Convergence criteria

Upcoming release

Scope of the convergence evaluation

The nonlinear convergence evaluation in numgeo is field-based. Every active physical field is assessed separately, because the unknowns and their conjugate residuals may differ substantially in magnitude and physical dimension. The active fields can comprise solid displacement \(\boldsymbol{u}\), pore-water pressure \(p^w\), pore-air pressure \(p^a\), water displacement \(\boldsymbol{w}\) and, where required by the element formulation, the volumetric variable \(J\) and mean pressure \(p_{mean}\).

Global convergence controls, in which all fields are combined into a single norm, are no longer evaluated. Overall convergence is achieved only when every active field with free equations has converged and all additional contact-iteration requirements are fulfilled. A field for which all equations are constrained does not contribute an equilibrium condition and is reported as inactive for the convergence evaluation.

The field-based evaluation uses three complementary measures:

• the residual or out-of-balance vector \(\boldsymbol{r}^{\alpha}\);
• the latest correction \(\boldsymbol{c}^{\alpha}\) relative to the accumulated solution increment \({}^{t}\Delta\boldsymbol{d}^{\alpha}\);
• the unbalanced energy of field \(\alpha\).

Residual equilibrium is mandatory. A correction or energy criterion is only a secondary indicator and cannot establish convergence by itself or in combination with another secondary criterion.

Norms and notation

The Euclidean norm is defined as

\[ \lVert \boldsymbol{x} \rVert_2 = \left(\sum_{j=1}^{n} x_j^2\right)^{1/2}. \]

The infinity norm is

\[ \lVert \boldsymbol{x} \rVert_\infty = \max_j |x_j|. \]

The superscript \(\alpha\) identifies a physical field. Quantities carrying the subscript \(free\) are evaluated only on unconstrained equations. Prescribed increments remain part of the deformation scale used by the correction and reference-energy criteria, but residual equilibrium is assessed only on free equations.

Solution-correction criterion

For field \(\alpha\), the largest absolute correction in Newton iteration \(i\) is

\[ c_{max}^{(i),\alpha} = \left\lVert \boldsymbol{c}^{(i),\alpha}\right\rVert_\infty, \]

and the largest accumulated solution increment is

\[ \Delta d_{max}^{(i),\alpha} = \left\lVert {}^{t}\Delta\boldsymbol{d}^{(i),\alpha}\right\rVert_\infty. \]

The relative correction criterion is fulfilled when

\[ c_{max}^{(i),\alpha} \leq \epsilon^{d,\alpha}_{act}\, \Delta d_{max}^{(i),\alpha}, \]

where \(\epsilon^{d,\alpha}_{act}\) is the currently active correction tolerance. It is equal to the user-defined or default value \(\epsilon^{d,\alpha}\) before relaxation and may be increased after the relaxation iteration described below.

An estimated-correction flag is also maintained by the convergence object. In the current implementation, the estimated maximum correction equals the latest maximum correction; consequently, it uses the same tolerance and does not provide an independent numerical measure.

Negligible increment and numerical stagnation

A very small response can make a relative correction ratio ill-conditioned. numgeo therefore identifies a numerically stagnant field after iteration zero when both

\[ \Delta d_{max}^{(i),\alpha} \leq \epsilon^{\Delta d,\alpha} \]

and

\[ c_{max}^{(i),\alpha} \leq \epsilon^{\Delta d,\alpha} \]

are fulfilled. Requiring both conditions prevents a field from being classified as stagnant while the current Newton correction is still significant.

Numerical stagnation does not establish convergence by itself. It activates two residual-based safeguards:

1. a relaxed relative residual criterion; and
2. a scale-aware absolute residual criterion.

The field is accepted through the stagnation route only if at least one of these residual checks is fulfilled.

