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Reattempting an increment because of trouble with element or material calculations
Abaqus/Standard may have trouble with the element calculations because of excessive distortion in largedisplacement problems or because of very large plastic strain increments. If this occurs and automatic time incrementation has been chosen, the increment will be attempted again with a time increment of D _ { H } times the current time increment, where you can define D _ { H } . By default, D _ { H } = 0 . 2 5 . If fixed time stepping has been chosen, the analysis will terminate with an error message.
Reattempting a diverging increment
Sometimes the increment is too large for the solution to converge at all—the initial state is outside the “radius of convergence” of the Newton method. This condition can be detected by observing the behavior of the largest residuals, r _ { \mathrm { m a x } } ^ { \alpha } . In some cases these will not decrease from iteration to iteration throughout an iteration sequence that leads to convergence, but we assume that, if they fail to decrease over two consecutive iterations, the iterations should be abandoned. Thus, if
\min \left(\left(r _ {\max} ^ {\alpha}\right) ^ {i}, \left(r _ {\max} ^ {\alpha}\right) ^ {i - 1}\right) > \left(r _ {\max} ^ {\alpha}\right) ^ {i - 2},
where i is the iteration counter, the iterations are abandoned. This check is first made after I _ { 0 } iterations following a solution discontinuity. You can define I _ { 0 } ; it must be at least 3. The default value of I _ { 0 } is 4. If fixed time stepping has been chosen, the analysis will terminate with an error message.
With automatic time stepping the increment is begun again, using a time increment of D _ { f } times the previous attempt, where you can define D _ { f } . By default, D _ { f } = 0 . 2 5 . This subdivision continues until a successful time increment is found or the minimum time increment allowed has failed, in which case the job ends with an error message. Using the line search algorithm with N ^ { l s } = 4 sometimes helps in such cases (see “Improving the efficiency of the solution by using the line search algorithm”).
Reattempting an increment when too many equilibrium iterations are required
In case quadratic convergence cannot be obtained, the logarithmic rate of convergence,
\ln \big (\big (r _ {\mathrm{max}} ^ {\alpha} \big) ^ {i} \big / \big (r _ {\mathrm{max}} ^ {\alpha} \big) ^ {i - 1} \big),
will often be maintained throughout the iteration process. This rate can be established during the early iterations. If convergence has not been achieved after I _ { R } or more iterations following a solution discontinuity, if automatic time incrementation has been selected, and if the slowest convergence rate over all fields suggests that more than I _ { C } total iterations subsequent to the last solution discontinuity are expected to be required, the increment is begun again with a time increment of D _ { C } times the one abandoned. If fixed time incrementation has been chosen, the iterations are continued; but if convergence is not achieved within I _ { C } iterations after the last solution discontinuity in the increment, the analysis will terminate with an error message.
You can define the values of I _ { R } , I _ { C } , and D _ { C } . By default, I _ { R } = 8 , I _ { C } = 1 6 , and D _ { C } { = } 0 . 5 .
Increasing or reducing the size of the time increment for efficiency
When automatic time incrementation is chosen, the effectiveness of the nonlinear equation solution is used in the selection of the next time increment (in addition to the time integration accuracy criteria discussed in “Time integration accuracy in transient problems,” Section 7.2.4). If no more than I _ { G } iterations are required in two consecutive increments, the time increment may be increased by a factor of D _ { D } . If an increment converges but takes more than I _ { L } iterations, the next time increment is reduced to D _ { B } times the current time increment. You can define the values of I _ { G } , I _ { L } , D _ { D } , and D _ { B } . By default, I _ { G } = 4 , I _ { L } = 1 0 , D _ { D } = 1 . 5 , and D _ { B } = 0 . 7 5 .
Extrapolation
At each increment after the first increment of a nonlinear analysis step Abaqus/Standard estimates the solution to the increment by extrapolating the solution from the previous increment (or increments). By default, 100% linear extrapolation is used (1% for the Riks method). Extrapolation is abandoned if
\Delta t _ {i} \leq D _ {E} \Delta t _ {i - 1},
where \Delta t _ { i } is the proposed new time increment, and \Delta t _ { i - 1 } is the last successful time increment. You can define the value of D _ { E } ; it is 0.1 by default.
You can turn this extrapolation scheme off for a particular step—see “Defining an analysis,” Section 6.1.2.
