Modified Newton-Raphson iteration

A number of iterative methods exist, as described in Appendix L. TK Solver uses a modified Newton-Raphson iterative procedure (see Sec. L.2), which is satis-fectory for a wide variety of problems. 

Equations (18) and (19) correspond to the Newton-Raphson iteration. If the left-hand side (i.e., [ ]) is not updated during the iteration, the iterative scheme is known as the modified Newton-Raphson iteration. 

Various procedures are available for accelerating the convergence of the modified Newton-Raphson iterations. Figure AIE.l shows the technique of computing individual acceleration factors, and <52 are known. Then, assuming a constant slope of the response curve, and from similar triangles, the value of <53 is computed ... 

The values of the functions Uf, V ,and Wf have to be calculated at discrete points and 10 elements with 6 collocation points per element were found to suffice. Rather than solve (15) and (16) separately it was found best to use collocation on these also, using four radial collocation points. This gives 459 simultaneous nonlinear algebraic equations which were solved by a modified Newton-Raphson Iteration. The modifications were the use of under-relaxation and less frequent evaluation of the Jacobian. 

Crisfield MA (1979) A faster modified Newton-Raphson iteration. Comput Methods Appl Mech Eng... 

Usually, modified Newton-Raphson methods with relaxation are applied. Additional iteration loops are necessary for the determination of the dynamic pressure losses in ducts and duct fittings. 

The quantities x, and yj are the iterants, whereas gi and g2 are formed exactly the way Equation 9.5 was developed. Two common methods for finding roots to nonlinear systems are (1) Newton-Raphson and (2) the modified Newton-Raphson. Both approaches are briefly discussed in the subsections below. 

The result (10.25) is a nonlinear equation. We can introduce a modified Newton-Raphson method (Owen and Hinton 1980) for solving (10.25). Then we rewrite (10.25), and (10.26)-( 10.29)by inttoducing variables with superscripts k and k — l, which implies the values of the variables at each iteration step, and we have... 

The simplest numerical method for a detailed geometrically and material nonlinear (GMN) analysis is the Newton-Raphson scheme (Crisfield 1979 Bathe 1995), which can be found in three forms (i) the full Newton-Raphson, which is the most accurate, but also the most time consuming, since the tangent stiffness of the structure has to be calculated and factorized within each iteration in the solution procedure (ii) the modified Newton-Raphson, which differs from the full Newton-Raphson in that the calculation and the factorization of the tangent stiffness matrix take place only in some iterations within each step, thus requiring in most cases a larger number of iterations per step but... 

The Newton-Raphson method may be computationally expensive in a multi dof problem because a new global stiffness matrix is used in each iterative step. In the Modified Newton-Raphson method, the same global stiffness matrix is used in all the iterative steps within an increment. This method requires more iterations to achieve convergence but each iteration is computed far more quickly. 

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