Mathematical Sciences: "Microwave Processing of Ceramic Materials"

提出了对陶瓷微波加工过程中出现的一类物理问题进行建模和分析的方法。这些问题产生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，控制热量的扩散和化学浓度的扩散，以及麦克斯韦方程的时间调和版本。这些方程以非线性方式耦合在一起，此外，在表面温度产生显著热辐射的某些应用中需要非线性热平衡。建议发展和使用在一定物理极限下的渐近方法来简化数学系统，并发展和使用数值方法来解决约简问题。该项目的目标是双重的。第一个是对基本物理机制的数学理解，这些机制使这些过程工作。这样的理解将给参与微波陶瓷加工的科学家和工程师提供基本的科学依据，以判断和解释他们的实验结果。第二个是渐近、数值和其他近似方法的发展和应用，这些方法是获得更现实和复杂过程的准确描述所必需的。这项研究的成功完成将使工程师和科学家能够设计出更有效的陶瓷烧结和连接工艺，从而生产出更高质量的材料。这也将有助于微波辅助气相沉积工艺的设计和实施，这在复合陶瓷材料的构建中非常有前途。

9623543 Kriegsmann It is proposed 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 together in a nonlinear fashion and, in addition, nonlinear heat balances are required in certain applications where surface temperatures produce significant thermal radiation. It is proposed 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 project 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 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.