4.1. Explicit Quasistatic Mode

BEGIN TIME CONTROL
  BEGIN TIME STEPPING BLOCK <string>time_block_name
    START TIME = <real>start_time_value
    BEGIN PARAMETERS FOR PRESTO REGION <string>region_name
      NUMBER OF QUASISTATIC TIME STEPS =
        <int>quasi_step_count
    END [PARAMETERS FOR PRESTO REGION <string>region_name]
  END [TIME STEPPING BLOCK <string>time_block_name]
  TERMINATION TIME = <real>termination_time
END [TIME CONTROL]

Explicit quasistatic mode provides an automated methodology to leverage explicit time stepping to efficiently solve quasistatic problems. Quasistatic mode is enabled by providing a non-zero input for the NUMBER OF QUASISTATIC TIME STEPS command for a solution time period. Explicit solution of a quasistatic analysis may prove to be more robust or faster than implicit solution in some cases.

Explicit quasistatic mode can be particularly helpful for analyses with contact. Analyses with several disjoint parts that are in static equilibrium only due to the contact and frictional forces between those may cause great difficulty for the standard implicit static solver. Explicit quasistatic mode can often solve these problems more robustly than the implicit solver. Additionally explicit quasistatic mode is well suited to analyses that involve buckling, strong material nonlinearity, and material failure or fracture. Such sharp nonlinearity cause substantial difficulty for standard implicit solution.

Activating explicit quasistatic mode has several effects on the analysis.

Implicit compatible defaults are used for element formulations for numerical parameters such as effective modulus, quadratic bulk viscosity, and linear bulk viscosity.
The analysis is automatically mass scaled (see the Sierra/SM User Manual) so that - steps will be taken in the current time period. As dynamic terms still have some meaning in explicit quasistatic analysis to function efficiently the size of the implied quasistatic time step should be significantly larger than the standard explicit time step. For example if the standard explicit time step is 1.0e-6 seconds and it is desired the load up quasi-statically over 100,000 steps then the time over which to perform this loading should be one second or greater. This would yield a quasistatic time step of 1.0e-5 seconds which is significantly larger than the standard explicit time step.
To minimize dynamic effects explicit quasistatic analysis should be run at a time step that exceeds the explicit time step. Sometimes it is convenient to greatly exceed the standard explicit time step. For stability reasons the time step is slowly increased from the standard explicit time step when entering a quasistatic solution period and slowly decreased to the standard explicit time step when exiting a quasistatic solution period.
A dynamic viscous damping term is included to accelerate convergence to a static equilibrium solution. This viscous term is automatically tuned based on the number of time steps being taken and the current model velocity.
A static residual norm is automatically computed and output to the global variable. Monitoring of this residual can indicate how closely the current solution approximates the target quasistatic state. The quasistatic residual is the \(L_2\) norm of the current nodal force imbalance divided by the reference force (reaction or external force sum). The explicit quasistatic residual has the same meaning as the implicit relative residual, generally a good solution will have a quasistatic residual of \(10^{-3}\) or below.

4.1.1. Usage Guidelines

It is important to note that quasistatic mode is a dynamic analysis; dynamic terms and effects will be present in the solution. However, if used properly, quasistatic mode will minimize these effects, yielding a solution with comparable accuracy to an equivalent implicit static analysis. Generally, the number of quasistatic explicit steps required to reach static equilibrium will be tens to hundreds of thousands, although a sharp estimate of the actual number of steps required is not available. Generally, dynamic effects are expected to reduce with increasing numbers of time steps. The quasistatic residual norm variable may be used to monitor static equilibrium at various steps of the analysis.

Quasistatic equilibrium is reached fastest on models with minimal potential low-mode vibrations. The most difficult types of analyses for quasistatic mode to solve include long, slender structural members that can vibrate at low frequencies.

Static equilibrium in explicit quasistatic mode is also sensitive to loading rate; more smoothly applied external loads will result in more rapid convergence to the quasistatic solution. It is also recommended that loads be held constant in the later part of the loading period. Ultimate quasistatic equilibrium can be reached in the contact load. An example showing recommended quasistatic mode loading is shown in Section 4.1.2.

Output results from quasistatic mode—particularly quantities such as reaction force and energy—are valid only when the model has reached static equilibrium. The mass scaling used by quasistatic mode can alter the results in the middle of a load step. Damping and other artificial forces may be arbitrarily large when the model is in motion in an intermediate state, while they tend to zero when a model reaches static equilibrium.

It is assumed that the ultimate material state obtained is strain path-independent as long as the strain path is monotonic. This condition holds for elastic models, \(J_2`\) plasticity models, soil foam, and most other commonly used models, but does not hold for rate-dependent models. Quasistatic mode will provide an answer for rate-dependent models, but a very large number of load steps may be required to ensure the loading is applied smoothly.

Although the smoothness of the strain paths and reduction of dynamic effects will improve with iterations, the most reliable way to detect possible material state overshoot error is to compare the result with a solution based on a larger number of quasistatic iterations. Current evidence indicates quasistatic material state integration errors reduce quadratically in iteration count.

4.1.2. Example Problem

The following input demonstrates the use of explicit quasistatic mode for a beam loaded by a pressure boundary condition.

begin function pfunc
  evaluate expression = cos_ramp(t, 0.0, 0.5)
end

begin time control
  begin time stepping block p1
    start time = 0.0
    begin parameters for presto region presto
      number of quasistatic time steps = 2000
    end
  end
  termination time = 1.0
end

begin presto region
  begin pressure
    surface = surface_1000
    function = pfunc
  end
end