An unconditionally stable algorithm for generalized thermoelasticity based on operator-splitting and time-discontinuous Galerkin finite element methods

An efficient time-stepping algorithm is proposed based on operator-splitting and the space–time discontinuous Galerkin finite element method for problems in the non-classical theory of thermoelasticity. The non-classical theory incorporates three models: the classical theory based on Fourier’s law of heat conduction resulting in a hyperbolic–parabolic coupled system, a non-classical theory of a fully-hyperbolic extension, and a combination of the two. The general problem is split into two contractive sub-problems, namely the mechanical phase and the thermal phase. Each sub-problem is discretized using the space–time discontinuous Galerkin finite element method. The sub-problems are stable which then leads to unconditional stability of the global product algorithm. A number of numerical examples are presented to demonstrate the performance and capability of the method

Recovery of a space-dependent vector source in thermoelastic systems

In this contribution, an inverse problem of determining a space-dependent vector source in a thermoelastic system of type-I, type-II and type-III is studied using information from a supplementary measurement at a fixed time. These thermoelastic systems consist of two equations that are coupled: a parabolic equation for the temperature [GRAPHICS] and a vectorial hyperbolic equation for the displacement [GRAPHICS] . In this latter one, the source is unknown, but solely space dependent. A spacewise-dependent additional measurement at the final time ensures that the inverse problem corresponding with each type of thermoelasticity has a unique solution when a damping term [GRAPHICS] (with [GRAPHICS] componentwise strictly monotone increasing) is present in the hyperbolic equation. Despite the ill-posed nature of these inverse problems, a stable iterative algorithm is proposed to recover the unknown source in the case that [GRAPHICS] is also linear. This method is based on a sequence of well-posed direct problems, which are numerically solved at each iteration, step by step, using the finite element method. The instability of the inverse source problem is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results support the theoretically obtained results

An approximation theory for the identification of linear thermoelastic systems

An abstract approximation framework and convergence theory for the identification of thermoelastic systems is developed. Starting from an abstract operator formulation consisting of a coupled second order hyperbolic equation of elasticity and first order parabolic equation for heat conduction, well-posedness is established using linear semigroup theory in Hilbert space, and a class of parameter estimation problems is then defined involving mild solutions. The approximation framework is based upon generic Galerkin approximation of the mild solutions, and convergence of solutions of the resulting sequence of approximating finite dimensional parameter identification problems to a solution of the original infinite dimensional inverse problem is established using approximation results for operator semigroups. An example involving the basic equations of one dimensional linear thermoelasticity and a linear spline based scheme are discussed. Numerical results indicate how the approach might be used in a study of damping mechanisms in flexible structures

On Stability of Hyperbolic Thermoelastic Reissner-Mindlin-Timoshenko Plates

In the present article, we consider a thermoelastic plate of Reissner-Mindlin-Timoshenko type with the hyperbolic heat conduction arising from Cattaneo's law. In the absense of any additional mechanical dissipations, the system is often not even strongly stable unless restricted to the rotationally symmetric case, etc. We present a well-posedness result for the linear problem under general mixed boundary conditions for the elastic and thermal parts. For the case of a clamped, thermally isolated plate, we show an exponential energy decay rate under a full damping for all elastic variables. Restricting the problem to the rotationally symmetric case, we further prove that a single frictional damping merely for the bending compoment is sufficient for exponential stability. To this end, we construct a Lyapunov functional incorporating the Bogovski\u{i} operator for irrotational vector fields which we discuss in the appendix.Comment: 27 page

Moore–Gibson–Thompson thermoelasticity

We consider a thermoelastic theory where the heat conduction is escribed by the Moore–Gibson–Thompson equation. In fact, this equation can be obtained after the introduction of a relaxation parameter in the Green–Naghdi type III model. We analyse the one- and three-dimensional cases. In three dimensions, we obtain the well-posedness and the stability of solutions. In one dimension, we obtain the exponential decay and the instability of the solutions depending on the conditions over the system of constitutive parameters.We also propose possible extensions for these theoriesPeer ReviewedPostprint (author's final draft