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

项目名称： 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

项目编号： No.11461037

项目类型： 地区科学基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 吕毅斌

作者单位： 昆明理工大学

项目金额： 36万元

中文摘要： 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。

中文关键词： 数值保角变换；模拟电荷法；有理逼近；广义本征值问题；流体分析

英文摘要： 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.

英文关键词： Numerical conformal mapping;Charge simulation method;Rational approximation;Generalized eigenvalue problem;Fluid analysis