利用類牛頓動力學方法求解非線性病態系統並應用於邊界值問題 本文基於純量形式之同倫法提出廣義動力學方法以求解非線性代數方程問題之新方法。為推導出純量形式之同倫法，首先將向量幾何中之向量同倫函數取其向量長度，並透過內積以取平方範數，之後藉由引入虛擬時間函數後，便可推導得廣義動力學方法。藉由引入轉換矩陣之概念，本文所提出廣義動力學方法可轉換成三種不同之方法分別為動力牛頓法、動力無Jacobian反矩陣法、及流形指數收斂演算法等三個方法。同時藉由廣義動力學方法，使用特定時間函數，吾人亦可推導出傳統牛頓法，此外本文廣義動力學方法亦具有極大潛力推導其它動力學方法。 本研究所提出之新方法可求解大型及各種工程問題所衍生出之非線性病態系統並應用於邊界值問題。除此之外，本研究所提出之方法於計算過程中不需再行求解Jacobian矩陣之反矩陣，因此具較高之數值穩定性，同時亦節省大量數值運算之時間。數值案例之分析結果顯示，本研究所提出之新方法具高效率以求解非線性之問題，同時亦可明顯提升分析精度與收斂性，對於病態系統之問題與病態初始值之問題亦具有相當好之處理能力。 關鍵字：動力學方法，純量同倫函數，擬時間積分函數，牛頓法，動力無Jacobian反矩陣法，流形指數收斂演算法。 下載本文(Download paper) 2011_cmes.2011.076.083 Paper No. : CMES201107181934 Dynamical Newton-Like Methods for Solving Ill-Conditioned Systems of Nonlinear Equations with Applications to Boundary Value Problems by Cheng-Yu Ku, Weichung Yeih, Chein-Shan Liu published in “CMES: Computer Modeling in Engineering & Sciences”. Tech Science Press grants you permission to post this soft-reprint on your web-site, as well as distribute it, in a reasonable way, solely for the purpose of academic/research exchanges with your immediate colleagues in your research community. ABSTRACT: In this paper, a general dynamical method based on the construction of a scalar homotopy function to transform a vector function of non-linear algebraic equations (NAEs) into a time-dependent scalar function by introducing a fictitious time-like variable is proposed. With the introduction of a transformation matrix, the proposed general dynamical method can be transformed into several dynamical Newton-like methods including the Dynamical Newton Method (DNM), the Dynamical Jacobian-Inverse Free Method (DJIFM) and the Manifold-Based Exponentially Convergent Algorithm (MBECA). From the general dynamical method, we can also derive the conventionalNewton method using a certain fictitious time-like function. The formulation presented in this paper demonstrates a variety of flexibility with the use of different transformation matrices to create other possible dynamical methods for solving NAEs. These three dynamical Newton-like methods are then adopted for the solution of ill-conditioned systems of nonlinear equations and applied to boundary value problems. Results reveal that taking advantages of the general dynamical method the proposed three dynamical Newton-like methods can improve the convergence and increase the numerical stability for solving NAEs especially for the system of nonlinear problems involving ill-conditioned Jacobian or poor initial values which cause convergence problems. Keywords: dynamical method, scalar homotopy function, fictitious time-like function,Newton’s method, dynamical Jacobian-inverse free method, manifold-based exponentially convergent algorithm.]]>
