A minimizing-movements approach to GENERIC systems

Ansgar Jüngel; Ulisse Stefanelli; Lara Trussardi; Ansgar Jüngel; Ulisse Stefanelli; Lara Trussardi

doi:10.3934/mine.2022005

Mathematics in Engineering

2022, Volume 4, Issue 1: 1-18. doi: 10.3934/mine.2022005

Previous Article Next Article

Research article Special Issues

A minimizing-movements approach to GENERIC systems

1.
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraß e 8-10, 1040 Wien, Austria
2.
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
3.
Vienna Research Platform on Accelerating Photoreaction Discovery, University of Vienna, Währingerstraß e 17, 1090 Wien, Austria
4.
Istituto di Matematica Applicata e Tecnologie Informatiche E. Magenes, via Ferrata 1, I-27100 Pavia, Italy

Received: 01 June 2020 Accepted: 10 May 2021 Published: 19 May 2021

We present a new time discretization scheme adapted to the structure of GENERIC systems. The scheme is based on incremental minimization and is therefore variational in nature. The GENERIC structure of the scheme provides stability and conditional convergence. We show that the scheme can be rigorously implemented in the classical case of the damped harmonic oscillator. Numerical evidence is collected, illustrating the performance of the method and, in particular, the conservation of the energy at the discrete level.
- damped harmonic oscillator,
- structure-preserving time discretization,
- GENERIC system
Citation: Ansgar Jüngel, Ulisse Stefanelli, Lara Trussardi. A minimizing-movements approach to GENERIC systems[J]. Mathematics in Engineering, 2022, 4(1): 1-18. doi: 10.3934/mine.2022005

Related Papers:

Abstract

We present a new time discretization scheme adapted to the structure of GENERIC systems. The scheme is based on incremental minimization and is therefore variational in nature. The GENERIC structure of the scheme provides stability and conditional convergence. We show that the scheme can be rigorously implemented in the classical case of the damped harmonic oscillator. Numerical evidence is collected, illustrating the performance of the method and, in particular, the conservation of the energy at the discrete level.

