A Port-Hamiltonian formulation of linear thermoelasticity and its mixed finite element discretization

References

• R. Rashad, F. Califano, A. J. van der Schaft, and S. Stramigioli, “Twenty years of distributed port-Hamiltonian systems: A literature review,” IMA J. Math. Control Inform., vol. 37, no. 4, pp. 1400–1422, 2020. DOI: 10.1093/imamci/dnaa018.
   Web of Science ®Google Scholar
• M. Kurula, H. Zwart, A. J. van der Schaft, and J. Behrndt, “Dirac structures and their composition on Hilbert spaces,” J. Math. Anal. Appl., vol. 372, no. 2, pp. 402–422, 2010. DOI: 10.1016/j.jmaa.2010.07.004.
   Web of Science ®Google Scholar
• A. Macchelli and C. Melchiorri, “Modeling and control of the Timoshenko beam. The distributed port Hamiltonian approach,” SIAM J. Control Optim., vol. 43, no. 2, pp. 743–767, 2004. DOI: 10.1137/S0363012903429530.
   Web of Science ®Google Scholar
• S. Aoues, F. L. Cardoso-Ribeiro, D. Matignon, and D. Alazard, “Modeling and control of a rotating flexible spacecraft: A port-Hamiltonian approach,” IEEE Trans. Contr. Syst. Technol., vol. 27, no. 1, pp. 355–362, 2019. DOI: 10.1109/TCST.2017.2771244.
   Web of Science ®Google Scholar
• A. Brugnoli, D. Alazard, V. Pommier-Budinger, and D. Matignon, “Port-Hamiltonian formulation and symplectic discretization of plate models. Part I: Mindlin model for thick plates,” Appl. Math. Modell., vol. 75, pp. 940–960, Nov. 2019. DOI: 10.1016/j.apm.2019.04.035.
   Web of Science ®Google Scholar
• A. Brugnoli, D. Alazard, V. Pommier-Budinger, and D. Matignon, “Port-Hamiltonian formulation and symplectic discretization of plate models. Part II: Kirchhoff model for thin plates,” Appl. Math. Modell., vol. 75, 961–981, Nov. 2019. DOI: 10.1016/j.apm.2019.04.036.
   Web of Science ®Google Scholar
• P. Kotyczka, “Structured discretization of the heat equation: Numerical properties and preservation of flatness,” in 23rd MTNS, Hong Kong, Jul. 16–20, 2018.
   Google Scholar
• P. Kotyczka, Numerical Methods for Distributed Parameter Port-Hamiltonian Systems, Technical University of Munich: TUM University Press, 2019.
   Google Scholar
• F. L. Cardoso-Ribeiro, D. Matignon, and V. Pommier-Budinger, “Port-Hamiltonian model of two-dimensional shallow water equations in moving containers,” IMA J. Math. Control Inform., vol. 37, no. 4, pp. 1348–1366, Jul. 2020. DOI: 10.1093/imamci/dnaa016.
   Web of Science ®Google Scholar
• R. Rashad, F. Califano, F. P. Schuller, and S. Stramigioli, “Port-Hamiltonian modeling of ideal fluid flow: Part I. Foundations and kinetic energy,” Journal of Geometry and Physics, pp. 104201, 2021a. DOI: 10.1016/j.geomphys. 2021.104201.
   Google Scholar
• R. Rashad, F. Califano, F. P. Schuller, and S. Stramigioli, “Port-Hamiltonian modeling of ideal fluid flow: Part II. Compressible and incompressible flow,” Journal of Geometry and Physics, pp. 104199, 2021b. DOI: 10.1016/j.geomphys.2021.104199.
   Google Scholar
• R. Altmann and P. Schulze, “A Port-Hamiltonian formulation of the Navier–Stokes equations for reactive flows,” Syst. Control Lett., vol. 100, pp. 51–55, 2017. DOI: 10.1016/j.sysconle.2016.12.005.
   Web of Science ®Google Scholar
• F. L. Cardoso-Ribeiro, D. Matignon, and L. Lefèvre, “A partitioned finite element method for power-preserving discretization of open systems of conservation laws,” IMA J. Math. Control Inform., vol. 12, 2020. DOI: 10.1093/imamci/dnaa038.
