Development of mathematical models and mathematical, computational framework for multi-media interaction processes

Abstract

This thesis presents development of mathematical models for multi-media interaction process using Eulerian description and associated computational infrastructure to obtain numerical solution of the initial value problems described by these mathematical models using finite element method. In the development of mathematical models for multi-media interaction processes the physics of solids, liquids and gases are described using conservation laws, appropriate constitutive equations and equations of state in Eulerian description. The use of conservation laws in Eulerian description for all media of an interaction process and the use of the same dependent variables in the resulting governing differential equations (GDEs) for solids, liquids and gases ensure that their interactions are intrinsic in the mathematical model. In the development of the constitutive equations and the equations of state, the same dependent variables are utilized as those in the conservation laws. The dependent variables of choice due to the Eulerian description (which is necessitated due to liquids and gases) are density, pressure, velocities, temperature, heat fluxes and stress deviations. When the mathematical models of the deforming matter for progressively increasing deformation are derived using conservation laws in Eulerian description, the constitutive equations must be derived using rate constitutive theories regardless of whether the deforming matter is solid or fluid. Thus complete mathematical description of the deforming matter is highly dependent on the appropriate choice of the specific constitutive equations. Assessment of the validity of various rate constitutive equations is an integral part of the present research. In this proposed approach, the physics of all interacting media of an interaction process are described by a single mathematical model (conservation laws) in the same dependent variables and hence their interactions are inherent in the mathematical model and require no further considerations. The resulting GDEs from these mathematical models are generally a system of non-linear partial differential equations in space coordinates and time. The hpk mathematical and computational finite element framework with space-time variationally consistent (STVC) integral forms is utilized to obtain the numerical solutions of the initial value problems described by the mathematical models. The proposed computational methodology permits higher order global differentiability approximations, ensures time accuracy of evolutions as well as unconditional stability of computations during the entire evolution. The methodology presented here for multi-media interaction processes is rather natural and lends itself naturally to accurate finite element computations in hpk framework when the integral forms are space-time variationally consistent (STVC). In most of the currently used methodologies, the interaction between the different media is established using constraint equations at the interfaces between the media. Thus, these approaches are error prone and the validity and accuracy of the computed solutions is highly dependent on the physics described by the constraint equations. In the proposed methodology, the constraint equations are completely eliminated.