# 7.2.1 CONVERGENCE AND TIME INTEGRATION CRITERIA: OVERVIEW Numerous control parameters are associated with the convergence and integration accuracy algorithms in Abaqus/Standard. These parameters are assigned default values that are chosen to optimize the accuracy and efficiency of the solution for a wide spectrum of nonlinear problems. You can change the solution control parameters, as described in the following sections: • A brief synopsis of the more important solution control parameters, together with a description of the circumstances in which they can be used effectively, is provided in “Commonly used control parameters,” Section 7.2.2. This section is likely to be the most useful for the general user and should be read first. • Abaqus/Standard incorporates an empirical algorithm designed to solve the equilibrium equations of nonlinear systems accurately and economically. The criteria used to establish convergence of nonlinear increments and the automatic adjustment of increment size based on the convergence rate are described in “Convergence criteria for nonlinear problems,” Section 7.2.3. • Abaqus/Standard allows you to choose “time integration accuracy parameters” in problems that have a physical time scale. The algorithms that use these parameters for automatically controlling time increment sizes are described in “Time integration accuracy in transient problems,” Section 7.2.4. • Abaqus/CFD allows you to choose the control parameters used in an Abaqus/CFD to Abaqus/Standard or to Abaqus/Explicit co-simulation to alleviate instability and mesh distortion during the analysis. # Modifying the default solution controls The default values for the solution control parameters need not be adjusted for most cases. You can reset them, however, within a step definition. Values given for the solution control parameters remain in effect for the remainder of the analysis or until they are reset. Input File Usage: \*CONTROLS The \*CONTROLS option can be repeated, if necessary, with different parameters. Abaqus/CAE Usage: Step module: Other→General Solution Controls→Edit: toggle on Specify # Resetting all default solution controls You can restore all solution control parameters to their default values. Input File Usage: \*CONTROLS, RESET Abaqus/CAE Usage: Step module: Other→General Solution Controls→Edit: toggle on Reset all parameters to their system-defined defaults # 7.2.2 COMMONLY USED CONTROL PARAMETERS Products: Abaqus/Standard Abaqus/CFD Abaqus/CAE # References • “Convergence and time integration criteria: overview,” Section 7.2.1 • \*CONTROLS • “Customizing general solution controls,” Section 14.15.1 of the Abaqus/CAE User’s Guide, in the HTML version of this guide # Overview Solution control parameters can be used to control: • nonlinear equation solution accuracy; • time increment adjustment; and • FSI stabilization and mesh distortion in an Abaqus/CFD to Abaqus/Standard or to Abaqus/Explicit co-simulation. These solution control parameters need not be changed for most analyses. In difficult cases, however, the solution procedure may not converge with the default controls or may use an excessive number of increments and iterations. After it has been established that such problems are not due to modeling errors, it may be useful to change certain control parameters. This section presents a brief synopsis of the more important solution control parameters, together with a description of the circumstances in which they can be used effectively. Values given for the solution control parameters remain in effect for the remainder of the analysis or until they are reset. You can restore all solution control parameters to their default values (see “Convergence and time integration criteria: overview,” Section 7.2.1). # Terminology In this section the word “flux” means the variable whose discretized equilibrium is being sought and for which the equilibrium equations may be nonlinear: force, moment, heat flux, concentration volumetric flux, or pore liquid volumetric flux. The word “field” refers to the basic variables of the system, such as the components of the displacement in a continuum stress analysis or temperature in a heat transfer analysis. The superscript refers to one such type of equation. The fields and corresponding fluxes available in Abaqus/Standard are listed in “Convergence criteria for nonlinear problems,” Section 7.2.3. Solution control parameters can be used to define tolerances for field equations. You can select the type of equation for which the solution control parameters are being defined, as shown in Table 7.2.2–1. The default tolerances can be reset if the analysis does not require high accuracy in the convergence criteria. Table 7.2.2–1 Selecting the field equation. The input file parameters are followed by the corresponding Abaqus/CAE conditions in parentheses.
Equilibrium equationInput file (Abaqus/CAE)DOF
All active fieldsFIELD=GLOBAL (Apply to all applicable fields)all
Force and bimomentFIELD=DISPLACEMENT (Displacement)1, 2, 3, 7
MomentFIELD=ROTATION (Rotation)4, 5, 6
Heat transferFIELD=TEMPERATURE (Temperature)11, 12, 13, ...
