Variational Integrators for Thermo‐Viscoelastic Discrete Systems

Abstract

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. Higher Order Variational IntegratorsThis work is a further development of an existing variational integrator (VI) for discrete thermo-elastic systems [1]. Again, the common notation for VIs [2] is adopted as well as the concept of thermacy [3]. Thermacy is also known as "thermal displacement", since its time derivative corresponds to the temperature ϑ =α. This formulation results in a natural definition of the entropy as "thermal momentum" and entropy flux as "thermal force". The variational principle for thermo-mechanically coupled systems [4] uses the Lagrangian L = T − ψ. It is the foundation for the numerical discretization.The construction of a variational integrator (VI) goes in two steps. Firstly, the state variables are approximated in time. Using Lagrangian polynomials of degree p the approximation of one time step h = t k+1 − t k readsSecondly, a time-step-wise quadrature rule of the action-integral is applied, here using Gauss-type integration of order gThe same numerical integration is applied to the virtual work and rearranged over the time grid of the approximationdefining the discrete generalized forces F d n . Finally, evaluation of the discrete Lagrangian L d (q 0 , q 1 . . . , q p+1 ) and the discrete generalized forces according to D'Alembert's principle leads to the position-momentum form for the time stepping scheme Model ProblemThe thermoelastic double pendulum is a minimalistic model, commonly used [5,6] for testing numerical time integrators. For brevity only the enhancements of the previous stage [1] are detailed here. The first one is the replacement of the thermo-elastic springs by the generalized Maxwell elements (spring in parallel to a series connection of spring and dash-pot) as illustrated in Fig. 1. Thus the free energy of each of these two Maxwell elements takes the form [7]