代用電荷法による非有界な多重連結領域の統一的な数値等角写像の方法 A Systematic Scheme of Numerical Conformal Mappings of Unbounded Multiply-connected Domains by the Charge Simulation Method

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平行な直線スリット,原点を中心とする同心円弧状の曲線スリット,原点を中心とする放射状の直線スリットをともなう複素平面の全体をそれぞれ平行スリット領域,円弧スリット領域,放射スリット領域と呼ぶ.本論文では,ポテンシャル問題の高精度高速解法として知られている代用電荷法を適用して,与えられたいくつかのJordan閉曲線の外側の非有界な多重連結領域から,平行スリット領域,円弧スリット領域,放射スリット領域への統一的な数値等角写像の方法を提案し,その有効性を数値実験的に検証する.具体的には,これらの等角写像の問題を1対の共役な調和関数を求める問題に帰着させ,それらの調和関数を複素対数ポテンシャルの1次結合で近似する.最終的には,すべての問題が同じ係数行列を持つ連立1次方程式に帰着し,電荷数の3乗の計算量を要する $LU$ 分解は1度行えばよい.近似写像関数は複素対数関数の数値計算に主値を用いて連続であり,座標のスケール変換に対して自然な不変性を示す.これらの等角写像は2次元ポテンシャル流解析への応用上重要な広く知られた問題であり,簡単で精度の高い近似写像関数の構成法は理論と応用をつなぐべきmissing linkであった.?vspace*{-2mm}The parallel,circular and radial slit domains are the entire planes with parallel rectilinear slits,circular slits concentric to the origin and radial slits pointing at the origin, respectively.We here present a simple method of numerical conformal mappings of an unbounded multiply-connected domain exterior to closed Jordan curves onto the parallel,circular and radial slit domains.These conformal mappings are familiar in science and engineering,and especially important in problems of two-dimensional potential flows around obstacleswith vortices and point-sources or sinks together with a uniform flow.But, no systematic method of computation has been established.We reduce the problems of conformal mapping to the Dirichlet problemwith a pair of conjugate harmonic functions and employ the charge simulation method,where the conjugate harmonic functions are approximatedby a linear combination of complex logarithmic potentials.The problems are finally reduced to a set of linear equations with a same coefficient matrix.The approximate mapping functions are continuous and analytic using the principal value of logarithmic function,and invariant to the scaling of the coordinate system.The numerical method for simple,accurate approximate mapping functions was a {\it missing link\/} between theory and applications of the conformal mappings.\vspace*{-6mm}

The parallel, circular and radial slit domains are the entire planes with parallel rectilinear slits, circular slits concentric to the origin and radial slits pointing at the orgin, respectively. We here present a simple method of numerical conformal mappings of an unbounded multiply-connected domain exterior to closed Jordan curves onto the parallel, circular and radial slit domains. These conformal mappings are familiar in science and engineering, and especially important in problems of two-dimensional potential flows around obstacles with vortices and point-sources or sinks together with a uniform flow. But, no systematic method of computation has been established. We reduce the problems of conformal mapping to the Dirichlet problem with a pair of conjugate harmonic functions and employ the charge simulation method, where the conjugate harmonic functions are approximated by a linear combination of complex logarithmic potentials. The problems are finally reduced to a set of linear equations with a same coefficient matrix. The approximate mapping functions are continuous and analytic using the principal value of logarithmic function, and invariant to the scaling of the coordinate system. The numerical method for simple, accurate approximate mapping functions was a missing link between theory and applications of the conformal mappings.

収録刊行物

  • 情報処理学会論文誌

    情報処理学会論文誌 42(3), 385-395, 2001-03-15

    一般社団法人情報処理学会

参考文献:  33件中 1-33件 を表示

被引用文献:  6件中 1-6件 を表示

各種コード

  • NII論文ID(NAID)
    110002725679
  • NII書誌ID(NCID)
    AN00116647
  • 本文言語コード
    JPN
  • 資料種別
    Journal Article
  • ISSN
    1882-7764
  • NDL 記事登録ID
    5701771
  • NDL 雑誌分類
    ZM13(科学技術--科学技術一般--データ処理・計算機)
  • NDL 請求記号
    Z14-741
  • データ提供元
    CJP書誌  CJP引用  NDL  NII-ELS  IPSJ 
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