Convergence Criteria for Nonlinear Problems

The fields and conjugate fluxes available in Abaqus/Standard are as follows:

Basic problem	Field	Conjugate flux
Stress analysis: force equilibrium	Displacement, $\mathbf{u}$	Force, $\mathbf{F}$
Stress analysis: moment equilibrium	Rotation, $\varphi$	Moment, $\mathbf{M}$
Stress analysis: analysis containing beams with warping	Warping, w	Bimoment, W
Heat transfer analysis	Temperature, $\theta$	Heat flux, q
Acoustic analysis (linear only)	Acoustic pressure, u	Rate of change of fluid volumetric flux
Pore liquid flow analysis	Pore liquid pressure, u	Pore liquid volumetric flux, q
Hydrostatic fluid modeling	Fluid pressure, p	Fluid volume, V
Mass diffusion analysis	Normalized concentration, $\varphi$	Mass concentration volumetric flux, Q
Piezoelectric analysis	Electrical potential, $\phi$	Electrical charge, q
Electric conduction analysis	Electrical potential, $\phi$	Electrical current, J
Mechanism analysis (connector elements with material flow degree of freedom)	Material flow	Material flux
Analysis containing C3D4H elements (all materials, except compressible hyperelastic elastomers and elastomeric foams).	Pressure Lagrange multiplier	Volumetric flux
Analysis containing C3D4H elements with compressible hyperelastic or hyperfoam materials.	Volumetric Lagrange multiplier	Pressure flux

In some cases the problem also involves constraint equations. In Abaqus/Standard the following constraints are included by using Lagrange multipliers:

Problem	Constraint variable	Constraint
Hybrid solid (except C3D4H elements)	Pressure stress	Volumetric strain compatibility
Hybrid beam	Axial force	Axial strain compatibility
Hybrid beam	Transverse shear force	Transverse shear strain compatibility
Distributing coupling	Force	Coupling displacement compatibility
Distributing coupling	Moment	Coupling rotation compatibility
Contact	Normal pressure	Surface penetration
Contact with Lagrange friction	Shear stress	Relative shear sliding

If the penalty method is used, the contact Lagrange multipliers may not be present.

In a general problem several (possibly nonlinear) coupled field equations of types $\alpha =1,2,\mathrm{\dots }N$ must be solved and several different (possibly nonlinear) constraints of type $j=1,2,\mathrm{\dots }K$ must be satisfied simultaneously. For example, in a structural problem in which hybrid beam elements are used, $\alpha =1$ might represent the displacement field and the equilibrium equations for the conjugate force and $\alpha =2$ might represent the rotation field and the equilibrium equations for the conjugate moment, while $j=1$ represents axial strain compatibility and $j=2$ represents transverse shear strain compatibility.

Each field, $\alpha$, that is active in the problem is tested for convergence of the field equations. The following measures are used in deciding if an increment has converged:

$r_{\max }^{\alpha }$

The largest residual in the balance equation for field $\alpha$.

$\mathrm{\Delta }{u}_{\max }^{\alpha }$

The largest change in a nodal variable of type $\alpha$ in the increment.

$c_{\max }^{\alpha }$

The largest correction to any nodal variable of type $\alpha$ provided by the current Newton iteration.

$e^j$

The largest error in a constraint of type j.

${\overline{q}}^{\alpha }\left(t\right)$

The instantaneous magnitude of the flux for field $\alpha$ at time t, averaged over the entire model (spatial average flux). This average is by default defined by the fluxes that the elements apply to their nodes and any externally defined fluxes:

$${\overline{q}}^{\alpha }\left(t\right)\stackrel{\mathrm{def}}{=}\frac{1}{{\sum }_{e=1}^E{\sum }_{n_e=1}^{N_e}{N}_{n_e}^{\alpha }+N_{ef}^{\alpha }}\left(\stackrel{E}{\sum _{e=1}}\stackrel{N_E}{\sum _{n_e=1}}\stackrel{N_{n_e}^{\alpha }}{\sum _{i=1}}{\left|q\right|}_{i,n_e}^{\alpha }+\stackrel{N_{ef}^{\alpha }}{\sum _{i=1}}{\left|q\right|}_i^{\alpha ,ef}\right).$$

Here, E is the number of elements in the model, $N_e$ is the number of nodes in element e, $N_{n_e}^{\alpha }$ is the number of degrees of freedom of type $\alpha$ at node $n_e$ of element e, ${\left|q\right|}_{i,n_e}^{\alpha }$ is the magnitude of the total flux component that element e applies at its ith degree of freedom of type $\alpha$ at its $n_e$th node at time t, $N_{ef}^{\alpha }$ is the number of external fluxes for field $\alpha$ (depends on element type, loading type, and number of loads applied to an element), and ${\left|q\right|}_i^{\alpha ,ef}$ is the magnitude of the ith external flux for field $\alpha$.

