10 time_steppers (release)
The time_stepping object contains properties that define the timestepping algorithms used to progress the procedure to solution. In static procedures it is common to use a continuation method to aid convergence to a solution and, even though these procedures do not have temporal components, we refer to the scale factor used by the continuation method as time.
Name 
 Type 
 Description 
 Default 
label 
 STRING 
 The user defined name for this definition. 
 
central_difference (alpha)  generalized_alpha (beta)  explicit_generalized_alpha (alpha)  continuation (beta)  implicit_midpoint (beta)  linear (release) 
 SUB OBJECT  SUB OBJECT  SUB OBJECT  SUB OBJECT  SUB OBJECT  SUB OBJECT 
 One of the given keywords must be defined to specify the time stepper method 

Example Usage:time_steppers: [ { label: "linear_statics", linear: { linear_equation_solver: "multi_frontal" } } ]
10.1 central_difference (alpha)
The central_difference object describes the parameters that will be used by a secondorder centraldifference time stepping approach.
Name 
 Type 
 Description 
 Default 
timestep_scale_factor [optional] (alpha) 
 DOUBLE 
 Scales the stable time increment computed by Coreform IGA. 
 0.9 
timestep_update_interval [optional] (alpha) 
 INTEGER 
 Frequency at which the time step is updated. Default is 1, meaning that the timestep is updated at each step. 
 1 
viscous_damping (alpha) [optional] 
 SUB OBJECT 
 See linked documentation 

10.1.1 viscous_damping (alpha)
Name 
 Type 
 Description 
 Default 
mass_damping_coefficient (alpha) 
 DOUBLE 
 The massproportional damping coefficient in Rayleigh damping. 
 0 
10.2 generalized_alpha (beta)
The generalized_alpha object defines parameters for the generalized alpha implicit dynamic time stepping method.
Name 
 Type 
 Description 
 Default 
nonlinear_equation_solver (beta)  predictor_multicorrector (beta) 
 STRING  SUB OBJECT 
 One of the given keywords must be defined to specify the type of generalized alpha time stepping 
 
spectral_radius (beta)  alpha_options (beta) 
 DOUBLE  SUB OBJECT 
 One of the given keywords must be defined to specify the generalized alpha parameters 
 
adaptivity (beta) [optional] 
 SUB OBJECT 
 See linked documentation 

10.2.1 nonlinear_equation_solver  predictor_multicorrector
One of the given keywords must be defined to specify the type of generalized alpha time stepping
Name 
 Type 
 Description 
 Default 
nonlinear_equation_solver (beta) 
 STRING 
 The user defined identifier of the nonlinear solver to use. 
 
 SUB OBJECT 
 See linked documentation 

10.2.1.1 predictor_multicorrector (beta)
This defines an explicit predictormulticorrector that uses lumped diagonal mass, damping, and stiffness matrices and a fixed number of nonlinear iterations.
Name 
 Type 
 Description 
 Default 
maximum_iterations (beta) 
 INTEGER 
 The maximum number of corrector iterations to take. (At least three recommended, better results with more.) 
 
relative_tolerance [optional] (beta) 
 DOUBLE 
 The relative convergence criterion used to determine whether to increase or decrease the time step when using adaptivity. 
 
absolute_tolerance [optional] (beta) 
 DOUBLE 
 The absolute convergence criterion used to determine whether to increase of decrease the time step when using adaptivity. 

10.2.2 spectral_radius  alpha_options
One of the given keywords must be defined to specify the generalized alpha parameters
Name 
 Type 
 Description 
 Default 
spectral_radius (beta) 
 DOUBLE 
 This is a single parameter from which all of the alpha_options values can be computed, and controls the amount of numerical dissipation in highfrequency modes while maintaining secondorder accuracy and unconditional stability for linear problems. In technical terms, it is the spectral radius of the method's steptostep amplification matrix for linear problems in the limit of time step going to infinity. The value must be in the range [0, 1], where 0 produces the greatest amount of damping and 1 is undamped. 
 
