Startseite Boundary update via resolvent for fluid–structure interaction
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Boundary update via resolvent for fluid–structure interaction

  • Martina Bukač EMAIL logo und Catalin Trenchea
Veröffentlicht/Copyright: 30. Juni 2020

Abstract

We propose a BOundary Update using Resolvent (BOUR) partitioned method, second-order accurate in time, unconditionally stable, for the interaction between a viscous incompressible fluid and a thin structure. The method is algorithmically similar to the sequential Backward Euler — Forward Euler implementation of the midpoint quadrature rule. (i) The structure and fluid sub-problems are first solved using a Backward Euler scheme, (ii) the velocities of fluid and structure are updated on the boundary via a second-order consistent resolvent operator, and then (iii) the structure and fluid sub-problems are solved again, using a Forward Euler scheme. The stability analysis based on energy estimates shows that the scheme is unconditionally stable. Error analysis of the semi-discrete problem yields second-order convergence in time. The two numerical examples confirm theoretical convergence analysis results and show an excellent agreement between the proposed partitioned scheme and the monolithic scheme.

JEL Classification: 65M12

Funding statement: The work of the first author was partially supported by the NSF under grants DMS 1912908, DMS 1619993, and DCSD 1934300. The work of the second author was partially supported by the AFOSR under grant FA 9550-16-1-0355 and the NSF under grant DMS 1522574.

Acknowledgment

We would like to thank Fasma Diele (Italian National Research Council, Bari) for helpful discussions on symplectic and geometric integration.

References

[1] S. Badia, F. Nobile, and C. Vergara, Fluid–structure partitioned procedures based on Robin transmission conditions, J.Comp. Phys., 227 (2008), 7027–7051.10.1016/j.jcp.2008.04.006Suche in Google Scholar

[2] S. Badia, F. Nobile, and C. Vergara, Robin-Robin preconditioned Krylov methods for fluid–structure interaction problems, Comput. Methods Appl. Mech. Engrg., 198 (2009), No. 33, 2768–2784.10.1016/j.cma.2009.04.004Suche in Google Scholar

[3] S. Badia, A. Quaini, and A. Quarteroni, Modular vs. non-modular preconditioners for fluid–structure systems with large added-mass effect, Comput. Methods Appl. Mech. Engrg., 197 (2008), No. 49, 4216–4232.10.1016/j.cma.2008.04.018Suche in Google Scholar

[4] H. Baek and G. Karniadakis, A convergence study of a new partitioned fluid–structure interaction algorithm based on fictitious mass and damping, J.Comp. Phys., 231 (2012), No. 2, 629–652.10.1016/j.jcp.2011.09.025Suche in Google Scholar

[5] J. Banks, W. Henshaw, and D. Schwendeman, An analysis of a new stable partitioned algorithm for FSI problems. Part I: Incompressible flow and elastic solids, J.Comp. Phys., 269 (2014), 108–137.10.1016/j.jcp.2014.03.006Suche in Google Scholar

[6] J. Banks, W. Henshaw, and D. Schwendeman, An analysis of a new stable partitioned algorithm for FSI problems. Part II: Incompressible flow and structural shells, J.Comp. Phys., 268 (2014), 399–416.10.1016/j.jcp.2014.03.004Suche in Google Scholar

[7] Y. Bazilevs, V.M. Calo, Hughes T.J.R, and Y. Zhang, Isogeometric fluid–structure interaction: theory algorithms and computations, Comput. Mech., 43 (2008), 3–37.10.1007/s00466-008-0315-xSuche in Google Scholar

[8] A. Bonito, R. Nochetto, and M. Pauletti, Dynamics of biomembranes: effect of the bulk fluid, Math. Model. Natur. Phenom., 6 (2011), No. 5, 25–43.10.1051/mmnp/20116502Suche in Google Scholar

[9] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.10.1007/978-0-387-70914-7Suche in Google Scholar

[10] M. Bukač, S. Čanić, R. Glowinski, J. Tambača, and A. Quaini, Fluid–structure interaction in blood flow capturing non-zero longitudinal structure displacement, J.Comp. Phys., 235 (2012), 515–541.10.1016/j.jcp.2012.08.033Suche in Google Scholar

[11] M. Bukac and B. Muha, Stability and convergence analysis of the extensions of the kinematically coupled scheme for the fluid–structure interaction, SIAM J. Numer. Anal, 54 (2016), No. 5, 3032–3061.10.1137/16M1055396Suche in Google Scholar

