Tham khảo tài liệu 'volume 20 - materials selection and design part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Gaussian elimination is used to solve Eq 45 one finds that it is generally necessary to store in computer memory approximately N nonzero coefficients which is impossible in problems with a large number of cells. Thus iterative methods are usually used to solve the matrix problem Eq 45. Iterative solution methods calculate a sequence of approximations q that converge to the solution q. The exact solution is not obtained but one stops calculating q w hen either the difference between successive iterates q 1 - q or the residual Aq - s is acceptably small. In the past popular iterative methods have been point-successive relaxation line-successive relaxation and methods based on approximate decomposition of matrix A into a product of lower and upper triangular matrices that can each be easily inverted Ref 30 . Recently these methods have largely been supplanted by two methods that have greatly reduced the computer time to solve implicit equations and thereby have made implicit methods more attractive. These more recent methods are conjugate-gradient methods Ref 41 and multigrid methods Ref 42 . When nonlinear finite-difference equations are solved the above iterative methods can be used in conjunction with Newton s method Ref 43 . A nonlinear difference approximation can be written F q o Eq 46 where F is a vector-valued function of the vector of unknowns q. If q is the approximation to the solution q after k Newton-iteration steps then q1 q7 q is obtained by solving the matrix equation Bq -F q Eq 47 The matrix ÔF ỡq is called the Jacobian matrix. Equation 47 is of the form of Eq 45 and can be solved by one of the iterative methods for linear equations. Thus solution for q involves using an iteration within an iteration. As in the solution of nonlinear equations for single variables convergence is sometimes accelerated by under-relaxation that is one takes qi 1 q q where q is the solution to Eq 47 and is an underrelaxation factor whose value lies between zero and one. .