Relative residual criterion

The residual norm is evaluated on the free equations of field \(\alpha\):

\[ R_2^{(i),\alpha} = \left\lVert \boldsymbol{r}^{(i),\alpha}_{free}\right\rVert_2. \]

The reference-force norm is recalculated in every iteration as

\[ F_{ref}^{(i),\alpha} = \max\left( \left\lVert\boldsymbol{f}^{\alpha}_{ext,free}\right\rVert_2, \left\lVert\boldsymbol{f}^{\alpha}_{dyn,free}\right\rVert_2, \left\lVert\boldsymbol{f}^{(i),\alpha}_{int,free}\right\rVert_2, \left\lVert\boldsymbol{f}^{t,\alpha}_{int,free}\right\rVert_2, \left\lVert\boldsymbol{f}^{(i-1),\alpha}_{int,free}\right\rVert_2 \right). \]

Here, \(\boldsymbol{f}^{t,\alpha}_{int}\) is the internal force at the start of the increment and \(\boldsymbol{f}^{(i-1),\alpha}_{int}\) is the internal force of the preceding iteration. The force contribution associated with Dirichlet boundary conditions is not included in the relative reference force because reaction forces may otherwise dominate the convergence scale.

The relative residual criterion is

\[ R_2^{(i),\alpha} \leq \epsilon^{r,\alpha}_{act}\, F_{ref}^{(i),\alpha}, \]

where \(\epsilon^{r,\alpha}_{act}\) is the currently active residual tolerance. Recomputing \(F_{ref}^{(i),\alpha}\) in every iteration prevents a temporary force spike from permanently loosening the convergence criterion.

When a field is numerically stagnant, an additional relaxed residual test is evaluated using

\[ R_2^{(i),\alpha} \leq 5\,\epsilon^{r,\alpha} F_{ref}^{(i),\alpha}. \]

The factor 5 used by this stagnation safeguard is fixed and is independent of the user-defined nonquadratic-convergence factor \(r^{\epsilon}\).

Scale-aware absolute residual safeguard

A purely relative residual criterion becomes ill-conditioned if both the residual and the reference force approach machine precision. In that case, their ratio can remain large although the absolute disequilibrium is numerically insignificant. To identify this situation without introducing a fixed dimensional tolerance, numgeo uses a component-wise numerical residual limit.

Let \(n_{\alpha}\) denote the number of equations belonging to field \(\alpha\), including constrained equations. A characteristic per-equation force scale is defined as

\[ F_{char}^{(i),\alpha} = \max\left( 1, \frac{\lVert\boldsymbol{f}^{\alpha}_{ext}\rVert_2}{\sqrt{n_{\alpha}}}, \frac{\lVert\boldsymbol{f}^{(i),\alpha}_{int}\rVert_2}{\sqrt{n_{\alpha}}}, \frac{\lVert\boldsymbol{f}^{\alpha}_{bc}\rVert_2}{\sqrt{n_{\alpha}}}, \frac{\lVert\boldsymbol{f}^{\alpha}_{dyn}\rVert_2}{\sqrt{n_{\alpha}}}, \frac{\lVert\boldsymbol{f}^{t,\alpha}_{int}\rVert_2}{\sqrt{n_{\alpha}}}, \frac{\lVert\boldsymbol{f}^{(i-1),\alpha}_{int}\rVert_2}{\sqrt{n_{\alpha}}} \right). \]

The lower bound of 1 is expressed in the force unit of the model. The component limit is

\[ R_{abs}^{(i),\alpha} = \epsilon_{mach}^{2/3} F_{char}^{(i),\alpha}, \]

where \(\epsilon_{mach}\) is the machine precision of the floating-point type used by numgeo. The safeguard is fulfilled when

\[ \max_{j\in free}\left|r_j^{(i),\alpha}\right| \leq R_{abs}^{(i),\alpha}. \]

The maximum residual component is used instead of the Euclidean norm so that the numerical limit does not become less restrictive as the number of equations increases.

For IEEE double precision and a unit characteristic force scale,

\[ \epsilon_{mach}^{2/3} \approx 3.67\cdot10^{-11}. \]

Accordingly, a fixed force tolerance of \(10^{-8}\) would be too large for a unit force scale and is not used. The dimensional absolute limit can exceed \(10^{-8}\) for models with very large characteristic forces, but it remains approximately \(3.67\cdot10^{-11}\) of the characteristic per-equation force scale and is considered only after numerical stagnation has been established.

