## 3. Examples

In this section, several representative analysis scenarios and their corresponding settings are described.

### 3.1. Highly nonlinear load-controlled stress analysis with dissipation

In stress analysis, a problem may be considered highly nonlinear if geometric instabilities, material nonlinearity such as degradation, or contact conditions are present.

analysis                    nonlinear
residue_function_type       artificial_damping
dissipated_energy_fraction  1e-6
step_size_init              1e-2
step_size_max               1e-1

Load control is enabled by default. For highly nonlinear problems, the full Newton method (default) is often more expedient than the accelerated modified Newton method. The default maximum number of Newton iterations (max_newton_iterations 50) should be sufficient in many cases. Lower values may cause many small time increments or even failure to converge, while higher values may lead to increased computation time. In case of very slow but reliable convergence, higher values (100-200) may be necessary.

When using a load-controlled strategy, a mechanism to dissipate energy must be enabled when instabilities occur. This is achieved with the residue_function_type artificial_damping option. The amount of damping is controlled with the dissipated_energy_fraction parameter. High values accelerate the analysis, increasing the increment sizes, but may influence the results on the other hand. Therefore, a compromise between analysis time and accuracy has to be found. While the parameter value is independent of the physical units, the necessary amount of damping depends on the problem. Thus, the default of 1e-4 may not be appropriate. It is recommended to vary this parameter in powers of 10 (e.g. make an analysis for 1e-7, 1e-6, 1e-5, 1e-4, 1e-3) and observe its effect.

When the problem is expected to be nonlinear from the start, an initial step size may be specified with the step_size_init option. The maximum step size should always be set, using the step_size_max option, for the reason that different load paths may be followed and collapse cannot be reliably predicted if the step size is too large (the solver may "jump over" collapse paths). The default of 1e-12 for the minimum step size is sufficient for most problems.

Another important option is max_divergences which may be used to trigger collapse by using smaller values (1 or 2) than the default. On the other hand, analyses involving material degradation may benefit from higher values (increasing the chances that the Newton iterations will converge).

### 3.2. Continuation post-buckling analysis

analysis                    nonlinear
increment_control_type      hyperplane
step_size_init              1e-2
step_size_lambda_max        1e-1

The option increment_control_type hyperplane enables (one of several types of) continuation analysis.

The maximum step size in "load"-direction can be controlled by means of the step_size_lambda_max option. Its counterpart is the step_size_dof_max option which controls the maximum step size in the DOF direction. The step_size_max option controls the arc-length of the increments.

### 3.3. Mildly nonlinear continuation analysis

analysis                    nonlinear
increment_control_type      hyperplane
correction_type             accelerated_newton
newton_method               delayed_modified
step_size_init              1

Continuation analysis is enabled with increment_control_type hyperplane. If the problem is only mildly nonlinear (for instance, if no limit points are expected), there is no need to limit the step size.

In this example, the delayed-modified Newton method is used in conjunction with acceleration via line search.

The nonlinear analysis is attempted in a single increment (step_size_init 1), which might not always work. In such cases, the default of 0.1 is more expedient.

analysis                    nonlinear
max_newton_iterations       100
max_divergences             10
tol_residuum                1e-3
tol_solution                1e-3

Load control is enabled by default. Due to the highly nonlinear contact conditions, the full Newton method (default) is often more expedient than the accelerated modified Newton method. Because the Newton iterations may converge slowly when contact is active, a high number of Newton iterations is allowed.

The maximum number of divergences is increased, this improves the chances that the Newton iterations will converge.

If the contact conditions are enforced via quadratic penalties, the tangent stiffness matrix may be, depending on the penalty factor, badly conditioned. In this case, the tolerances for convergence may need to be increased using the tol_residuum and tol_solution options.

### 3.5. Mildly nonlinear load-controlled analysis

analysis                    nonlinear
correction_type             accelerated_newton
newton_method               delayed_modified

Nonlinear heat-transfer analysis is often well-behaved and therefore may be carried out with the above settings.

Load-controlled analysis is enabled by default. Because the problem is considered mildly nonlinear, there is no need to limit the step size, and the delayed-modified Newton method (or the modified Newton method) can used in conjunction with acceleration via line search.

Depending on the amount of detail required, the maximum step size may be limited. Depending on the accuracy required, the tolerances for convergence may be modified.