Analysis of a 3D nonlinear, moving boundary problem describing fluid-mesh-shell interaction
Authors:
Sunčica Čanić, Marija Galić and Boris Muha
Journal:
Trans. Amer. Math. Soc. 373 (2020), 6621-6681
MSC (2010):
Primary 74F10, 35D30; Secondary 74K25
DOI:
https://doi.org/10.1090/tran/8125
Published electronically:
July 8, 2020
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Abstract | References | Similar Articles | Additional Information
Abstract: We consider a nonlinear, moving boundary, fluid-structure interaction problem between a time-dependent incompressible, viscous fluid flow, and an elastic structure composed of a cylindrical shell supported by a mesh of elastic rods. The fluid flow is modeled by the time-dependent Navier-Stokes equations in a three-dimensional cylindrical domain, while the lateral wall of the cylinder is modeled by the two-dimensional linearly elastic Koiter shell equations coupled to a one-dimensional system of conservation laws defined on a graph domain, describing a mesh of curved rods. The mesh-supported shell allows displacements in all three spatial directions. Two-way coupling based on kinematic and dynamic coupling conditions is assumed between the fluid and composite structure, and between the mesh of curved rods and Koiter shell. Problems of this type arise in many applications, including blood flow through arteries treated with vascular prostheses called stents. We prove the existence of a weak solution to this nonlinear, moving boundary problem by using the time discretization via a Lie operator splitting method combined with an Arbitrary Lagrangian-Eulerian approach, and a nontrivial extension of the Aubin-Lions-Simon compactness result to problems on moving domains.
- [1] Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
- [2] Stuart S. Antman, Nonlinear problems of elasticity, 2nd ed., Applied Mathematical Sciences, vol. 107, Springer, New York, 2005. MR 2132247
- [3] George Avalos, Irena Lasiecka, and Roberto Triggiani, Higher regularity of a coupled parabolic-hyperbolic fluid-structure interactive system, Georgian Math. J. 15 (2008), no. 3, 403–437. MR 2466407
- [4] George Avalos and Roberto Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discrete Contin. Dyn. Syst. Ser. S 2 (2009), no. 3, 417–447. MR 2525761, https://doi.org/10.3934/dcdss.2009.2.417
- [5] Viorel Barbu, Zoran Grujić, Irena Lasiecka, and Amjad Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, Fluids and waves, Contemp. Math., vol. 440, Amer. Math. Soc., Providence, RI, 2007, pp. 55–82. MR 2359449, https://doi.org/10.1090/conm/440/08476
- [6] Viorel Barbu, Zoran Grujić, Irena Lasiecka, and Amjad Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana Univ. Math. J. 57 (2008), no. 3, 1173–1207. MR 2429089, https://doi.org/10.1512/iumj.2008.57.3284
- [7] H. Beirão da Veiga, On the existence of strong solutions to a coupled fluid-structure evolution problem, J. Math. Fluid Mech. 6 (2004), no. 1, 21–52. MR 2027753, https://doi.org/10.1007/s00021-003-0082-5
- [8] Muriel Boulakia, Existence of weak solutions for the motion of an elastic structure in an incompressible viscous fluid, C. R. Math. Acad. Sci. Paris 336 (2003), no. 12, 985–990 (English, with English and French summaries). MR 1993967, https://doi.org/10.1016/S1631-073X(03)00235-8
- [9] Muriel Boulakia, Existence of weak solutions for an interaction problem between an elastic structure and a compressible viscous fluid, J. Math. Pures Appl. (9) 84 (2005), no. 11, 1515–1554 (English, with English and French summaries). MR 2181459, https://doi.org/10.1016/j.matpur.2005.08.004
- [10] Martina Bukač, Sunčica Čanić, and Boris Muha, A nonlinear fluid-structure interaction problem in compliant arteries treated with vascular stents, Appl. Math. Optim. 73 (2016), no. 3, 433–473. MR 3498934, https://doi.org/10.1007/s00245-016-9343-7
- [11]
