Nonlinear Convergence Criteria

In order to assess whether the nonlinear process has converged, a number of convergence criteria are available. These criteria and respective tolerances can be selected on the NLPARM Bulk Data Entry card.

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.

Relative Error in Displacements

The relative error in displacements (printed in the convergence summary as EUI) is calculated as:(1)

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:(2)

‖ 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:(3)

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:(4)

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:(5)

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:(6)

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.