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研究生:蔡尚融
研究生(外文):Shang-Rong Cai
論文名稱:以平行Newton-Krylov-Schwarz演算法解Poisson-Boltzmann方程式的有限元素解在膠體科學上的應用
論文名稱(外文):Parallel Newton-Krylov-Schwarz Algorithms for Finite Element Solution of Three Dimensional Poisson-Boltzmann Equations with Applications in Colloidal Science
指導教授:黃楓南
指導教授(外文):Feng-Nan Hwang
學位類別:碩士
校院名稱:國立中央大學
系所名稱:數學研究所
學門:數學及統計學門
學類:數學學類
論文出版年:2008
畢業學年度:96
語文別:英文
論文頁數:28
中文關鍵詞:非精確牛頓法overlapped Schwarz preconditioning有限元素法膠體科學三維模擬Poisson-Boltzmann 方程式
外文關鍵詞:Poisson-Boltzmann equation3D simulationcolloidal scienceinexact Newtonoverlapped Schwarz preconditioningfinite element methodparallel processing
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運用平行的Newton-Krylov-Schwarz 演算法,求大型稀鬆非線性方程式組的解, 此非線性系統是介由有限元素法,作離散化在三維的Poisson-Boltzmann 方程式; 於膠質科學的應用中,做帶電膠質微粒在電解液中的三維數值模擬。Poisson-Boltzmann 方程式, 為描述帶電膠體粒子於電解液中,其電位能分佈情形的方程式。並進行關於平行效能的研究, 討論使用 LU 和 ILU 作為不同的preconditioner 的條件下作比較,其結果顯示, 在使用32 顆處理器時,可達的58 % 效率。
We employ the Newton-Krylov-Schwarz algorithms for solving a large sparse nonlinear system of equations arising from the finite element discretization of three dimensional Poisson-Boltzmann equation (PBE) in the application in colloidal science. The method do the numerical simulation in three dimensional space for the charged colloidal particles in a electrolyte. The PBE is used to describe the distribution of electrostatic potential in a colloidal system. We validate our code by computing the electrostatic forces of their interactions on the charged colloidal particles, and the results agree with other published data. we also conduct parallel performance
study on a parallel machine, and the result shows that our code reachs 58% efficiency up to 32 processors.
List of Tables ii
List of Figures iii
Notations iv
1 Introduction 1
2 Poisson-Boltzmann model in a symmetric electrolyte 4
3 Finite element method for Poisson-Boltzmann equation 7
4 Newton-Krylov-Schwarz algorithm 9
4.1 Inexact Newton method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2 Krylov iterative methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.3 Overlapping Schwarz preconditioner . . . . . . . . . . . . . . . . . . . . . . . . . 10
5 Software developments 12
5.1 Toolkits and workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.2 Particles Interaction Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6 Numerical Results 17
6.1 Simulation domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6.2 Two Isolated Charged Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.3 Two Charged Particles in a Charged Cylindrical Pore . . . . . . . . . . . . . . . 21
6.4 Parameters study and parallel performance . . . . . . . . . . . . . . . . . . . . . 24
7 Conclusions 26
Bibliography 27
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