RUI: Efficient Numerical Methods for Axisymmetric Problems

轴对称问题是在三维（3D）域上定义的问题，该问题相对于轴是对称的。这些问题出现在生物医学工程、电磁学和光学等领域的各种应用中。利用柱坐标和傅里叶级数分解，可以将轴对称问题简化为一系列二维问题。二维问题对应的离散问题明显小于三维问题对应的离散问题，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，因此在开发适合这些加权二维问题的数值方法时需要特别注意。在这个项目中，我们将在坚实的数学支持下研究有效的数值技术，这些技术可以应用于轴对称问题，包括各种癌症治疗中出现的问题。该项目的第一个目标是对轴对称H（旋度）和H（div）问题进行多网格分析，包括轴对称时调和麦克斯韦方程。对于轴对称H（旋度）和H（div）问题的多重网格，以前已经在假设数据与旋转变量无关的情况下进行了研究，但在大多数应用中并非如此。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有意义的应用。本科生将参与评估多重网格的性能，设计可用于这些癌症治疗的高效天线。此外，该项目还将为研究一般数据下的轴对称问题提供新的数学工具。本课题的第二个目标是利用P1有限元方法研究具有轴对称数据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，因此这将是一个新的和有意义的问题。PI还将与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”活动。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.