## 2. MDL Commands

In addition to the MDL command analysis dynamic_nonlinear, the following commands may be specified in the case block.

correction_termination_test flux_normalised|dof_and_residue|normalised_dof_and_residue

Specifies the type of test which indicates whether the newton iterations have sufficiently converged. Default is flux_normalised.

The flux_normalised correction termination test computes the residuum normalised by the energy flux for each type of field (displacements, rotations, Lagrange multipliers, etc.) separately. The flux_normalised test has additional options, see Section 5.

The dof_and_residue correction termination test tests the L2-norm residuum against tol_residuum and the L2-norm of the solution changes w.r.t. the previous Newton iteration against tol_solution. The following tolerances are used: tol_residuum (default 1e-3) and tol_solution (default 1e-3).

The normalised_dof_and_residue correction termination test divides the L2-norm residuum by that of the first Newton iteration. The same is done for the L2-norm of the solution changes w.r.t. the previous Newton iteration. The following tolerances are used: tol_residuum (default 1e-3) and tol_solution (default 1e-3).

gradients istep

Controls the computation of the gradients (strains, stresses, heat transfer, etc.) for the current case. If istep is set to 0 (default) no gradients are computed and saved, i.e. only DOF solutions will be found on the database. If istep is set to -1, the gradients for the last converged step of at stage or the linear solution step are computed and saved. If istep is set to a positive value, the gradients will be computed at each istep step of at stage as well as for the last step of a stage.

line_search yes|no

Specifies whether line search should be used. Default is no. This line search method satisfies the strong Wolfe condition. When the cost of calculating the first variation is significantly lower than the cost of calculating the second variation, it is recommended to activate line search, as it improves the probability that the Newton iterations converge.

max_divergences v

Defines the maximum number of consecutive divergences before the step is reduced during a load step (optional). Divergence occurs when the energy of the current correction is larger than the energy of the preceding correction. Default is 4.

max_newton_iterations v

Maximum number of Newton iterations per step. Default is 50.

max_steps v

Maximum number of steps (increments) per stage. Default is 99999.

multistep_integration_order

Defines the integration order of the implicit transient integration scheme. Currently supported values are 1 (Euler backward difference formula), 2, 3, 4, 5, or 6. The solver is unconditionally nonlinear-stable for values of 1 and 2. Default is 2.

newton_method modified|delayed_modified|conventional

Specifies what kind of Newton method should be used. Default is conventional.

newton_periodic_update v

For the modified and delayed modified Newton methods, this option specifies the maximum number of consecutive iterations without re-factorization. When this number is reached, a re-factorization will be performed, and the counter is reset. Note that for high values of newton_periodic_update and depending on the newton_method setting, several increments may be performed until this number of Newton iterations is reached. Default is 100.

residue_function_type order_n|rayleigh_damping|artificial_damping

Specifies the residue method. Default is order_n, for which the MDL command multistep_integration_order defines the order of the implicit time-integration.

The artificial_damping method works like in the static nonlinear solver; the damping is of the Rayleigh viscous type. Inertia effects are neglected, and the problem becomes a first-order ODE. The global viscosity matrix is generated in function either of the dissipated_energy_fraction option or the rayleigh_alpha and rayleigh_beta options. In the first case, the global viscosity matrix is a scaled instance of the mass matrix, such that at the first increment, the ratio of the dissipated energy to the total energy corresponds to the value given in dissipated_energy_fraction. In the second case, the global viscosity matrix is constructed in the same way as for rayleigh_damping (see below).

For the rayleigh_damping (second-order) method, the global viscosity matrix C is constructed from the global mass matrix M and the global linear stiffness matrix K in the following way:

C = α M + β K

The coefficient α can be specified with the MDL command rayleigh_alpha, and likewise, the coefficient β can be specified with the MDL command rayleigh_beta. Note that these coefficients can be determined in the same way as explained in Section 4 for the static nonlinear solver.

step_size_init v

The step size at the beginning of the stage. Default is 0.1.

step_size_max v

The maximum step size during the stage. Default is 1.0.

step_size_min v

The minimum step size during the stage. Default is 1e-12.

tol_dynamic v

The dynamic nonlinear solver controls the step size with a local error estimator, which is obtained with Milne's method. The option tol_dynamic defines the maximum permitted absolute error of a degree of freedom during one time step. Default is 1e-5. The step size is adapted accordingly to this criteria. Setting tol_dynamic to a large value will inactivate the transient integration error control.

tol_residuum v

The error tolerance value for the residuum ('equilibrium') error; used by the Newton correction termination test. The default depends on the correction_termination_test option.

tol_solution v

The error tolerance value for the dof ('displacement') error; used by the Newton correction termination test. The default depends on the correction_termination_test option.

tol_work v

The error tolerance value for the work error; used by the Newton correction termination test. The default depends on the correction_termination_test option.