_CR6","doi-asserted-by":"publisher","first-page":"159","DOI":"10.2140\/jomms.2021.16.159","volume":"16","author":"Y Guan","year":"2021","unstructured":"Guan Y, Dong L, Atluri SN (2021a) A new meshless Fragile Points Method (FPM) with minimum unknowns at each point, for flexoelectric analysis under two theories with crack propagation, I: theory and implementation. J Mech Mater Struct 16(2):159\u2013195","journal-title":"J Mech Mater Struct"},{"issue":"2","key":"3188_CR7","doi-asserted-by":"publisher","first-page":"197","DOI":"10.2140\/jomms.2021.16.197","volume":"16","author":"Y Guan","year":"2021","unstructured":"Guan Y, Dong L, Atluri SN (2021b) A new meshless Fragile Points Method (FPM) with minimum unknowns at each point, for flexoelectric analysis under two theories with crack propagation, II: validation and discussion. J Mech Mater Struct 16(2):197\u2013223","journal-title":"J Mech Mater Struct"},{"key":"3188_CR8","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1016\/j.enganabound.2021.09.018","volume":"134","author":"D Haghighi","year":"2022","unstructured":"Haghighi D, Abbasbandy S, Shivanian E, Dong L, Atluri SN (2022) The Fragile Points Method (FPM) to solve two-dimensional hyperbolic telegraph equation using point stiffness matrices. Eng Anal Bound Elem 134:11\u201321","journal-title":"Eng Anal Bound Elem"},{"key":"3188_CR9","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1016\/j.enganabound.2022.09.036","volume":"146","author":"D Haghighi","year":"2023","unstructured":"Haghighi D, Abbasbandy S, Shivanian E (2023) Study of the fragile points method for solving two-dimensional linear and nonlinear wave equations on complex and cracked domains. Eng Anal Bound Elem 146:44\u201355","journal-title":"Eng Anal Bound Elem"},{"issue":"4","key":"3188_CR10","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1002\/cnm.736","volume":"21","author":"J Li","year":"2005","unstructured":"Li J (2005) Application of radial basis meshless methods to direct and inverse biharmonic boundary value problems. Commun Numer Methods Eng 21(4):169\u2013182","journal-title":"Commun Numer Methods Eng"},{"issue":"3","key":"3188_CR11","doi-asserted-by":"publisher","first-page":"1993","DOI":"10.1002\/num.22638","volume":"37","author":"J Li","year":"2021","unstructured":"Li J, Cheng Y (2021) Barycentric rational method for solving biharmonic equation by depression of order. Numer Methods Partial Differ Equ 37(3):1993\u20132007","journal-title":"Numer Methods Partial Differ Equ"},{"issue":"2","key":"3188_CR12","doi-asserted-by":"publisher","first-page":"737","DOI":"10.1016\/j.apm.2010.07.030","volume":"35","author":"X Li","year":"2011","unstructured":"Li X, Zhu J, Zhang S (2011) A meshless method based on boundary integral equations and radial basis functions for biharmonic-type problems. Appl Math Model 35(2):737\u2013751","journal-title":"Appl Math Model"},{"issue":"8","key":"3188_CR13","doi-asserted-by":"publisher","first-page":"1790","DOI":"10.1080\/00207160.2013.862525","volume":"91","author":"M Li","year":"2014","unstructured":"Li M, Amazzar G, Naji A, Chen CS (2014) Solving biharmonic equation using the localized method of approximate particular solutions. Int J Comput Math 91(8):1790\u20131801","journal-title":"Int J Comput Math"},{"key":"3188_CR14","doi-asserted-by":"publisher","first-page":"120","DOI":"10.1016\/j.camwa.2020.09.023","volume":"88","author":"YC Liu","year":"2021","unstructured":"Liu YC, Fan CM, Yeih W, Ku CY, Chu CL (2021) Numerical solutions of two-dimensional Laplace and biharmonic equations by the localized Trefftz method. Comput Math Appl 88:120\u2013134","journal-title":"Comput Math Appl"},{"issue":"10\u201311","key":"3188_CR15","doi-asserted-by":"publisher","first-page":"1746","DOI":"10.1080\/00207160802647365","volume":"86","author":"N Mai-Duy","year":"2009","unstructured":"Mai-Duy N, Ho-Minh D, Tran-Cong T (2009) A Galerkin approach incorporating integrated radial basis function networks for the solution of 2D biharmonic equations. Int J Comput Math 86(10\u201311):1746\u20131759","journal-title":"Int J Comput