[12] M. Bukač, I. Yotov, and P. Zunino, An operator splitting approach for the interaction between a fluid and a multilayered poroelastic structure, Numer. Meth. Partial Difer. Equ., 31 (2015), No. 4, 1054–1100.10.1002/num.21936Suche in Google Scholar

[13] E. Burman and M. Fernández, Stabilization of explicit coupling in fluid–structure interaction involving fluid incompressibility, Comput. Methods Appl. Mech. Engrg., 198 (2009), 766–784.10.1016/j.cma.2008.10.012Suche in Google Scholar

[14] E. Burman and M. Fernández, An unfitted Nitsche method for incompressible fluid–structure interaction using overlapping meshes, Comput. Methods Appl. Mech. Engrg., 279 (2014), 497 – 514.10.1016/j.cma.2014.07.007Suche in Google Scholar

[15] P. Causin, J. F. Gerbeau, and F. Nobile, Added-mass effect in the design of partitioned algorithms for fluid–structure problems, Comput. Methods Appl. Mech. Engrg., 194 (2005), No. 42-44, 4506–4527.10.1016/j.cma.2004.12.005Suche in Google Scholar

[16] C. M. Colciago, S. Deparis, and A. Quarteroni, Comparisons between reduced order models and full 3D models for fluid–structure interaction problems in haemodynamics, J. Comp. Appl. Math., 265 (2014), 120–138.10.1016/j.cam.2013.09.049Suche in Google Scholar

[17] W. G Dettmer and D. Perić, A new staggered scheme for fluid–structure interaction, Int. J. Numer. Methods Engrg., 93 (2013), No. 1, 1–22.10.1002/nme.4370Suche in Google Scholar

[18] D. R. Durran, Numerical Methods for Fluid Dynamics with Applications to Geophysics, 2nd ed., Texts in Applied Mathematics, Vol. 32, Springer, New York, 2010.10.1007/978-1-4419-6412-0Suche in Google Scholar

[19] L. Failer and T. Wick, Adaptive time-step control for nonlinear fluid–structure interaction, J.Comp. Phys., 366 (2018), 448–477.10.1016/j.jcp.2018.04.021Suche in Google Scholar

[20] M. Fernández, Incremental displacement-correction schemes for incompressible fluid–structure interaction: stability and convergence analysis, Numerische Mathematik, 123 (2013), No. 1, 21–65.10.1007/s00211-012-0481-9Suche in Google Scholar

[21] M. Fernández, J-F. Gerbeau, and C. Grandmont, A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid, Int. J. Numer. Methods Engrg., 69 (2007), No. 4, 794–821.10.1002/nme.1792Suche in Google Scholar

[22] M. Fernández and J. Mullaert, Convergence and error analysis for a class of splitting schemes in incompressible fluid–structure interaction, IMA J. Numer. Anal., 36 (2016), No. 4, 1748–1782.10.1093/imanum/drv055Suche in Google Scholar

[23] A. Figueroa, S. Baek, C. Taylor, and J. Humphrey, A computational framework for fluid–solid-growth modeling in cardiovascular simulations, Comput. Methods Appl. Mech. Engrg., 198 (2009), No. 45, 3583–3602.10.1016/j.cma.2008.09.013Suche in Google Scholar PubMed PubMed Central

[24] C. Figueroa, I. Vignon-Clementel, K. Jansen, T. Hughes, and C. Taylor, A coupled momentum method for modeling blood flow in three-dimensional deformable arteries, Comput. Methods Appl. Mech. Engrg., 195 (2006), No. 41-43, 5685–5706.10.1016/j.cma.2005.11.011Suche in Google Scholar

[25] D. Forti, M. Bukac, A. Quaini, S. Canic, and S. Deparis, A Monolithic Approach to Fluid–Composite Structure Interaction, J. Sci. Comp., (2016), 1–26.10.1007/s10915-017-0363-5Suche in Google Scholar

[26] D. F. Griffiths and J. M. Sanz-Serna, On the scope of the method of modified equations, SIAM J. Sci. Stat. Comp., 7 (1986), No. 3, 994–1008.10.1137/0907067Suche in Google Scholar

[27] D. F. Griffiths and D. J. Higham, Numerical Methods for Ordinary Differential Equations. Initial Value Problems, Springer Undergraduate Mathematics Series, Springer-Verlag, 2010.10.1007/978-0-85729-148-6Suche in Google Scholar

