# Nonlinear resolution type¶

For nonlinear structural mechanics simulations the resolution of the nonlinear equation system has to be selected. Currently two very similiar methods are available:

• Newton method:

The Newton method is a general resolution method of root finding problems. For a general nonlinear functional $$F$$ that depends on an unknown vector $$x$$ the Newton method can be summarized as follows: Find a series $$x^n$$ of approximations that converges towards the true solution $$x$$. Starting with a current iterate $$x^n$$ the next is found by linearizing the functional $$F$$ around $$x^n$$ and find the root $$x^{n+1}$$ of this linearization.

Written as mathematical expressions it reads as:

$$$\label{ref1} 0 = F(x^{n+1}) \approx F(x^n) + F'(x^n)\cdot (x^{n+1}-x^n)$$$$$$\label{ref2} x^{n+1} = x^n - [F'(x^n)]^{-1} \cdot F(x^n)$$$
• Newton-Krylov method:

This is a modified version of the Newton method, especially designed for iterative solvers. It can only be selected if the equation solver is an iterative solver.

When defining the Newton metod (or Newton-Krylov method) several parameters can be adjusted, but in most cases the standard settings should be a good choice. The most importan are the following ones:

• Convergence criteria type:

You can choose between a relative and an absolute convergence criteria for the Newton method. For the absolute criterion the convergence is achieved if the maximum residual of all degrees of freedom of a given Newton iteration is lower than the given tolerance. For the relative criteria the same is checked for the maximum relative residual (maximum absolute residual divided by the external force). The standard tolerance for the relative and absolute criteria ($$10^{-6}$$) is chosen to be really strict. Increasing it up to $$10^{-4}$$ may speed up the solution time significantly and may be reasonable in most cases, but it may also lead to wrong results in some situations.

Hint

If no external forces are applied the relative residual may become singular and convergence can not be reached. In this case switching to an absolute criteria is recommended.

• Prediction matrix:

This setting specifies which matrix should be used in the prediction phase of the Newton algorithm. A good choice leads to a better starting point and thus to a faster convergence. In most cases the standard selection of tangent matrix should be prefered.

Hint

In nonlinear analyses with a plastic material, the convergence may fail when unloading in the plastic zone occurs and a tangent prediction is used. In these situations choosing the elastic matrix for the prediction should be preferred.

• Jacobian matrix:

Select which stiffnes matrix should be used for computing the Jacobian of the Newton method. Selecting the tangent matrix allows a Full Newton approach (by also setting both reactualisation rates to 1), whereas the leastic matrix leads to a Quasi-Newton approach. The maximum number of iterations setting specifies allows the maximum number of Newton iterations per time increment. If this value is reached, the increment is considered non-converging. If an automatic timestepping is used, the time-increment is reduced in order to reach convergence. For a manual timestepping the simulation run will fail.