Convergence of strain constraints in hybrid elements
Strain constraint convergence in “hybrid” elements is checked by comparing the largest error in each strain constraint, e ^ { j } , with an absolute tolerance for the corresponding error, T ^ { j } . The magnitudes of these errors are reported in the message (.msg) file after each iteration as “compatibility errors.” For example, the volumetric compatibility error is a measure of the accuracy with which the incompressibility constraint is satisfied. Since nonlinearity in constraint equations is generally reflected in the field equations in the same problem, no attempt is made to estimate convergence rates in these constraint equations: we assume that the measures of convergence rate in the field equations are sufficient.
You can define the T ^ { j } \ : ( T ^ { v o l } , T ^ { a x i a l } , and T ^ { t s h e a r } ) . By default, all of the T ^ { j } = 1 0 ^ { - 5 } .
Input File Usage: *CONTROLS, PARAMETERS=CONSTRAINTS
T ^ {v o l}, T ^ {a x i a l}, T ^ {t s h e a r}
Abaqus/CAE Usage: Step module: Other→General Solution Controls→Edit: toggle on Specify: Constraint Equations
Severe discontinuity iterations
Abaqus/Standard distinguishes between regular, equilibrium iterations (in which the solution varies smoothly) and severe discontinuity iterations (SDIs) in which abrupt changes in stiffness occur. By default, Abaqus/Standard will continue to iterate until the severe discontinuities are sufficiently small (or no severe discontinuities occur) and the equilibrium (flux) tolerances are satisfied. For more information
on the criteria used for the severe discontinuity checks, see “Severe discontinuities in Abaqus/Standard” in “Defining an analysis,” Section 6.1.2. Alternatively, Abaqus/Standard will continue to iterate until no severe discontinuities occur and the equilibrium (flux) tolerances are satisfied. This more traditional method can cause convergence difficulties if the contact conditions are only weakly determined and contact “chattering” occurs or if a large number of severe discontinuity iterations are required to settle the contact conditions.
You can define the contact and slip compatibility tolerance, the soft contact compatibility tolerance for low pressure, and the contact force error tolerance.
Input File Usage: *CONTROLS, PARAMETERS=CONSTRAINTS , , , , , , , T ^ { c o n t } , T ^ { s o f t } , , , T ^ { c f e }
Abaqus/CAE Usage: Step module: Other→General Solution Controls→Edit: toggle on Specify: Constraint Equations Defining the contact force error tolerance is not supported in Abaqus/CAE.
Severe discontinuity iterations in implicit dynamic analysis
In implicit dynamic analysis, the average time of all contact changes in the increment is estimated and the time incrementation is interrupted to solve impact equations at that time. With augmented Lagrange or penalty constraint enforcement methods or with softened contact, no contact constraints are imposed when impact equations are solved. However, if the contact constraints are not satisfied within given tolerances, a severe discontinuity iteration is forced. See “Intermittent contact/impact,” Section 2.4.2 of the Abaqus Theory Guide, for details on intermittent contact in dynamic problems.
Controlling the number of severe discontinuity iterations
By default, Abaqus applies sophisticated criteria involving changes in penetration, changes in the residual force, and the number of severe discontinuities from one iteration to the next to determine whether iteration should be continued or terminated. Hence, it is in principle not necessary to limit the number of severe discontinuity iterations. This makes it possible to run contact problems that require large numbers of contact changes without having to change the control parameters. It is still possible to set a limit, I _ { S } ^ { c } , , for the maximum number of severe discontinuity iterations; by default, I _ { S } ^ { c } = 5 0 , which in practice should always be more than the actual number of iterations in an increment.
Input File Usage: *CONTROLS, PARAMETERS=TIME INCREMENTATION , , , , , ,
Abaqus/CAE Usage: Step module: Other→General Solution Controls→Edit: toggle on Specify: Time Incrementation; click More to see additional data tables
Controlling the number of severe discontinuity iterations when severe discontinuities always force iterations
In this case a limit, I _ { S } , is placed on the number of iterations caused by severe discontinuities in an increment. If more than I _ { S } iterations are required for severe discontinuities, the increment is started over with a time increment size of D _ { S } times the abandoned increment size (for automatic
time incrementation). If fixed time incrementation was chosen, the analysis terminates with an error message. You can define the values of I _ { S } and D _ { S } . By default, I _ { S } = 1 2 and D _ { S } = 0 . 2 5 .
