Time integration accuracy in transient problems

Abaqus/Standard usually uses automatic time stepping schemes for the solution of transient problems. Factors influencing the increment size for transient problems include convergence aspects related to the degree of geometric, material, and contact nonlinearity (which also influence non-transient problems and are discussed in Convergence criteria for nonlinear problems) and the ability of the time integration operator to accurately resolve variations in the accelerations, velocities, and displacements over an increment. This section discusses tolerance parameters and adjustments to the time increment size related to the latter aspect.

The following topics are discussed:

Time incrementation parameters and adjustment criteria
Avoiding small changes to the time increment size during implicit integration procedures

Time incrementation parameters and adjustment criteria

Table 1 lists tolerance parameters available for specific analysis procedures. Descriptions of time integrators for the transient procedure types and, in the case of implicit dynamics, discussion of additional factors influencing the time increment size related to accuracy of time integration are provided in the respective sections referenced in Table 1.

Table 1. Time integration accuracy measures for various procedures.
Procedure	Accuracy measure $(S^{J})$	Tolerance $(T^{J})$
Implicit dynamics (Implicit dynamic analysis using direct integration)	Half-increment residual	Half-increment residual tolerance
Transient heat transfer analysis (Uncoupled heat transfer analysis)	Temperature increment, $Δ θ$	$Δ θ_{m a x}$
Consolidation analysis (Coupled pore fluid diffusion and stress analysis)	Pore pressure increment, $Δ u_{w}$	$Δ u_{w}^{m a x}$
Creep and viscoelastic material behavior (Rate-dependent plasticity: creep and swelling)	$({{\dot{\bar{ε}}}^{c r} \|}_{t + Δ t} - {{\dot{\bar{ε}}}^{c r} \|}_{t}) Δ t$	Creep tolerance

In any transient analysis where automatic time incrementation is used, some of these tolerances, $T^{J}$ , $J = 1, 2, \dots$ , will be active. Corresponding measures of the integration accuracy, $S^{J}$ , will be calculated for each increment in the step. Abaqus/Standard will use these values to adjust the time incrementation using the criteria described in this section. The smallest time increment required by all criteria is used if more than one accuracy measure is active.

Reducing the time increment size

If $S^{J} > T^{J}$ for any control, J, that is active in the step, the time increment $Δ t$ is too large to satisfy that time integration accuracy requirement. The increment is, therefore, begun again with a time increment of

D_{A} (\frac{T^{J}}{S^{J}}) Δ t,

where you can define the value of $D_{A}$ . By default, $D_{A}$ = 0.85.

Input File Usage

CONTROLS, PARAMETERS=TIME INCREMENTATION
first data line
 , , ,  $D_{A}$

Abaqus/CAE Usage

Step module: OtherGeneral Solution ControlsEdit: toggle on Specify: Time Incrementation; click More to see additional data tables

Increasing the time increment size

If at the current time increment, $Δ t$ ,

Δ t {(\frac{S^{J}}{Δ t})}_{i} < W_{G} T^{J}

for all J in each of $I_{T}$ consecutive increments, i, and if no cut-back has occurred within those increments because of nonlinearity, the next time increment will be increased to

min (D_{G} Δ t_{p}, D_{M} Δ t) .

You can define the values of $I_{T}$ , $W_{G}$ , and $D_{G}$ . By default, $I_{T}$ = 3, $W_{G}$ = 0.75, and $D_{G}$ = 0.8. $Δ t_{p}$ is the proposed new time increment, which is defined as

Δ t_{p} = (\frac{T^{J}}{S^{J} / Δ t})

for transient heat transfer and transient mass diffusion problems and which is defined as

Δ t_{p} = I_{T} (\frac{T^{J}}{\sum_{i = 1}^{I_{T}} {(S^{J} / Δ t)}_{i}})

for other transient problems.

A limit, $D_{M}$ , is placed on the time increment increase factor. The default value of $D_{M}$ depends on the type of analysis:

$D_{M}^{d y n}$ = 1.25 for dynamic analysis
$D_{M}^{d i f f}$ = 2.0 for diffusion-dominated processes: creep, transient heat transfer, coupled temperature-displacement, soils consolidation, and transient mass diffusion
$D_{M}$ = 1.5 for all other cases

You can redefine $D_{M}$ for each analysis type.

If the problem is nonlinear, the time increment may be restricted by the rate of convergence of the nonlinear equations. The time incrementation controls used with nonlinear problems are described in Convergence criteria for nonlinear problems.

Input File Usage

CONTROLS, PARAMETERS=TIME INCREMENTATION
 , , , , , , , , ,  $I_{T}$ 
 , , , , , , ,  $W_{G}$ 
 $D_{G}$ ,  $D_{M}$ ,  $D_{M}^{d y n}$ ,  $D_{M}^{d i f f}$

Abaqus/CAE Usage

Step module: OtherGeneral Solution ControlsEdit: toggle on Specify: Time Incrementation; click More to see additional data tables

Avoiding small changes to the time increment size during implicit integration procedures

In linear transient problems when Abaqus/Standard uses implicit integration, the system of equations must be reformed and decomposed whenever the time increment changes even though the stiffness matrix does not change. Therefore, to reduce the number of increments at which the system matrix changes, Abaqus/Standard makes use of the factor $D_{L}$ , where

D_{L} = m i n (\frac{Δ t_{p}}{D_{M} Δ t}) .

The definition of $D_{L}$ results in the following inequality between the proposed and the current time increments:

Δ t_{p} \geq D_{L} D_{M} Δ t .

Based on this inequality the time increment is allowed to increase only when its value computed by the criteria described earlier in this section, or computed using the value of PNEWDT specified in certain user subroutines (UMAT, for example), is greater than or equal to $D_{L} D_{M} Δ t$ . The default value of $D_{L}$ is 1.0, but you can redefine it to be a smaller number. Reducing $D_{L}$ to a value less than 1.0 allows the time increment to increase by a factor that is smaller than $D_{M}$ , thereby forcing a time increment change, even if the change is small. Otherwise, the solution continues with the same $Δ t$ .

Input File Usage

CONTROLS, PARAMETERS=TIME INCREMENTATION
first data line
second data line
 , , , ,  $D_{L}$

Abaqus/CAE Usage

Step module: OtherGeneral Solution ControlsEdit: toggle on Specify: Time Incrementation; click More to see additional data tables