   Google Scholar
• R. C. Kirby and T. T. Kieu, “Symplectic-mixed finite element approximation of linear acoustic wave equations,” Numer. Math., vol. 130, no. 2, pp. 257–291, Jun. 2015. DOI: 10.1007/s00211-014-0667-4.
   Web of Science ®Google Scholar
• G. Avalos and I. Lasiecka, “Boundary controllability of thermoelastic plates via the free boundary conditions,” SIAM J. Control Optim., vol. 38, no. 2, pp. 337–383, 2000. DOI: 10.1137/S0363012998339836.
   Web of Science ®Google Scholar
• S. W. Hansen and B. Y. Zhang, “Boundary control of a linear thermoelastic beam,” J. Math. Anal. Appl., vol. 210, no. 1, pp. 182–205, 1997. DOI: 10.1006/jmaa.1997.5437.
   Web of Science ®Google Scholar
• M. A. Biot, “Thermoelasticity and irreversible thermodynamics,” J. Appl. Phys., vol. 27, no. 3, pp. 240–253, 1956. DOI: 10.1063/1.1722351.
   Web of Science ®Google Scholar
• D. E. Carlson, “Linear thermoelasticity,” in Linear Theories of Elasticity and Thermoelasticity: Linear and Nonlinear Theories of Rods, Plates, and Shells, C. Truesdell, Ed. Berlin, Heidelberg: Springer, 1973, pp. 297–345. DOI: 10.1007/978-3-662-39776-3_.2.
   Google Scholar
• R. B. Hetnarski and M. R. Eslami, Thermal Stresses: Advanced Theory and Applications, vol. 158. Springer Netherlands, Dordrecht: Springer, 2009.
   Google Scholar
• R. Abeyaratne, Lecture Notes on the Mechanics of Elastic Solids. Volume II: Continuum Mechanics, 1st ed., Cambridge, MA: American Society of Mechanical Engineers, 2012.
   Google Scholar
• C. Beattie, V. Mehrmann, H. Xu, and H. Zwart, “Linear port-Hamiltonian descriptor systems,” Math. Control Signals Syst., vol. 30, no. 4, p. 17, 2018. DOI: 10.1007/s00498-018-0223-3.
   Google Scholar
• V. Mehrmann and R. Morandin, “Structure-preserving discretization for port-Hamiltonian descriptor systems,” in 2019 IEEE 58th Conference on Decision and Control (CDC), Nice (France), 2019, pp. 6863–6868. DOI: 10.1109/CDC40024.2019.
   Google Scholar
• V. Duindam, A. Macchelli, S. Stramigioli, and H. Bruyninckx, Modeling and Control of Complex Physical Systems, Springer-Verlag Berlin Heidelberg, 2009.
   Google Scholar
• A. Serhani, G. Haine, and D. Matignon, “Anisotropic heterogeneous n-D heat equation with boundary control and observation: I. Modeling as port-Hamiltonian system,” IFAC-PapersOnLine, vol. 52, no. 7, pp. 51–56, 2019. 3rd IFAC Workshop on Thermodynamic Foundations for a Mathematical Systems Theory TFMST 2019. DOI: 10.1016/j.ifacol.2019.07.009.
   Google Scholar
• A. Serhani, G. Haine, and D. Matignon, “Anisotropic heterogeneous n-D heat equation with boundary control and observation: II. Structure-preserving discretization,” IFAC-PapersOnLine, vol. 52, no. 7, pp. 57–62, 2019. 3rd IFAC Workshop on Thermodynamic Foundations for a Mathematical Systems Theory TFMST 2019. DOI: 10.1016/j.ifacol.2019.07.010.
   Google Scholar
• A. Brugnoli, “A Port-Hamiltonian formulation of flexible structures. Modelling and structure-preserving finite element discretization,” Ph.D. thesis, Université de Toulouse, ISAE-SUPAERO, France, 2020.
   Google Scholar
• P. Chadwick, “On the propagation of thermoelastic disturbances in thin plates and rods,” J. Mech. Phys. Solids, vol. 10, no. 2, pp. 99–109, 1962. DOI: 10.1016/0022-5096(62)90013-3.
