基于有理逼近的数值保角变换模拟电荷法及其流体分析研究

With a strong vitality, conformal mapping theory has been widely applied in the fluid dynamics and many other fields. Computing conformal mapping is a very difficult question, but by the charge simulation method, may be transformed into simple, high-precision approximate mapping function, which greatly simplifies the computational complexity. In recent years, Mathematical researchers have achieved many good results on the studies of the charge simulation method for the numerical conformal mapping. However, the charge simulation method merely gives the location of the charge points through experience. However, calculation results with high-precision only can be achieved through using the amount of charge points. As for this situation, based on rational approximation theory, the project is intended to build charge calculation modes that can automatically determine the charge points, and calculate a few charge points through solving the high-precision method of the generalized eigenvalue problem. Then we propose new method with high-speed and high-precision, which can calculate the conformal mapping by using a few charge points. This project is to verify the validity of new method through the conformal mapping of the typical graphical numerical experiments. Then the new method is applied to the fluid dynamics field to calculate the vortex of the fluid analysis with high-speed and high-precision so as to verify the practicality of the new method, providing the basis for the new method to be widely applied in the electromagnetic theory and other areas. Therefore, this project has high academic value.

保角变换理论在流体力学等许多领域中有着广泛的应用，具有强大的生命力。计算保角变换是个很困难的问题，但通过模拟电荷法，可以构成表现简单、精度高的近似变换函数，从而大大简化计算的复杂度。近年来，数学研究者对数值保角变换计算的模拟电荷法的研究已经取得了许多好的结果。但是，模拟电荷法只能通过经验给定模拟电荷点的位置，只有利用大量的模拟电荷才能得到高精度的计算结果。针对这个情况，本项目基于有理逼近理论，建立自动决定模拟电荷点的电荷求解理论模型，利用求解广义本征值问题的高精度算法计算少数模拟电荷点，进而提出利用少数模拟电荷点计算保角变换的高速高精度的新算法。本项目通过一些典型图形保角变换的数值实验验证新算法的有效性，然后将新算法应用到流体力学领域中，高速高精度地计算流体分析中的各种涡流以验证新算法的实用性，为新算法在电磁理论等其他领域里更加广泛地应用提供理论依据，具有较高的学术研究价值。

保角变换理论在流体力学等许多领域中有着广泛的应用。近年来数学研究者对数值保角变换计算的模拟电荷法的研究做了很多工作。本项目主要目标是针对数值保角变换的模拟电荷法中的电荷求解理论模型，在广义本征值问题上建立求解模拟电荷的高精度算法，进而构造高精度的近似保角变换函数，发展行之有效的新算法。项目组采用对数值保角变换模拟电荷法中的电荷求解问题转化为求解约束方程的问题，阐述了约束方程随着电荷数量增多的性质变化情况，然后应用有理逼近等高精度地求解约束方程。本项目所发展的算法改良了约束方程的性质，构造了高精度的近似保角变换函数，得到了高精度的保角变换计算效果。本项目取得了一系列的成果，包括：1）在单连通区域数值保角变换模拟电荷法研究上，项目组研究了数值保角变换计算的LMS算法及基于BiCR算法的数值保角变换计算法。其中关键性的途径是采用改良约束方程中模拟电荷点的矩阵，提高了模拟电荷的计算精度，进而改善了保角变换计算结果。2）在多连通区域数值保角变换模拟电荷法的研究上，项目组研究了多连通区域中双连通区域数值保角变换模拟电荷法的约束方程并发展了有理逼近迭代法和GMRES法等，高速高精度地计算了双连通区域数值保角变换。3）在流体分析中涡流计算的等高线等研究上，项目组进行了流体分析中涡流计算的等高线数值模拟及保角变换的图像处理和相关数学公式编辑器的研究，验证了新算法的实用性，为新算法在流体力学等领域里的应用提供了理论依据。项目组成员发表期刊及会议论文14篇，申请了3项计算机软件著作权，完成了预期研究目标。