 SUB OBJECT 
 See linked documentation 

10.2.2.1 alpha_options (beta)
Generalized alpha parameters. Setting alpha_f = 1.0 and alpha_m = 1.0 recovers the Newmark method. Secondorder accuracy requires gamma = 1/2 + alpha_m  alpha_f and beta = 1/4 (1 + alpha_m  alpha_f)^2. Unconditional stability requires alpha_m >= alpha_f >= 1/2.
Name 
 Type 
 Description 
 Default 
alpha_f (beta) 
 DOUBLE 
 The alpha_f parameter to the generalizedalpha algorithm. 
 
alpha_m (beta) 
 DOUBLE 
 The alpha_m parameter to the generalizedalpha algorithm. 
 
beta (beta) 
 DOUBLE 
 The beta parameter to the generalizedalpha algorithm. 
 
gamma (beta) 
 DOUBLE 
 The gamma parameter to the generalizedalpha algorithm. 

10.2.3 adaptivity (beta)
Options for adaptive (pseudo)time stepping in implicit methods and the explicit predictor–multicorrector method.
Name 
 Type 
 Description 
 Default 
maximum_time_step (beta) 
 DOUBLE 
 The largest allowed time step, usually based on accuracy needs of an application. 
 
minimum_time_step (beta) 
 DOUBLE 
 The smallest allowed time step, to prevent stagnation. Reaching this step size terminates the analysis. Must be less than half the initial time step to accommodate halfsteps for selfstarting. 
 
decrease_factor (beta) 
 DOUBLE 
 Factor by which time step is scaled after a step fails to converge. Typically substantially less than one (e.g., 0.5), to avoid repeated consecutive failures. 
 
increase_factor (beta) 
 DOUBLE 
 Factor by which time step is scaled after successful convergence of a step. Typically only slightly greater than one (e.g., 1.125), to avoid immediately failing and cutting back again. 

Example Usage:adaptivity: { maximum_time_step: 0.1, minimum_time_step: 0.00001, decrease_factor: 0.25, increase_factor: 1.5 }
10.3 explicit_generalized_alpha (alpha)
The explicit_generalized_alpha object describes the parameters that will be used by an explicit secondorder time stepping approach with tunable numerical dissipation for highfrequency modes.
Name 
 Type 
 Description 
 Default 
spectral_radius (alpha) 
 DOUBLE 
 This parameter governs how much numerical dissipation is added to suppress highfrequency modes of the problem. The value must be in the range [0, 1], where 0 produces the greatest amount of damping in the generalizedalpha algorithm and 1 is undamped. 
 
timestep_scale_factor [optional] (alpha) 
 DOUBLE 
 The solver estimates a maximum stable time step and a time step at which numerical dissipation is controlled by spectral_radius for the highestfrequency mode. For spectral_radius = 1, we have , and for spectral_radius < 1, we have . If the timestep_scale_factor is , the solver will select a time step of . 
 0.9 
timestep_update_interval [optional] (alpha) 
 INTEGER 
 Frequency at which the time step is updated. Default is 1, meaning that the timestep is updated at each step. 
 1 
viscous_damping (alpha) [optional] 
 SUB OBJECT 
 See linked documentation 

10.3.1 viscous_damping (alpha)
Name 
 Type 
 Description 
 Default 
mass_damping_coefficient (alpha) 
 DOUBLE 
 The massproportional damping coefficient in Rayleigh damping. 
 0 
10.4 continuation (beta)
An implicit statics time stepping approach based on numerical continuation. Many nonlinear procedures are unable to be solved in a single nonlinear step (e.g. a single Newton iteration) as the true solution is outside of the radius of convergence for the nonlinear solver centered at the initial state. The continuation method breaks up the nonlinear procedure from a single step into multiple substeps. A scaling parameter is created that describes the distance from the initial state to the final state, often this parameter is referred to as "time" even though this method ignores mass contributions. At each of these substeps (timesteps) the original nonlinear procedure is redefined as a simpler nonlinear procedure that increments from the previous timestep’s computed solution to the solution at the current timestep. With this approach, each of these substeps describes a procedure is lessnonlinear than the original procedure (fundamental theorem of calculus) and is thus more likely for the nonlinear solver to converge to a solution.
Name 
 Type 
 Description 
 Default 
nonlinear_equation_solver (beta) 
 STRING 
 The user defined identifier of the nonlinear solver to use. 
 