Unbalanced-energy criterion

The unbalanced energy of field \(\alpha\) is evaluated on the free equations as

\[ E_r^{(i),\alpha} = \left| \left(\boldsymbol{r}^{(i),\alpha}_{free}\right)^T \boldsymbol{c}^{(i),\alpha}_{free} \right|. \]

The reference energy is calculated at iteration zero using the accumulated field increment and the average internal force between the start of the increment and iteration zero:

\[ E_{ref}^{\alpha} = \left| \left({}^{t}\Delta\boldsymbol{d}^{(0),\alpha}\right)^T \frac{1}{2} \left( \boldsymbol{f}^{t,\alpha}_{int} + \boldsymbol{f}^{(0),\alpha}_{int} \right) \right|. \]

The reference energy includes all equations of the field so that prescribed increments contribute to the deformation scale. The relative energy criterion is

\[ E_r^{(i),\alpha} \leq \epsilon^{e,\alpha}_{act} E_{ref}^{\alpha}. \]

The energy may additionally be classified as negligible after iteration zero if

\[ E_r^{(i),\alpha} \leq \epsilon^{\Delta e,\alpha}. \]

Energy is a secondary criterion. Neither the relative nor the negligible-energy criterion can establish convergence without a residual-equilibrium criterion.

The scalar products are evaluated using scaled arithmetic so that overflow is detected and converted into a controlled cutback rather than causing an unhandled floating-point exception.

Relaxation for nonquadratic convergence

The relaxation iteration is

\[ n_{rel} = \max\left( 1, \min\left( \left\lfloor\frac{n_{max}}{2}\right\rfloor, 12 \right) \right), \]

where \(n_{max}\) is the maximum number of equilibrium iterations allowed for the current step.

For ordinary nonlinear steps and \(i\geq n_{rel}\), the active tolerances are

\[ \epsilon^{d,\alpha}_{act}=r^{\epsilon}\epsilon^{d,\alpha}, \qquad \epsilon^{r,\alpha}_{act}=r^{\epsilon}\epsilon^{r,\alpha}, \qquad \epsilon^{e,\alpha}_{act}=r^{\epsilon}\epsilon^{e,\alpha}. \]

The default value is \(r^{\epsilon}=2\). For *Reduction steps, the correction tolerance is relaxed by \(r^{\epsilon}\), the residual tolerance is relaxed by the fixed factor 5, and the energy tolerance remains unchanged:

\[ \epsilon^{d,\alpha}_{act}=r^{\epsilon}\epsilon^{d,\alpha}, \qquad \epsilon^{r,\alpha}_{act}=5\epsilon^{r,\alpha}, \qquad \epsilon^{e,\alpha}_{act}=\epsilon^{e,\alpha}. \]

Relaxation changes the active tolerances only. It does not alter the requirement that residual equilibrium and a secondary criterion must both be fulfilled.

Acceptance of a field

For every active field with free equations, numgeo accepts convergence through one of the following routes:

1. numerical stagnation and the scale-aware absolute residual safeguard;
2. numerical stagnation and the relaxed relative residual criterion;
3. the active relative residual criterion and the relative correction criterion;
4. the active relative residual criterion and the estimated-correction criterion;
5. the active relative residual criterion and the relative energy criterion;
6. the active relative residual criterion and the negligible-energy criterion;
7. the active relative residual criterion and the negligible increment/correction criterion.

The following combinations are explicitly insufficient:

• correction and energy without residual equilibrium;
• negligible increment/correction without relative or absolute residual equilibrium;
• a residual criterion without any secondary criterion, except for the two specifically defined stagnation routes above.

The complete increment is accepted only if all active fields converge and the contact iteration, where applicable, is also converged.

Divergence, non-finite values and enforced cutback

The correction, accumulated increment, residual, external force, internal force, boundary force, dynamic force and stored internal-force vectors are checked for NaN and infinity before convergence measures are evaluated. A non-finite value is reported together with the affected quantity, global degree of freedom and node where available. The increment is rejected and a cutback is enforced even when NOCUTBACK is active.

