{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:21:33Z","timestamp":1760235693845,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2021,9,9]],"date-time":"2021-09-09T00:00:00Z","timestamp":1631145600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"ERC programme H2020","award":["ERC-2017-COG - MultiphysMicroCaps"],"award-info":[{"award-number":["ERC-2017-COG - MultiphysMicroCaps"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>An innovative data-driven model-order reduction technique is proposed to model dilute micrometric or nanometric suspensions of microcapsules, i.e., microdrops protected in a thin hyperelastic membrane, which are used in Healthcare as innovative drug vehicles. We consider a microcapsule flowing in a similar-size microfluidic channel and vary systematically the governing parameter, namely the capillary number, ratio of the viscous to elastic forces, and the confinement ratio, ratio of the capsule to tube size. The resulting space-time-parameter problem is solved using two global POD reduced bases, determined in the offline stage for the space and parameter variables, respectively. A suitable low-order spatial reduced basis is then computed in the online stage for any new parameter instance. The time evolution of the capsule dynamics is achieved by identifying the nonlinear low-order manifold of the reduced variables; for that, a point cloud of reduced data is computed and a diffuse approximation method is used. Numerical comparisons between the full-order fluid-structure interaction model and the reduced-order one confirm both accuracy and stability of the reduction technique over the whole admissible parameter domain. We believe that such an approach can be applied to a broad range of coupled problems especially involving quasistatic models of structural mechanics.<\/jats:p>","DOI":"10.3390\/e23091193","type":"journal-article","created":{"date-parts":[[2021,9,9]],"date-time":"2021-09-09T12:20:37Z","timestamp":1631190037000},"page":"1193","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Data-Driven Space-Time-Parameter Reduced-Order Model with Manifold Learning for Coupled Problems: Application to Deformable Capsules Flowing in Microchannels"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2790-9914","authenticated-orcid":false,"given":"Toufik","family":"Boubehziz","sequence":"first","affiliation":[{"name":"Biomechanics and Bioengineering Laboratory (UMR CNRS 7338), Universit\u00e9 de Technologie de Compi\u00e8gne CNRS, Alliance Sorbonne Universit\u00e9, 60203 Compi\u00e8gne, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3294-8093","authenticated-orcid":false,"given":"Carlos","family":"Quesada-Granja","sequence":"additional","affiliation":[{"name":"Biomechanics and Bioengineering Laboratory (UMR CNRS 7338), Universit\u00e9 de Technologie de Compi\u00e8gne CNRS, Alliance Sorbonne Universit\u00e9, 60203 Compi\u00e8gne, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7727-3846","authenticated-orcid":false,"given":"Claire","family":"Dupont","sequence":"additional","affiliation":[{"name":"Biomechanics and Bioengineering Laboratory (UMR CNRS 7338), Universit\u00e9 de Technologie de Compi\u00e8gne CNRS, Alliance Sorbonne Universit\u00e9, 60203 Compi\u00e8gne, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pierre","family":"Villon","sequence":"additional","affiliation":[{"name":"Laboratoire Roberval, Universit\u00e9 de Technologie de Compi\u00e8gne, Alliance Sorbonne Universit\u00e9, 60203 Compi\u00e8gne, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0854-4670","authenticated-orcid":false,"given":"Florian","family":"De Vuyst","sequence":"additional","affiliation":[{"name":"Laboratoire de Math\u00e9matiques Appliqu\u00e9es de Compi\u00e8gne (EA 2222), Universit\u00e9 de Technologie de Compi\u00e8gne, Alliance Sorbonne Universit\u00e9, 60203 Compi\u00e8gne, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anne-Virginie","family":"Salsac","sequence":"additional","affiliation":[{"name":"Biomechanics and Bioengineering Laboratory (UMR CNRS 7338), Universit\u00e9 de Technologie de Compi\u00e8gne CNRS, Alliance Sorbonne Universit\u00e9, 60203 Compi\u00e8gne, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1162\/neco.1989.1.2.270","article-title":"A learning algorithm for continually running fully Recurrent Neural Networks","volume":"1","author":"Williams","year":"1989","journal-title":"Neural Comput."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","article-title":"Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations","volume":"378","author":"Raissi","year":"2019","journal-title":"J. Comput. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.cma.2015.03.018","article-title":"Dynamic data-driven reduced-order models","volume":"291","author":"Peherstorfer","year":"2015","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1016\/j.cma.2016.03.025","article-title":"Data-driven operator inference for nonintrusive projection-based model reduction","volume":"306","author":"Peherstorfer","year":"2016","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"085101","DOI":"10.1063\/1.5113494","article-title":"A deep learning enabler for nonintrusive reduced order modeling of fluid flows","volume":"31","author":"Pawar","year":"2019","journal-title":"Phys. Fluids"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"868","DOI":"10.1016\/j.cma.2016.12.033","article-title":"A parameterized non-intrusive reduced order model and error analysis for general time-dependent nonlinear partial differential equations and its applications","volume":"317","author":"Xiao","year":"2017","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_7","unstructured":"Cordier, L. (2008). Proper Orthogonal Decomposition: An overview. Lecture Series 2008-01 on Post-Processing of Experimental and Numerical Data, Von Karman Institute for Fluid Dynamics."