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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2026, Vol. 43 ›› Issue (2): 260-272.doi: 10.3969/j.issn.1005-3085.2026.02.004cstr: 32411.14.cjem.CN61-1269/O1.2026.02.004

• • 上一篇    下一篇

多连通区域到螺旋狭缝单位圆盘的共形映射计算法

赵  鑫1,2,  吕毅斌1,  伍  康1   

  1. 1. 昆明理工大学理学院，昆明 650500

    2. 新疆工业学院通识学院，和田 848000

  • 收稿日期:2023-06-21 接受日期:2024-06-23 出版日期:2026-04-15 发布日期:2026-06-15
  • 通讯作者: 吕毅斌 E-mail: lybwyz@126.com
  • 基金资助:
    国家自然科学基金 (12461079)；新疆工业学院新青育苗 (硕士科研启动项目) (2025XQYM006).

Numerical Conformal Mapping of Multiply Connected Regions onto Unit Discs with Spiral Slits

ZHAO Xin1,2,  LÜ Yibin1, WU Kang1   

  1. 1. Faculty of Science, Kunming University of Science and Technology, Kunming 650500
    2. School of General Education, Xinjiang University of Technology, Hetian 848000
  • Received:2023-06-21 Accepted:2024-06-23 Online:2026-04-15 Published:2026-06-15
  • Contact: Y. Lü. E-mail address: lybwyz@126.com
  • Supported by:
    The National Natural Science Foundation of China (12461079); the Xinjiang University of Technology 2025 Annual ``Xin Qing Yu Miao" Program (2025XQYM006).

摘要：

共形映射在流体力学等领域发挥着重要作用，但在复杂区域内其解析表达式验证难以获取。基于模拟电荷法，提出了将多连通区域映射到带有对数螺旋狭缝的单位圆盘的数值共形映射计算方法。首先，该方法使用Laplace方程基本解的线性组合近似共形映射函数中的待定函数。其次，根据边值条件建立关于该线性组合中系数的约束方程组，并利用广义极小残差法求解约束方程组得到模拟电荷，进而构造了高精度的近似共形映射函数。根据问题域内的复势可以由正则域上的复势经过共形映射得到，进一步模拟了有界多连通区域内螺旋点涡的绕流。数值实验验证了所提算法的有效性，并给出了有界多连通区域内螺旋点涡的流线模拟。

关键词: 多连通区域, 模拟电荷法, 广义极小残差法, 数值共形映射, 螺旋点涡

Abstract:

Conformal mapping plays an important role in fluid such as fluid mechanics, yet its analytical expression is difficult to derive for complex domains. This paper proposes a numerical method based on the charge simulation method for calculating conformal mappings from the multiply connected regions onto disk with logarithmic spiral slit domains. First, the proposed method approximates the undetermined function in the conformal mapping function by using the linear combination of the fundamental solutions of the Laplace equation. Second, we establish the constraint equation system about the coefficients in the linear combination according to the boundary value conditions. The generalized minimal residual (GMRES(m)) method is used to solve the constraint equation system and obtain the charge, which constructs a high-precision approximate conformal mapping function. Because the complex potential in the problem domain can be derived from that in the regular region via conformal mapping, the flow around of the spiral point vortex in the bounded multiply connected region is further simulated. Numerical experiments show the effectiveness of the proposed algorithm and simulate the spiral point vortex in the bounded multiply connected region.

Key words: multiply connected regions, charge simulation method, GMRES(m) method, numerical conformal mapping, spiral point vortex

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