A sustained energetic-divergence check is also applied. If the current unbalanced energy exceeds five times the larger of the initial-energy and negligible-energy scales, and simultaneously exceeds the preceding iteration by at least 25 %, a divergence counter is increased. A cutback is requested after four consecutive iterations satisfying both conditions. The counter is reset as soon as the growth conditions are no longer fulfilled.

Default field tolerances

The default user-adjustable tolerances are:

dof	\(\epsilon^{d}\)	\(\epsilon^{e}\)	\(\epsilon^{r}\)	\(\epsilon^{\Delta d}\)	\(\epsilon^{\Delta e}\)	\(r^{\epsilon}\)
u	\(1\cdot10^{-2}\)	\(1\cdot10^{-2}\)	\(1\cdot10^{-2}\)	\(1\cdot10^{-6}\)	\(1\cdot10^{-5}\)	2
w	\(1\cdot10^{-2}\)	\(1\cdot10^{-2}\)	\(1\cdot10^{-2}\)	\(1\cdot10^{-5}\)	\(1\cdot10^{-5}\)	2
pw	\(1\cdot10^{-2}\)	\(1\cdot10^{-2}\)	\(1\cdot10^{-2}\)	\(1\cdot10^{-3}\)	\(1\cdot10^{-10}\)	2
pa	\(1\cdot10^{-2}\)	\(1\cdot10^{-2}\)	\(1\cdot10^{-2}\)	\(1\cdot10^{-3}\)	\(1\cdot10^{-10}\)	2

The internally controlled mixed-element fields J and pmean use \(10^{-2}\) for the correction, residual and energy ratios, \(10^{-5}\) for the negligible increment and negligible energy, and \(r^{\epsilon}=2\).

The absolute residual safeguard has no user-defined force tolerance. Its limit is generated automatically from machine precision and the current characteristic force scale.

Convergence reporting

The message file (.msg) reports, for every active field:

• the unbalanced energy and active energy tolerance;
• the maximum correction, its node and the active correction tolerance;
• the residual and reference-force norms and the active residual tolerance;
• the external, current internal, boundary, dynamic, start-of-increment and preceding-iteration force norms;
• the maximum residual component, characteristic per-equation force scale and absolute component limit whenever the stagnation safeguard is evaluated;
• the exact acceptance or rejection route;
• non-finite and overflow diagnostics.

The existing .conv file retains the format

step inc attempt iter rhs_ratio inc_ratio rhs_criterion inc_criterion

for compatibility with the convergence viewer. If the absolute stagnation safeguard is active, equivalent ratios are written so that the legacy viewer displays the actual solver decision consistently. The legacy reporting interface does not provide the attempt number; this column therefore remains equal to 1.

A field-resolved file <jobname>.conv.fields is additionally written with one record per active field:

step inc iter field residual_ratio correction_ratio energy_ratio force_criterion correction_criterion energy_criterion max_abs_residual abs_component_limit status

The status values are:

• 1: field accepted;
• 0: field rejected;
• -1: non-finite value or arithmetic overflow;
• 2: field has no free equations and does not participate in equilibrium evaluation.

If an incremental solution strategy based on iterative methods is to be effective, realistic criteria should be used for the termination of the iteration. At the end of each iteration, the solution obtained should be checked to see whether it has converged within preset tolerances or whether the iteration is diverging. The convergence tolerance determines the speed and accuracy of a calculation. If the criterion is too coarse, the solution may be quite inaccurate. On the other hand, a criterion which is too tight results in unnecessary computations. Similarly, an ineffective divergence check can terminate the iteration when the solution is not actually diverging or force the iteration to search for an unattainable solution.

In general, three types of convergence criteria are used to terminate the iterations:

• Criteria based on the residuum \(\boldsymbol{r}\)
• Criteria based on the solution \(\boldsymbol{d}\)
• Criteria based on the ''unbalanced energy''

In numgeo we offer the user two approaches to evaluate the convergence of the solution:

• Dof-based Controls

  Evaluate convergence for each physical field individually (Default)

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  read more...

