# A Newton-Krylov method for nonlinear problems

*by N. Tardieu, EDF R&D / AMA*

For solving nonlinear problems, the Newton method is traditionally used. This is for example in the case for the nonlinear statics and dynamics operators STAT_NON_LINE and DYNA_NON_LINE. A new method, called the Newton-Krylov, was implanted in these operators.

The Newton-Krylov is part of the family of the inexact Newton’s methods, whose principle is to replace the condition K_{n} δu_{n} = R(u_{n}) to be checked by the increment of the solution δu by the weaker condition ||K_{n} δu_{n} - R(u_{n})|| ≤ η_{n} ||R(u_{0})||, where η_{n} is called the "forcing term". The Newton-Krylov is based on the use of an iterative method (also called a Krylov method) to implement this weaker condition. To do this, the value of the forcing-term is used as an adaptive convergence criterion for the Krylov method.

This method is very suited for problems with many degrees of freedom as for the modeling of the Moisture Separator Reheater which has 1 million degrees of freedom in large strain and elasto-plasticity. Using the Newton-Krylov method associated with an iterative solver, the CPU time is reduced by half compared to the use of the Newton’s method associated with a direct solver.

This work is the subject of a publication: *A Newton-Krylov method for non-linear mechanics*, N. Tardieu and E. Cheignon, European Journal of Computational Mechanics, to appear.