Port-Hamiltonian modelling of fluid dynamics models with variable cross-section

Authors

Harshit Bansal, Hans Zwart, Laura Iapichino, Wil Schilders, Nathan van de Wouw

Abstract

Many single- and multi-phase fluid dynamical systems are governed by non-linear evolutionary equations. A key aspect of these systems is that the fluid typically flows across spatially and temporally varying cross-sections. We, first, show that not any choice of state-variables may be apt for obtaining a port-Hamiltonian realization under spatially varying cross-section. We propose a modified choice of the state-variables and then represent fluid dynamical systems in port-Hamiltonian representations. We define these port-Hamiltonian representations under spatial variation in the cross-section with respect to a new proposed state-dependent and extended Stokes- Dirac structure. Finally, we account for temporal variations in the cross-section and obtain a suitable structure that respects key properties, such as, for instance, the property of dissipation inequality.

Keywords

multi-phase; non-linear; evolutionary equations; varying cross-sections; port-Hamiltonian; Stokes-Dirac structure; dissipation inequality

Citation

• Journal: IFAC-PapersOnLine
• Year: 2021
• Volume: 54
• Issue: 9
• Pages: 365–372
• Publisher: Elsevier BV
• DOI: 10.1016/j.ifacol.2021.06.095
• Note: 24th International Symposium on Mathematical Theory of Networks and Systems MTNS 2020- Cambridge, United Kingdom

BibTeX

@article{Bansal_2021,
  title={{Port-Hamiltonian modelling of fluid dynamics models with variable cross-section}},
  volume={54},
  ISSN={2405-8963},
  DOI={10.1016/j.ifacol.2021.06.095},
  number={9},
  journal={IFAC-PapersOnLine},
  publisher={Elsevier BV},
  author={Bansal, Harshit and Zwart, Hans and Iapichino, Laura and Schilders, Wil and van de Wouw, Nathan},
  year={2021},
  pages={365--372}
}

Download the bib file

References

• Aarsnes, (2014)
• 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
• Bansal, H. et al. Port-Hamiltonian formulation of two-phase flow models. Systems & Control Letters vol. 149 104881 (2021) – 10.1016/j.sysconle.2021.104881
• Chaturantabut, S., Beattie, C. & Gugercin, S. Structure-Preserving Model Reduction for Nonlinear Port-Hamiltonian Systems. SIAM Journal on Scientific Computing vol. 38 B837–B865 (2016) – 10.1137/15m1055085
• 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
• Jacob, (2012)
• Kotyczka, Weak form of Stokes-Dirac structures and geometric discretization of port-Hamiltonian systems. Journal of Computational Physics (2018)
• Le Gorrec, Y., Zwart, H. & Maschke, B. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM Journal on Control and Optimization vol. 44 1864–1892 (2005) – 10.1137/040611677
• LeVeque, Finite Volume Methods for Hyperbolic Problems. (2002)
• Macchelli, A., van der Schaft, A. J. & Melchiorri, C. Port Hamiltonian formulation of infinite dimensional systems I. Modeling. 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601) 3762-3767 Vol.4 (2004) doi:10.1109/cdc.2004.1429324 – 10.1109/cdc.2004.1429324
• Martins, V. D. S., Maschke, B. & Gorrec, Y. L. Hamiltonian approach to the stabilization of systems of two conservation laws. IFAC Proceedings Volumes vol. 43 581–586 (2010) – 10.3182/20100901-3-it-2016.00095
• Maschke, B. M. & van der Schaft, A. J. Port-Controlled Hamiltonian Systems: Modelling Origins and Systemtheoretic Properties. IFAC Proceedings Volumes vol. 25 359–365 (1992) – 10.1016/s1474-6670(17)52308-3
• Maschke, B. M. & van der Schaft, A. J. FROM CONSERVATION LAWS TO PORT-HAMILTONIAN REPRESENTATIONS OF DISTRIBUTED-PARAMETER SYSTEMS. IFAC Proceedings Volumes vol. 38 483–488 (2005) – 10.3182/20050703-6-cz-1902.00735
• Matignon, D. & Hélie, T. A class of damping models preserving eigenspaces for linear conservative port-Hamiltonian systems. European Journal of Control vol. 19 486–494 (2013) – 10.1016/j.ejcon.2013.10.003
• Mehrmann, V. & Morandin, R. Structure-preserving discretization for port-Hamiltonian descriptor systems. 2019 IEEE 58th Conference on Decision and Control (CDC) 6863–6868 (2019) doi:10.1109/cdc40024.2019.9030180 – 10.1109/cdc40024.2019.9030180
• Lordejani, S. N. et al. Modeling and Numerical Implementation of Managed-Pressure-Drilling Systems for the Assessment of Pressure-Control Systems. SPE Drilling & Completion vol. 35 598–619 (2020) – 10.2118/201108-pa
• Pasumarthy, R. & van der Schaft, A. J. Achievable Casimirs and its implications on control of port-Hamiltonian systems. International Journal of Control vol. 80 1421–1438 (2007) – 10.1080/00207170701361273
• Sankar, D. S. Pulsatile Flow of a Two‐Fluid Model for Blood Flow through Arterial Stenosis. Mathematical Problems in Engineering vol. 2010 (2010) – 10.1155/2010/465835
• Trang VU, N. M., LEFEVRE, L. & MASCHKE, B. Port-Hamiltonian formulation for systems of conservation laws: application to plasma dynamics in Tokamak reactors. IFAC Proceedings Volumes vol. 45 108–113 (2012) – 10.3182/20120829-3-it-4022.00016
• 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
• van der Schaft, A. Port-Hamiltonian Modeling for Control. Annual Review of Control, Robotics, and Autonomous Systems vol. 3 393–416 (2020) – 10.1146/annurev-control-081219-092250
• van der Schaft, A. & Maschke, B. Generalized port-Hamiltonian DAE systems. Systems & Control Letters vol. 121 31–37 (2018) – 10.1016/j.sysconle.2018.09.008
• van der Schaft, A. & Maschke, B. Dirac and Lagrange Algebraic Constraints in Nonlinear Port-Hamiltonian Systems. Vietnam Journal of Mathematics vol. 48 929–939 (2020) – 10.1007/s10013-020-00419-x
• Zhou, W., Hamroun, B., Gorrec, Y. L. & Couenne, F. Infinite Dimensional Port Hamiltonian Representation of reaction diffusion processes. IFAC-PapersOnLine vol. 48 476–481 (2015) – 10.1016/j.ifacol.2015.05.119