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Variational Integrators for Thermo‐Viscoelastic Discrete Systems
Author(s) -
Kern Dominik,
Romero Ignacio,
Groß Michael
Publication year - 2015
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201510018
Subject(s) - variational integrator , variational principle , discretization , mathematics , virtual work , mathematical analysis , principle of least action , calculus of variations , hamilton's principle , classical mechanics , action (physics) , integrator , equations of motion , physics , finite element method , quantum mechanics , voltage , thermodynamics
Variational integrators are modern time‐integration schemes based on a discretization of the underlying variational principle. In this paper, Hamilton's principle is approximated by an action sum, whose vanishing variation results in discrete Euler‐Lagrange equations or, equivalently, in discrete evolution equations for the position and momentum. In order to include the viscous and thermal virtual work (mechanical and thermal virtual dissipation), Hamilton's principle is extended by D'Alembert terms, which account for the time evolution equation of the internal variable and Fourier's law. From this variational formulation, variational integrators using different orders of approximation of the state variables as well as of the quadrature of the action integral are constructed and compared. A thermo‐viscoelastic double pendulum comprised of two discrete masses connected by generalized Maxwell elements, and subject to heat conduction between them serves as a discrete model problem. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)