J. Butany, K. Carmichael, S. W. Leong, and M. J. Collins,
Coronary artery stents: identification and evaluation,
Journal of clinical pathology 58(8):795-804, 2005. - [12] Sunčica Čanić, New mathematics for next-generation stent design, SIAM News 52 (2019), no. 3, 1, 3. MR 3967610
- [13] Sunčica Čanić, Marija Galić, Matko Ljulj, Boris Muha, Josip Tambača, and Yifan Wang, Analysis of a linear 3D fluid-mesh-shell interaction problem, Z. Angew. Math. Phys. 70 (2019), no. 2, Paper No. 44, 38. MR 3914588, https://doi.org/10.1007/s00033-019-1087-1
- [14] Sunčica Čanić, Matea Galović, Matko Ljulj, and Josip Tambača, A dimension-reduction based coupled model of mesh-reinforced shells, SIAM J. Appl. Math. 77 (2017), no. 2, 744–769. MR 3640634, https://doi.org/10.1137/16M1088181
- [15] Sunčica Čanić and Josip Tambača, Cardiovascular stents as PDE nets: 1D vs. 3D, IMA J. Appl. Math. 77 (2012), no. 6, 748–770. MR 2999136, https://doi.org/10.1093/imamat/hxs047
- [16] T. Chacón Rebollo, V. Girault, F. Murat, and O. Pironneau, Analysis of a coupled fluid-structure model with applications to hemodynamics, SIAM J. Numer. Anal. 54 (2016), no. 2, 994–1019. MR 3482396, https://doi.org/10.1137/140991509
- [17] Antonin Chambolle, Benoît Desjardins, Maria J. Esteban, and Céline Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, J. Math. Fluid Mech. 7 (2005), no. 3, 368–404. MR 2166981, https://doi.org/10.1007/s00021-004-0121-y
- [18] C. H. Arthur Cheng, Daniel Coutand, and Steve Shkoller, Navier-Stokes equations interacting with a nonlinear elastic biofluid shell, SIAM J. Math. Anal. 39 (2007), no. 3, 742–800. MR 2349865, https://doi.org/10.1137/060656085
- [19] C. H. Arthur Cheng and Steve Shkoller, The interaction of the 3D Navier-Stokes equations with a moving nonlinear Koiter elastic shell, SIAM J. Math. Anal. 42 (2010), no. 3, 1094–1155. MR 2644917, https://doi.org/10.1137/080741628
- [20] Igor Chueshov and Tamara Fastovska, On interaction of circular cylindrical shells with a Poiseuille type flow, Evol. Equ. Control Theory 5 (2016), no. 4, 605–629. MR 3603250, https://doi.org/10.3934/eect.2016021
- [21] I. Chueshov and I. Ryzhkova, On the interaction of an elastic wall with a Poiseuille-type flow, Ukrainian Math. J. 65 (2013), no. 1, 158–177. MR 3104890, https://doi.org/10.1007/s11253-013-0771-0
- [22] Igor Chueshov and Iryna Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models, System modeling and optimization, IFIP Adv. Inf. Commun. Technol., vol. 391, Springer, Heidelberg, 2013, pp. 328–337. MR 3409574, https://doi.org/10.1007/978-3-642-36062-6_33
- [23] Philippe G. Ciarlet, Mathematical elasticity. Vol. I, Studies in Mathematics and its Applications, vol. 20, North-Holland Publishing Co., Amsterdam, 1988. Three-dimensional elasticity. MR 936420
- [24] Philippe G. Ciarlet, Mathematical elasticity. Vol. III, Studies in Mathematics and its Applications, vol. 29, North-Holland Publishing Co., Amsterdam, 2000. Theory of shells. MR 1757535
- [25] P. G. Ciarlet and A. Roquefort, Justification of a two-dimensional nonlinear shell model of Koiter’s type, Chinese Ann. Math. Ser. B 22 (2001), no. 2, 129–144. MR 1835394, https://doi.org/10.1142/S0252959901000139
- [26] C. Conca, F. Murat, and O. Pironneau, The Stokes and Navier-Stokes equations with boundary conditions involving the pressure, Japan. J. Math. (N.S.) 20 (1994), no. 2, 279–318. MR 1308419, https://doi.org/10.4099/math1924.20.279
- [27] Carlos Conca, Jorge San Martín H., and Marius Tucsnak, Motion of a rigid body in a viscous fluid, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 6, 473–478 (English, with English and French summaries). MR 1680008, https://doi.org/10.1016/S0764-4442(99)80193-1
- [28] Daniel Coutand and Steve Shkoller, Motion of an elastic solid inside an incompressible viscous fluid, Arch. Ration. Mech. Anal. 176 (2005), no. 1, 25–102. MR 2185858, https://doi.org/10.1007/s00205-004-0340-7
