解非线性方程的自动调节阻尼法

解非线性方程组的一般方法是将其线性化,形成各种形式的迭代程序进行数值近似计算.对于复杂强非线性问题,在迭代过程中往往不易收敛,甚至数值失稳而发散.不能满足工程要求.常规的牛顿法及改进的牛顿法均未彻底解决这一问题,因而使得复杂强非线性问题的数值模拟计算受到了限制.本文提出一种新的方法---自动调节阻尼法,是对带阻尼因子的牛顿法的进一步改进.引进阻尼因子向量,在迭代过程中,通过判断与调整,不断地自动调节阻尼因子向量,引用有效收敛系数与加速系数,改善对赋初值的要求,加速求解的迭代过程,保证了复杂强非线性方程求解的稳定性.采用这一新的方法,已成功地数值模拟了飞机中的一些复杂的传热问题,可进一步推广用于非线性流动、传热、结构动力响应等各种复杂强非线性的工程问题的数值模拟计算.

The Auto_Adjustable Damping Method for Solving Nonlinear Equations

The general approach for solving the nonlinear equations is linearizing the equations and forming various iterative procedures,then executing the numerical simulation.For the strongly nonlinear problems,the solution obtained in the iterative process is always difficult,even divergent due to the numerical instability.It can not fulfill the engineering requirements.Newton's method and its variants can not settle this problem.As a result,the application of numerical simulation for the strongly nonlinear problems is limited.An auto_adjustable damping method has been presented in this paper.This is a further improvement of Newton's method with damping factor.A set of vector of damping factor is introduced.This set of vector can be adjusted continuously during the iterative process in accordance with the judgement and adjustment.An effective convergence coefficient and quichening coefficient are employed to relax the restricted requirements for the initial values and to shorten the iterative process.Then,the numerical stability will be ensured for the solution of complicated strongly nonlinear equations.Using this method,some complicated strongly nonlinear heat transfer problems in airplanes and aeroengines have been numerically simulated successfully.It can be used for the numerical simulation of strongly nonlinear problems in engineering such as nonlinear hydrodynamics and aerodynamics,heat transfer and structural dynamic response etc.