Nonlinear Convergence Criteria

There are two options available to control nonlinear convergence criteria and respective tolerances, NLPARM Bulk Data Entry and NLCTRL Bulk Data Entry. The basic principle in assessing nonlinear convergence is to compare an error measure of the solution with a pre-determined tolerance level. When the error falls below the specified tolerance, the problem is considered converged. In a case of multiple, simultaneous convergence criteria, all criteria need to be satisfied for the solution to be converged.

Detailed convergence check information for implicit nonlinear subcases, increments, and Newton-Raphson iterations are printed in the _nl.out file. Additional options for checking Nonlinear Analysis run information and monitoring progress of Nonlinear Implicit Analysis jobs are listed in Runtime Monitoring in the User Guide.

Relative Error in Displacements

The relative error in displacements (printed in the convergence summary as EUI) is calculated as:

$E_{U} = k \frac{‖ A \cdot Δ u ‖}{‖ A \cdot u ‖}$

Where, $A$ is a normalizing vector consisting of square roots of diagonal elements of stiffness matrix $K$ , $A_{i} = \sqrt{K_{i i}}$ and the vector norm is calculated as:

$‖ A \cdot u ‖ = \sum_{i} | A_{i} u_{i} |$

$k = \frac{q}{1 - q}$ for small displacement nonlinear analysis and the value of $q$ is calculated as:

$q$ is a contraction factor that corrects the increment of solution $Δ u_{n}$ to better represent the actual error in the small displacement nonlinear solution. It is expressed as:

$q = \frac{‖ Δ u_{n} ‖}{‖ Δ u_{n - 1} ‖}$

In order to stabilize the behavior of $q$ in practical computations, it is updated iteratively according to the formula:

$q_{n} = \frac{2}{3} \frac{‖ Δ u_{n} ‖}{‖ Δ u_{n - 1} ‖} + \frac{1}{3} q_{n - 1}$

Starting from initial value

$q_{1} = 0.99$ .

Note: The contraction factor is meaningful when the solution is close to having converged - it then reasonably estimates the actual error remaining in the small displacement nonlinear solution.

$k = 1$ for large displacement nonlinear analysis.

Relative Error in Terms of Loads

The relative error in terms of loads (printed in convergence summary as EPI) measures the relative strength of the residual, and $R$ is calculated as:

$E_{P} = \frac{‖ R \cdot u ‖}{‖ f \cdot u ‖}$

The load vector $f$ in this formula includes nodal reactions due to specified displacements.

Relative Error in Terms of Work

The relative error in terms of work (printed in convergence summary as EWI) measures the relative change in solution energy, and is calculated as:

$E_{W} = \frac{‖ R \cdot Δ u ‖}{‖ f \cdot u ‖}$

Note: The above norms only measure the error of the nonlinear iterative process. Their values do not represent the accuracy of the finite element solution, only the fact that the nonlinear process has converged properly.