Mathematical Sciences: Microwave Processing of Ceramic Materials

研究人员提出建模和分析一类 微波处理陶瓷中出现的物理问题。 这些问题引起了具有挑战性的非线性初始 边值问题，其数学陈述 涉及一个或多个非线性抛物方程， 热量的扩散和化学浓度的扩散， 和时谐版本的麦克斯韦方程。 这些 方程以非线性方式耦合，此外， 在某些应用中需要非线性热平衡 其中表面温度产生显著的热辐射。 研究者建议开发和使用渐近方法， 由某些物理限制决定，以简化数学 系统，并开发和使用数值方法来解决 减少问题。 提议者的目标是双重的。 首先是 对基本物理机制的数学理解， 让流程发挥作用。 这样的理解， 科学家和工程师参与微波处理， 陶瓷的基本科学依据， 解释他们的实验结果。 二是 发展和应用渐近，数值，和其他 获得准确描述所需的近似方法 更真实和复杂的过程。 成功 完成拟议的研究将使工程师和 科学家设计更有效的烧结和连接 生产高质量陶瓷的方法 材料. 这也将有助于设计和实施 微波辅助气相沉积工艺， 在复合陶瓷材料的构建中具有广阔的应用前景。

The investigator proposes to model and analyze a class of physical problems arising in microwave processing of ceramics. These problems give rise to challenging nonlinear initial boundary value problems, the mathematical statements of which involve one or more nonlinear parabolic equations, governing the diffusion of heat and the diffusion of a chemical concentration, and the time-harmonic version of Maxwell's equations. These equations are coupled in a nonlinear fashion and, in addition, nonlinear heat balances are required in certain applications where surface temperatures produce significant thermal radiation. The investigator proposes to develop and use asymptotic methods, dictated by certain physical limits, to simplify the mathematical systems and to develop and use numerical methods to solve the reduced problems. The goals of the proposer are twofold. The first is the mathematical understanding of the basic physical mechanisms which make the processes work. Such an understanding would give scientists and engineers involved in microwave processing of ceramics the basic scientific underpinnings by to judge and interpret their experimental findings. The second is the development and application of asymptotic, numerical, and other approximate methods required to obtain an accurate description of more realistic and complicated processes. The successful completion of the proposed research would allow engineers and scientists to design more efficient sintering and joining processes for ceramics which would produce higher quality materials. It would also help in the design and implementation of microwave assisted vapor deposition processes which are highly promising in the construction of composite ceramic materials.