Input File Usage: *CONTROLS, PARAMETERS=TIME INCREMENTATION
\begin{array}{c}, \ldots , \ldots , I _ {S} \\ , \ldots , D _ {S} \end{array}
Abaqus/CAE Usage: Step module: Other→General Solution Controls→Edit: toggle on Specify: Time Incrementation; click More to see additional data tables
Improving the efficiency of the solution by using the line search algorithm
Abaqus/Standard provides the option of including a “line search” algorithm. The purpose of the line search is to improve the robustness of the Newton or quasi-Newton methods. By default, the line search is active only for steps that use the quasi-Newton method. During equilibrium iterations where residuals are large, the line search algorithm scales the correction to the solution by a line search scale factor, s ^ { l s } . An iterative process is used to find the value of \textit { s } ^ { l s } that minimizes the component of the residual vector in the direction of the correction vector; this component is called y ^ { j } , where j is the line search iteration number. Each line search iteration requires one pass through the Abaqus/Standard element loop but does not require any operations using the global stiffness matrix.
It is usually sufficient to determine s ^ { l s } only to modest accuracy. There are several controls used to limit this accuracy. \mathbf { A } maximum of \mathbf { \hat { \boldsymbol { j } } } = N ^ { l s } line search iterations are performed. There is a limit on the allowable range of s ^ { l s } :
s _ {m i n} ^ {l s} \leq s ^ {l s} \leq s _ {m a x} ^ {l s}.
The line search ceases when
| y ^ {j} | \leq f _ {s} ^ {l s} | y ^ {0} |,
where y ^ { 0 } is evaluated before the first equilibrium iteration. The residual reduction factor at which the line search ceases, f _ { s } ^ { l s } , is typically set to a rather loose tolerance. The line search algorithm will also cease when the change in s ^ { l s } provided by a line search iteration is less than \eta ^ { l s } times s ^ { l s } .
You canmethod, and the values of with the qua N ^ { l s } , s _ { m a x } ^ { l s } , s _ { m i n } ^ { l s } , f _ { s } ^ { l s } \eta ^ { l s } . By defauto a nonzer N ^ { l s } = 0 with the Newtonactivate the line N ^ { l s } { = } 5 N ^ { l s } search algorithm or to zero to forcibly deactivparameters are = 1.0, = 0.0001, f _ { s } ^ { l s } = 0 . 2 5 rch. , and \eta ^ { l s } = 0 . 1 0 es for the additional line search. These defaults are chosen to achieve modest accuracy for the line search scale factor, while minimizing the additional cost of line search iterations. More agressive line searching can be beneficial in some simulations, especially when many nonlinear iterations and/or cutbacks are needed to resolve sharp discontinuities in the solution. In these cases you could try allowing more line search iterations ( N ^ { l s } = 1 0 ) and requiring more accuracy in the line search scale factor ( \ \eta ^ { l s } = 0 . 0 1 ) . This may result in more line search iterations but fewer nonlinear iterations and cutbacks and an overall reduction in solution cost.
Input File Usage: *CONTROLS, PARAMETERS=LINE SEARCH
N ^ {l s}, s _ {m a x} ^ {l s}, s _ {m i n} ^ {l s}, f _ {s} ^ {l s}, \eta^ {l s}
Abaqus/CAE Usage: Step module: Other→General Solution Controls→Edit: toggle on Specify: Line Search Control
7.2.4 TIME INTEGRATION ACCURACY IN TRANSIENT PROBLEMS
Products: Abaqus/Standard Abaqus/CAE
References
• “Convergence and time integration criteria: overview,” Section 7.2.1
• “Implicit dynamic analysis using direct integration,” Section 6.3.2
• “Uncoupled heat transfer analysis,” Section 6.5.2
• “Coupled pore fluid diffusion and stress analysis,” Section 6.8.1
• “Rate-dependent plasticity: creep and swelling,” Section 23.2.4
• *CONTROLS
• “Customizing general solution controls,” Section 14.15.1 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
Overview
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,” Section 7.2.3) 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.
Time incrementation parameters and adjustment criteria
Table 7.2.4–1 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 7.2.4–1.
