Port-Hamiltonian formulations of poroelastic network models

Authors

R. Altmann, V. Mehrmann, B. Unger

Abstract

ABSTRACT We investigate an energy-based formulation of the two-field poroelasticity model and the related multiple-network model as they appear in geosciences or medical applications. We propose a port-Hamiltonian formulation of the system equations, which is beneficial for preserving important system properties after discretization or model-order reduction. For this, we include the commonly omitted second-order term and consider the corresponding first-order formulation. The port-Hamiltonian formulation of the quasi-static case is then obtained by (formally) setting the second-order term zero. Further, we interpret the poroelastic equations as an interconnection of a network of submodels with internal energies, adding a control-theoretic understanding of the poroelastic equations.

Citation

• Journal: Mathematical and Computer Modelling of Dynamical Systems
• Year: 2021
• Volume: 27
• Issue: 1
• Pages: 429–452
• Publisher: Informa UK Limited
• DOI: 10.1080/13873954.2021.1975137

BibTeX

@article{Altmann_2021,
  title={{Port-Hamiltonian formulations of poroelastic network models}},
  volume={27},
  ISSN={1744-5051},
  DOI={10.1080/13873954.2021.1975137},
  number={1},
  journal={Mathematical and Computer Modelling of Dynamical Systems},
  publisher={Informa UK Limited},
  author={Altmann, R. and Mehrmann, V. and Unger, B.},
  year={2021},
  pages={429--452}
}

