Convergence criteria
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:
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Dof-based Controls
Evaluate convergence for each physical field individually (Default)
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Global Controls
Evaluate convergence without distinguishing between different fields
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:
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
General considerations
In general, convergence criteria take the following form:
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.
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
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:
where
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\(c^{(i),\alpha,\text{max}}\) 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.
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\(\epsilon^{d,\alpha}\) 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
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:
where \(n^{iter}_{max}\) is the maximum number of iterations of the current step. The relaxed tolerances are then given as:
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:
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.
The internal energy is integrated using the trapezoidal rule:
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
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 |
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
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:
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 \(\Delta \boldsymbol{d}^{(i)}\):
Default values
The default values for the global convergence criteria are:
| \(\epsilon^{d}\) | \(\epsilon^e\) | \(\epsilon^{r}\) |
|---|---|---|
| \(1 \cdot 10^{-2}\) | \(5 \cdot 10^{-3}\) | \(5 \cdot 10^{-3}\) |