# 4.11. Implicit Dynamic Time Integration

Sierra/SM has the ability to perform implicit solution of both quasistatic and dynamic problems. In quasistatic calculations, the solution for static equilibrium is obtained at each step, ignoring the inertial mass and acceleration terms of the equations of motion. In dynamic problems, the inertial terms are included.

## 4.11.1. Implicit Dynamics

    BEGIN IMPLICIT DYNAMICS
      ACTIVE PERIODS = <string list>period_names
      USE {HHT|GENERALIZED ALPHA INTEGRATION}(HHT)
      ALPHA = <real>alpha(-0.1) [DURING <string list>period_names]
      BETA = <real>beta(0.3025) [DURING <string list>period_names]
      GAMMA = <real>gamma(0.6) [DURING <string list>period_names]
      CONTACT TIMESTEP = {ON|OFF}(ON) [DURING <string list>period_names]
    END [IMPLICIT DYNAMICS]

The `IMPLICIT DYNAMICS}` command block enables implicit dynamics and contains commands to control the behavior of implicit time integration. This command block is placed in the `ADAGIO REGION` command block. Formerly, it was necessary to create a new procedure for switching between implicit dynamics and quasistatics in an analysis. Now, these different analysis types can all be done within a single region, and the command lines in the `IMPLICIT DYNAMICS` command block can be used to enable or disable implicit dynamics for specific time periods within a single region. Section 4.11.1.1 through Section 4.11.1.4 describe the command lines in this command block.

### 4.11.1.1. Active Periods

    ACTIVE PERIODS = <string list>period_names

If the `IMPLICIT DYNAMICS` command block is present in the Adagio region, it is active for all periods by default. Implicit dynamics can be activated for specific time periods with the `ACTIVE PERIODS` command line. See Section 2.6 for more information about this optional command line.

### 4.11.1.2. Use HHT|Generalized Alpha Integration

    USE HHT|GENERALIZED ALPHA INTEGRATION

Adagio supports the HHT and generalized alpha algorithms for implicit time integration. Implementation follows the theory outlined in the Sierra/SD Theory Manual.

### 4.11.1.3. HHT Parameters

    ALPHA = <real>alpha(-0.1) [DURING <string list>period_names]
    BETA = <real>beta(0.3025) [DURING <string list>period_names]
    GAMMA = <real>gamma(0.6) [DURING <string list>period_names]

HHT integration parameters may be modified via the `ALPHA`, `BETA`, and `GAMMA` command lines. These parameters can all be specified for the entire analysis, or they can vary by solution period by appending them with the optional `DURING` specification (see Section 4.2.1).

- The `ALPHA` command line specifies \(\alpha\), which is the dissipation factor applied to the internal force vector. The value of `alpha` controls numerical damping and must always be less than or equal to zero. Its default value of -0.1 results in slight numerical damping of modes with frequencies higher than can be resolved with a given time step. To maintain second-order accuracy, the value of `alpha` should not be less than \(-\frac{1}{3}\).

- The `BETA` command line specifies the stability parameter \(\beta\) for time integrators in the Newmark family. The default value of `beta` is 0.3025. It should have a value of \(0.25(1-\alpha)^2\) to maintain second-order accuracy.

- The `GAMMA` command line specifies the dissipation factor \(\gamma\) for Newmark time integrators. The default value of `gamma` is 0.6. It should have a value of \(0.5-\alpha\) to maintain second-order accuracy.

The HHT time integrator [[1]] is used to integrate the semi-discrete equations of motion in time. The behavior of the HHT integrator is controlled with three parameters: \(\alpha\), \(\beta\), and \(\gamma\). With proper selection of these parameters, the HHT integrator is unconditionally stable. If \(\alpha=0\), the HHT integrator reduces to the Newmark method [[2]], for which unconditional stability requires \(2\beta \geq \gamma \geq 1/2\). The trapezoidal rule is recovered if \(\alpha=0\), \(\beta=0.25\), and \(\gamma=0.5\). For a detailed discussion of the theory of implicit time integrators, see [[3]].

### 4.11.1.4. Implicit Dynamic Timestep Cutback for Contact

    CONTACT TIMESTEP = ON|OFF(ON)
      [DURING <string list>period_names]

When running an implicit dynamics simulation, errors can sometimes occur when too large of a time step is taken for solving a problem involving contact. For example blocks move completely through each other during a single time step contact may be missed or solved inaccurately. The `CONTACT TIMESTEP` command line specifies whether the time step can be cut back for better integration of contact impact. The default is `ON` which allows for the time step to be cut. This parameter can also vary by period by appending the command lines with the optional `DURING` specification (see Section 4.2.1).

[1]

H.M. Hilber, T.J.R. Hughes, and R.L. Taylor. Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Engineering and Structural Dynamics, 5:283–292, 1977.

[2]

N.M. Newmark. A method of computation for structural dynamics. Journal of Engineering Mechanics, ASCE, 85(EM3):67–94, 1959.

[3]

T.J.R. Hughes. The Finite Element Method–Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Inc, Englewood Cliffs, NJ, 1987.