Implicit dynamic analysis using direct integration

Time integration methods

Abaqus/Standard uses the Hilber-Hughes-Taylor time integration by default unless you specify that the application type is quasi-static. The Hilber-Hughes-Taylor operator is an extension of the Newmark $\beta$-method. Numerical parameters associated with the Hilber-Hughes-Taylor operator are tuned differently for moderate dissipation and transient fidelity applications (as discussed later in this section). The backward Euler operator is used by default if the application classification is quasi-static.

These time integration operators are implicit, which means that the operator matrix must be inverted and a set of simultaneous nonlinear dynamic equilibrium equations must be solved at each time increment. This solution is done iteratively using Newton's method. The principal advantage of these operators is that they are unconditionally stable for linear systems; there is no mathematical limit on the size of the time increment that can be used to integrate a linear system. An unconditionally stable integration operator is of great value when studying structural systems because a conditionally stable integration operator (such as that used in the explicit method) can lead to impractically small time steps and, therefore, a computationally expensive analysis.

Marching through a simulation with a finite time increment size generally introduces some degree of numerical damping. This damping differs from the material damping discussed in Material damping (and in many cases these two forms of damping will work well together). The amount of damping associated with the time integration varies among the operator types (for example, the backward Euler operator tends to be more dissipative than the Hilber-Hughes-Taylor operator) and in many cases (such as with the Hilber-Hughes-Taylor operator) depends on settings of numerical parameters associated with the operator. The ability of the operator to effectively treat contact conditions is often of considerable importance with respect to their usefulness. For example, some changes in contact conditions can result in “negative damping” (nonphysical energy source) for many time integrators, which can be very undesirable.

It is possible to override the time integrator implied by the application-type classification; for example, you can perform a moderate dissipation dynamic analysis using the backward Euler integrator. Changing the default integrator is not generally recommended but may be useful in special cases.

DYNAMIC, TIME INTEGRATOR=HHT-TF

DYNAMIC, TIME INTEGRATOR=HHT-MD

DYNAMIC, TIME INTEGRATOR=BWE

Additional control over integrator parameters

Additional user controls enable modifications to settings of numerical parameters associated with the Hilber-Hughes-Taylor operator (see Hilber, Hughes, and Taylor (1977) for descriptions of the numerical parameters). The default parameter settings depend on the specified application type, as indicated in Table 1 (see Czekanski, El-Abbasi, and Meguid (2001) for the basis of these settings).

These parameters can be adjusted or modified individually if the Hilber-Hughes-Taylor operator is being used. If the default settings of these parameters correspond to the transient fidelity settings shown in Table 1 and you explicitly modify the $\alpha$ parameter alone, the other parameters will be adjusted automatically to $\beta =1/4\cdot {\left(1-\alpha \right)}^2$ and $\gamma =1/2-\alpha$. This relation provides control of the numerical damping associated with the time integrator while preserving desirable characteristics of the integrator. The numerical damping grows with the ratio of the time increment to the period of vibration of a mode. Negative values of $\alpha$ provide damping; whereas $\alpha =0$ results in no damping (energy preserving) and is exactly the trapezoidal rule (sometimes called the Newmark $\beta$-method, with $\beta =1/4$ and $\gamma =1/2$). The setting $\alpha =-1/3$ provides the maximum numerical damping. It gives a damping ratio of about 6% when the time increment is 40% of the period of oscillation of the mode being studied. Allowable values of $\alpha$, $\beta$, and $\gamma$ are: $-\frac{1}{2}\le \alpha \le 0$, $\beta >0$, $\gamma \ge \frac{1}{2}$.

DYNAMIC, ALPHA=$\alpha$, BETA=$\beta$, GAMMA=$\gamma$

Step module: Create Step: General: Dynamic, Implicit: Other: Alpha: Specify: $\alpha$

Default incrementation schemes

Automatic time incrementation is used by default for nonlinear dynamic procedures. The main factors used to control adjustments to the time increment size for an implicit dynamic procedure are the convergence behavior of the Newton iterations and the accuracy of the time integration. The time increment size may vary considerably during an analysis. Details of the time increment control algorithm depend on the type of dynamic application you are studying.

The following factors are considered by default in the time increment control algorithm if you specify a quasi-static–type application (the same factors control the time increment size for purely static analyses):

• The time increment size is reduced if an increment appears to be diverging or if the convergence rate is slow.

• The time increment size is fairly aggressively increased if rapid convergence occurs in previous increments.

