With modified Newton iterations the current tangent stiffness matrix is replaced with a previous stiffness matrix, say from the beginning of the increment. This reduces the numerical cost for each iteration since the inversion of the tangent stiffness matrix is not required for every iteration. Three common forms of modified Newton-Raphson are:

K_T⁰ method: The initial stiffness matrix is used exclusively

K_T¹ method: The stiffness matrix is updated on the first iteration of each increment only

K_T² method: The stiffness matrix is updated on the first and second iterations of each increment

If arc-length is to be used with modified methods, it is advisable to ensure that the stiffness is calculated at the beginning of the increment at least. The K_T¹ procedure is shown in the following figure.

The convergence rate of modified Newton iterations is not quadratic and the procedure often diverges. However, when coupled with the line search procedure it forms an iteration algorithm that is particularly suitable for structures exhibiting extreme material nonlinearity. Newton-Raphson iteration is more effective for geometrically nonlinear problems than modified Newton iteration.

The choice of standard or modified Newton-Raphson procedures is problem and resource dependent, although with the swiftly advancing computer technology, issues of computational resources are becoming increasingly less important and the standard Newton-Raphson is generally recommended - particularly for geometrically nonlinear analyses.

A comparative table of the two methods is as follows…