Fluid-Structure Port-Hamiltonian Model for Incompressible Flows in Tubes with Time Varying Geometries

Authors

Luis A. Mora, Le Gorrec Yann, Hector Ramirez, Juan Yuz

Abstract

A simple and scalable finite-dimensional model based on the port-Hamiltonian framework is proposed to describe the fluid–structure interaction in tubes with time-varying geometries. For this purpose, the moving tube wall is described by a set of mass-spring-damper systems while the fluid is considered as a one-dimensional incompressible flow described by the average momentum dynamics in a set of incompressible flow sections. To couple these flow sections small compressible volumes are defined to describe the pressure between two adjacent fluid sections. The fluid-structure coupling is done through a power-preserving interconnection between velocities and forces. The resultant model includes external inputs for the fluid and inputs for external forces over the mechanical part that can be used for control or interconnection purposes. Numerical examples show the accordance of this simplified model with finite-element models reported in the literature.

Citation

• Journal: Mathematical and Computer Modelling of Dynamical Systems
• Year: 2020
• Volume: 26
• Issue: 5
• Pages: 409–433
• Publisher: Informa UK Limited
• DOI: 10.1080/13873954.2020.1786841

BibTeX

@article{Mora_2020,
  title={{Fluid-Structure Port-Hamiltonian Model for Incompressible Flows in Tubes with Time Varying Geometries}},
  volume={26},
  ISSN={1744-5051},
  DOI={10.1080/13873954.2020.1786841},
  number={5},
  journal={Mathematical and Computer Modelling of Dynamical Systems},
  publisher={Informa UK Limited},
  author={Mora, Luis A. and Yann, Le Gorrec and Ramirez, Hector and Yuz, Juan},
  year={2020},
  pages={409--433}
}

