On the integration of the arbitrary Lagrangian-Eulerian concept and non-equilibrium thermodynamics

The aim of this treatise is to present a harmonious mathematical formulation of an explicit moving mesh method that can be used as a basis for many numerical techniques. In most cases a moving mesh is only used to include arbitrary motions and deformations of a geometry into the simulation of a physical problem. The innovative part of this research is to develop a mechanism that controls the motion of interior mesh points by detecting and tracking the location of physical phenomena. The motion of points in the explicit moving mesh method is based on a less known perspective of the Arbitrary Lagrangian-Eulerian (ALE) concept. This formulation provides a degree of freedom to relocate the interior mesh points of the computational domain. The main objective of the approach presented in this treatise is to move mesh points such that they are concentrated in regions with physical phenomena. This interaction between the moving mesh and physics is based on the concept of entropy production with the so-called irreversibility sensor. Hence, a large part of this treatise is focussed on a proper mathematical formulation of non-equilibrium thermodynamics. Hidden within the above topics is a less obvious issue: continuum physics is a field theory that applies to every point in a domain, while thermodynamics makes statements about complete systems. This issue materializes especially with the mathematical formulations for the change of a volume, where the corresponding field density is equal to one. Furthermore, the time derivative of a domain integration with time-dependent integration limits introduces additional terms that have to be accounted for. This treatise starts therefore with a thorough introduction into kinematics with an emphasis on a proper description of configurations and motions. The postulate of spatial collocation is the basis of the Eulerian formulation, where there is a clear separation between material and its spatial location. This introduces the distinction between the reference and spatial approaches, including the corresponding field representations and time derivatives. All these concepts are applied in the derivation of the Geometric Conservation Law (GCL) and Reynolds transport theorem. The ALE concept is based on the postulate of arbitrary collocation, which is an extension of the postulate of spatial collocation. Due to the extension there are two reference configurations: one for the material domain (viz. physical problem) and one for an arbitrary moving domain (viz. computational domain). The postulate of arbitrary collocation states that at the considered time the current configurations of both domains are collocated in space. The classical application of the ALE concept is to transform the material domain to the reference configuration of the computational domain. Since a reference configuration does not change with respect to time it is not necessary to update the corresponding spatial discretization (computational mesh). Consequently it is not possible to refine the spatial discretization at locations with physical phenomena, which would increase the accuracy of the numerical approximation. Therefore another, more general perspective of the ALE concept is introduced, where the integration limits of the considered domain are upgraded at every time step. This requires the derivation of a new set of kinematic relations for fields (local formulation) and complete domains (integral formulation). The result is a generalised version of Reynolds transport theorem that is valid for arbitrary moving domains. A summary of all formulations within the ALE concept is given and it is shown how both the Lagrangian and Eulerian approaches are captured by this single concept. The kinematics that are introduced with the ALE concept are extended to kinetics with the postulate of conservation laws. With the definition of a production term it is possible to derive the framework of a master balance law. For a combination of these laws and the ALE concept it is convenient to introduce the arbitrary convection operator and the apparent compressibility, which is the local formulation for the time derivative of a volume. Based on these concepts a relation can be derived between the various reference time derivatives that are introduced by the ALE concept. The local formulation of a master balance law is generalised to the corresponding variational formulation, because this forms the mathematical basis of many numerical techniques (for instance the Finite Element Method (FEM)). Various local formulations of the GCL are studied as an example of a conservation law. This study clarifies the issue of the time derivative of a field density that is equal to one by making a distinction between geometrical and substantial field densities. A (continuum) multi-physics approach means in this treatise a combination of various physical phenomena in the same mathematical framework, viz. a system of master balance laws. The physical principles of conservation of matter (including chemical reactions), conservation of momentum and conservation of energy are recapitulated within this framework. The explicit moving mesh method is implemented in this framework by additional master balance laws for the geometrical velocity, which is the degree of freedom obtained with the ALE concept. The innovative idea is to use the gradient of entropy production, here denoted as irreversibility sensor, as ‘driving force’ for the geometrical ve- locity. The constitutive relations for geometrical stress tensor are a modelling degree of freedom, which determines whether the computational mesh behaves analogous to an elastic material or a fluid. An important requirement for these models is that they provide a steady state mechanism when there are physical irreversibilities. The mathematical formulation of classical irreversible thermodynamics is based on the so-called local equilibrium hypothesis. This hypothesis does not comply with the innovative concept of the irreversibility sensor that is introduced in this treatise. Hence, a new mathematical formulation for irreversible or non-equilibrium thermodynamics is required. An essential step in the approach of this treatise is the definition of a separate state space for non-equilibrium thermodynamics besides the Newtonian space-time continuum. In 1973 Robert Hermann proposed to base this state space on the mathematical concept of a non-degenerating contact 1-form. For a contact 1-form there exists a mathematical theorem for the change of a potential, which is identified as the Gibbs fundamental relation in non-equilibrium thermodynamics. Implementation of concepts such as thermodynamic state variables or the first law of thermodynamics is straightforward. Equilibrium thermodynamics is here a special case of non-equilibrium thermodynamics, which is a contradistinction of the more conventional approach to extend equilibrium states with non-equilibrium phenomena. This treatise proposes a new, unique formulation for the second law of thermodynamics that has a sound foundation in contact geometry and is a modern interpretation of Carath´eodory’s inaccessibility criterion. Both the first and second law of thermodynamics are used to identify some of the coordinates in the non-equilibrium Hermann state space. This state space is defined for complete thermodynamic systems, so there is no spatial information available nor is a diffusive flux defined. An innovative approach is proposed to obtain a local formulation for extensive thermodynamic quantities in the Newtonian space-time continuum. The postulate for a (non-equilibrium) extensive flux is used to transform thermodynamic laws to equivalent relations that can be applied in continuum physics (e.g. see the theorem of continuum thermodynamics). Hence, a local formulation for the entropy production is obtained that can be used as an irreversibility sensor in an explicit moving mesh method. Furthermore, a unique conclusion is that the local formulation of the entropy production can be identified as the non-equilibrium implementation of the Gibbs-Duhem equation.