• Global Controls

  Evaluate convergence without distinguishing between different fields

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  read more...

Recommended

Per default numgeo uses dof-based controls to evaluate convergence of the iterative solution process - which is also the recommended approach.

Norms and notation

In many cases, the Euclidean (\(l_2\)) norm is used to calculate the comparative quantities to assess convergence:

\[ \lVert \sqcup \rVert_2 = \sqrt{\sqcup_{i}^T\sqcup_{i}} = \left( \sum_{i_{dof}}^{n_{dof}} \sqcup_{i_{dof}}^2 \right)^{\frac{1}{2}}. \]

By using the Euclidean norm, the mean error over all degrees of freedom is controlled. In some cases, the infinite norm \(\lvert \sqcup \vert_\infty\) is used instead of the Euclidean norm, which is defined as follows

\[ \lvert \sqcup \vert_\infty = \text{max} \left(|\sqcup|\right) = \text{max} \left( \text{abs} \left(\sqcup,\alpha \right) \right) \]

General considerations

In general, convergence criteria take the following form:

\[ \sqcup^\alpha \leq \epsilon s^\alpha \]

Here, \(\sqcup^\alpha\) is a quantity related to field \(\alpha\) that determines whether the system is in equilibrium, i.e., whether convergence is achieved. Typically, \(\sqcup^\alpha\) is calculated as the Euclidean or infinite norm of the solution increment, the force (or flux), or the energy of field \(\alpha\). The parameter \(\epsilon\) represents the desired accuracy, and \(s^\alpha\) is a scaling factor that ensures the convergence criterion is independent of the problem's dimensions. Generally, \(s^\alpha\) is of the same type as \(\sqcup^\alpha\), but there is no universal consensus on its exact definition, and different programs use various versions.

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Dof-based controls

In some cases, especially when dealing with multi-field problems, the flux component is composed of different components \(\boldsymbol{f} = [\boldsymbol{f}^\alpha]^T\) which are conjugated to the \(\alpha\)th variable in the problem. Analogous, the residuum reads \(\boldsymbol{r} = [\boldsymbol{r}^\alpha]^T\). For the example of a three-phase porous media consisting of the solid matrix (\(s\)), the pore water (\(w\)) and the pore air (\(a\)), the variables to be sought of are the solid displacement \(\boldsymbol{u}\), the pore water pressure (\(p^w\)) and the pore air pressure (\(p^a\)), i.e. \(\alpha = \{s,w,a\}\). The vector of solution thus reads \(\boldsymbol{d} = [\boldsymbol{d}^s,\boldsymbol{d}^w,\boldsymbol{d}^a]^T = [\boldsymbol{u}, \boldsymbol{p}^w, \boldsymbol{p}^a]^T\). The flux vector and the residual now read \(\boldsymbol{f}=[\boldsymbol{f}^s,\boldsymbol{f}^w,\boldsymbol{f}^a]^T\) and \(\boldsymbol{r}=[\boldsymbol{r}^s,\boldsymbol{r}^w,\boldsymbol{r}^a]^T\), respectively.

Using the global convergence controls might not be a suitable choice due to the (very) different magnitudes of the different degrees of freedom (e.g. displacements in the order of cm, and pore water pressure in the order of several hundred kPa). In such cases controlling the accuracy of the solution based on each individual type of degree of freedom (e.g. solid displacements or pore water pressure) might be more appropriate. This is especially useful in multi-field solutions.

The dof-based convergence evaluation in numgeo uses all three types of convergence criteria based on

• the residuum \(\boldsymbol{r}^\alpha\)
• the solution \(\boldsymbol{d}^\alpha\)
• the ''unbalanced energy'' of field \(\alpha\)

The criteria are evaluated for each active field separately. A field is accepted if one of the admissible combinations listed in subsection Acceptance of dof-based convergence is fulfilled. In some cases quadratic convergence of the iterations is not possible. In these cases, we use a temporary relaxation of selected convergence criteria as described in subsection Nonquadratic convergence. Notice that if the response for field \(\alpha\) is very small, the negligible response criteria may dominate the convergence evaluation. Therefore, great caution is required in the choice of the criterion for negligible response \(\epsilon^{\Delta d,\alpha}\). A graphical representation of some of the quantities used to evaluate convergence are given in Figure 1

Figure 1: A simplified graphical illustration depicting key quantities in the solution process for the response of a non-linear spring with a single degree of freedom to applied loads.

