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RUI: Efficient Numerical Methods for Axisymmetric Problems

RUI：轴对称问题的高效数值方法

基本信息

批准号：
1913050
负责人：
Minah Oh
金额：
$ 10万
依托单位：
James Madison University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1913050&HistoricalAwards=false
关键词：
RUI Efficient Numerical Methods Axisymmetric

项目摘要

An axisymmetric problem is a problem defined on a three-dimensional (3D) domain that is symmetric with respect to an axis. These problems arise in various applications in the field of biomedical engineering, electromagnetism, and optics. An axisymmetric problem can be reduced to a sequence of two-dimensional (2D) problems by using cylindrical coordinates and a Fourier series decomposition. A discrete problem corresponding to a 2D problem is significantly smaller than a discrete problem corresponding to a 3D one, so such dimension reduction is an attractive feature considering computation time. The resulting 2D problem, however, is posed in weighted function spaces and is mathematically quite different from the analogous "standard" 2D problems, so special care is required when developing numerical methods that are well-fit for these weighted 2D problems. In this project, we will study efficient numerical techniques with solid mathematical support that can be applied to axisymmetric problems including those that arise in the treatment of various cancer treatments. The first goal of this project is to perform multigrid analysis for axisymmetric H(curl) and H(div) problems with general data including the axisymmetric time harmonic Maxwell equations. Multigrid for axisymmetric H(curl) and H(div) problems have been studied previously under the assumption that the data is independent of the rotational variable, which is not the case for most applications. Therefore, this project will bring new results for general axisymmetric problems with meaningful applications in Hepatic Microwave Ablation, an alternate treatment to liver, breast, bone, and lung cancer. Undergraduate students will be a part of evaluating the performance of multigrid in designing efficient antennas that can be used for these cancer treatments. Furthermore, this project will provide new mathematical tools to study axisymmetric problems with general data as well. The second goal of this project is to study axisymmetric state-constrained elliptic optimal control problems with axisymmetric data by using P1 finite element methods. There are very few studies done on axisymmetric optimal control problems, so this will be new and significant. The PI will also run a "Numerical Analysis Day" for local high school students with her undergraduate students from the James Madison University Association for Women in Mathematics Student Chapter.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

轴对称问题是在相对于轴对称的三维 (3D) 域上定义的问题。这些问题出现在生物医学工程、电磁学和光学领域的各种应用中。通过使用柱坐标和傅里叶级数分解，轴对称问题可以简化为一系列二维 (2D) 问题。与 2D 问题相对应的离散问题明显小于与 3D 问题相对应的离散问题，因此考虑到计算时间，这种降维是一个有吸引力的特征。然而，由此产生的二维问题是在加权函数空间中提出的，并且在数学上与类似的“标准”二维问题有很大不同，因此在开发非常适合这些加权二维问题的数值方法时需要特别小心。在这个项目中，我们将研究具有坚实数学支持的高效数值技术，这些技术可应用于轴对称问题，包括在各种癌症治疗中出现的问题。该项目的第一个目标是使用包括轴对称时间调和麦克斯韦方程在内的一般数据对轴对称 H(curl) 和 H(div) 问题进行多重网格分析。轴对称 H(curl) 和 H(div) 问题的多重网格之前已经在假设数据独立于旋转变量的情况下进行了研究，但大多数应用情况并非如此。因此，该项目将为一般轴对称问题带来新的结果，并在肝脏微波消融（肝癌、乳腺癌、骨癌和肺癌的替代治疗）中具有有意义的应用。本科生将参与评估多重网格在设计可用于这些癌症治疗的高效天线方面的性能。此外，该项目还将提供新的数学工具来研究一般数据的轴对称问题。该项目的第二个目标是利用P1有限元方法研究轴对称数据下的轴对称状态约束椭圆最优控制问题。关于轴对称最优控制问题的研究很少，因此这将是新的且有意义的。 PI 还将与她来自詹姆斯麦迪逊大学女性数学学生分会的本科生一起为当地高中生举办“数值分析日”。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。