Mathematical Sciences: On a Class of Degenerate Systems Arising from the Electrical Heating of Conductors

数学科学：关于由导体电加热引起的一类简并系统

• 批准号: 9424448
• 负责人: Xiangsheng Xu
• 金额: $ 3万
• 依托单位: University of Arkansas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-06-01 至 1997-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9424448&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Class; Degenerate; Systems

9424448 Xu Many industrial processes involve the electrical heating of a conductor. A typical example is the electrical spot welding of metal sheets in the manufacturing of such products as automobile bodies, tanks, and armored vehicles. The mathematical modeling of such processes often leads to systems of degenerate partial differential equations with diffusive and convective couplings. The study of these systems leads to deep and intriguing questions in nonlinear analysis. In these considerations, the presence of strong nonlinearities and degeneracy gives rise to very rich structures of solutions. Our recent research findings indicate that solutions of these systems display new phenomena that cannot be incorporated into the classical variational formulation, and a new notion of a capacity solution is, therefore, introduced to study these systems. In this proposal, we propose to study the finer properties of capacity solutions, such as regularity, partial regularity, large time behavior, and other related questions. The proposed research is important from the point of view of industrial applications. The mathematical results we wish to obtain can lead to a more efficient production line, a new or better product. For example, in the welding of metal sheets, one of our objectives is to find out how the welding time is related to the material properties and the current applied. This information is critical to the automation of the welding procedure.

小行星9424448 许多工业过程涉及导体的电加热。 一个典型的例子是在汽车车身、坦克和装甲车辆等产品的制造中对金属板进行电点焊。这种过程的数学建模往往导致系统的退化偏微分方程的扩散和对流耦合。对这些系统的研究引出了非线性分析中一些深刻而有趣的问题。在这些考虑中，强非线性和简并性的存在会产生非常丰富的解结构。 我们最近的研究结果表明，这些系统的解决方案显示新的现象，不能纳入经典的变分提法，和一个新的概念的容量解决方案，因此，引入研究这些系统。在这个提议中，我们提出研究容量解的更精细的性质，如正则性、部分正则性、大时间行为以及其他相关问题。 从工业应用的角度来看，所提出的研究是重要的。我们希望获得的数学结果可以导致更有效的生产线，新的或更好的产品。例如，在金属板的焊接中，我们的目标之一是找出焊接时间与材料特性和施加的电流之间的关系。这些信息对于焊接过程的自动化至关重要。