{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:04:50Z","timestamp":1760148290178,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,20]],"date-time":"2023-04-20T00:00:00Z","timestamp":1681948800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001807","name":"Fapesp","doi-asserted-by":"publisher","award":["2019\/24915-4"],"award-info":[{"award-number":["2019\/24915-4"]}],"id":[{"id":"10.13039\/501100001807","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>It was recently proved that the knowledge of the transverse displacement of a nanoplate in an open subset of its mid-plane, measured for any interval of time, allows for the unique determination of the spatial components {fm(x,y)}m=1M of the transverse load \u2211m=1Mgm(t)fm(x,y), where M\u22651 and {gm(t)}m=1M is a known set of linearly independent functions of the time variable. The nanoplate mechanical model is built within the strain gradient linear elasticity theory, according to the Kirchhoff\u2013Love kinematic assumptions. In this paper, we derive a reconstruction algorithm for the above inverse source problem, and we implement a numerical procedure based on a finite element spatial discretization to approximate the loads {fm(x,y)}m=1M. The computations are developed for a uniform rectangular nanoplate clamped at the boundary. The sensitivity of the results with respect to the main parameters that influence the identification is analyzed in detail. The adoption of a regularization scheme based on the singular value decomposition turns out to be decisive for the accuracy and stability of the reconstruction.<\/jats:p>","DOI":"10.3390\/axioms12040398","type":"journal-article","created":{"date-parts":[[2023,4,20]],"date-time":"2023-04-20T05:35:46Z","timestamp":1681968946000},"page":"398","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Reconstructing Loads in Nanoplates from Dynamic Data"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2248-6422","authenticated-orcid":false,"given":"Alexandre","family":"Kawano","sequence":"first","affiliation":[{"name":"Escola Polit\u00e9cnica, University of S\u00e3o Paulo, Av. Prof. Luciano Gualberto, 380-Butant\u00e3, S\u00e3o Paulo 05508-010, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2186-7688","authenticated-orcid":false,"given":"Antonino","family":"Morassi","sequence":"additional","affiliation":[{"name":"Polytechnic Department of Engineering and Architecture, University of Udine, Via Cotonificio 114, 33100 Udine, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1045","DOI":"10.1002\/mma.8565","article-title":"Inverse load identification in vibrating nanoplates","volume":"46","author":"Kawano","year":"2023","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1477","DOI":"10.1016\/S0022-5096(03)00053-X","article-title":"Experiments and theory in strain gradient elasticity","volume":"51","author":"Lam","year":"2003","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1016\/j.compstruct.2017.06.040","article-title":"A review of continuum mechanics models for size-dependent analysis of beams and plates","volume":"177","author":"Thai","year":"2017","journal-title":"Compos. Struct."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4053","DOI":"10.1109\/TED.2011.2165544","article-title":"Gauge factor tuning, long-term stability, and miniaturization of nanoelectromechanical carbon-nanotube sensors","volume":"58","author":"Helbling","year":"2011","journal-title":"IEEE Trans. Electron Devices"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Hurst, A.M., Lee, S., Petrone, N., Weert, J.V.D., Zande, A.M.V.D., and Hone, J. (2013, January 16\u201320). A transconductive graphene pressure sensor. Proceedings of the 2013 Transducers & Eurosensors XXVII: The 17th International Conference on Solid-State Sensors, Actuators and Microsystems (2013), Barcelona, Spain.","DOI":"10.1109\/Transducers.2013.6626834"},{"key":"ref_6","first-page":"1","article-title":"Frontiers of nanosensor technology","volume":"103","author":"Khanna","year":"2009","journal-title":"Sens. Transducers"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/j.sna.2011.05.021","article-title":"Non-invasive intraocular pressure monitoring with a contact lens engineered with a nanostructured polymeric sensing film","volume":"170","author":"Laukhin","year":"2011","journal-title":"Sens. Actuators A Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2055","DOI":"10.1081\/PDE-120016136","article-title":"Uniqueness along a line for an inverse wave source problem","volume":"27","author":"Cheng","year":"2002","journal-title":"Commun. Partial. Differ. Equ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1007\/s10255-020-0917-4","article-title":"Uniqueness to some inverse source problems for the wave equation in unbounded domains","volume":"36","author":"Hu","year":"2020","journal-title":"Acta Math. Appl. Sin."