adaptivity (beta) [optional] 
 SUB OBJECT 
 See linked documentation 

Example Usage:continuation: { nonlinear_equation_solver: "newton_raphson", adaptivity: { maximum_time_step: 0.1, minimum_time_step: 0.00001, decrease_factor: 0.25, increase_factor: 1.5 } }
10.4.1 adaptivity (beta)
Options for adaptive (pseudo)time stepping in implicit methods and the explicit predictor–multicorrector method.
Name 
 Type 
 Description 
 Default 
maximum_time_step (beta) 
 DOUBLE 
 The largest allowed time step, usually based on accuracy needs of an application. 
 
minimum_time_step (beta) 
 DOUBLE 
 The smallest allowed time step, to prevent stagnation. Reaching this step size terminates the analysis. Must be less than half the initial time step to accommodate halfsteps for selfstarting. 
 
decrease_factor (beta) 
 DOUBLE 
 Factor by which time step is scaled after a step fails to converge. Typically substantially less than one (e.g., 0.5), to avoid repeated consecutive failures. 
 
increase_factor (beta) 
 DOUBLE 
 Factor by which time step is scaled after successful convergence of a step. Typically only slightly greater than one (e.g., 1.125), to avoid immediately failing and cutting back again. 

Example Usage:adaptivity: { maximum_time_step: 0.1, minimum_time_step: 0.00001, decrease_factor: 0.25, increase_factor: 1.5 }
10.5 implicit_midpoint (beta)
An implicit dynamics time stepper using the midpoint rule as the time integration scheme. The implicit midpoint rule includes no numerical dissipation, and is equivalent to the generalized_alpha method with spectral_radius = 1. In simulations of slow dynamics using large time steps, it is recommended to instead use generalized_alpha with spectral_radius < 1, which will apply damping to underresolved modes.
Name 
 Type 
 Description 
 Default 
nonlinear_equation_solver (beta) 
 STRING 
 The user defined identifier of the nonlinear solver to use. 
 
adaptivity (beta) [optional] 
 SUB OBJECT 
 See linked documentation 

10.5.1 adaptivity (beta)
Options for adaptive (pseudo)time stepping in implicit methods and the explicit predictor–multicorrector method.
Name 
 Type 
 Description 
 Default 
maximum_time_step (beta) 
 DOUBLE 
 The largest allowed time step, usually based on accuracy needs of an application. 
 
minimum_time_step (beta) 
 DOUBLE 
 The smallest allowed time step, to prevent stagnation. Reaching this step size terminates the analysis. Must be less than half the initial time step to accommodate halfsteps for selfstarting. 
 
decrease_factor (beta) 
 DOUBLE 
 Factor by which time step is scaled after a step fails to converge. Typically substantially less than one (e.g., 0.5), to avoid repeated consecutive failures. 
 
increase_factor (beta) 
 DOUBLE 
 Factor by which time step is scaled after successful convergence of a step. Typically only slightly greater than one (e.g., 1.125), to avoid immediately failing and cutting back again. 

Example Usage:adaptivity: { maximum_time_step: 0.1, minimum_time_step: 0.00001, decrease_factor: 0.25, increase_factor: 1.5 }
10.6 linear (release)
This method defines the procedure to be linear in nature and will solve the procedure in a single linear solve. The linearization of the procedure is determined at the initial state. Loads and boundary conditions are evaluated at final value of the procedure’s interval.
Name 
 Type 
 Description 
 Default 
linear_equation_solver 
 STRING 
 The user defined identifier of the linear solver to use. 

Example Usage:linear: { linear_equation_solver: "multi_frontal" }