Hydrostatic fluidFIELD=HYDROSTATIC FLUID PRESSURE (Hydrostatic Fluid Pressure)8
Pore fluid pressureFIELD=PORE FLUID PRESSURE (Pore Fluid Pressure)8
Mass diffusionFIELD=CONCENTRATION (Concentration)11
Electrical conductionFIELD=ELECTRICAL POTENTIAL (Electrical Potential)9
Mechanism analysis (connector elements with material flow degree of freedom)FIELD=MATERIAL FLOW (Unsupported)10
Equilibrium equationInput file (Abaqus/CAE)DOF
Analysis containing C3D4H elements (all materials, except compressible hyperelastic elastomers and elastomeric foams).FIELD=PRESSURE LAGRANGE MULTIPLIER (Unsupported)N/A
Analysis containing C3D4H elements with compressible hyperelastic or hyperfoam materials.FIELD=VOLUMETRIC LAGRANGE MULTIPLIER (Unsupported)N/A
The most significant solution control parameters for field equation tolerances— $R _ { n } ^ { \alpha } , \ C _ { n } ^ { \alpha } , \tilde { q } _ { 0 } ^ { \alpha }$ , and $\tilde { q } _ { u } ^ { \alpha }$ —may have to be modified in cases where the residuals are large relative to the fluxes or in cases where the incremental solution is essentially zero. Input File Usage: \*CONTROLS, PARAMETERS=FIELD, FIELD=field Abaqus/CAE Usage: Step module: Other→General Solution Controls→Edit: toggle on Specify: Field Equations: Apply to all applicable fields or Specify individual fields: field # Modifying the residual control $R _ { n } ^ { \alpha }$ is the convergence criterion for the ratio of the largest residual to the corresponding average flux norm, $\tilde { q } ^ { \alpha }$ , for convergence. $\tilde { q } ^ { \alpha }$ is defined in “Convergence criteria for nonlinear problems,” Section 7.2.3. The default value is $R _ { n } ^ { \alpha } = 5 \times 1 0 ^ { - 3 }$ , which is rather strict by engineering standards but in all but exceptional cases will guarantee an accurate solution to complex nonlinear problems. The value for this ratio can be increased to a larger number if some accuracy can be sacrificed for computational speed. # Modifying the solution correction control $C _ { n } ^ { \alpha }$ is the convergence criterion for the ratio of the largest solution correction to the largest corresponding incremental solution value. The default value is $C _ { n } ^ { \alpha } = 1 0 ^ { - 2 }$ . In addition to sufficiently small residuals, Abaqus/Standard requires that the largest correction to the solution value be small in comparison to the largest corresponding incremental solution value. Some analyses may not require such accuracy, thus permitting this ratio to be increased. To avoid testing the magnitude of the solution correction, you can set $C _ { n } ^ { \alpha }$ to 1.0. # Specifying the average flux $\tilde { q } ^ { \alpha }$ is the value of average flux used by Abaqus/Standard for checking residuals. The default value is the time average flux calculated by Abaqus/Standard, as defined in “Convergence criteria for nonlinear problems,” Section 7.2.3. You may, however, define a constant value, $\tilde { q } _ { u } ^ { \alpha }$ , for the average flux, in which case $\tilde { q } ^ { \alpha } = \tilde { q } _ { u } ^ { \alpha }$ throughout the step. You may wish to use absolute tolerances for your residual checks. The absolute tolerance value is then equal to the product of the average flux, $\tilde { q } _ { u } ^ { \alpha }$ , and the ratio $R _ { n } ^ { \alpha }$ . # Modifying the initial time average flux $\tilde { q } _ { 0 } ^ { \alpha }$ is the initial value of the time average flux for the current step. The default value is the time average flux from the previous step or $1 0 ^ { - 2 }$ if this is Step 1. Redefining $\tilde { q } _ { 0 } ^ { \alpha }$ is sometimes helpful when a coupled problem is analyzed and some of the fields in the problem are not active in the first step; for example, if a static step is carried out before a fully coupled thermal-stress step. Redefinition of $\tilde { q } _ { 0 } ^ { \alpha }$ can also be useful if the first step is essentially a null step; for example, in a contact problem before any contact occurs, the initial fluxes (forces) generated are zero. In such cases $\tilde { q } _ { 0 } ^ { \alpha }$ should be given as a typical flux magnitude that will occur when field first becomes active. The initial value of $\tilde { q } ^ { \alpha }$ is retained until an iteration is completed for which $\bar { q } ^ { \alpha } > \epsilon ^ { \alpha } \tilde { q } ^ { \alpha }$ , at which time we redefine $\tilde { q } ^ { \alpha } = \bar { q } ^ { \alpha }$ . This new $\tilde { q } ^ { \alpha }$ can become less than $\tilde { q } _ { 0 } ^ { \alpha }$ . The criterion for zero flux compared to $\tilde { q } ^ { \alpha }$ is $\bar { q } ^ { \alpha } \leq \tilde { q } ^ { \alpha } \cdot \epsilon ^ { \alpha }$ (see “Convergence criteria for nonlinear problems,” Section 7.2.3). If you specify the average flux, $\tilde { q } _ { u } ^ { \alpha }$ , directly, the value given for $\tilde { q } _ { 0 } ^ { \alpha }$ is ignored. # Abaqus/Standard output The controls in effect for an analysis are listed in the data (.dat) and message (.msg) files. Nondefault controls are marked by \*\*\*. For example, specifying the following controls:
Field Equation $R_n^\alpha$ $C_n^\alpha$ $\tilde{q}_0^\alpha$ $\tilde{q}_u^\alpha$ $R_P^\alpha$ $\epsilon^\alpha$
Displacement0.011.010.0--1.E-4
Rotation0.022.020.02.E3--
would result in the following output: CONVERGENCE TOLERANCE PARAMETERS FOR FORCE
*** CRIT. FOR RESIDUAL FORCE FOR A NONLINEAR PROBLEM1.000E-02
*** CRITERION FOR DISP. CORRECTION IN A NONLINEAR PROBLEM1.00
*** INITIAL VALUE OF TIME AVERAGE FORCE10.0
AVERAGE FORCE IS TIME AVERAGE FORCE
ALT. CRIT. FOR RESIDUAL FORCE FOR A NONLINEAR PROBLEM2.000E-02
*** CRIT. FOR ZERO FORCE RELATIVE TO TIME AVRG. FORCE1.000E-04
CRIT. FOR DISP. CORRECTION WHEN THERE IS ZERO FLUX1.000E-03
CRIT. FOR RESIDUAL FORCE WHEN THERE IS ZERO FLUX1.000E-08
FIELD CONVERSION RATIO1.00
CONVERGENCE TOLERANCE PARAMETERS FOR MOMENT
*** CRIT. FOR RESIDUAL MOMENT FOR A NONLINEAR PROBLEM2.000E-02
*** CRIT. FOR ROTATION CORRECTION IN A NONLINEAR PROBLEM2.00
*** INITIAL VALUE OF TIME AVERAGE MOMENT20.0
*** USER DEFINED VALUE OF AVERAGE MOMENT NORM2.000E+03
ALT. CRIT. FOR RESID. MOMENT FOR A NONLINEAR PROBLEM2.000E-02
CRIT. FOR ZERO MOMENT RELATIVE TO TIME AVRG. MOMENT1.000E-05
CRIT. FOR ROTATION CORRECTION WHEN ZERO FLUX1.000E-03
CRIT. FOR RESIDUAL MOMENT WHEN ZERO FLUX1.000E-08
FIELD CONVERSION RATIO1.00
# Controlling the time incrementation scheme Solution control parameters can be used to alter both the convergence control algorithm and the time incrementation scheme. The time incrementation parameters $I _ { 0 }$ and $I _ { R }$ are the most significant since they have a direct effect on convergence. They may have to be modified if convergence is (initially) nonmonotonic or if convergence is nonquadratic. Nonmonotonic convergence may occur if various nonlinearities interact; for example, the combination of friction, nonlinear material behavior, and geometric nonlinearity may lead to nonmonotonically decreasing residuals. Nonquadratic convergence will occur if the Jacobian is not exact, which may occur for complex material models. It may also occur if the Jacobian is nonsymmetric but the symmetric equation solver is used. In that case the unsymmetric equation solver should be specified for the step (see “Defining an analysis,” Section 6.1.2). Input File Usage: $\scriptstyle * { \mathrm { C O N T R O L S , P A R A M E T E R S } } = { \mathrm { T I M E ~ I N C R E M E N T A T I O N } }$ Abaqus/CAE Usage: Step module: Other→General Solution Controls→Edit: toggle on Specify: Time Incrementation # Specifying the equilibrium iteration for a residual check $I _ { 0 }$ is the number of equilibrium iterations after which the check is made that the residuals are not increasing in two consecutive iterations. The default value is $I _ { 0 } ~ = ~ 4$ . If the initial convergence is nonmonotonic, it may be necessary to increase this value. # Specifying the equilibrium iteration for a logarithmic rate of convergence check $I _ { R }$ is the number of equilibrium iterations after which the logarithmic rate of convergence check begins. The default value is $I _ { R } = 8$ . In cases where convergence is nonquadratic and this cannot be corrected by using the unsymmetric equation solver for the step, the logarithmic convergence check should be eliminated by setting this parameter to a high value. # Avoiding premature cutbacks in difficult analyses Sometimes it is useful to increase both $I _ { 0 }$ and $I _ { R }$ . For example, in a difficult analysis involving both friction and the concrete material model, it may be helpful to set $I _ { 0 } = 8$ and $I _ { R } = 1 0$ to avoid premature cutbacks of the time increment. These two parameters can be raised to more appropriate values for severely discontinuous problems by increasing them individually. # Automatically setting the time incrementation parameters You can automatically set the parameters described above to the values $I _ { 0 } = 8$ and $I _ { R } = 1 0$ . In this case any values that you specified previously for $I _ { 0 }$ and $I _ { R }$ are overridden. However, if $I _ { 0 }$ and $I _ { R }$ are specified multiple times in a step with different solution control settings, the last definition will be used. Input File Usage: \*CONTROLS, ANALYSIS=DISCONTINUOUS Abaqus/CAE Usage: Step module: Other→General Solution Controls→Edit: toggle on Specify: Time Incrementation: Discontinuous analysis # Improving solution efficiency in a problem that involves a high coefficient of friction The solution efficiency can sometimes be improved in an analysis that involves a high coefficient of friction by automatically setting the time incrementation parameters and using the unsymmetric equation solver. # Abaqus/Standard output The controls in effect for an analysis are listed in the data (.dat) and message (.msg) files. Nondefault controls are marked by \*\*. For example, specifying the time incrementation parameters $I _ { 0 } { = } 7$ and $I _ { R } { = } 1 0$ would result in the following output:
TIMEINCREMENTATION CONTROL PARAMETERS:
***FIRST EQUIL. ITERATION FOR CONSECUTIVE DIVERGENCE CHECK7
***EQUIL. ITER. AT WHICH LOG. CONVERGENCE RATE CHECK BEGINS10
EQUIL. ITER. AFTER WHICH ALTERNATE RESIDUAL IS USED9
MAXIMUM EQUILIBRIUM ITERATIONS ALLOWED16
EQUIL. ITERATION COUNT FOR CUT-BACK IN NEXT INCREMENT10
MAX EQUIL. ITERS IN TWO INCREMENTS FOR TIME INC. INCREASE4
MAXIMUM ITERATIONS FOR SEVERE DISCONTINUITIES12
MAXIMUM CUT-BACKS ALLOWED IN AN INCREMENT5
MAX DISCON. ITERS IN TWO INCS FOR TIME INC. INCREASE6
CUT-BACK FACTOR AFTER DIVERGENCE0.250
CUT-BACK FACTOR FOR TOO SLOW CONVERGENCE0.500
CUT-BACK FACTOR AFTER TOO MANY EQUILIBRIUM ITERATIONS0.750
# Activating the “line search” algorithm In strongly nonlinear problems the Newton algorithms used in Abaqus/Standard may sometimes diverge during equilibrium iteration. The line search algorithm (discussed in “Improving the efficiency of the solution by using the line search algorithm” in “Convergence criteria for nonlinear problems,” Section 7.2.3) detects these situations automatically and applies a scale factor to the computed solution correction, which helps to prevent divergence. The line search algorithm is particularly useful when the quasi-Newton method (see “Solution method” in “Convergence criteria for nonlinear problems,” Section 7.2.3) is used. By default, the line search algorithm is enabled only during steps where the quasi-Newton method is used. Set the maximum number of line search iterations, $N ^ { l s }$ , to a reasonable value (such as 5) to activate the line search procedure or to zero to forcibly deactivate the line search. Input File Usage: $\scriptstyle * { \mathrm { C O N T R O L S } } , { \mathrm { P A R A M E T E R S = L I N E ~ S E A R C H } }$ Nls Abaqus/CAE Usage: Step module: Other→General Solution Controls→Edit: toggle on Specify: Line Search Control: $N ^ { l s }$ # Defining tolerances for constraint equations Solution control parameters can be used to set tolerances for constraint equations. You can set strain compatibility tolerances for hybrid elements, displacement and rotation compatibility tolerances for distributing coupling constraints (specified as surface-based constraints or using DCOUP2D/DCOUP3D elements), and compatibility tolerances for softened contact. See “Convergence criteria for nonlinear problems,” Section 7.2.3, for details. # Controlling the solution accuracy in direct cyclic analysis Solution control parameters can be used in direct cyclic analysis to specify when to impose the periodicity conditions and to set tolerances for stabilized state and plastic ratchetting detections. Input File Usage: \*CONTROLS, TYPE=DIRECT CYCLIC $I _ { P I } , C R _ { n } ^ { \alpha } , \ C U _ { n } ^ { \alpha } , C R _ { 0 } ^ { \alpha } , \ C U _ { 0 } ^ { \alpha }$ Abaqus/CAE Usage: Step module: Other→General Solution Controls→Edit: toggle on Specify: Direct Cyclic: $I _ { P I } , C R _ { n } ^ { \alpha } , \ C U _ { n } ^ { \alpha } , C R _ { 0 } ^ { \alpha } , \ C U _ { 0 } ^ { \alpha }$ # Imposing the periodicity condition You can specify the iteration number at which the periodicity condition is first imposed, $I _ { P I }$ . The default value is $I _ { P I } = 1$ , in which case the periodicity condition is imposed for all iterations from the beginning of an analysis. This solution control parameter rarely needs to be reset from its default value. # Defining tolerances for stabilized state and plastic ratchetting detections You can specify the stabilized state detection criteria, $C R _ { n } ^ { \alpha }$ and $C U _ { n } ^ { \alpha } . \ C R _ { n } ^ { \alpha }$ is the maximum allowable ratio of the largest residual coefficient on any terms in the Fourier series to the corresponding average flux norm, and $C U _ { n } ^ { \alpha }$ is the maximum allowable ratio of the largest correction to the displacement coefficient on any terms in the Fourier series to the largest displacement coefficient. The default values are $C R _ { n } ^ { \alpha }$ $= 5 \times 1 0 ^ { - 3 }$ and $C U _ { n } ^ { \alpha } = 5 \times 1 0 ^ { - 3 }$ . The solution converges to a stabilized state if both these criteria are satisfied. If plastic ratchetting occurs, the shape of the stress-strain curves remains unchanged but the mean value of the plastic strain over a cycle continues to shift from one iteration to the next. In that case it is desirable to use separate tolerances for the constant term in the Fourier series to detect the plastic ratchetting. You can also specify the plastic ratchetting detection criteria, and . is the maximum allowable ratio of the largest residual coefficient on the constant term in the Fourier series to the corresponding average flux norm, and $C U _ { 0 } ^ { \alpha }$ is the maximum allowable ratio of the largest correction to the displacement coefficient on the constant term in the Fourier series to the largest displacement coefficient. The default values are $C R _ { 0 } ^ { \alpha } = 5 \times 1 0 ^ { - 3 }$ and $C U _ { 0 } ^ { \alpha } = 5 \times 1 0 ^ { - 3 }$ . Plastic ratchetting is expected if the residual coefficients and the corrections to the displacement coefficients on any of the periodic terms are within the tolerances set by $C R _ { n } ^ { \alpha }$ and $C U _ { n } ^ { \alpha }$ , respectively, but the maximum residual coefficient on the constant term and the maximum correction to the displacement coefficient on the constant term exceed the tolerances set by $C R _ { 0 } ^ { \alpha }$ and $C U _ { 0 } ^ { \alpha }$ , respectively. # Abaqus/Standard output The controls in effect for an analysis are listed in the data (.dat) and message (.msg) files. Nondefault controls are marked by \*\*. For example, specifying the following controls:
$I_{PI}$ $CR_{n}^{\alpha}$ $CU_{n}^{\alpha}$ $CR_{0}^{\alpha}$ $CU_{0}^{\alpha}$
51.0E-41.0E-41.0E-41.E-4
would result in the following output: STABILIZED STATE AND PLASTIC RATCHETTING DETECTION PARAMETERS FOR FORCE
** CRIT. FOR RESI. COEFF. ON ANY FOURIER TERMS1.0E-04
** CRIT. FOR CORR. TO DISP. COEFF. ON ANY FOURIER TERMS1.0E-04
** CRIT. FOR RESI. COEFF. ON CONSTANT FOURIER TERM1.0E-04
** CRIT. FOR CORR. TO DISP. COEFF. ON CONST. FOURIER TERM1.0E-04
PERIODICITY CONDITION CONTROL PARAMETER:
**ITERATION NUMBER AT WHICH PERIODICITY CONDITION
**STARTS TO IMPOSE
5 # Controlling the solution accuracy and mesh quality in a deforming-mesh analysis with Abaqus/CFD Solution control parameters can be used to control the mesh motion and to maintain the mesh quality in deforming-mesh problems involving moving boundaries or deforming geometries. They can also be used to control FSI stabilization when performing Abaqus/CFD to Abaqus/Standard or to Abaqus/Explicit co-simulation. # Controlling mesh smoothing and FSI stabilization When the implicit algorithm (default) for mesh smoothing is used, you can specify the number of iterations before performing a convergence check, the maximum number of iterations, the FSI penalty