Modified Newton Algorithm

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This command is used to construct a ModifiedNewton algorithm object, which uses the modified newton-raphson algorithm to solve the nonlinear residual equation. The command is of the following form:

algorithm ModifiedNewton <-secant> <-initial>

-secant	optional flag to indicate to use secant stiffness iterations.
-initial	optional flag to indicate to use initial stiffness iterations.

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NOTES:

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THEORY:

The theory for the ModifiedNewton method is similar to that for the Newton-Raphson method. The difference is that the tangent at the initial guess is used in the iterations, instead of the current tangent. The Modified Newmark method is thus an iterative method in which, starting at a good initial guess <math>U_0</math> we keep iterating until <math>\Delta U</math> is small enough using the following:

<math> \Delta U = - K_0^{-1}R(U_n),\!</math>

<math> U_{n+1} = U_n + \Delta U\,\!</math>

where:

<math>K_0 = \frac{\partial R(U_0)}{\partial U}\,\!</math>

The advantage of this method over the regular Newton method, is that the system Jacobian is formed only once at the start of the step and factored only once if a direct solver is used. The drawback of this method is that it requires more iterations than Newton's method.

note: when -initial flag is provided <math>K_0</math> is Jacobian from undeformed configuration.

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Code Developed by: fmk