矩形到任意多边形区域的Schwarz-Christoffel变换数值解法  

Numerical Solution of Schwarz-Christoffel Transformation From Rectangles to Arbitrary Polygonal Domains

作　　者：王玉风 姬安召 崔建斌 WANG Yufeng;JI Anzhao;CUI Jianbin(School of Energy Engineering,Longdong University,Qingyang,Gansu 745000,P.R.China)

机构地区：[1]陇东学院能源工程学院,甘肃庆阳745000

出　　处：《应用数学和力学》2019年第1期75-88,共14页Applied Mathematics and Mechanics

基　　金：甘肃省自然科学基金(1606RJZM092; 17JR5RM355);甘肃省高等学校科研项目(2017B-61)

摘　　要：运用Schwarz-Christoffel变换方法,建立多边形区域到带状区域共形映射数学模型.对于模型中的约束条件和奇异积分问题,根据Riemann(黎曼)原理,建立复参数与实参数互逆变换,消除非线性系统的约束条件;经过合理积分路径的确定,模型中的奇异积分转化为Gauss-Jacobi(高斯-雅可比)型积分;采用Levenberg-Marquardt算法对非线性系统模型进行求解.根据第一类椭圆函数性质,建立了矩形区域到带状区域共形映射数学模型,通过复参数椭圆函数的计算,得到矩形边界与带状区域边界的关系.最后,对8点对称多边形区域与27点不规则条带状区域计算,将不规则封闭区域边界映射到矩形区域边界,矩形区域内的正交网格,通过变换之后在多边形区域内依然满足正交性,为研究不规则区域到规则区域映射的数值计算奠定基础.With the Schwarz-Christoffel transformation method,a mathematical model of conformal mapping from polygonal domains to strip domains was established. For constraint conditions and singular integral problems in the model,the reciprocal transformation between complex parameters and real parameters was conducted based on the R iemann principle,which eliminates constraint conditions of the nonlinear system. By means of reasonable integration paths,the singular integral in the model was transformed into the Gauss-Jacobi integral,and the nonlinear system model was solved with the Levenberg-Marquardt algorithm. According to the first class elliptic function characteristics,the mathematical model of conformal mapping from rectangular domains to strip domains was built,and the relationship between the rectangular boundary and the strip boundary was obtained through calculation of the complex parameter elliptic function. At last,an 8-point polygonal domain and a 27-point irregular strip domain were calculated to map the irregular closed domain boundary to the rectangular domain boundary. The orthogonal grid in the rectangular domain still meets orthogonality in the polygonal domain after mapping. This study provides a foundation for numerical calculation of mapping from irregular domains to regular ones.

关 键 词：Schwarz-Christoffel变换 复参数椭圆函数 LEVENBERG-MARQUARDT算法 矩形区域 多边形区域 

分 类 号：O242.1[理学—计算数学]