- [29] Daniel Coutand and Steve Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Ration. Mech. Anal. 179 (2006), no. 3, 303–352. MR 2208319, https://doi.org/10.1007/s00205-005-0385-2
- [30] B. Desjardins and M. J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal. 146 (1999), no. 1, 59–71. MR 1682663, https://doi.org/10.1007/s002050050136
- [31] B. Desjardins, M. J. Esteban, C. Grandmont, and P. Le Tallec, Weak solutions for a fluid-elastic structure interaction model, Rev. Mat. Complut. 14 (2001), no. 2, 523–538. MR 1871311, https://doi.org/10.5209/rev_REMA.2001.v14.n2.17030
- [32]
J. Donea,
Arbitrary Lagrangian-Eulerian finite element methods,
Computational methods for transient analysis North-Holland, Amsterdam, 1983. - [33] Q. Du, M. D. Gunzburger, L. S. Hou, and J. Lee, Analysis of a linear fluid-structure interaction problem, Discrete Contin. Dyn. Syst. 9 (2003), no. 3, 633–650. MR 1974530, https://doi.org/10.3934/dcds.2003.9.633
- [34] Eduard Feireisl, On the motion of rigid bodies in a viscous incompressible fluid, J. Evol. Equ. 3 (2003), no. 3, 419–441. Dedicated to Philippe Bénilan. MR 2019028, https://doi.org/10.1007/s00028-003-0110-1
- [35] Giovanni P. Galdi and Ana L. Silvestre, Existence of time-periodic solutions to the Navier-Stokes equations around a moving body, Pacific J. Math. 223 (2006), no. 2, 251–267. MR 2221027, https://doi.org/10.2140/pjm.2006.223.251
- [36] Giovanni P. Galdi, On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications, Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam, 2002, pp. 653–791. MR 1942470
- [37] Giovanni P. Galdi, Mathematical problems in classical and non-Newtonian fluid mechanics, Hemodynamical flows, Oberwolfach Semin., vol. 37, Birkhäuser, Basel, 2008, pp. 121–273. MR 2410706, https://doi.org/10.1007/978-3-7643-7806-6_3
- [38] G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, 2nd ed., Springer Monographs in Mathematics, Springer, New York, 2011. Steady-state problems. MR 2808162
- [39] Roland Glowinski, Finite element methods for incompressible viscous flow, Handbook of numerical analysis, Vol. IX, Handb. Numer. Anal., IX, North-Holland, Amsterdam, 2003, pp. 3–1176. MR 2009826
- [40] Céline Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM J. Math. Anal. 40 (2008), no. 2, 716–737. MR 2438783, https://doi.org/10.1137/070699196
- [41] Céline Grandmont and Matthieu Hillairet, Existence of global strong solutions to a beam-fluid interaction system, Arch. Ration. Mech. Anal. 220 (2016), no. 3, 1283–1333. MR 3466847, https://doi.org/10.1007/s00205-015-0954-y
- [42] Georges Griso, Asymptotic behavior of structures made of curved rods, Anal. Appl. (Singap.) 6 (2008), no. 1, 11–22. MR 2380884, https://doi.org/10.1142/S0219530508001031
- [43] Giovanna Guidoboni, Roland Glowinski, Nicola Cavallini, and Suncica Canic, Stable loosely-coupled-type algorithm for fluid-structure interaction in blood flow, J. Comput. Phys. 228 (2009), no. 18, 6916–6937. MR 2567876, https://doi.org/10.1016/j.jcp.2009.06.007
- [44] Thomas J. R. Hughes, Wing Kam Liu, and Thomas K. Zimmermann, Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Comput. Methods Appl. Mech. Engrg. 29 (1981), no. 3, 329–349. MR 659925, https://doi.org/10.1016/0045-7825(81)90049-9
- [45] Mladen Jurak and Josip Tambača, Derivation and justification of a curved rod model, Math. Models Methods Appl. Sci. 9 (1999), no. 7, 991–1014. MR 1710272, https://doi.org/10.1142/S0218202599000452
- [46] Mladen Jurak and Josip Tambača, Linear curved rod model. General curve, Math. Models Methods Appl. Sci. 11 (2001), no. 7, 1237–1252. MR 1848199, https://doi.org/10.1142/S0218202501001318
- [47] W. T. Koiter, On the foundations of the linear theory of thin elastic shells. I, II., Nederl. Akad. Wetensch. Proc. Ser. B 73 (1970), 169-182; ibid 73 (1970), 183–195. MR 0280050
