## 3. Examples

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

### 3.1. Dynamic buckling stress analysis with Rayleigh damping

ebc 1
value  0. dof [UX UY UZ RX RY RZ] nodeset nodes_left
value  0. dof [   UY UZ RX RY RZ] nodeset nodes_right
value -1. dof [UX               ] nodeset nodes_right
end
end

(max_displacement=3.)
(duration=0.5)
case 1
analysis                   dynamic_nonlinear
stage                      2 sfactor (duration)
end

stage 2
ebc                        1 sfactor (max_displacement/duration)
step_size_init             (0.1*duration)
tol_dynamic                1e-2
residue_function_type      rayleigh_damping
rayleigh_alpha             123.6
rayleigh_beta              0.00008074
end

case 1
shell_intersection_angle 10
end

In this example, a panel is compressed in the x-direction by 3mm within 0.5s. If the analysis duration is different from the value of 1 (in time units, usually seconds), a separate stage has to be defined, stage 2 in the present example. The duration of the analysis is given with the MDL command

stage 2 sfactor f

The amplitude of any specified boundary conditions in the stage must be inversely-scaled by the analysis duration, this is done with the sfactor option. On the other hand, the sizes for the minimum, maximum, and initial step are given in absolute values. Hence, step_size_init should always be defined.

The dynamic tolerance (tol_dynamic) option has a significant influence on both accuracy and numerical effort. A large dynamic tolerance leads to relatively large time step sizes, requiring many Newton iterations per increment. Using a dynamic tolerance that yields smaller time step sizes which require only one or two Newton iterations is not only more accurate but often more effective. In some cases, a modified Newton method may further reduce the computational effort. Conversely, when the dynamic tolerance is set to an overly conservative value, the resulting time step size may become extremely small in the presence of high-frequency vibrations and fast movements.

When geometric instabilities are present, Rayleigh damping should be selected (by setting the residue_function_type option to rayleigh_damping and by setting the values for rayleigh_alpha and rayleigh_beta) to ensure an effective solution process.

For highly nonlinear problems, the full Newton method (default) is often more expedient than the modified Newton method. For mildly nonlinear problems and for problems where the time step size is predominantly limited by the time integration error, a modified Newton method may be more effective. 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.

### 3.2. Dynamic buckling stress analysis with visco-elastic materials

(max_displacement=3.)
(duration=0.05)
case 1
analysis                   dynamic_nonlinear
stage                      2 sfactor (duration)
end

stage 2
ebc                        1 sfactor (max_displacement/duration)
step_size_init             (0.1*duration)
tol_dynamic                1e-4
end

In this example, the options for Rayleigh damping are omitted since all dissipation should come from the visco-elastic material. The dynamic tolerance is crucial for the accuracy of the time integration: If the time step sizes are too large, the numerical damping which is caused by the time integration error may be larger than the material damping.

### 3.3. Quasi-static buckling stress analysis with artificial damping

case 1
analysis                    dynamic_nonlinear
ebc                         1
tol_dynamic                 1e-3
residue_function_type       artificial_damping
dissipated_energy_fraction  1e-3
end

The dynamic solver can also be applied to load-controlled quasi-static analysis. In this case, the analysis duration is irrelevant, and a separate stage does not need to be defined. Also in this case, the dynamic tolerance controls the time step size and is important to computational efficiency and accuracy.

Energy is dissipated with artificial damping; the residue_function_type artificial_damping option also tells the solver to neglect inertia forces. 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.

### 3.4. Mildly nonlinear heat-transfer analysis

(one_day=84600)
(duration=365*one_day)
case 1
analysis                    dynamic_nonlinear
stage                       2 sfactor (duration)
end

stage 2
dof_init                    1
ebc                         1 sfunction '1'
multistep_integration_order 4
newton_method               delayed_modified
step_size_min               (one_day)
step_size_init              (one_day)
step_size_max               (one_day)
tol_dynamic                 1e6
end

In this example, an initial temperature field is given (dof_init), and a constant temperature is applied at the boundary (ebc 1 sfunction '1'). If the problem is well behaved, it can be solved with a higher-order BDF that is more accurate but not unconditionally stable, and a modified Newton method can be used.

The analysis duration is one year (365 days), and the time step size is fixed to one day (84600 seconds). A very large dynamic tolerance is used to inactivate the solver's error estimator.