***resolution newton#
Description#
The default algorithm is the newton method, which can be used as a
full updated Newton-Raphson algorithm, or as a modified Newton-Raphson
method depending on the *algorithm choice in the sequence
definitions.
Note that the finite element method calculates the following residual:
where \(\bf F_{ext}\) are terms of the discretized weak form of the problem variational statement due to externally applied forces, \(\bf F^i_{int}\) are the terms due to “internal forces” (e.g. the \(\int_V \delta\ten \varepsilon:\ten \sigma dV\) virtual work term in mechanics), and \(\bf R^i\) is the residual imbalance at an iteration \(i\) due to the non-linearity of the problem. Convergence of a loading step is achieved when a measure of this residual falls below a desired magnitude.
A truncated Taylor series is used to find the adjustment to the problem variables (degrees of freedom):
Where \(\bf K\) is a stiffness matrix. Note that The stiffness can be an approximate measure of the real stiffness at a give time, and the solution can still be found. Typically however, any approximations to the stiffness other than the real algorithmic consistent tangent will result in increased iterations, and a reduction in the convergence “radius.”
Syntax#
The Newton-Raphson method provides a quadratic convergence: the residual
of the problem to be solved can not theoretically increase. Because
real problems usually do not fall into the Newton-Raphson assumptions,
one may observe that the residual increases during iterations. In this
case, use the **line_search option to activate the line search
procedure. The idea behind it is that if the residual increases, it
means that the magnitude of the current Newton-Raphson correction is too
large: a search is done along the descent direction to find the largest
magnitude so that the residual decreases (one show that this optimum
value always exists).
***resolution newton
\(~\,\) **line_search