## 2. MDL Commands

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

correction_type newton|accelerated_newton

Specifies the type of correction in a given Newton iteration. Default is newton.

The newton correction type only makes use of the residuum and the tangent matrix (updated or not) to compute the correction.

The accelerated_newton correction type is meaningful with a modified Newton method. It monitors whether the sequence of Newton iterations converges (whether the residuum decreases), and if not (divergence), invokes the line search algorithm to find a better solution. With line search, it is often possible to find a solution outside the convergence region. Thus, when using the MDL commands newton_method modified or newton_method delayed_modified, the accelerated Newton correction can converge for larger time steps than the normal Newton correction type.

correction_termination_test flux_normalised|regularised|dof_and_residue|normalised_dof_and_residue

Specifies the type of test which indicates whether the Newton iterations of the correction phase have sufficiently converged. The tests make use of some or all of the MDL commands tol_residuum, tol_solution, and tol_work. 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 regularised correction termination test regularizes the residuum, the solution changes, and the energy change, with the goal of making the test independent of the units. Thus, changing the units of an FE model (e.g. [kg,m,s]) to another (consistent) unit system (e.g. [t,mm,s]) should not require modification of the tolerances. The following tolerances are used: tol_residuum (default 1e-3), tol_solution (default 1e-3), and tol_work (default 1e-7).

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.

gradients_only yes|no

Specifies whether only gradients (and reaction forces) should be computed, while the solution vector is defined solely by the the initial conditions, and the boundary conditions and the current load increment. Default is no.

increment_control_type load|state|hyperplane|hyperelliptic|local_hyperelliptic

Specifies the increment constraint strategy to use. The increment constraint strategy serves two purposes: It defines the increment constraint for the predictor phase and the corrector phase, and it computes the predictor solution from the solution of the last converged step. This predictor solution is then used as initial solution to the correction phase.

The following strategies are available: load means load-controlled. To avoid singularities at limit points in post-buckling analysis, residue_function_type artificial_damping can be used.

A state-controlled strategy is used with state.

The other strategies are of the continuation type. For the hyperelliptic and local_hyperelliptic strategies, the parameters hyperelliptic_a and hyperelliptic_b define the size of the ellipse.

The default strategy is load.

For more details on the kinds of constraints, refer to Section 6.2.

init_step_number v

The initial step cycle number. Default is 1.

line_search yes|no

Specifies whether line search should be used for the accelerated_newton method (see the MDL command correction_type). The 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 have line search enabled, as it improves the probability that the Newton iterations converge. Default is yes.

max_divergences v

Defines the maximum number of consecutive divergences during the correction phase of a load step. Default is 4. When this number is reached, the correction phase is terminated unsuccessfully, and the step increment is re-started with a reduced step size.

max_newton_iterations v

Specifies the maximum number of Newton iterations per increment. Default is 50.

max_steps v

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

newton_method conventional|modified|delayed_modified

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

The conventional method performs factorization for the predictor and for each Newton iteration, and thus is a full Newton method.

For the modified method, factorization of the tangent matrix is performed when the correction phase has terminated unsuccessfully. Hence, several increments may be performed without any factorization.

The delayed_modified method performs a factorization for the predictor and for the first Newton iteration.

Note that the artificial_damping option for the MDL command residue_function_type enforces factorization each time the step size changes. For the modified and delayed_modified Newton methods, is is recommended to use the correction_type accelerated_newton option.

newton_periodic_update v

For the modified and delayed modified, non-accelerated 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 default|artificial_damping|scaled

Specifies the type of residuum (first variation or internal forces vector) and its derivative (second variation or tangent matrix).

The default type computes the first and second variation without any modifications.

In a load-controlled post-buckling analysis involving collapse, the artificial_damping type is necessary for the Newton iterations to converge. It adds viscous forces to the residuum vector (first variation). The velocities are computed from the difference of the solution to the last converged solution, divided by the step size. To the tangent matrix (second variation), the viscosity matrix divided by the step size is added. With artificial damping, singularity of the tangent stiffness matrix at limit points can be avoided and energy can be dissipated during mode jumps. See also Section 4.

The scaled type scales the DOFs by their respective diagonal element in the tangent matrix. This may be beneficial in continuation analysis (e.g. increment_control_type local_hyperelliptic). Due to the scaling, it is recommended to use the default correction termination test.

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. In continuation analysis, this is the size along the load path.

step_size_min v

The minimum step size during the stage. Default is 1e-12. In continuation analysis, this is the size along the load path.

step_size_lambda_max v

The maximum lambda step size during the stage in a continuation method analysis. Default is 1.0.

step_size_lambda_min v

The minimum lambda step size during the stage in a continuation method analysis. Default is 1e-12.

step_size_dof_max v

The maximum degrees of freedoms step size during the stage in a continuation method analysis. Default is 1.0.

step_size_dof_min v

The minimum degrees of freedoms step size during the stage in a continuation method analysis. Default is 1e-12.

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 ('solution') error; used by the Newton correction termination test. The default value on the correction_termination_test option.

tol_work v

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