RUI: Efficient Numerical Methods for Axisymmetric Problems

轴对称问题是定义在相对于轴对称的三维（3D）域上的问题。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。一个轴对称的问题可以减少到一系列的二维（2D）的问题，通过使用圆柱坐标和傅立叶级数分解。对应于2D问题的离散问题明显小于对应于3D问题的离散问题，因此考虑到计算时间，这种降维是有吸引力的特征。然而，由此产生的二维问题，是在加权函数空间，是从类似的“标准”的二维问题数学上有很大的不同，所以需要特别小心时，开发的数值方法，很好地适合这些加权二维问题。在这个项目中，我们将研究有效的数值技术与坚实的数学支持，可以应用于轴对称问题，包括那些出现在各种癌症治疗的治疗。 这个项目的第一个目标是执行多重网格分析的轴对称H（旋度）和H（div）的一般数据，包括轴对称时间谐波麦克斯韦方程组的问题。多重网格的轴对称H（旋度）和H（div）问题已经研究了以前的假设下，数据是独立的旋转变量，这是不是大多数应用程序的情况下。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（肝脏、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，本计画也将提供新的数学工具来研究一般资料下的轴对称问题。 本计画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型最优控制问题。目前对轴对称最优控制问题的研究还很少，因此这将是一个新的和有意义的问题。PI还将为当地高中学生举办“数值分析日”，她的本科生来自詹姆斯麦迪逊大学数学学生协会。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.