Numerical Conformal Mappings From Multiply Connected Regions Onto Annular Domains With Slits

ZHAO Xin; LÜ Yibin

doi:10.21656/1000-0887.440134

Volume 45 Issue 2

Feb. 2024

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2024 > 45(2): 245-252

ZHAO Xin, LÜ Yibin. Numerical Conformal Mappings From Multiply Connected Regions Onto Annular Domains With Slits[J]. Applied Mathematics and Mechanics, 2024, 45(2): 245-252. doi: 10.21656/1000-0887.440134

Citation:

PDF( 2700 KB)

Numerical Conformal Mappings From Multiply Connected Regions Onto Annular Domains With Slits

doi: 10.21656/1000-0887.440134

ZHAO Xin,
LÜ Yibin^,

Faculty of Science, Kunming University of Science and Technology, Kunming 650500, P.R.China

Received Date: 2023-05-03
Rev Recd Date: 2023-11-06
Publish Date: 2024-02-01

Abstract

Abstract

A numerical method was proposed based on the charge simulation method for calculating conformal mappings from the bounded high connectivity regions onto unit annular domains with logarithmic spiral slits. The bi-conjugate residual (BiCR) method was used to solve the constraint equation system acquired with the Dirichlet boundary conditions, obtain the simulated charge, and further construct a high-precision approximate conformal mapping function. Numerical examples show the effectiveness of the proposed method applied to simulate flow over the spiral point vortex in the bounded high connectivity domains.
- conformal mapping,
- charge simulation method,
- bounded high connectivity regions,
- BiCR method,
- spiral point vortex

FullText(HTML)

References(34)