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1137\/130932715","article-title":"A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems","volume":"57","author":"Benner","year":"2015","journal-title":"SIAM Rev."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1309","DOI":"10.1007\/s40430-014-0259-3","article-title":"Practical implementation aspects of Galerkin reduced order models based on Proper Orthogonal Decomposition for Computational Fluid Dynamics","volume":"37","author":"Silva","year":"2015","journal-title":"J. Bras. Soc. Mech. Sci. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1137\/S0895479803423925","article-title":"Model Reduction of MIMO Systems via Tangential Interpolation","volume":"26","author":"Gallivan","year":"2004","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/s11831-011-9064-7","article-title":"A short review on model order reduction based on Proper Generalized Decomposition","volume":"18","author":"Chinesta","year":"2011","journal-title":"Arch. Comput. Methods Eng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"53","DOI":"10.3166\/ejcm.19.53-64","article-title":"On the Use of Proper Generalized Decompositions for Solving the Multidimensional Chemical Master Equation","volume":"19","author":"Chinesta","year":"2010","journal-title":"Rev. Europ\u00e9enne M\u00e9canique Num\u00e9rique\/Eur. J. Comput. Mech."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/j.cma.2011.11.018","article-title":"Proper Generalized Decomposition based dynamic data-driven control of thermal processes","volume":"213\u2013216","author":"Ghnatios","year":"2012","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1016\/j.crma.2004.08.006","article-title":"An \u2018empirical interpolation\u2019 method: Application to efficient reduced-basis discretization of partial differential equations","volume":"339","author":"Barrault","year":"2004","journal-title":"Comptes Rendus Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2737","DOI":"10.1137\/090766498","article-title":"Nonlinear model reduction via discrete empirical interpolation","volume":"32","author":"Chaturantabut","year":"2010","journal-title":"SIAM J. Sci. Comp."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jcp.2014.01.011","article-title":"Non-linear model reduction for the Navier-Stokes equations using residual DEIM method","volume":"263","author":"Xiao","year":"2014","journal-title":"J. Comp. Phys."},{"key":"ref_17","unstructured":"Lappano, E., Naets, F., Desmet, W., Mundo, D., and Nijman, E. (2016, January 19\u201321). A greedy sampling approach for the projection basis construction in parametric model order reduction for structural dynamics models. Proceedings of the 27th International Conference on Noise and Vibration Engineering, Leuven, Belgium."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"825","DOI":"10.3166\/reef.11.825-867","article-title":"An Introduction to Moving Least Squares Meshfree Methods","volume":"11","author":"Breitkopf","year":"2002","journal-title":"Rev. Europ\u00e9enne El\u00e9ments Finis"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2029","DOI":"10.2514\/1.J050233","article-title":"Toward real-time computational-fluid-dynamics-based aeroelastic computations using a database of reduced-order information","volume":"48","author":"Amsallem","year":"2010","journal-title":"AIAA J."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1025","DOI":"10.1002\/nme.2540","article-title":"Reduced-order modeling of parameterized PDEs using time\u2013space-parameter principal component analysis","volume":"80","author":"Audouze","year":"2009","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"170","DOI":"10.3109\/1061186X.2014.971327","article-title":"Microencapsulation of therapeutic bispecific antibodies producing cells: Immunotherapeutic organoids for cancer management","volume":"23","author":"Compte","year":"2015","journal-title":"J. Drug Target."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"946","DOI":"10.1002\/btpr.226","article-title":"Progress technology in microencapsulation methods for cell therapy","volume":"25","author":"Rabanel","year":"2009","journal-title":"Biotechnol. Prog."