References

[1]	L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 19 (1995), 191–246.
[2]	L. Ambrosio, N. Gigli, G. Savaré, Gradient flows in metric spaces and in the space of probability measures, 2 Eds., Basel: Birkhäuser Verlag, 2008.
[3]	F. Auricchio, E. Boatti, A. Reali, U. Stefanelli, Gradient structures for the thermomechanics of shape-memory materials, Comput. Methods Appl. Mech. Engrg., 299 (2016), 440–469. doi: 10.1016/j.cma.2015.11.005
[4]	A. Bacho, E. Emmrich, A. Mielke, An existence result and evolutionary $\Gamma$-convergence for perturbed gradient systems, J. Evol. Equ., 19 (2019), 479–522. doi: 10.1007/s00028-019-00484-x
[5]	P. Betsch, M. Schiebl, GENERIC-based formulation and discretization of initial boundary value problems for finite strain thermoelasticity, Comput. Mech., 65 (2020), 503–531. doi: 10.1007/s00466-019-01781-5
[6]	P. Betsch, M. Schiebl, Energy–momentum–entropy consistent numerical methods for large strain thermoelasticity relying on the GENERIC formalism, Internat. J. Numer. Methods Engrg., 119 (2019), 1216–1244.
[7]	H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Amsterdam: North-Holland, 1973.
[8]	E. Celledoni, V. Grimm, R. McLachlan, D. McLaren, D. O'Neale, B. Owren, et al., Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field" method, J. Comput. Phys., 231 (2012), 6770–6789. doi: 10.1016/j.jcp.2012.06.022
[9]	S. Conde Martín, P. Betsch, J. C. García Orden, A temperature based thermodynamically consistent integration scheme for discrete thermo-elastodynamics, Commun. Nonlinear Sci. Numer. Simul., 32 (2016), 63–80. doi: 10.1016/j.cnsns.2015.08.006
[10]	S. Conde Martín, J. C. García Orden, On energy–entropy– momentum integration methods for discrete thermo-viscoelastodynamics, Comput. Struct., 181 (2017), 3–20. doi: 10.1016/j.compstruc.2016.05.010
[11]	E. De Giorgi, A. Marino, M. Tosques, Problems of evolution in metric spaces and maximal decreasing curve, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 68 (1980), 180–187.
[12]	P. Dondl, T. Frenzel, A. Mielke, A gradient system with a wiggly energy and relaxed EDP-convergence, ESAIM Control Optim. Calc. Var., 25 (2019), 68. doi: 10.1051/cocv/2018058
[13]	M. H. Duong, Formulation of the relativistic heat equation and the relativistic kinetic Fokker-Planck equations using GENERIC, Phys. A, 439 (2015), 34–43. doi: 10.1016/j.physa.2015.07.019
[14]	M. H. Duong, M. A. Peletier, J. Zimmer, GENERIC formalism of a Vlasov–Fokker–Planck equation and connection to large-deviation principles, Nonlinearity, 26 (2013), 2951–2971. doi: 10.1088/0951-7715/26/11/2951
[15]	J. C. García Orden, I. Romero, Energy-Entropy-Momentum integration of discrete thermo-visco-elastic dynamics, Eur. J. Mech. Solids, 32 (2012), 76–87. doi: 10.1016/j.euromechsol.2011.09.007
[16]	O. Gonzalez, Time integration and discrete Hamiltonian systems, J. Nonlin. Sci., 6 (1996), 449–467. doi: 10.1007/BF02440162
[17]	O. Gonzalez, Exact energy-momentum conserving algorithms for general models in nonlinear elasticity, Comput. Methods Appl. Mech. Engrg., 190 (2000), 1763–1783. doi: 10.1016/S0045-7825(00)00189-4
[18]	M. Grmela, H. C. Öttinger, Dynamics and thermodynamics of complex fluids. I. Development of a general formalism, Phys. Rev. E, 56 (1997), 6620–6632.
[19]	E. Hairer, C. Lubich, G. Wanner, Geometric numerical integration, 2 Eds., Berlin: Springer, 2006.
[20]	M. Hoyuelos, GENERIC framework for the Fokker-Planck equation, Phys. A, 442 (2016), 350–358. doi: 10.1016/j.physa.2015.09.035
[21]	M. Hütter, T. A. Tervoort, Finite anisotropic elasticity and material frame indifference from a nonequilibrium thermodynamics perspective, J. Non-Newton. Fluid., 152 (2008), 45–52. doi: 10.1016/j.jnnfm.2007.10.009
[22]	M. Hütter, B. Svendsen, On the formulation of continuum thermodynamic models for solids as general equations for non-equilibrium reversible–irreversible coupling, J. Elast., 104 (2011), 357–368. doi: 10.1007/s10659-011-9327-4
[23]	M. Hütter, B. Svendsen, Thermodynamic model formulation for viscoplastic solids as general equations for nonequilibrium reversible–irreversible coupling, Contin. Mech. Thermodyn., 24 (2012), 211–227. doi: 10.1007/s00161-011-0232-7
[24]	A. Jelić, M. Hütter, H. C. Öttinger, Dissipative electromagnetism from a nonequilibrium thermodynamics perspective, Phys. Rev. E, 74 (2006), 041126. doi: 10.1103/PhysRevE.74.041126
[25]	A. Jüngel, U. Stefanelli, L. Trussardi, Two structure-preserving time discretizations for gradient flows, Appl. Math. Optim., 80 (2020), 733–764.