Solution increment

To limit the maximum error for every type of degrees-of-freedom (e.g. displacements, pore pressures,...), rather than the mean error over all degrees of freedom, the following expression is used:

\[ \label{eq:convergence-solution} \frac{c^{(i),\alpha,\text{max}}}{^{t}\Delta d^{(i),\alpha,\text{max}}} \leq \epsilon^{d,\alpha}, \]

where

• \(c^{(i),\alpha,\text{max}}\) is the largest change in solution (in absolute value) in iteration \(i\):

  \[ c^{(i),\alpha,\text{max}} = \Big\lvert \boldsymbol{c}^{(i),\alpha} \Big\rvert_\infty \]

• \(^{t}\Delta d^{(i),\text{max}}\) is the largest change in solution (in absolute value) in the present increment, i.e. the solution at the end of iteration \(i\) of the present increment minus the solution at the start of the increment:

  \[ ^{t}\Delta d^{(i),\alpha,\text{max}} = \Big\lvert ^{t}\Delta d^{(i),\alpha} \Big\rvert_\infty \]

  Note that \(^{t}\Delta d^{(i),\alpha}\) accounts for changes in the solution resulting from the prescribed Dirichlet boundary conditions.

• \(\epsilon^{d,\alpha}\) is the tolerance criteria used to check for convergence.

Evaluating the accuracy of the solution based on Eq. (\(\ref{eq:convergence-solution}\)) bears the problem, that if the response for field \(\alpha\) is very small, Eq. (\(\ref{eq:convergence-solution}\)) will not be satisfied although the contribution of field \(\alpha\) to the overall solution is negligible. Therefore, numgeo also evaluates whether the solution increment or the correction itself is negligible. In the present implementation this criterion is active after the first iteration of an increment and is fulfilled if

\[ ^{t}\Delta d^{(i),\alpha,\text{max}} \leq \epsilon^{\Delta d,\alpha} ~~~~\text{or}~~~~ c^{(i),\alpha,\text{max}} \leq \epsilon^{\Delta d,\alpha}. \]

Therein, \(\epsilon^{\Delta d,\alpha}\) is the criterion for negligible response for field \(\alpha\).

Nonquadratic convergence

numgeo versions before 2026.06:

In some cases quadratic convergence of the iterations is not possible. To avoid unnecessary iteration, numgeo uses looser tolerance criteria for the solution increment and the unbalanced energy of the dof-based solution controls after \(n^{iter}_{nc}\) iterations. The value of \(n^{iter}_{nc}\) is defined as:

\[ n^{iter}_{nc}=\min\left(\dfrac{n^{iter}_{max}}{2},12\right), \]

where \(n^{iter}_{max}\) is the maximum number of iterations of the current step. The relaxed tolerances are then given as:

\[ ^{*}\epsilon^{d,\alpha} = r^\epsilon \epsilon^{d,\alpha} ~~~~\text{and}~~~~ ^{*}\epsilon^{e,\alpha} = r^\epsilon \epsilon^{e,\alpha} \]

therein, \(r^\epsilon > 1\) is a relaxation factor, which can be prescribed by the user (default \(r^\epsilon = 2\)).

