American Institute of Mathematical Sciences

Advanced
January  2009, 11(1): 109-130. doi: 10.3934/dcdsb.2009.11.109

Discrete dynamics of complex bodies with substructural dissipation: Variational integrators and convergence

Matteo Focardi 1, and  Paolo Maria Mariano 2,

1. 

Dipartimento di Matematica "U. Dini", Università di Firenze, viale Morgagni 67/A, I-50139 Firenze, Italy
2. 

DICeA, Università di Firenze, via Santa Marta 3, I-50139 Firenze, Italy

Received  December 2007 Revised  March 2008 Published  November 2008

For the linearized setting of the dynamics of complex bodies we construct variational integrators and prove their convergence by making use of BV estimates on the rate fields. We allow for peculiar substructural inertia and internal dissipation, all accounted for by a d'Alembert-Lagrange-type principle.
Keywords: dynamical systems, asynchronous variational integrators, convergence, Complex bodies.
Mathematics Subject Classification: Primary: 37M15, 74A30, 74A99.
Citation: Matteo Focardi, Paolo Maria Mariano. Discrete dynamics of complex bodies with substructural dissipation: Variational integrators and convergence. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 109-130. doi: 10.3934/dcdsb.2009.11.109
[1]

Oscar E. Fernandez, Anthony M. Bloch, P. J. Olver. Variational Integrators for Hamiltonizable Nonholonomic Systems. Journal of Geometric Mechanics, 2012, 4 (2) : 137-163. doi: 10.3934/jgm.2012.4.137
[2]

Leonardo Colombo, Fernando Jiménez, David Martín de Diego. Variational integrators for mechanical control systems with symmetries. Journal of Computational Dynamics, 2015, 2 (2) : 193-225. doi: 10.3934/jcd.2015003
[3]

El Houcein El Abdalaoui, Sylvain Bonnot, Ali Messaoudi, Olivier Sester. On the Fibonacci complex dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2449-2471. doi: 10.3934/dcds.2016.36.2449
[4]

Cédric M. Campos, Sina Ober-Blöbaum, Emmanuel Trélat. High order variational integrators in the optimal control of mechanical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4193-4223. doi: 10.3934/dcds.2015.35.4193
[5]

Pedro L. García, Antonio Fernández, César Rodrigo. Variational integrators for discrete Lagrange problems. Journal of Geometric Mechanics, 2010, 2 (4) : 343-374. doi: 10.3934/jgm.2010.2.343
[6]

Werner Bauer, François Gay-Balmaz. Variational integrators for anelastic and pseudo-incompressible flows. Journal of Geometric Mechanics, 2019, 11 (4) : 511-537. doi: 10.3934/jgm.2019025
[7]

Xin Li, Wenxian Shen, Chunyou Sun. Invariant measures for complex-valued dissipative dynamical systems and applications. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2427-2446. doi: 10.3934/dcdsb.2017124
[8]

Yong Chen, Hongjun Gao, María J. Garrido–Atienza, Björn Schmalfuss. Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 79-98. doi: 10.3934/dcds.2014.34.79
[9]

J. Gwinner. On differential variational inequalities and projected dynamical systems - equivalence and a stability result. Conference Publications, 2007, 2007 (Special) : 467-476. doi: 10.3934/proc.2007.2007.467
[10]

Xianfeng Ma, Ercai Chen. Pre-image variational principle for bundle random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 957-972. doi: 10.3934/dcds.2009.23.957
[11]

S. Aubry, G. Kopidakis, V. Kadelburg. Variational proof for hard Discrete breathers in some classes of Hamiltonian dynamical systems. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 271-298. doi: 10.3934/dcdsb.2001.1.271
[12]

Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809
[13]

Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129
[14]

Mats Vermeeren. Modified equations for variational integrators applied to Lagrangians linear in velocities. Journal of Geometric Mechanics, 2019, 11 (1) : 1-22. doi: 10.3934/jgm.2019001
[15]

Claude Le Bris, Frédéric Legoll. Integrators for highly oscillatory Hamiltonian systems: An homogenization approach. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 347-373. doi: 10.3934/dcdsb.2010.13.347
[16]

Y. A. Li, P. J. Olver. Convergence of solitary-wave solutions in a perturbed bi-hamiltonian dynamical system ii. complex analytic behavior and convergence to non-analytic solutions. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 159-191. doi: 10.3934/dcds.1998.4.159
[17]

Mariano Giaquinta, Paolo Maria Mariano, Giuseppe Modica. A variational problem in the mechanics of complex materials. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 519-537. doi: 10.3934/dcds.2010.28.519
[18]

Jianfeng Feng, Mariya Shcherbina, Brunello Tirozzi. Dynamical behaviour of a large complex system. Communications on Pure & Applied Analysis, 2008, 7 (2) : 249-265. doi: 10.3934/cpaa.2008.7.249
[19]

Micol Amar, Andrea Braides. A characterization of variational convergence for segmentation problems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 347-369. doi: 10.3934/dcds.1995.1.347
[20]

Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences