La sèrie de seminaris CoMe (Computational Mechanics Group Seminar Series) presenta "Symplectic, entropy-momentum conserving time integrators for thermo-elastic systems". Condueix la sessió Pablo L. Mata Almonacid.
This work is devoted to the formulation of a new class of time integrators for finite-dimensional and completely integrable thermoelastic systems with symmetries which are able to conduct heat. We consider an arrangement of N mases connected by M thermo-elastic springs. The spatial position of the masses is described by means a vector of generalizes displacements and each thermo-elastic spring is assigned with a thermal displacement such that its time derivative corresponds to the temperature. A first step toward the construction of such integrators consists in taking into account the Hamiltonian structure of the reversible adiabatic problem to apply standard techniques in variational integration. In order to allow for the conduction of small amounts of heat, the balance equation describing the conservation of the entropy in each spring has to be modified including an (small) entropy flux consistent with the Fourier law. In this way, heat conduction is treated as a non-Hamiltonian perturbation of an integrable system. A discrete form of D’Alembert principle is used to formulate a new class of second-order accurate, momentum-conserving time integrators which reduce to a family of symplectic Runge-Kutta methods in the non-conductive limit. Numerical evidence shows that proposed algorithms are thermodynamically consistent in the following sense: (i) total entropy of the system always grows due to an algebraic property of the method and (ii) they show an excellent energy behavior which may be related to the ability of the method to reproduce the long term structure of the solutions.