Unbalanced force

For the dof-based control by means of ''unbalanced force'', the Euclidean norm of the residuum \(\lVert \boldsymbol{r}^{(i),\alpha} \rVert_2\) at iteration \((i)\) is compared to the maximum Euclidean norm of the external forces \(^{t+\Delta t}\boldsymbol{f}^\alpha_{ext}\), the internal forces \(^{t+\Delta t}\boldsymbol{\tilde{f}}^\alpha_{int}\) and the inertia forces \(^{t+\Delta t}\boldsymbol{f}^\alpha_{dyn}\) of field \(\alpha\) at the end of the current increment. In the implementation, the residual and the reference force vectors are evaluated on the unconstrained degrees of freedom of the field only:

\[ \Big\lVert ^{t}\boldsymbol{r}^{(i),\alpha}_{free} \Big\rVert_2 \leq \epsilon^{r,\alpha} s^{tol} \max \left( \Big\lVert~ ^{t+\Delta t} \boldsymbol{f}^\alpha_{ext,free} ~\Big\rVert_2, \Big\lVert~ ^{t+\Delta t} \boldsymbol{\tilde{f}}^\alpha_{int,free} ~\Big\rVert_2, \Big\lVert ~ ^{t+\Delta t} \boldsymbol{f}^\alpha_{dyn,free} ~\Big\rVert_2 \right). \]

Therein, \(\epsilon^{r,\alpha}\) is the criterion for negligible unbalance force and \(s^{tol}\) is a step-dependent tolerance scaling factor. The force vector associated with Dirichlet boundary conditions is evaluated for output purposes, but is not used as reference force for the convergence check.

For numgeo versions later than 06.2026:

If the solution increment or the correction itself is negligible (see Solution Increment), a relaxed tolerance is used: \(\epsilon^{r,\alpha}_{rel}=r^\epsilon\epsilon^{r,\alpha}\). Where \(r^\epsilon\) is per default 2 (5 in *Reduction steps) but can be changed by the user (see Nonquadratic convergence).

Unbalanced energy

The energy convergence criterion measures the energy flow to the system resulting from the residual, which is like an error in energy. As a reference energy, the energy resulting from the internal forces is used.

\[ \frac{\Big\lvert \left({^t}\boldsymbol{r}^{(i),\alpha}_{free} \right)^T \boldsymbol{c}^{(i),\alpha}_{free} \Big\rvert}{ \Big\lvert \Delta {^t}W^{i=0,\alpha}_{int} \Big\rvert } \leq s^{tol}\epsilon^{e,\alpha}, \]

The internal energy is integrated using the trapezoidal rule:

\[ \Delta {^t}W^{(i=0),\alpha}_{int} = \dfrac{1}{2}\left( ^{t_0}\boldsymbol{f}^\alpha_{int} + ^{t}\boldsymbol{f}^{(i=0),\alpha}_{int} \right)^T \Delta \boldsymbol{d}^{(i=0),\alpha} \]

In the implementation, the unbalance energy in the numerator is evaluated on the unconstrained degrees of freedom of the field only. The reference energy in the denominator is evaluated using the field degrees of freedom of the increment, so that prescribed increments contribute to the deformation scale.

Similar to the control of the solution increment, the unbalance energy of field \(\alpha\) may be treated as negligible if

\[ \Big\lvert \left({^t}\boldsymbol{r}^{(i),\alpha}_{free} \right)^T \boldsymbol{c}^{(i),\alpha}_{free} \Big\rvert \leq \epsilon^{\Delta e,\alpha}. \]

Therein, \(\epsilon^{\Delta e,\alpha}\) is the criterion for negligible unbalance energy for field \(\alpha\).

For numgeo versions later than 06.2026:

If the solution increment or the correction itself is negligible (see Solution Increment), a relaxed tolerance is used: \(\epsilon^{e,\alpha}_{rel}=r^\epsilon\epsilon^{e,\alpha}\). Where \(r^\epsilon\) is per default 2 (5 in *Reduction steps) but can be changed by the user (see Nonquadratic convergence).

Acceptance of dof-based convergence

For each active field, numgeo first evaluates the individual force, solution-increment, estimated-correction, energy, negligible-increment and negligible-energy flags. The field is then accepted if one of the admissible combinations is fulfilled.

numgeo versions before 2026.06:

• solution increment, force and energy
• negligible solution increment or negligible correction
• solution increment and energy
• estimated correction and energy
• solution increment and negligible energy
• estimated correction and negligible energy
• solution increment and force
• estimated correction and force
• energy and force
• negligible energy and force

A single force criterion, a single energy criterion, a single solution-increment criterion or a single negligible-energy criterion is not sufficient. Notice that the negligible solution-increment/correction criterion is a special case in this implementation and is accepted before the two-criterion combinations are evaluated.

numgeo versions later than 06.2026:

If the negligible solution-increment/correction criterion is fulfilled, it does not by itself define convergence. Instead, it initiates a relaxed evaluation of the force and energy criteria using the relaxed tolerances defined above. In *Reduction steps, the default relaxation factor is increased to account for slow convergence near the limit state.