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/j.jcis.2010.11.038","article-title":"Comparison between measurements of elasticity and free amino group content of ovalbumin microcapsule membranes: Discrimination of the cross-linking degree","volume":"355","author":"Chu","year":"2011","journal-title":"J. Colloid Interface Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"063008","DOI":"10.1103\/PhysRevE.87.063008","article-title":"Characterizing the membrane properties of capsules flowing in a square-section microfluidic channel: Effects of the membrane constitutive law","volume":"87","author":"Hu","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"4561","DOI":"10.1039\/c4sm00349g","article-title":"Mechanical characterization of cross-linked serum albumin microcapsules","volume":"10","author":"Deschamps","year":"2014","journal-title":"Soft Matter"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.piutam.2015.03.013","article-title":"Characterization of Capsule Membrane Properties using a Microfluidic Photolithographied Channel: Consequences of Tube Non-squareness","volume":"16","author":"Salsac","year":"2015","journal-title":"Procedia IUTAM"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1381","DOI":"10.1007\/s00396-016-3885-8","article-title":"Characterization of the mechanical properties of cross-linked serum albumin microcapsules: Effect of size and protein concentration","volume":"294","author":"Gubspun","year":"2016","journal-title":"Colloid Polym. Sci."},{"key":"ref_28","first-page":"686","article-title":"Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations","volume":"378","author":"Deschamps","year":"2016","journal-title":"J. Fluid Mech."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1018","DOI":"10.1177\/1081286520977602","article-title":"Diffuse approximation for identification of the mechanical properties of microcapsules","volume":"26","author":"Quesada","year":"2021","journal-title":"Math. Mech. Solids"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1017\/jfm.2016.603","article-title":"Motion of a spherical capsule in branched tube flow with finite inertia","volume":"806","author":"Wang","year":"2016","journal-title":"J. Fluid Mech."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1016\/j.medengphy.2017.06.040","article-title":"Deformability- and size-based microcapsule sorting","volume":"48","author":"Vesperini","year":"2017","journal-title":"Med. Eng. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1017\/jfm.2018.414","article-title":"Path selection of a spherical capsule in a microfluidic branched channel: Towards the design of an enrichment device","volume":"849","author":"Wang","year":"2018","journal-title":"J. Fluid Mech."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1017\/jfm.2018.829","article-title":"Sorting same-size red blood cells in deep deterministic lateral displacement devices","volume":"859","author":"Biros","year":"2019","journal-title":"J. Fluid Mech."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3722","DOI":"10.1039\/D0SM02249G","article-title":"Sorting of capsules according to their stiffness: From principle to application","volume":"17","author":"Vesperini","year":"2021","journal-title":"Soft Matter"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"103193","DOI":"10.1016\/j.jfluidstructs.2020.103193","article-title":"Real-time prediction of the deformation of microcapsules using Proper Orthogonal Decomposition","volume":"101","author":"Quesada","year":"2021","journal-title":"J. Fluids Struct."},{"key":"ref_36","unstructured":"Savignat, J.M. (2000). Approximation Diffuse Hermite et Ses Applications. [Ph.D. Thesis, \u00c9cole Nationale Sup\u00e9rieure des Mines de Paris]."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"987","DOI":"10.1007\/s00158-013-0942-5","article-title":"Towards a space reduction approach for efficient structural shape optimization","volume":"48","author":"Raghavan","year":"2013","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1017\/jfm.2011.462","article-title":"Flow of a spherical capsule in a pore with circular or square cross-section","volume":"705","author":"Hu","year":"2012","journal-title":"J. Fluid Mech."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Wang, X., Merlo, A., Dupont, D., Salsac, A.V., and Barth\u00e8s-Biesel, D. (2021). Characterization of the mechanical properties of microcapsules with a reticulated membrane: Comparison of microfluidic and microrheometric approaches. Flow, in press.","DOI":"10.1017\/flo.2021.8"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"829","DOI":"10.1002\/nme.2859","article-title":"Coupling of finite element and boundary integral methods for a capsule in a Stokes flow: Numerical Methods for A Capsule in A Stokes Flow","volume":"83","author":"Walter","year":"2010","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/j.cocis.2010.07.001","article-title":"Modeling the motion of capsules in flow","volume":"16","year":"2011","journal-title":"Curr. Opin. Colloid Interface Sci."},{"key":"ref_42","unstructured":"Bauer, F.L. (1971). Singular value decomposition and least squares solutions. Linear Algebra, Springer."},{"key":"ref_43","unstructured":"Buhmann, M.D. (2003). Radial Basis Functions, Cambridge University Press. Cambridge Monographs on Applied and Computational Mathematics."},{"key":"ref_44","unstructured":"Dubuisson, M.P., and Jain, A. (1994, January 9\u201313). A modified Hausdorff distance for object matching. Proceedings of the 12th International Conference on Pattern Recognition, Jerusalem, Israel."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/9\/1193\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:59:51Z","timestamp":1760165991000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/9\/1193"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,9]]},"references-count":44,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2021,9]]}},"alternative-id":["e23091193"],"URL":"https:\/\/doi.org\/10.3390\/e23091193","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2021,9,9]]}}}