Div-curl systems and Variational principles

Div-curl 系统和变分原理

• 批准号: 0808115
• 负责人: Giles Auchmuty
• 金额: $ 9万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-08-01 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0808115&HistoricalAwards=false
• 关键词: Div; curl; systems; Variational; principles

This project will support the investigator's analysis of certain basic systems of equations of interest in science and engineering. Our primary topic will be boundary value problems for div-curl system on bounded regions in space. Both Maxwell's equations for an electromagnetic field and some problems in fluid mechanics are governed by this system with certain associated boundary conditions. There still are many open questions about these systems and their solutions. The proposed research centers on how to describe the mathematical properties, and approximation, of solutions of these boundary value problems. This information is needed for the development of good algorithms for the computation of these fields or flows -- and should have wide applicability for the numerical modeling of devices and equipment.Mathematically, these div-curl systems are over-determined systems of equations that only have solutions when certain compatibility conditions hold -- and often require extra data in addition to standard boundary conditions. These extra conditions include both natural analytical conditions and some subtle conditions that arise from geometrical considerations. They sometimes are known to experts on these subjects, but their statements have previously only been vaguely stated, since the precise versions require considerable geometrical and analytical detail. They have been carefully described in recent papers of the PI and his collaborators. This proposal is to support the implementation, and further development, of these results in some models that arise in applications. The proposed research will use the variational characterization of the solutions to obtain sharp results about the finite energy solutions of these systems in a number of important geometrical situations. We also intend to investigate the development of good methods for the numerical approximation of the solutions in various different geometrical configurations. A particular interest will be the analysis of the solution of certain models of ferromagnetic bodies and electrostatic discharges.

该项目将支持研究人员对科学和工程领域感兴趣的某些基本方程组的分析。我们的主要主题是空间有界区域上 div-curl 系统的边值问题。 电磁场的麦克斯韦方程和流体力学中的一些问题都由具有某些相关边界条件的系统控制。关于这些系统及其解决方案仍然存在许多悬而未决的问题。拟议的研究重点是如何描述这些边值问题的解的数学属性和近似值。这些信息是开发计算这些场或流的良好算法所必需的，并且应该对装置和设备的数值建模具有广泛的适用性。从数学上讲，这些 div-curl 系统是超定方程组，只有在满足某些兼容性条件时才有解，并且除了标准边界条件之外，通常还需要额外的数据。这些额外条件包括自然分析条件和出于几何考虑而产生的一些微妙条件。有时这些主题的专家都知道它们，但它们的陈述以前只是含糊其辞，因为精确的版本需要大量的几何和分析细节。 PI 及其合作者最近的论文中对它们进行了仔细的描述。该提案旨在支持这些结果在应用中出现的一些模型中的实施和进一步开发。拟议的研究将利用解的变分表征来获得有关这些系统在许多重要几何情况下的有限能量解的清晰结果。我们还打算研究开发各种不同几何配置中解的数值近似的良好方法。特别感兴趣的是对铁磁体和静电放电的某些模型的解决方案的分析。