Development of Mathematical Models and Computational Framework for Multi-physics Interaction Processes

摘要

This paper presents development of mathematical models for multi-physics interaction processes in which 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 choice 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 mathematic model. In the development of the constitutive equations and the equations of state, the same dependent variables are also utilized as those in the conservation laws. The dependent variables of choice due to the Eulerian description (which is necessary for liquids and gases) are density, pressure, velocities, temperature, heat fluxes and stress deviations. For solid, liquids and gases the development of constitutive equations is based on rate constitutive equations utilizing convected time derivatives of the chosen stress and strain measures. The resulting GDEs from these mathematical models are 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 associated initial value problems. The proposed computational methodology permits higher order global differentiability approximations, and 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 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 solution is highly dependent on the physics described by the constraint equations. In the proposed methodology, the constraint equations are completely eliminated.