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References

• Fluid-Structure Interaction. Lecture Notes in Computational Science and Engineering (Springer Berlin Heidelberg, 2006). doi:10.1007/3-540-34596-5 – 10.1007/3-540-34596-5
• Bukač, M., Čanić, S., Glowinski, R., Muha, B. & Quaini, A. A modular, operator‐splitting scheme for fluid–structure interaction problems with thick structures. International Journal for Numerical Methods in Fluids vol. 74 577–604 (2013) – 10.1002/fld.3863
• Bukač, M., Čanić, S. & Muha, B. A partitioned scheme for fluid–composite structure interaction problems. Journal of Computational Physics vol. 281 493–517 (2015) – 10.1016/j.jcp.2014.10.045
• Ghigo, A. R., Fullana, J.-M. & Lagrée, P.-Y. A 2D nonlinear multiring model for blood flow in large elastic arteries. Journal of Computational Physics vol. 350 136–165 (2017) – 10.1016/j.jcp.2017.08.039
• Chouly, F., Van Hirtum, A., Lagrée, P.-Y., Pelorson, X. & Payan, Y. Numerical and experimental study of expiratory flow in the case of major upper airway obstructions with fluid–structure interaction. Journal of Fluids and Structures vol. 24 250–269 (2008) – 10.1016/j.jfluidstructs.2007.08.004
• Rasani, M. R., Inthavong, K. & Tu, J. Y. Three-Dimensional Fluid-Structure Interaction Modeling of Expiratory Flow in the Pharyngeal Airway. IFMBE Proceedings 467–471 (2011) doi:10.1007/978-3-642-21729-6_118 – 10.1007/978-3-642-21729-6_118
• Thomson, S. L., Mongeau, L. & Frankel, S. H. Aerodynamic transfer of energy to the vocal folds. The Journal of the Acoustical Society of America vol. 118 1689–1700 (2005) – 10.1121/1.2000787
• Šidlof, P., Horáček, J. & Řidký, V. Parallel CFD simulation of flow in a 3D model of vibrating human vocal folds. Computers & Fluids vol. 80 290–300 (2013) – 10.1016/j.compfluid.2012.02.005
• Jiang, W., Zheng, X. & Xue, Q. Computational Modeling of Fluid–Structure–Acoustics Interaction during Voice Production. Frontiers in Bioengineering and Biotechnology vol. 5 (2017) – 10.3389/fbioe.2017.00007
• Donea, J., Huerta, A., Ponthot, J. ‐Ph. & Rodríguez‐Ferran, A. ArbitraryLagrangian–Eulerian Methods. Encyclopedia of Computational Mechanics (2004) doi:10.1002/0470091355.ecm009 – 10.1002/0470091355.ecm009
• Wong, K. K. L., Thavornpattanapong, P., Cheung, S. C. P. & Tu, J. Numerical Stability of Partitioned Approach in Fluid-Structure Interaction for a Deformable Thin-Walled Vessel. Computational and Mathematical Methods in Medicine vol. 2013 1–10 (2013) – 10.1155/2013/638519
• Causin, P., Gerbeau, J. F. & Nobile, F. Added-mass effect in the design of partitioned algorithms for fluid–structure problems. Computer Methods in Applied Mechanics and Engineering vol. 194 4506–4527 (2005) – 10.1016/j.cma.2004.12.005
• Lozovskiy, A., Olshanskii, M. A. & Vassilevski, Y. V. Analysis and assessment of a monolithic FSI finite element method. Computers & Fluids vol. 179 277–288 (2019) – 10.1016/j.compfluid.2018.11.004
• Papadakis, G. Coupling 3D and 1D fluid–structure‐interaction models for wave propagation in flexible vessels using a finite volume pressure‐correction scheme. Communications in Numerical Methods in Engineering vol. 25 533–551 (2009) – 10.1002/cnm.1212
• John, V. Finite Element Methods for Incompressible Flow Problems. Springer Series in Computational Mathematics (Springer International Publishing, 2016). doi:10.1007/978-3-319-45750-5 – 10.1007/978-3-319-45750-5
• Mora, L. A., Ramírez, H., Yuz, J. I. & Gorrec, Y. L. A Scalable port-Hamiltonian Model for Incompressible Fluids in Irregular Geometries. IFAC-PapersOnLine vol. 52 102–107 (2019) – 10.1016/j.ifacol.2019.08.018
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
• Kotyczka P.. Proceedings of the 8th International Workshop on Multidimensional Systems (nDS13), VDE (2013)
• Altmann, R. & Schulze, P. A port-Hamiltonian formulation of the Navier–Stokes equations for reactive flows. Systems & Control Letters vol. 100 51–55 (2017) – 10.1016/j.sysconle.2016.12.005
• Sch M.. 1889–1896 (2010)
• Cardoso-Ribeiro, F. L., Matignon, D. & Pommier-Budinger, V. A port-Hamiltonian model of liquid sloshing in moving containers and application to a fluid-structure system. Journal of Fluids and Structures vol. 69 402–427 (2017) – 10.1016/j.jfluidstructs.2016.12.007
• Duindam, V., Macchelli, A., Stramigioli, S. & Bruyninckx, H. Modeling and Control of Complex Physical Systems. (Springer Berlin Heidelberg, 2009). doi:10.1007/978-3-642-03196-0 – 10.1007/978-3-642-03196-0
• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. (2014) doi:10.1561/9781601987877 – 10.1561/9781601987877
• Olver, P. J. Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics (Springer New York, 1986). doi:10.1007/978-1-4684-0274-2 – 10.1007/978-1-4684-0274-2
• Bird R.B.. Introductory Transport Phenomena (2015)
• P. M. Gresho and R. L. Sani, Incompressible Flow and the Finite Element Method, Volume 1: Advection-Diffusion and Isothermal Laminar Flow, John Wiley & Sons, Inc,  New York, USA, 1998.
• Panton, R. L. Incompressible Flow. (2013) doi:10.1002/9781118713075 – 10.1002/9781118713075
• Mulley, R. Flow of Industrial Fluids. (2004) doi:10.1201/9781420038286 – 10.1201/9781420038286
• Pérez-García, J., Sanmiguel-Rojas, E. & Viedma, A. New coefficient to characterize energy losses in compressible flow at T-junctions. Applied Mathematical Modelling vol. 34 4289–4305 (2010) – 10.1016/j.apm.2010.05.005
• R. Brodkey and H. Hershey, Transport Phenomena: A Unified Approach, Chemical Engineering Series, McGraw Hill International, New York, 1988.
• Murdock, J. W. Fundamental Fluid Mechanics for the Practicing Engineer. (CRC Press, 2018). doi:10.1201/9781315274065 – 10.1201/9781315274065
• Degroote, J., Haelterman, R., Annerel, S., Bruggeman, P. & Vierendeels, J. Performance of partitioned procedures in fluid–structure interaction. Computers & Structures vol. 88 446–457 (2010) – 10.1016/j.compstruc.2009.12.006