Analyses for moderate dissipation-type applications also use these same factors, as well as a default upper bound on the time increment size equal to one-tenth of the step duration.

The following factors are considered by default in the time increment control algorithm if you specify a transient fidelity–type application:

• The time increment size is reduced if an increment appears to be diverging or if the convergence rate is slow.

• The time increment size is reduced if changes in contact status are detected during the first attempt of processing an increment. The new increment size is set such that the end of the increment corresponds to the average time of the contact status changes that were detected with the previous increment size. (In such cases an additional very small time increment is used to enforce compatibility of velocities and accelerations across active contact interfaces.)

• The time increment size is reduced if the half-increment residual (out-of-balance force) halfway through a time increment exceeds the half-increment residual tolerance, which is 10,000 times the time average force for a contact analysis or 1000 times the time average force for an analysis without contact.

• The time increment is gradually increased if rapid convergence occurs in previous increments.

• The upper bound for the time increment size is equal to 1/100 of the step duration.

Intermittent contact/impact

The second and third factors described in the preceding list often result in very small time increment sizes for contact simulations that are performed as a transient fidelity application (and the time increment size tends to remain small due to the fourth factor). This problem can be avoided by specifying a different application type or by using more detailed user controls, as discussed below.

General settings for the time increment controls

A high level user control over which factors are considered by the time increment control algorithm can be used to override the defaults implied by the specified application type for the analysis. Regardless of the application type you have specified, you can enforce time increment controls associated with either quasi-static applications or transient fidelity applications.

DYNAMIC, INCREMENTATION=AGGRESSIVE

DYNAMIC, INCREMENTATION=CONSERVATIVE

Controlling the half-increment residual

Controls associated with the half-increment residual tolerance are provided for tuning the time incrementation. These controls are intended for advanced users and typically do not need to be modified.

DYNAMIC, NOHAF

DYNAMIC, HALFINC SCALE FACTOR=scale factor

DYNAMIC, HAFTOL=tolerance

Step module: Create Step: General: Dynamic, Implicit: Incrementation: toggle on Suppress half-increment residual calculation

Step module: Create Step: General: Dynamic, Implicit: Incrementation: Half-increment Residual: Specify scale factor: scale factor

Step module: Create Step: General: Dynamic, Implicit: Incrementation: Half-increment Residual: Specify value: tolerance

Controlling incrementation involving contact

By default, specifying a transient fidelity application typically results in reduced time increment sizes upon changes in contact status. An extra time increment with a very small size is subsequently performed to enforce compatibility of velocities and accelerations across active contact interfaces. Direct user control over these incrementation aspects is available.

DYNAMIC, IMPACT=NO

DYNAMIC, IMPACT=AVERAGE TIME

DYNAMIC, IMPACT=CURRENT TIME

Direct time incrementation

You may directly specify the time increment size to be used. This approach is not generally recommended but may be useful in special cases. The analysis will terminate if convergence tolerances are not satisfied within the maximum number of iterations allowed.

It is possible to ignore convergence tolerances: the solution to an increment is accepted after the specified maximum number of iterations allowed even if convergence tolerances are not satisfied. Ignoring convergence tolerances can result in highly nonphysical results and is not recommended except by analysts with a thorough understanding of how to interpret results obtained this way.

DYNAMIC, DIRECT

DYNAMIC, DIRECT=NO STOP

Step module: Create Step: General: Dynamic, Implicit: Incrementation: Fixed

Step module: Create Step: General: Dynamic, Implicit: Other: Accept solution after reaching maximum number of iterations

Default amplitude for loads

Loads such as applied forces or pressures are ramped on by default if you have selected the quasi-static application classification; such ramping tends to enhance robustness because the load increment size is proportional to the time increment size. For example, if the Newton iterations are not able to converge for a particular time increment size, the automatic time incrementation algorithm will reduce the time increment size and restart the Newton iterations with a smaller load incremental considered.

For the other application classifications the dynamic procedure applies loads with a step function by default such that the full load is applied in the first increment of the step (regardless of the time increment size) and the load magnitude remains constant over each step. Thus, if the first increment is unable to converge with the original time increment size, reducing the time increment will not reduce the load increment by default. In some cases the convergence behavior will still improve upon reducing the time increment because the regularizing effect of inertia on the integration operators is inversely proportional to the square of the time increment size. See Defining an analysis for more information on default amplitude types for the various procedures and how to override the default.