References

[1]	MURASHIGE S, CHOI W. Parasitic capillary waves on small-amplitude gravity waves with a linear shear current[J]. Journal of Marine Science and Engineering, 2021, 9(11): 1217. doi: 10.3390/jmse9111217
[2]	NASSER M, VUORINEN M. Computation of conformal invariants[J]. Applied Mathematics and Computation, 2021, 389(2): 125617.
[3]	孙烨丽, 沈璐璐, 杨博. 功能梯度板中Griffith裂纹尖端应力场的三维解析研究[J]. 应用数学和力学, 2021, 42(1): 36-48. doi: 10.21656/1000-0887.410143 SUN Yeli, SHEN Lulu, YANG Bo. 3D analytical solutions of stress fields at Griffith crack tips in functionally graded plates[J]. Applied Mathematics and Mechanics, 2021, 42(1): 36-48. (in Chinese) doi: 10.21656/1000-0887.410143
[4]	曾祥太, 吕爱钟. 含有非圆形双孔的无限平板中应力的解析解研究[J]. 力学学报, 2019, 51(1): 170-181. https://www.cnki.com.cn/Article/CJFDTOTAL-LXXB201901018.htm ZENG Xiangtai, LV Aizhong. Analytical stress solution research on an infinte plate contianing two non-circular holes[J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(1): 170-181. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-LXXB201901018.htm
[5]	李岩松, 陈寿根. 寒区非圆形隧道冻胀力的解析解[J]. 力学学报, 2020, 52(1): 196-207. https://www.cnki.com.cn/Article/CJFDTOTAL-LXXB202001018.htm LI Yansong, CHEN Shougen. Analytical solution of frost heaving force in non-circular cold region tunnels[J]. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(1): 196-207. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-LXXB202001018.htm
[6]	HAKULA H, NASYROV S, VUORINEN M. Conformal moduli of symmetric circular quadrilaterals with cusps[J]. Electronic Transactions on Numerical Analysis Etna, 2021, 54: 460-482. doi: 10.1553/etna_vol54s460
[7]	CONSTANTIN A, STRAUSS W, VARVARUCA E. Large-amplitude steady downstream water waves[J]. Communications in Mathematical Physics, 2021, 387(1): 237-266. doi: 10.1007/s00220-021-04178-9
[8]	SYMM G T. An integral equation method in conformal mapping[J]. Numerische Mathematik, 1966, 9(3): 250-258. doi: 10.1007/BF02162088
[9]	SYMM G T. Numerical mapping of exterior domains[J]. Numerische Mathematik, 1967, 10(5): 437-445. doi: 10.1007/BF02162876
[10]	SYMM G T. Conformal mapping of doubly-connected domains[J]. Numerische Mathematik, 1969, 13(5): 448-457. doi: 10.1007/BF02163272
[11]	AMANO K. Numerical conformal mappings of exterior domains based on the charge simulation method[J]. Information Processing Society of Japan Journal, 1988, 29(1): 62-72.
[12]	伍康, 吕毅斌, 石允龙, 等. 有界多连通区域数值保角变换的GMRES(m)法[J]. 应用数学和力学, 2022, 43(9): 1026-1033. doi: 10.21656/1000-0887.420305 WU Kang, LV Yibin, SHI Yunlong, et al. The GMRES(m) method for numerical conformal mapping of bounded multi-connected domains[J]. Applied Mathematics and Mechanics, 2022, 43(9): 1026-1033. (in Chinese) doi: 10.21656/1000-0887.420305
[13]	AMANO K, OKANO D. Numerical conformal mappings onto the canonical slit domains[J]. Theoretical and Applied Mechanics Japan, 2012, 60: 317-332.
[14]	AMANO K. A charge simulation method for numerical conformal mapping onto circular and radial slit domains[J]. SIAM Journal on Scientific Computing, 1998, 19(4): 1169-1187. doi: 10.1137/S1064827595294307
[15]	OKANO D, OGATA H, AMANO K, et al. Numerical conformal mappings of bounded multiply connected domains by the charge simulation method[J]. Journal of Computational and Applied Mathematics, 2003, 159(1): 109-117. doi: 10.1016/S0377-0427(03)00572-7
[16]	NASSER M M S. A boundary integral equation for conformal mapping of bounded multiply connected regions[J]. Computational Methods and Function Theory, 2009, 9(1): 127-143. doi: 10.1007/BF03321718
[17]	NASSER M M S, KALMOUN E M. Application of integral equations to simulate local fields in carbon nanotube reinforced composites[R/OL]. 2019[2023-11-06]. https://arxiv.org/pdf/1910.10614.pdf.
[18]	PORTER R M. An interpolating polynomial method for numerical conformal mapping[J]. SIAM Journal on Scientific and Statistical Computing, 2001, 23(3): 1027-1041. doi: 10.1137/S1064827599355256
[19]	HAKULA H, QUACH T, RASILA A. Conjugate function method and conformal mappings in multiply connected domains[J]. SIAM Journal on Scientific and Statistical Computing, 2019, 41(3): 1753-1776. doi: 10.1137/17M1124164
[20]	KOEBE P. Abhandlungen zur theorie der konformen abbildung Ⅳ: abbildung mehrfach zusamme-nhängender schlichter bereiche auf schlitzbereiche[J]. Acta Mathematica, 1916, 41: 305-344. doi: 10.1007/BF02422949
[21]	CROWDY D G, MARSHALL J S. Conformal mappings between canonical multiply connected domains[J]. Computational Methods and Function Theory, 2006, 6(1): 59-76. doi: 10.1007/BF03321118
[22]	KOKKINOS C A, PAPAMICHAEL N, SLDERIDIS A B. An orthonormalization method for the approximate conformal mapping of multiply-connected domains[J]. IMA Journal of Numerical Analysis, 1990, 10(3): 343-359. doi: 10.1093/imanum/10.3.343
[23]	MAYO A. Rapid methods for the conformal mapping of multiply connected regions[J]. Journal of Computational and Applied Mathematics, 1986, 14(1/2): 143-153.
[24]	REICHEL L. A fast method for solving certain integral equations of the first kind with application to conformal mapping[J]. Journal of Computational and Applied Mathematics, 1986, 14(1/2): 125-142.
[25]	SANGAWI A W K, MURID A H M, NASSER M M S. Linear integral equations for conformal mapping of bounded multiply connected regions onto a disk with circular slits[J]. Applied Mathematics and Computation, 2011, 218(5): 2055-2068.
[26]	WEN G C. Conformal Mappings and Boundary Value Problems[M]. American Mathematical Society, 1992.
[27]	SOGABE T, SUGIHARA M, ZHANG S L. An extension of the conjugate residual method to nonsymmetric linear systems[J]. Journal of Computational and Applied Mathematics, 2008, 226(1): 103-113.
[28]	MUROTA K. Comparison of conventional and "invariant" schemes of fundamental solutions method for annular domains[J]. Japan Journal of Industrial and Applied Mathematics, 1995, 12(1): 61-85. doi: 10.1007/BF03167382
[29]	NEILL D R, HAYATDAVOODI M, ERTEKIN R C. On solitary wave diffraction by multiple, inline vertical cylinders[J]. Nonlinear Dynamics, 2018, 91(2): 975-994. doi: 10.1007/s11071-017-3923-1
[30]	YAO W, JAIMAN R K. Stability analysis of the wake-induced vibration of tandem circular and square cylinders[J]. Nonlinear Dynamics, 2019, 95(1): 13-28. doi: 10.1007/s11071-018-4547-9
[31]	ZHANG Z, ZHANG X, GE Y. Motion-induced vortex shedding and lock-in phenomena of a rectangular section[J]. Nonlinear Dynamics, 2020, 102(4): 2267-2280. doi: 10.1007/s11071-020-06080-w
[32]	ROSSOW V J. Lift enhancement by an externally trapped vortex[J]. Journal of Aircraft, 1978, 15(9): 77-672.
[33]	LENTINK D, DICKSON W B, VAN LEEUWEN J L, et al. Leading-edge vortices elevate lift of autorotating plant seeds[J]. Science, 2009, 324(5933): 1438-1440. doi: 10.1126/science.1174196
[34]	吕毅斌, 赖富明, 王樱子, 等. 基于GMRES(m)法的双连通区域数值保角变换的计算法[J]. 数学杂志, 2016, 36(5): 1028-1034. https://www.cnki.com.cn/Article/CJFDTOTAL-SXZZ201605017.htm LV Yibin, LAI Fuming, WANG Yingzi, et al. The GMRES(m) method for numerical conformal mapping of doubly-connected domain[J]. Journal of Mathematics, 2016, 36(5): 1028-1034. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-SXZZ201605017.htm