${\tilde{q}}^{\alpha }\left(t\right)$

An overall time-averaged value of the typical flux for field $\alpha$ so far during this step including the current increment. Normally, ${\tilde{q}}^{\alpha }\left(t\right)$ is defined as ${\overline{q}}^{\alpha }$ averaged over all the increments in the step in which ${\overline{q}}^{\alpha }$ is nonzero. The ${\overline{q}}^{\alpha }$ for the current increment is recalculated after every iteration of the current increment.

$${\tilde{q}}^{\alpha }\left(t\right)\stackrel{\mathrm{def}}{=}\frac{1}{N_t}\stackrel{N_t}{\sum _{i=1}}{\overline{q}}^{\alpha }\left({t|}_i\right),$$

where $N_t$ is the total number of increments so far in the step, including the current increment, in which ${\overline{q}}^{\alpha }\left({t|}_i\right)>{ϵ}^{\alpha }{\tilde{q}}^{\alpha }\left({t|}_i\right)$. Here ${\overline{q}}^{\alpha }\left({t|}_i\right)$ is the value of ${\overline{q}}^{\alpha }$ at increment i and ${ϵ}^{\alpha }$ is a small number. The default for ${ϵ}^{\alpha }$ is 10⁻⁵, but in rare cases, you can change this default.

Alternatively, you can define a value for the average flux in the step, ${\tilde{q}}_u^{\alpha }$. In this case, ${\tilde{q}}^{\alpha }\left(t\right)={\tilde{q}}_u^{\alpha }$ throughout the step.

At the start of the step, ${\tilde{q}}^{\alpha }$ is normally the value from the previous step (except for Step 1, when ${\tilde{q}}^{\alpha }={10}^{-2}$ by default). Alternatively, you can define an initial value for the time average flux, ${\tilde{q}}_0^{\alpha }$, as described in Modifying the Initial Time Average Flux. ${\tilde{q}}^{\alpha }$ retains its initial value until an iteration is completed for which ${\overline{q}}^{\alpha }>{ϵ}^{\alpha }{\tilde{q}}^{\alpha }$, at which time we redefine ${\tilde{q}}^{\alpha }={\overline{q}}^{\alpha }$. (If ${\tilde{q}}_u^{\alpha }$ is defined, the value defined for ${\tilde{q}}_0^{\alpha }$ is ignored.)

${\tilde{q}}_{\max }^{\alpha }$

The time-averaged value of the largest flux corresponding to the field $\alpha$ during this step, excluding the current increment.

$q_{\max }^{\alpha }$

The largest flux corresponding to the field $\alpha$ during the current iteration.

The time-averaged value of the flux (${\tilde{q}}^{\alpha }\left(t\right)$) is computed from the spatial average of the flux (${\overline{q}}^{\alpha }\left(t\right)$) at various instants in time. In some situations where only a small part of the model is active (the fluxes over the rest of the model are zero or very small), the spatial average of a flux over the entire model can be very small when compared to the spatial average over the active part of the model. Over a period of time this can result in a small value for the time-averaged value of the flux and in turn may lead to a convergence criterion that is very strict by engineering standards. To avoid such an excessively strict convergence criterion, Abaqus/Standard uses an algorithm to determine the active parts of a model at any given instant.

During an iteration any flux $\left|q_i^{\alpha }\left(t\right)\right|<{ϵ}_l^{\alpha }{\tilde{q}}_{\max }^{\alpha }$ is treated as inactive, and the corresponding degree of freedom is also marked inactive. ${\tilde{q}}_{\max }^{\alpha }$ is the time-averaged value of the largest flux in the model during the current step. The default value of ${ϵ}_l^{\alpha }$ is 10⁻⁵; you can redefine this parameter.

At the end of an iteration the largest flux in the model during the current iteration ($q_{\max }^{\alpha }$) is compared with the time-averaged value of the largest flux (${\tilde{q}}_{\max }^{\alpha }$). If $q_{\max }^{\alpha }\ge 0.1{\tilde{q}}_{\max }^{\alpha }$, the spatial average is computed over only the active parts of the model; if $q_{\max }^{\alpha }<0.1{\tilde{q}}_{\max }^{\alpha }$, all inactive parts of the model are reclassified as active and the spatial average is computed over the entire model. The appropriate spatial average of the flux obtained in this manner is then used to compute the time-averaged flux ${\tilde{q}}^{\alpha }\left(t\right)$ that is used in the convergence criterion. Setting ${ϵ}_l^{\alpha }=0$ forces the spatial averages of a flux to be always computed over the entire model.

If you specify a value for the average flux in the step, ${\tilde{q}}_u^{\alpha }$, ${\tilde{q}}^{\alpha }\left(t\right)={\tilde{q}}_u^{\alpha }$ throughout the step.

Most nonlinear engineering calculations will be sufficiently accurate if the error in the residuals is less than $\frac{1}{2}$%. Therefore, Abaqus/Standard normally uses

as the residual check, where you can define $R_n^{\alpha }$ (it is 0.005 by default). If this inequality is satisfied, convergence is accepted if the largest correction to the solution, $c_{\max }^{\alpha }$, is also small compared to the largest incremental change in the corresponding solution variable, $\mathrm{\Delta }{u}_{\max }^{\alpha }$,

or if the magnitude of the largest correction to the solution that would occur with one more iteration, estimated as

satisfies the same criterion:

You can define $C_n^{\alpha }$; the default value is 10⁻².