Table 7.2.4–1 Time integration accuracy measures for various procedures.
| Procedure | Accuracy measure ( $S^{J}$ ) | Tolerance ( $T^{J}$ ) |
| Implicit dynamics (“Implicit dynamic analysis using direct integration,” Section 6.3.2) | Half-increment residual | Half-increment residual tolerance |
| Transient heat transfer analysis (“Uncoupled heat transfer analysis,” Section 6.5.2) | Temperature increment, $\Delta\theta$ | $\Delta\theta_{max}$ |
| Consolidation analysis (“Coupled pore fluid diffusion and stress analysis,” Section 6.8.1) | Pore pressure increment, $\Delta u_w$ | $\Delta u_w^{max}$ |
| Creep and viscoelastic material behavior (“Rate-dependent plasticity: creep and swelling,” Section 23.2.4) | $(\dot{\bar{\varepsilon}}^{cr}|_{t+\Delta t}-\dot{\bar{\varepsilon}}^{cr}|_t)\Delta t$ | Creep tolerance |
In any transient analysis where automatic time incrementation is used, some of these tolerances, T ^ { J } , J = 1 , 2 , . . . , 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 \Delta t is too large to satisfy that time integration accuracy requirement. The increment is, therefore, begun again with a time increment of
D _ {A} \left(\frac {T ^ {J}}{S ^ {J}}\right) \Delta t,
where you can define the value of D _ { A } . By default, D _ { A } = 0 . 8 5 .
Input File Usage: *CONTROLS, PARAMETERS=TIME INCREMENTATION first data line , , D _ { A }
Abaqus/CAE Usage: Step module: Other→General Solution Controls→Edit: toggle on Specify: Time Incrementation; click More to see additional data tables
Increasing the time increment size
If at the current time increment, \Delta t . ,
\Delta t \left(\frac {S ^ {J}}{\Delta t}\right) _ {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} \Delta t _ {p}, D _ {M} \Delta t).
You can define the values of I _ { T } , W _ { G } , and D _ { G } . By default, I _ { T } = 3 , W _ { G } = 0 . 7 5 , and D _ { G } = 0 . 8 . \ \Delta t _ { p } is the proposed new time increment, which is defined as
\Delta t _ {p} = \left(\frac {T ^ {J}}{S ^ {J} / \Delta t}\right)
for transient heat transfer and transient mass diffusion problems and which is defined as
\Delta t _ {p} = I _ {T} \left(\frac {T ^ {J}}{\sum_ {i = 1} ^ {I _ {T}} \left(S ^ {J} / \Delta t\right) _ {i}}\right)
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 . 2 5
• D _ { M } ^ { d i f f } = 2 . 0 for diffusion-dominated processes: creep, transient heat transfer, coupled temperaturedisplacement, 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,” Section 7.2.3.
Input File Usage: *CONTROLS, PARAMETERS=TIME INCREMENTATION
\begin{array}{l} \text {, , , , , , , , , , ,} I _ {T} \\ \text {, , , , , , , ,} W _ {G} \\ D _ {G}, D _ {M}, D _ {M} ^ {d y n}, D _ {M} ^ {d i f f} \end{array}
Abaqus/CAE Usage: Step module: Other→General Solution Controls→Edit: 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} = \min \left(\frac {\Delta t _ {p}}{D _ {M} \Delta t}\right).
The definition of D _ { L } results in the following inequality between the proposed and the current time increments:
\Delta t _ {p} \geq D _ {L} D _ {M} \Delta 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 } \Delta 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 \Delta t . .
Input File Usage: *CONTROLS, PARAMETERS=TIME INCREMENTATION first data line second data line , , , ,
Abaqus/CAE Usage: Step module: Other→General Solution Controls→Edit: toggle on Specify: Time Incrementation; click More to see additional data tables
Part IV: Analysis Techniques
• Chapter 8, “Analysis Techniques: Introduction”
• Chapter 9, “Analysis Continuation Techniques”
• Chapter 10, “Modeling Abstractions”
• Chapter 11, “Special-Purpose Techniques”
• Chapter 12, “Adaptivity Techniques”
• Chapter 13, “Optimization Techniques”
• Chapter 14, “Eulerian Analysis”
• Chapter 15, “Particle Methods”
• Chapter 16, “Sequentially Coupled Multiphysics Analyses”
• Chapter 17, “Co-simulation”
• Chapter 18, “Extending Abaqus Analysis Functionality”
• Chapter 19, “Design Sensitivity Analysis”
• Chapter 20, “Parametric Studies”