Stable algorithms for generalized thermoelasticity based on operator-splitting and time-discontinuous Galerkin finite element methods

Doctoral Thesis


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University of Cape Town

This thesis deals with the theoretical and numerical analysis of coupled problems in thermoelasticity. Of particular interest are models that support propagation of thermal energy as waves, rather than the usual mechanism by diffusion. The thesis consists of two parts. The first deals with the non-classical, linear thermoelastic model first proposed and developed by Green and Naghdi in the years between 1991 and 1995, as a possible alternative that potentially removes the shortcomings of the standard Fourier based model. The non-classical theory incorporates three models: the classical model 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. An efficient staggered time-stepping algorithm is proposed based on operator-splitting and the time-discontinuous Galerkin finite element method for the non-classical, linear thermoelastic model. The coupled problem is split into two contractive sub-problems, namely, the mechanical phase and thermal phase, on the basis of an entropy controlling mechanism. In the mechanical phase temperature is allowed to vary so as to ensure the entropy remains constant, while the thermal phase is a purely non-classical heat conduction problem in a fixed configuration. Each sub-problem is discretized using the time-discontinuous Galerkin finite element method, resulting in stable time-stepping sub-algorithms. A global stable algorithm is obtained by combining the algorithms for the sub-problems by way of a product method. A number of numerical examples are presented to demonstrate the performance and capability of the method. The second part of this work concerns the formulation of a thermodynamically consistent generalized model of nonlinear thermoelasticity, whose linearization about a natural reference configuration includes the theory of Green and Naghdi. The generalized model is based on the fundamental laws of continuum mechanics and thermodynamics, and is realized through two basic assumptions: The first is the inclusion into the state space of a vector field, which is known as the thermal displacement, and is a time primitive of the absolute temperature. The second is that the heat flux vector is additively split into two parts, which are referred to as the energetic and dissipative components of the heat flux vector. The application of the Coleman-Noll procedure leads to find constitutive relations for the stress, entropy, and energetic component of the heat flux as derivatives of the free energy function. Furthermore, a Clausius-Duhem-type inequality is assumed on a constitutive relation for the dissipative component of the heat flux vector to ensure thermodynamic consistency. A Lyapunov function is obtained for the generalized problem with finite strains; this serves as the basis for the stability analysis of the numerical methods designed for generalized thermoelasticity at finite strains. Due to the lack of convexity of the elastic potential in the finite strain case, a direct extension of the time-discontinuous formulation from the linear to the finite strain case does not guarantee stability. For this reason, various numerical formulations both in monolithic and staggered approaches with fully or partially time-discontinuity assumptions are presented in the framework of the space-time methods. The stability of each of the numerical algorithms is thoroughly analysed. The capability of the newly formulated generalized model of thermoelasticity in predicting various expected features of non-Fourier response is illustrated by a number of numerical examples. These also serve to demonstrate the performance of the space-time Galerkin method in capturing fine solution features.