The superscripts i, $i-1$, and $i-2$ refer to the iteration number, and ${\left(r_{\max }^{\alpha }\right)}^0$ refers to the largest residual in field $\alpha$ at the start of the first iteration of the increment. See Commonly Used Control Parameters for more details on specifying $C_n^{\alpha }$.

In some cases there may be zero flux in the equations of type $\alpha$ anywhere in the model during some increments. Zero flux is defined as ${\overline{q}}^{\alpha }\le {ϵ}^{\alpha }{\tilde{q}}^{\alpha }$, where, as discussed earlier, ${ϵ}^{\alpha }$ has a default value of 10⁻⁵ and the solution for field $\alpha$ is accepted if $r_{\max }^{\alpha }\le {ϵ}^{\alpha }{\tilde{q}}^{\alpha }$. If not, $c_{\max }^{\alpha }$ is compared to $\mathrm{\Delta }{u}_{\max }^{\alpha }$, and convergence for field $\alpha$ is accepted when $c_{\max }^{\alpha }\le C_{ϵ}^{\alpha }\mathrm{\Delta }{u}_{\max }$. The default value of $C_{ϵ}^{\alpha }$ is 10⁻³; you can redefine this parameter.

Cases may arise where more than one field is active in the model yet there is negligible response in some of the fields in some increments. If some type of physical conversion factor, $f_{\beta }^{\alpha }$, exists between active fields $\alpha$ and $\beta$, ${\tilde{q}}^{\alpha }$ in the above paragraph can be replaced by $f_{\beta }^{\alpha }{C}_f{\tilde{q}}^{\beta }$ for those particular increments where ${\tilde{q}}^{\alpha }$ is deemed too small (${\overline{q}}^{\alpha }\le {\tilde{q}}^{\alpha }<f_{\beta }^{\alpha }{C}_f{\tilde{q}}^{\beta }$) to be used realistically as part of the convergence criteria for field $\alpha$. An example of $f_{\beta }^{\alpha }$ is a characteristic length to convert between force and moment.

Here, $f_{\beta }^{\alpha }$ is a factor calculated by Abaqus/Standard based on the problem definition and the fields involved and $C_f$ is a field conversion ratio that you can define. The default value for $C_f$ is 1.0. Currently, this concept is used only for converting between the fields associated with forces and moments, when $f_{\beta }^{\alpha }$ represents a characteristic element length.

Linear cases do not require more than one equilibrium iteration per increment. If

for all $\alpha$, the increment is considered to be linear.

You can define $R_l^{\alpha }$; it is intended to be very small. The default value of $R_l^{\alpha }$ is 10⁻⁸. Any case that passes such a stringent comparison of the largest residual with the average flux magnitude in each field is considered linear and does not require further iteration. If this requirement is satisfied at some iteration after the first, the solution is accepted without any check on the size of the correction to the solution.

In some cases quadratic convergence of the iterations is not possible because the Jacobian of the Newton scheme is approximated. If after $I_P$ iterations the convergence rate is only linear, Abaqus/Standard uses a looser tolerance,

as the residual check. This tolerance modification is not applied when the quasi-Newton method is used, since it is normal for this method to require a larger number of iterations to converge.

You can define $R_p^{\alpha }$, which is 2 × 10⁻² by default. You can also define $I_P$ (by default, $I_P=9$; see Controlling Iteration).

Convergence also requires that

Iteration continues until both criteria are satisfied for all active fields or the increment is abandoned.

When the active field is the displacement, the convergence criterion requiring the largest displacement correction to be small relative to the maximum displacement increment ($c_{\max }^{\alpha }\le C_n^{\alpha }\mathrm{\Delta }{u}_{\max }^{\alpha }$) is ignored when the maximum displacement increment itself is very small, as defined by $\mathrm{\Delta }{u}_{max}^{\alpha }<{ϵ}_d^{\alpha }{f}_{\beta }^{\alpha }$, where $f_{\beta }^{\alpha }$ is the characteristic element length. The default value for ${ϵ}_d^{\alpha }$ is 10⁻⁸; you can redefine this parameter.

Abaqus/Standard may have trouble with the element calculations because of excessive distortion in large-displacement 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, $D_H=0.25$. If fixed time stepping has been chosen, the analysis will terminate with an error message.

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, $r_{\max }^{\alpha }$. 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

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, $D_f=0.25$. 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 $N^{ls}=4$ sometimes helps in such cases (see Improving the Efficiency of the Solution by Using the Line Search Algorithm).

In case quadratic convergence cannot be obtained, the logarithmic rate of convergence,

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 $\alpha$ 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, $I_R=8$, $I_C=16$, and $D_C$=0.5.

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). 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, $I_G=4$, $I_L=10$, $D_D=1.5$, and $D_B=0.75$.

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

where $\mathrm{\Delta }{t}_i$ is the proposed new time increment, and $\mathrm{\Delta }{t}_{i-1}$ 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.

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 for details on intermittent contact in dynamic problems.

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, $I_S^c=50$, which in practice should always be more than the actual number of iterations in an increment.

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, $I_S=12$ and $D_S=0.25$.