- [48] Igor Kukavica and Amjad Tuffaha, Solutions to a fluid-structure interaction free boundary problem, Discrete Contin. Dyn. Syst. 32 (2012), no. 4, 1355–1389. MR 2851902, https://doi.org/10.3934/dcds.2012.32.1355
- [49] Igor Kukavica, Amjad Tuffaha, and Mohammed Ziane, Strong solutions for a fluid structure interaction system, Adv. Differential Equations 15 (2010), no. 3-4, 231–254. MR 2588449
- [50] Daniel Lengeler and Michael R\ocirc{u}žička, Weak solutions for an incompressible Newtonian fluid interacting with a Koiter type shell, Arch. Ration. Mech. Anal. 211 (2014), no. 1, 205–255. MR 3147436, https://doi.org/10.1007/s00205-013-0686-9
- [51] Julien Lequeurre, Existence of strong solutions to a fluid-structure system, SIAM J. Math. Anal. 43 (2011), no. 1, 389–410. MR 2765696, https://doi.org/10.1137/10078983X
- [52] Boris Muha and Suncica Canić, Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls, Arch. Ration. Mech. Anal. 207 (2013), no. 3, 919–968. MR 3017292, https://doi.org/10.1007/s00205-012-0585-5
- [53] Boris Muha and Sunčica Čanić, A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: a constructive existence proof, Commun. Inf. Syst. 13 (2013), no. 3, 357–397. MR 3226949, https://doi.org/10.4310/CIS.2013.v13.n3.a4
- [54] Boris Muha and Sunčica Čanić, Existence of a solution to a fluid-multi-layered-structure interaction problem, J. Differential Equations 256 (2014), no. 2, 658–706. MR 3121710, https://doi.org/10.1016/j.jde.2013.09.016
- [55] Boris Muha and Sunčica Čanić, Fluid-structure interaction between an incompressible, viscous 3D fluid and an elastic shell with nonlinear Koiter membrane energy, Interfaces Free Bound. 17 (2015), no. 4, 465–495. MR 3450736, https://doi.org/10.4171/IFB/350
- [56] Boris Muha and Sunčica Čanić, Existence of a weak solution to a fluid-elastic structure interaction problem with the Navier slip boundary condition, J. Differential Equations 260 (2016), no. 12, 8550–8589. MR 3482692, https://doi.org/10.1016/j.jde.2016.02.029
- [57] Boris Muha and Sunčica Čanić, A generalization of the Aubin-Lions-Simon compactness lemma for problems on moving domains, J. Differential Equations 266 (2019), no. 12, 8370–8418. MR 3944259, https://doi.org/10.1016/j.jde.2018.12.030
- [58] Michael R\ocirc{u}žička, Multipolar materials, Workshop on the Mathematical Theory of Nonlinear and Inelastic Material Behaviour (Darmstadt, 1992) Bonner Math. Schriften, vol. 239, Univ. Bonn, Bonn, 1993, pp. 53–64. MR 1290373
- [59] J. Tambača, M. Kosor, S. Čanić, and D. Paniagua, Mathematical modeling of vascular stents, SIAM J. Appl. Math. 70 (2010), no. 6, 1922–1952. MR 2596508, https://doi.org/10.1137/080722618
- [60] Igor Velčić, Nonlinear weakly curved rod by Γ-convergence, J. Elasticity 108 (2012), no. 2, 125–150. MR 2948053, https://doi.org/10.1007/s10659-011-9358-x
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Additional Information
Sunčica Čanić
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720
Email:
canics@berkeley.edu
Marija Galić
Affiliation:
Department of Mathematics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia
Email:
marijag5@math.hr
Boris Muha
Affiliation:
Department of Mathematics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia
MR Author ID:
936441
Email:
borism@math.hr
DOI:
https://doi.org/10.1090/tran/8125
Received by editor(s):
August 11, 2019
Received by editor(s) in revised form:
February 7, 2020
Published electronically:
July 8, 2020
Additional Notes:
The first author was supported in part by the U.S. National Science Foundation under grants DMS-1853340 and DMS-1613757. The second and third author were supported in part by the Croatian Science Foundation (Hrvatska zaklada za znanost) grant number IP-2018-01-3706 and by the Croatia-USA bilateral grant “Fluid-elastic structure interaction with the Navier slip boundary condition”.
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