[28] G. Guidoboni, R. Glowinski, N. Cavallini, and S. Čanić, Stable loosely-coupled-type algorithm for fluid–structure interaction in blood flow, J.Comp. Phys., 228 (2009), No. 18, 6916–6937.10.1016/j.jcp.2009.06.007Suche in Google Scholar

[29] A. Guzel and W. Layton, Time filters increase accuracy of the fully implicit method, BIT Numer. Math., 58 (2018), No. 2, 301–315.10.1007/s10543-018-0695-zSuche in Google Scholar

[30] A. Guzel and C. Trenchea, The Williams step increases the stability and accuracy of the hoRA time filter, Appl. Numer. Math., 131 (2018), 158–173.10.1016/j.apnum.2018.05.003Suche in Google Scholar

[31] A. Guzel and C. Trenchea, The Williams step increases the stability and accuracy of the hoRA time filter, Appl. Numer. Math., 131 (2018), 158–173.10.1016/j.apnum.2018.05.003Suche in Google Scholar

[32] E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, Vol. 31, Springer, Heidelberg, 2010.Suche in Google Scholar

[33] M. Heil, A. Hazel, and J. Boyle, Solvers for large-displacement fluid–structure interaction problems: segregated versus monolithic approaches, Comput. Mech., 43 (2008), No. 1, 91–101.10.1007/s00466-008-0270-6Suche in Google Scholar

[34] W. Hundsdorfer and J. Verwer, Numerical Solution of Time-Dependent Advection–Diffusion–Reaction Equations, Springer Series in Computational Mathematics, Vol. 33, Springer-Verlag, Berlin, 2003.10.1007/978-3-662-09017-6Suche in Google Scholar

[35] W. Layton, Y. Li, and C. Trenchea, Recent developments in IMEX methods with time filters for systems of evolution equations, J. Comp. Appl. Math., 299 (2016), 50–67.10.1016/j.cam.2015.09.038Suche in Google Scholar

[36] R. J. LeVeque, Numerical Methods for Conservation Laws, 2-nd ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992.10.1007/978-3-0348-8629-1Suche in Google Scholar

[37] M Lukáčová-Medvid'ová, G Rusnáková, and A Hundertmark-Zaušková, Kinematic splitting algorithm for fluid–structure interaction in hemodynamics, Comput. Methods Appl. Mech. Engrg., 265 (2013), 83–106.10.1016/j.cma.2013.05.025Suche in Google Scholar

[38] O. Oyekole, C. Trenchea, and M. Bukac, A Second-Order in Time Approximation of Fluid-Structure Interaction Problem, SIAM J. Numer. Anal, 56 (2018), No. 1, 590–613.10.1137/17M1140054Suche in Google Scholar

[39] C. S. Peskin, Numerical analysis of blood flow in the heart, J.Comp. Phys., 25 (1977), No. 3, 220–252.10.1016/0021-9991(77)90100-0Suche in Google Scholar

[40] A. Quaini and A Quarteroni, A semi-implicit approach for fluid–structure interaction based on an algebraic fractional step method, Math. Models Methods Appl. Sci., 17 (2007), No. 6, 957–985.10.1142/S0218202507002170Suche in Google Scholar

[41] R. D. Ruth, A canonical integration technique, IEEE Trans. Nucl. Sci. (1983), 2669–2671.10.1109/TNS.1983.4332919Suche in Google Scholar

[42] R. Van Loon, P. D. Anderson, J. De Hart, and F. P. T. Baaijens, A combined fictitious domain/adaptive meshing method for fluid–structure interaction in heart valves, Int. J. Numer. Methods Fluids, 46 (2004), No. 5, 533–544.10.1002/fld.775Suche in Google Scholar

[43] Z. Wang, N. Wood, and X. Xu, A viscoelastic fluid–structure interaction model for carotid arteries under pulsatile flow, J. Numer. Methods Biomed. Engrg, 31 (2015), No. 5.10.1002/cnm.2709Suche in Google Scholar

[44] R. F. Warming and B. J. Hyett, The modified equation approach to the stability and accuracy analysis of finite-difference methods, J.Comp. Phys., 14 (1974), 159–179.10.1016/0021-9991(74)90011-4Suche in Google Scholar

[45] Y. Yu, H. Baek, and G. Karniadakis, Generalized fictitious methods for fluid–structure interactions: analysis and simulations, J.Comp. Phys., 245 (2013), 317–346.10.1016/j.jcp.2013.03.025Suche in Google Scholar

Published Online: 2020-06-30
Published in Print: 2021-03-26

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 26.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/jnma-2019-0081/pdf
Button zum nach oben scrollen