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References

• 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
• Rashad, R., Califano, F., van der Schaft, A. J. & Stramigioli, S. Twenty years of distributed port-Hamiltonian systems: a literature review. IMA Journal of Mathematical Control and Information vol. 37 1400–1422 (2020) – 10.1093/imamci/dnaa018
• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. (2014) doi:10.1561/9781601987877 – 10.1561/9781601987877
• Egger, H. Structure preserving approximation of dissipative evolution problems. Numerische Mathematik vol. 143 85–106 (2019) – 10.1007/s00211-019-01050-w
• Serhani, A., Matignon, D. & Haine, G. A Partitioned Finite Element Method for the Structure-Preserving Discretization of Damped Infinite-Dimensional Port-Hamiltonian Systems with Boundary Control. Lecture Notes in Computer Science 549–558 (2019) doi:10.1007/978-3-030-26980-7_57 – 10.1007/978-3-030-26980-7_57
• Beattie, C. & Gugercin, S. Structure-preserving model reduction for nonlinear port-Hamiltonian systems. IEEE Conference on Decision and Control and European Control Conference 6564–6569 (2011) doi:10.1109/cdc.2011.6161504 – 10.1109/cdc.2011.6161504
• 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
• Ciarlet P.G., Mathematical Elasticity. Vol. I (1988)
• Gugercin S., Proceedings 48th IEEE Conference on Decision and Control (2009)
• Polyuga, R. V. & van der Schaft, A. Structure Preserving Moment Matching for Port-Hamiltonian Systems: Arnoldi and Lanczos. IEEE Transactions on Automatic Control vol. 56 1458–1462 (2011) – 10.1109/tac.2011.2128650
• Wolf, T., Lohmann, B., Eid, R. & Kotyczka, P. Passivity and Structure Preserving Order Reduction of Linear Port-Hamiltonian Systems Using Krylov Subspaces. European Journal of Control vol. 16 401–406 (2010) – 10.3166/ejc.16.401-406
• Kotyczka, P. & Lefèvre, L. Discrete-time port-Hamiltonian systems based on Gauss-Legendre collocation. IFAC-PapersOnLine vol. 51 125–130 (2018) – 10.1016/j.ifacol.2018.06.035
• 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
• Beattie, C., Mehrmann, V., Xu, H. & Zwart, H. Linear port-Hamiltonian descriptor systems. Mathematics of Control, Signals, and Systems vol. 30 (2018) – 10.1007/s00498-018-0223-3
• van der Schaft, A. J. Port-Hamiltonian Differential-Algebraic Systems. Surveys in Differential-Algebraic Equations I 173–226 (2013) doi:10.1007/978-3-642-34928-7_5 – 10.1007/978-3-642-34928-7_5
• 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
• 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
• Jacob, B. & Zwart, H. J. Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces. (Springer Basel, 2012). doi:10.1007/978-3-0348-0399-1 – 10.1007/978-3-0348-0399-1
• Kato, T. Perturbation Theory for Linear Operators. Classics in Mathematics (Springer Berlin Heidelberg, 1995). doi:10.1007/978-3-642-66282-9 – 10.1007/978-3-642-66282-9
• Lions J.-L., Non-homogeneous Boundary Value Problems and Applications. Vol. I (1972)
• 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
• Biot, M. A. General Theory of Three-Dimensional Consolidation. Journal of Applied Physics vol. 12 155–164 (1941) – 10.1063/1.1712886
• DETOURNAY, E. & CHENG, A. H.-D. Fundamentals of Poroelasticity. Analysis and Design Methods 113–171 (1993) doi:10.1016/b978-0-08-040615-2.50011-3 – 10.1016/b978-0-08-040615-2.50011-3
• Egger H., Math. Comput. Simulat.
• Showalter, R. E. Diffusion in Poro-Elastic Media. Journal of Mathematical Analysis and Applications vol. 251 310–340 (2000) – 10.1006/jmaa.2000.7048
• Altmann, R., Maier, R. & Unger, B. Semi-explicit discretization schemes for weakly coupled elliptic-parabolic problems. Mathematics of Computation vol. 90 1089–1118 (2021) – 10.1090/mcom/3608
• Zeidler E., Nonlinear Functional Analysis and Its Applications IIa: Linear Monotone Operators (1990)
• Zoback M.D., Reservoir Geomechanics (2010)
• Robert Altmann, R. A., Eric T. Chung, E. T. C., Roland Maier, R. M., Daniel Peterseim, D. P. & Sai-Mang Pun, S.-M. P. Computational Multiscale Methods for Linear Heterogeneous Poroelasticity. Journal of Computational Mathematics vol. 38 41–57 (2020) – 10.4208/jcm.1902-m2018-0186
• Brown, D. L. & Vasilyeva, M. A Generalized Multiscale Finite Element Method for poroelasticity problems I: Linear problems. Journal of Computational and Applied Mathematics vol. 294 372–388 (2016) – 10.1016/j.cam.2015.08.007
• Fu, S. et al. Computational multiscale methods for linear poroelasticity with high contrast. Journal of Computational Physics vol. 395 286–297 (2019) – 10.1016/j.jcp.2019.06.027
• Biot, M. A. Thermoelasticity and Irreversible Thermodynamics. Journal of Applied Physics vol. 27 240–253 (1956) – 10.1063/1.1722351
• Målqvist, A. & Persson, A. A generalized finite element method for linear thermoelasticity. ESAIM: Mathematical Modelling and Numerical Analysis vol. 51 1145–1171 (2017) – 10.1051/m2an/2016054
• Sobey I., Int. J. Numer. Anal. Model., Series B (2012)
• Vardakis, J. C. et al. Investigating cerebral oedema using poroelasticity. Medical Engineering & Physics vol. 38 48–57 (2016) – 10.1016/j.medengphy.2015.09.006
• Carman, P. C. Permeability of saturated sands, soils and clays. The Journal of Agricultural Science vol. 29 262–273 (1939) – 10.1017/s0021859600051789
• Cao Y., Discrete Cont. Dyn.-B (2013)
• Altmann, R., Maier, R. & Unger, B. A semi‐explicit integration scheme for weakly‐coupled poroelasticity with nonlinear permeability. PAMM vol. 20 (2021) – 10.1002/pamm.202000061
• TULLY, B. & VENTIKOS, Y. Cerebral water transport using multiple-network poroelastic theory: application to normal pressure hydrocephalus. Journal of Fluid Mechanics vol. 667 188–215 (2010) – 10.1017/s0022112010004428
• Egger, H., Kugler, T., Liljegren-Sailer, B., Marheineke, N. & Mehrmann, V. On Structure-Preserving Model Reduction for Damped Wave Propagation in Transport Networks. SIAM Journal on Scientific Computing vol. 40 A331–A365 (2018) – 10.1137/17m1125303
• Simeon B., Numerische Simulation Gekoppelter Systeme von Partiellen und Differential-algebraischen Gleichungen der Mehrkörperdynamik (2000)
• Brenan K.E., Numerical Solution of Initial-value Problems in Differential-algebraic Equations (1996)
• Kunkel, P. & Mehrmann, V. Differential-Algebraic Equations. EMS Textbooks in Mathematics (2006) doi:10.4171/017 – 10.4171/017
• Bunse-Gerstner, A., Byers, R., Mehrmann, V. & Nichols, N. K. Feedback design for regularizing descriptor systems. Linear Algebra and its Applications vol. 299 119–151 (1999) – 10.1016/s0024-3795(99)00167-6
• Singular Control Systems. Lecture Notes in Control and Information Sciences (Springer-Verlag, 1989). doi:10.1007/bfb0002475 – 10.1007/bfb0002475
• Mehl C., Linear Algebra Appl. (2020)