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1632","DOI":"10.1137\/080725635","article-title":"Solving inverse source problems using observability. Applications to the Euler-Bernoulli plate equation","volume":"48","author":"Alves","year":"2009","journal-title":"SIAM J. Control. Optim."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1088\/0266-5611\/22\/1\/021","article-title":"Force inversion in floating plate dynamics","volume":"22","author":"Dempsey","year":"2006","journal-title":"Inverse Probl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"744","DOI":"10.1002\/mma.3518","article-title":"A unique continuation result for the plate equation and an application","volume":"39","author":"Arat","year":"2016","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1515\/jiip.1996.4.5.437","article-title":"Determination of forces in vibrations of beams and plates by pointwise and line observations","volume":"4","author":"Yamamoto","year":"1996","journal-title":"J. Inverse Ill-Posed Probl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"085002","DOI":"10.1088\/0266-5611\/29\/8\/085002","article-title":"Uniqueness in the determination of vibration sources in rectangular Germain\u2013Lagrange plates using displacement measurements over line segments with arbitrary small length","volume":"29","author":"Kawano","year":"2013","journal-title":"Inverse Probl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1137\/0330054","article-title":"Exact semi-internal control of an Euler-Bernoulli equation","volume":"30","author":"Kim","year":"1992","journal-title":"SIAM J. Control. Optim."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/0020-7683(68)90036-X","article-title":"On first strain-gradient theories in linear elasticity","volume":"4","author":"Mindlin","year":"1968","journal-title":"Int. J. Solids Struct."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1524","DOI":"10.1002\/zamm.201600107","article-title":"The modified indeterminate couple stress model: Why Yang et al.\u2019s arguments motivating a symmetric couple stress tensor contain a gap and why the couple stress tensor may be chosen symmetric nevertheless","volume":"97","author":"Munch","year":"2017","journal-title":"ZAMM J. Appl. Math. Mech. Z. Angew. Math. Mech."},{"key":"ref_18","unstructured":"Burnett, D.S. (1987). Finite Element Analysis, Addison-Wesley Publishing Company."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.ijengsci.2014.11.002","article-title":"On pull-in instabilities of microcantilevers","volume":"87","author":"Rahaeifard","year":"2015","journal-title":"Int. J. Eng. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1137\/090763524","article-title":"Uniqueness and nonuniqueness results for a certain class of almost periodic distributions","volume":"43","author":"Kawano","year":"2011","journal-title":"SIAM J. Math. Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2506","DOI":"10.1137\/19M1262322","article-title":"Detecting a prey in a spider orb web","volume":"79","author":"Kawano","year":"2019","journal-title":"SIAM J. Appl. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2297","DOI":"10.1137\/20M1372792","article-title":"Detecting a prey in a spider orb-web from in-plane vibration","volume":"81","author":"Kawano","year":"2021","journal-title":"SIAM J. Appl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Kaipio, J., and Somersalo, E. (2004). Statistical and Computational Inverse Problems, Springer. Applied Mathematical Sciences.","DOI":"10.1007\/b138659"},{"key":"ref_24","unstructured":"Vieweg, F., and Braunschweig, S. (1987). Stable Solution of Inverse Problems, Springer."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Kirsch, A. (2021). An Introduction to the Mathematical Theory of Inverse Problems, Springer. [3rd ed.].","DOI":"10.1007\/978-3-030-63343-1"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"045001","DOI":"10.1088\/0266-5611\/28\/4\/045001","article-title":"Local analysis of inverse problems: H\u00f6lder stability and iterative reconstruction","volume":"28","author":"Qiu","year":"2012","journal-title":"Inverse Probl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/4\/398\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:19:45Z","timestamp":1760123985000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/4\/398"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,20]]},"references-count":26,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["axioms12040398"],"URL":"https:\/\/doi.org\/10.3390\/axioms12040398","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,4,20]]}}}