Newton
The NewtonRaphson algorithm uses the Newton-Raphson
algorithm to solve the nonlinear residual equation. The Newton-Raphson
method is the most widely used and most robust method for solving
nonlinear algebraic equations. The command is of the following form:
algorithm Newton < -initial > < -initialThenCurrent >|
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optional flag to indicate to use initial stiffness iterations |
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optional flag to indicate to use initial stiffness on first step, then use current stiffness for subsequent steps |
References
Theory
The Newton-Raphson method is an iterative method where, starting at a good initial guess \(x_0\,\!\) we keep iterating until our convergence criteria is met with the following:
\[\Delta x = - \frac{df(x_n)}{dx}^{-1}f(x_n)\,\!\]
\[ x_{n+1} = x_n + \Delta x\,\!\]
The method is generalized to n unknowns by replacing the above scalar equations with matrix ones.
\[R(U_n+\Delta x) = R(U_n)+\frac{\partial R(U_n)}{\partial U} \Delta U + O(\Delta U ^2) \,\!\]
The matrix \(\frac{\partial R(U_n)}{\partial U}\,\!\) is called the system Jacobian matrix and will be denoted \(K\):
\[K = \frac{\partial R(U_n)}{\partial U}\,\!\]
resulting in our iterative procedure where starting from a good initial guess we iterate until our convergence criteria is met with the following:
\[ \Delta U = - K^{-1}R(U_n),\!\]
\[ U_{n+1} = U_n + \Delta U\,\!\]
Code developed by: fmk