Default values

The default values for the dof based convergence criteria are:

dof	\(\epsilon^{d}\)	\(\epsilon^e\)	\(\epsilon^{r}\)	\(\epsilon^{\Delta d}\)	\(\epsilon^{\Delta e}\)	\(r^\epsilon\)
u	\(1 \cdot 10^{-2}\)	\(1 \cdot 10^{-2}\)	\(1 \cdot 10^{-2}\)	\(1 \cdot 10^{-6}\)	\(1 \cdot 10^{-5}\)	2
w	\(1 \cdot 10^{-2}\)	\(1 \cdot 10^{-2}\)	\(1 \cdot 10^{-2}\)	\(1 \cdot 10^{-5}\)	\(1 \cdot 10^{-5}\)	2
pw	\(1 \cdot 10^{-2}\)	\(1 \cdot 10^{-2}\)	\(1 \cdot 10^{-2}\)	\(1 \cdot 10^{-3}\)	\(1 \cdot 10^{-10}\)	2
pa	\(1 \cdot 10^{-2}\)	\(1 \cdot 10^{-2}\)	\(1 \cdot 10^{-2}\)	\(1 \cdot 10^{-3}\)	\(1 \cdot 10^{-10}\)	2

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Global convergence controls

We evaluate the global (overall) convergence without distinguishing between different fields \(\alpha\) (e.g. solid displacement and pore water pressure). This measure gives an idea of the overall (mean) quality of the solution, however, especially in multi-field solutions, this measure could shadow non-converged fields or lead to non-convergence due to the different fields with significantly different magnitudes.

Unbalanced energy

\[ \frac{\left| \left({^t}\boldsymbol{r}^{(i)} \right)^T \boldsymbol{c}^{(i)} \right| }{ \left| \left({^t}\boldsymbol{r}^{(i=0)} \right)^T \Delta \boldsymbol{d}^{(i)} \right| } \leq \epsilon^{e}, \]

Unbalanced forces

For the global convergence control, the Euclidean norm of the residuum \(\boldsymbol{r}\) at iteration \((i)\) is compared to a load norm, which is defined as the maximum of the external \(\boldsymbol{f}_{ext}\), internal \(\boldsymbol{f}^{(i)}_{int}\) and inertia \(\boldsymbol{M^{(i)}\ddot{d}^{(i)}}\) contributions to the residuum:

\[ \lVert ^{t}\boldsymbol{r}^{(i)} \rVert_2 \leq \epsilon^r \text{max} \big( \lVert ^{t}\boldsymbol{f}_{ext} \rVert_2~,~ \lVert ^{t}\boldsymbol{f}_{int} \rVert_2~,~ \lVert ^{t}\boldsymbol{M\ddot{d}} \rVert_2 \big). \]

Therein, \(\epsilon^{r}\) is the criterion for negligible unbalance force. Note that in most cases, the external load vector is only calculated at the beginning of the current load step and does not change during the iterative procedure.

Solution increment

The third global solution control checks if the \(L_2\) norm of the correction \(\boldsymbol{c}^{(i)}\) is smaller than a predefined fraction \(\epsilon^d\) of the L\(_2\) norm of the solution increment \(\Delta \boldsymbol{d}^{(i)}\):

\[ \boldsymbol{c}^{(i)} \leq \epsilon^d \Delta \boldsymbol{d}^{(i)} \]

Default values

The default values for the global convergence criteria are:

\(\epsilon^{d}\)	\(\epsilon^e\)	\(\epsilon^{r}\)
\(1 \cdot 10^{-2}\)	\(5 \cdot 10^{-3}\)	\(5 \cdot 10^{-3}\)