Math"},{"issue":"6","key":"3188_CR16","doi-asserted-by":"publisher","first-page":"707","DOI":"10.1002\/(SICI)1098-2426(199611)12:6<707::AID-NUM4>3.0.CO;2-W","volume":"12","author":"RK Mohanty","year":"1996","unstructured":"Mohanty RK, Pandey PK (1996) Difference methods of order two and four for systems of mildly nonlinear biharmonic problems of the second kind in two space dimensions. Numer Methods Partial Differ Equ 12(6):707\u2013717","journal-title":"Numer Methods Partial Differ Equ"},{"issue":"3","key":"3188_CR17","doi-asserted-by":"publisher","first-page":"1003","DOI":"10.1002\/num.21855","volume":"30","author":"L Mu","year":"2014","unstructured":"Mu L, Junping W, Ye X (2014) Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes. Numer Methods Partial Differ Equ 30(3):1003\u20131029","journal-title":"Numer Methods Partial Differ Equ"},{"issue":"4","key":"3188_CR18","doi-asserted-by":"publisher","first-page":"1497","DOI":"10.1002\/num.22361","volume":"35","author":"L Mu","year":"2019","unstructured":"Mu L, Ye X, Zhang S (2019) Development of a P2 element with optimal L2 convergence for biharmonic equation. Numer Methods Partial Differ Equ 35(4):1497\u20131508","journal-title":"Numer Methods Partial Differ Equ"},{"key":"3188_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-09205-7","volume-title":"Partial differential equations in mechanics 2: the biharmonic equation","author":"AP Selvadurai","year":"2000","unstructured":"Selvadurai AP (2000) Partial differential equations in mechanics 2: the biharmonic equation. Poisson\u2019s equation, Springer Science and Business Media"},{"issue":"8","key":"3188_CR20","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1515\/zna-2015-0100","volume":"70","author":"E Shivanian","year":"2015","unstructured":"Shivanian E (2015) A meshless method based on radial basis and spline interpolation for 2-D and 3-D inhomogeneous biharmonic BVPs. Zeitschrift f\u00fcr Naturforschung A 70(8):673\u2013682","journal-title":"Zeitschrift f\u00fcr Naturforschung A"},{"key":"3188_CR21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40314-020-01175-x","volume":"39","author":"E Shivanian","year":"2020","unstructured":"Shivanian E, Abbasbandy S (2020) Pseudospectral meshless radial point interpolation for generalized biharmonic equation in the presence of Cahn\u2013Hilliard conditions. Comput Appl Math 39:1\u201318","journal-title":"Comput Appl Math"},{"issue":"8","key":"3188_CR22","doi-asserted-by":"publisher","first-page":"1736","DOI":"10.1002\/nme.6914","volume":"123","author":"K Wang","year":"2022","unstructured":"Wang K, Shen B, Li M, Dong L, Atluri SN (2022) A Fragile Points Method, with an interface debonding model, to simulate damage and fracture of U-notched structures. Int J Numer Methods Eng 123(8):1736\u20131759","journal-title":"Int J Numer Methods Eng"},{"issue":"1","key":"3188_CR23","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1002\/num.1690090102","volume":"9","author":"WS Yousif","year":"1993","unstructured":"Yousif WS, Evans DJ (1993) Explicit block iterative method for the solution of the biharmonic equation. Numer Methods Partial Differ Equ 9(1):1\u201312","journal-title":"Numer Methods Partial Differ Equ"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03188-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03188-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03188-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,19]],"date-time":"2025-05-19T14:10:48Z","timestamp":1747663848000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03188-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,3]]},"references-count":23,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2025,7]]}},"alternative-id":["3188"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03188-w","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2025,4,3]]},"assertion":[{"value":"8 November 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 February 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 March 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 April 2025","order":4,"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 conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"209"}}