User Area > Advice
Although the NewtonRaphson iteration procedure is stable and converges
quadratically (provided the initial estimate is reasonably close to the solution), it has
the disadvantage that the tangent stiffness matrix requires computationally expensive
inversion during each iteration. Also, it may fail to converge when extreme material
nonlinearities are present in a structure. For this case modified NewtonRaphson iteration
procedures may be more effective.
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 NewtonRaphson are:
 K_{T}^{0} method: The initial stiffness matrix is used exclusively
 K_{T}^{1} method: The stiffness matrix is updated on the first iteration
of each increment only
 K_{T}^{2} method: The stiffness matrix is updated on the first and
second iterations of each increment
If arclength 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_{T}^{1}
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. NewtonRaphson iteration is more
effective for geometrically nonlinear problems than modified
Newton iteration.
The choice of standard or modified NewtonRaphson 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
NewtonRaphson is generally recommended  particularly for geometrically nonlinear
analyses.
A comparative table of the two methods is as follows…
NewtonRaphson 
Modified
NewtonRaphson 
Iterative stiffness matrix updates 
Less frequent stiffness matrix updates 
Converges rapidly (quadratically) 
Converges more slowly (linearly) 
Generally requires few iterations per increment to converge 
Generally requires more iterations per increment to
converge 
Computationally expensive per iteration 
Computationally inexpensive per iteration 
May fail under extreme material nonlinearity (e.g. brittle
cracking) 
May be essential for extreme material nonlinearity 

May be assisted using additional "iterative
acceleration" techniques 
