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Development of Mathematical Models and Mathematical, Computational Framework for Multi-media Interaction Processes.

机译：多媒体交互过程的数学模型和数学计算框架的开发。

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摘要

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 [1--3] 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 nonlinear 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.
机译：本文提出了使用欧拉描述和相关计算基础设施的多媒体交互过程数学模型的开发方法，以利用有限元方法获得这些数学模型描述的初值问题的数值解。在用于多媒体相互作用过程的数学模型的开发中，使用守恒定律，适当的本构方程和欧拉描述中的状态方程来描述固体，液体和气体的物理特性。在欧拉描述中对相互作用过程的所有介质使用守恒定律，并在固体，液体和气体的控制微分方程（GDE）中使用相同的因变量，以确保它们的相互作用在数学模型中是固有的。在本构方程和状态方程的发展中，利用了与守恒定律相同的因变量。由于欧拉描述（由于液体和气体而必须），因此选择的因变量是密度，压力，速度，温度，热通量和应力偏差。当使用欧拉描述中的守恒定律推导出用于逐步变形的变形物的数学模型时，无论变形物是固体还是流体，都必须使用速率本构理论[1--3]来导出本构方程。因此，对变形物质的完整数学描述高度依赖于特定本构方程的适当选择。评估各种速率本构方程的有效性是本研究的组成部分。在这种提出的方​​法中，相互作用过程中所有相互作用介质的物理性质都由一个在相同因变量中的单一数学模型（守恒定律）描述，因此它们的相互作用是该数学模型固有的，无需进一步考虑。这些数学模型中的GDE通常是一个在空间坐标和时间上都是非线性偏微分方程的系统。利用具有时空变化一致（STVC）积分形式的hpk数学和计算有限元框架来获得数学模型描述的初值问题的数值解。所提出的计算方法允许更高阶的全局微分逼近，确保演化的时间准确性以及整个演化过程中计算的无条件稳定性。这里介绍的用于多媒体交互过程的方法是很自然的，并且当积分形式是时空变化一致的（STVC）时，它自然适用于hpk框架中的精确有限元计算。在大多数当前使用的方法中，交互使用约束方程在介质之间的接口处建立不同介质之间的关系。因此，这些方法容易出错，并且所计算解的有效性和准确性高度依赖于约束方程式描述的物理学。在提出的方法中，完全消除了约束方程。

著录项

  • 作者

    Ma, Yongting.;

  • 作者单位

    University of Kansas.;

  • 授予单位 University of Kansas.;
  • 学科 Applied Mathematics.;Engineering Mechanical.
  • 学位 Ph.D.
  • 年度 2011
  • 页码 228 p.
  • 总页数 228
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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