Energie-Entropie-konsistente Zeitintegratoren für die nichtlineare Thermoviskoelastodynamik

This work deals with an energy-entropy-consistent simulation of a thermoviscoelastic model problem and continuum. Both systems are described by the Poissonian variables - linear momentum, configuration, entropy and internal variable. The linear momentum and the configuration as independent variables lead to the equations of motion as differential equations of first order. The thermal evolution equation is derived by the second law of thermodynamics. The heat flux is described by Fourier's law. The equations of motion and the thermal evolution equation are linked through the constitutive equation of the internal energy. A viscous evolution equation, as fourth equation, is necessary to describe the viscous deformation behavior. This equation is based on deformation-like internal variables and a fourth order compliance tensor, which is restricted to the one-dimensional case for the model problem. The internal dissipation is given by a quadratic form of the viscous Mandel stress.

The four differential equations of first order are transformed by the refined General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) format into a matrix-vector notation. This format is called in the following enhanced GENERIC format. The enhanced GENERIC format yields with the related degeneracy conditions structure preservation properties for an isolated system. An isolated system is defined as an adiabatic system, which does not do mechanical work. These properties are in addition to a constant linear and angular momentum, the constant total energy, an increasing total entropy and a decreasing Lyapunov function. The last one is a stability criterion for thermoviscoelastic systems.

New is furthermore the consistent embedding of external mechanical and thermal loads. These external loads affect the aforementioned preservation properties of isolated systems. In this case the related balance equations of the system (consistency properties) are considered.

The discretization in time is done for the model problem and the continuum with two different integrators. On the one hand the midpoint-rule and on the other hand the enhanced TC (Thermodynamically Consistent) integrator is used. The enhanced TC integrator is constructed such, that the underlying enhanced GENERIC format reflects the algorithmic properties after the discretization in time, for an isolated system. For a continuum a discretization in space is necessary, which is given by the Finite-Element-Method. A projection of the test function of the thermal evolution equation is necessary for an energy consistent discretization. Furthermore the energy consistency can only be guaranteed using the enhanced TC integrator, which leads to enhanced stability.

The external mechanical and thermal loads are included with fixed bearings, external mechanical loads, thermal constraints and external thermal loads. The Lagrangian multipliers are used to fulfill the thermal constraint. These Lagrangian multipliers are well known for constraining the motion of a system.

The enhanced GENERIC format will be extended for the model problem after the discretization with the external loads. In contrast to that, the enhanced GENERIC format for the continuum, which is here given in the strong evolution equations, contains the external loads. This yields the necessary weak evolution equations for the solution of the system.

The consistency properties are shown for representative numerical examples with different boundary conditions.

The coupled mechanical system under consideration is formulated in terms of the Poissonian variables. This leads to a monolithic solution with the Newton-Raphson method. Therefore, two Newton-Raphson methods are necessary, one to resolve the viscous internal variable on local (element) level and another one to resolve the equations of motion, the thermal evolution equation and if applicable the equation of projection on global level. This is called a multi-level Newton-Raphson method.