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Modified Newton-Raphson Methods

Although the Newton-Raphson 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 Newton-Raphson 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 Newton-Raphson are:

KT0 method: The initial stiffness matrix is used exclusively
KT1 method: The stiffness matrix is updated on the first iteration of each increment only
KT2 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 KT1 procedure is shown in the following figure.

Nonlinear_Modified_Newton.gif (4350 bytes)

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.

Selection and Implications

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…


Modified Newton-Raphson

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


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