   Web of Science ®Google Scholar
• R. Lifshitz and M. L. Roukes, “Thermoelastic damping in micro- and nanomechanical systems,” Phys. Rev. B, vol. 61, no. 8, pp. 5600–5609, 2000. DOI: 10.1103/PhysRevB.61.5600.
   Web of Science ®Google Scholar
• J. E. Lagnese, Boundary Stabilization of Thin Plates, Society for Industrial and Applied Mathematics, University City, Philadelphia, 1989. DOI: 10.1137/1.9781611970821.
   Google Scholar
• J. G. Simmonds, “ Major simplifications in a current linear model for the motion of a thermoelastic plate,” Q. Appl. Math., vol. 57, no. 4, pp. 673–679,1999. DOI: 10.1090/qam/1724299.
   Web of Science ®Google Scholar
• A. N. Norris, “Dynamics of thermoelastic thin plates: A comparison of four theories,” J. Therm. Stress., vol. 29, no. 2, pp. 169–195, 2006. DOI: 10.1080/01495730500257482.
   Web of Science ®Google Scholar
• W. Nowacki, Thermoelasticity, Oxford, UK: Pergamon Press, 1962.
   Google Scholar
• G. Cohen and S. Fauqueux, “Mixed spectral finite elements for the linear elasticity system in unbounded domains,” SIAM J. Sci. Comput., vol. 26, no. 3, pp. 864–884, 2005. DOI: 10.1137/S1064827502407457.
   Web of Science ®Google Scholar
• Z. Weng, X. Feng, and P. Huang, “A new mixed finite element method based on the Crank–Nicolson scheme for the parabolic problems,” Appl. Math. Modell., vol. 36, no. 10, pp. 5068–5079, 2012. DOI: 10.1016/j.apm.2011.12.044.
   Web of Science ®Google Scholar
• D. Arnold and J. Lee, “Mixed methods for elastodynamics with weak symmetry,” SIAM J. Numer. Anal., vol. 52, no. 6, pp. 2743–2769, 2014. DOI: 10.1137/13095032X.
   Web of Science ®Google Scholar
• S. Memon, N. Nataraj, and A. K. Pani, “An a posteriori error analysis of mixed finite element Galerkin approximations to second order linear parabolic problems,” SIAM J. Numer. Anal., vol. 50, no. 3, pp. 1367–1393, 2012. DOI: 10.1137/100782760.
   Web of Science ®Google Scholar
• V. I. Danilovskaya, “Thermal stresses in an elastic half-space due to sudden heating of its boundary,” Prikladnaya Matematika i Mech., vol. 14, no. 316–318, 1950. (in Russian)
   Google Scholar
• M. Balla, “Analytical study of the thermal shock problem of a half-space with various thermoelastic models,” Acta Mech., vol. 89, no. 1–4, 73–92, Mar. 1991. DOI: 10.1007/BF01171248.
   Web of Science ®Google Scholar
• E. Rabizadeh, A. Saboor Bagherzadeh, and T. Rabczuk, “Goal-oriented error estimation and adaptive mesh refinement in dynamic coupled thermoelasticity,” Comput. Struct., vol. 173, pp. 187–211, 2016. DOI: 10.1016/j.compstruc.2016.05.024.
   Web of Science ®Google Scholar
• F. Rathgeber, D. A. Ham, L. Mitchell, M. Lange, F. Luporini, A. T. T. McRae, G. T. Bercea, G. R. Markall, and P. H. J. Kelly, “Firedrake: Automating the finite element method by composing abstractions,” ACM Trans. Math. Softw., vol. 43, no. 3, pp. 1–27, 2017. DOI: 10.1145/2998441.
   Web of Science ®Google Scholar
• F. R. de Hoog, J. H. Knight, and A. N. Stokes, “An improved method for numerical inversion of Laplace transforms,” SIAM J. Sci. Stat. Comput., vol. 3, no. 3, pp. 357–366, 1982. DOI: 10.1137/0903022.
   Web of Science ®Google Scholar