[26]	R. C. Kraaij, A. Lazarescu, C. Maes, M. Peletier, Fluctuation symmetry leads to GENERIC equations with non-quadratic dissipation, Stochastic Process. Appl., 130 (2020), 139–170. doi: 10.1016/j.spa.2019.02.001
[27]	R. Kraaij, A. Lazarescu, C. Maes, M. Peletier, Deriving GENERIC from a generalized fluctuation symmetry, J. Stat. Phys., 170 (2018), 492–508. doi: 10.1007/s10955-017-1941-5
[28]	M. Krüger, M. Groß, P. Betsch, An energy–entropy-consistent time stepping scheme for nonlinear thermo-viscoelastic continua, ZAMM Z. Angew. Math. Mech., 96 (2016), 141–178. doi: 10.1002/zamm.201300268
[29]	M. Liero, A. Mielke, M. A. Peletier, D. R. M. Renger, On microscopic origins of generalized gradient structures, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 1–35.
[30]	R. McLachlan, G. Quispel, N. Robidoux, Geometric integration using discrete gradients, Phil. Trans. R. Soc. Lond. A, 357 (1999), 1021–1045. doi: 10.1098/rsta.1999.0363
[31]	A. Mielke, Formulation of thermoelastic dissipative material behavior using GENERIC, Contin. Mech. Thermodyn., 23 (2011), 233–256. doi: 10.1007/s00161-010-0179-0
[32]	A. Mielke, Dissipative quantum mechanics using GENERIC, In: Proc. of the conference on recent trends in dynamical systems, Springer, 2013,555–585.
[33]	A. Mielke, A. Montefusco, M. Peletier, Exploring families of energy-dissipation landscapes via tilting–three types of EDP convergence, Contin. Mech. Thermodyn., 33 (2021), 611–637. doi: 10.1007/s00161-020-00932-x
[34]	A. Mielke, R. Rossi, G. Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 18 (2012), 36–80. doi: 10.1051/cocv/2010054
[35]	A. Montefusco, F. Consonni, G. Beretta, Essential equivalence of the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) and steepest-entropy-ascent models of dissipation for nonequilibrium thermodynamics, Phys. Rev. E, 91 (2015), 042138. doi: 10.1103/PhysRevE.91.042138
[36]	B. S. Mordukhovich, Variational analysis and generalized differentiation. I. Basic theory, Berlin: Springer-Verlag, 2006.
[37]	H. C. Öttinger, Beyond equilibrium thermodynamics, New Jersey: John Wiley, 2005.
[38]	H. C. Öttinger, Generic integrators: structure preserving time integration for thermodynamic systems, J. Non-Equil. Thermody., 43 (2018), 89–100. doi: 10.1515/jnet-2017-0034
[39]	D. Portillo, J. C. García Orden, I. Romero, Energy–entropy– momentum integration schemes for general discrete non-smooth dissipative problems in thermomechanics, Internat. J. Numer. Methods Engrg., 112 (2017), 776–802. doi: 10.1002/nme.5532
[40]	L. Portinale, U. Stefanelli, Penalization via global functionals of optimal-control problems for dissipative evolution, Adv. Math. Sci. Appl., 28 (2019), 425–447.
[41]	G. Quispel, D. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A-Math. Theor., 41 (2008), 045206. doi: 10.1088/1751-8113/41/4/045206
[42]	T. Roche, R. Rossi, U. Stefanelli, Stability results for doubly nonlinear differential inclusions by variational convergence, SIAM J. Control Optim., 52 (2014), 1071–1107. doi: 10.1137/130909391
[43]	I. Romero, Thermodynamically consistent time-stepping algorithms for non-linear thermomechanical systems, Internat. J. Numer. Methods Engrg., 79 (2009), 706–732. doi: 10.1002/nme.2588
[44]	I. Romero, Algorithms for coupled problems that preserve symmetries and the laws of thermodynamics. Part I: monolithic integrators and their application to finite strain thermoelasticity, Comput. Methods Appl. Mech. Engrg., 199 (2010), 1841–1858. doi: 10.1016/j.cma.2010.02.014
[45]	I. Romero, Algorithms for coupled problems that preserve symmetries and the laws of thermodynamics. Part II: fractional step methods, Comput. Methods Appl. Mech. Engrg., 199 (2010), 2235–2248. doi: 10.1016/j.cma.2010.03.016
[46]	R. Rossi, G. Savaré, {Gradient flows of non convex functionals in Hilbert spaces and applications}, ESAIM Control Optim. Calc. Var., 12 (2006), 564–614. doi: 10.1051/cocv:2006013
[47]	R. Rossi, A, Mielke, G. Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008), 97–169.
[48]	S. Sato, T. Matsuo, H. Suzuki, D. Furihata, A Lyapunov-type theorem for dissipative numerical integrators with adaptive time-stepping, SIAM J. Numer. Anal., 53 (2015), 2505–2518. doi: 10.1137/140996719
[49]	J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65–96.
[50]	Y. Suzuki, M. Ohnawa, GENERIC formalism and discrete variational derivative method for the two-dimensional vorticity equation, J. Comput. Appl. Math., 296 (2016), 690–708. doi: 10.1016/j.cam.2015.10.018

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)