Analysis of a 3D nonlinear, moving boundary problem describing fluid-mesh-shell interaction

Čanić, Sunčica; Galić, Marija; Muha, Boris

doi:10.1090/tran/8125

Analysis of a 3D nonlinear, moving boundary problem describing fluid-mesh-shell interaction
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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.

References

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
Stuart S. Antman, Nonlinear problems of elasticity, 2nd ed., Applied Mathematical Sciences, vol. 107, Springer, New York, 2005. MR 2132247
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, DOI 10.1515/GMJ.2008.403
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, DOI 10.3934/dcdss.2009.2.417
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, DOI 10.1090/conm/440/08476
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, DOI 10.1512/iumj.2008.57.3284
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, DOI 10.1007/s00021-003-0082-5
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, DOI 10.1016/S1631-073X(03)00235-8
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, DOI 10.1016/j.matpur.2005.08.004
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, DOI 10.1007/s00245-016-9343-7
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.
Sunčica Čanić, New mathematics for next-generation stent design, SIAM News 52 (2019), no. 3, 1, 3. MR 3967610
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, DOI 10.1007/s00033-019-1087-1
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, DOI 10.1137/16M1088181
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, DOI 10.1093/imamat/hxs047
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, DOI 10.1137/140991509
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, DOI 10.1007/s00021-004-0121-y
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, DOI 10.1137/060656085
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, DOI 10.1137/080741628
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, DOI 10.3934/eect.2016021
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, DOI 10.1007/s11253-013-0771-0
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, DOI 10.1007/978-3-642-36062-6_{3}3
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
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
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, DOI 10.1142/S0252959901000139
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, DOI 10.4099/math1924.20.279
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, DOI 10.1016/S0764-4442(99)80193-1
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, DOI 10.1007/s00205-004-0340-7
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, DOI 10.1007/s00205-005-0385-2
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, DOI 10.1007/s002050050136
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, DOI 10.5209/rev_{R}EMA.2001.v14.n2.17030
J. Donea, Arbitrary Lagrangian-Eulerian finite element methods, Computational methods for transient analysis North-Holland, Amsterdam, 1983.
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, DOI 10.3934/dcds.2003.9.633
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, DOI 10.1007/s00028-003-0110-1
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, DOI 10.2140/pjm.2006.223.251
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
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, DOI 10.1007/978-3-7643-7806-6_{3}
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, DOI 10.1007/978-0-387-09620-9
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
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, DOI 10.1137/070699196
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, DOI 10.1007/s00205-015-0954-y
Georges Griso, Asymptotic behavior of structures made of curved rods, Anal. Appl. (Singap.) 6 (2008), no. 1, 11–22. MR 2380884, DOI 10.1142/S0219530508001031
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, DOI 10.1016/j.jcp.2009.06.007
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, DOI 10.1016/0045-7825(81)90049-9
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, DOI 10.1142/S0218202599000452
Mladen Jurak and Josip Tambača, Linear curved rod model. General curve, Math. Models Methods Appl. Sci. 11 (2001), no. 7, 1237–1252. MR 1848199, DOI 10.1142/S0218202501001318
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
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, DOI 10.3934/dcds.2012.32.1355
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
Daniel Lengeler and Michael Růžička, 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, DOI 10.1007/s00205-013-0686-9
Julien Lequeurre, Existence of strong solutions to a fluid-structure system, SIAM J. Math. Anal. 43 (2011), no. 1, 389–410. MR 2765696, DOI 10.1137/10078983X
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, DOI 10.1007/s00205-012-0585-5
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, DOI 10.4310/CIS.2013.v13.n3.a4
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, DOI 10.1016/j.jde.2013.09.016
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, DOI 10.4171/IFB/350
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, DOI 10.1016/j.jde.2016.02.029
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, DOI 10.1016/j.jde.2018.12.030
Michael Růžička, 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
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, DOI 10.1137/080722618
Igor Velčić, Nonlinear weakly curved rod by $\Gamma$-convergence, J. Elasticity 108 (2012), no. 2, 125–150. MR 2948053, DOI 10.1007/s10659-011-9358-x