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Early termination criteria

numgeo allows the termination of an analysis step once a user-defined stress or strain criterion is reached. This is useful for analyses in which the final state of a step is not known a priori. Typical examples are displacement-controlled loading paths that should stop once a prescribed stress or strain level is reached.

The criterion is evaluated at the integration points of all elements belonging to the selected element set. A step is terminated as soon as the criterion is fulfilled at least at one integration point.

Supported quantities

The termination criterion is defined by a scalar quantity \(q\) evaluated at the integration points. Currently, stress and strain components are supported.

For stresses, the following quantities can be used:

Input quantity Compared quantity
S11 \(\sigma_{11}\)
S22 \(\sigma_{22}\)
S33 \(\sigma_{33}\)
S12 \(\sigma_{12}\)
S23 \(\sigma_{23}\)
S13 \(\sigma_{13}\)

For strains, the corresponding quantities are:

Input quantity Compared quantity
E11 \(\varepsilon_{11}\)
E22 \(\varepsilon_{22}\)
E33 \(\varepsilon_{33}\)
E12 \(\varepsilon_{12}\)
E23 \(\varepsilon_{23}\)
E13 \(\varepsilon_{13}\)

The termination value prescribed by the user is denoted by \(q^\mathrm{t}\).

Reference values

At the beginning of a step, numgeo stores the reference value of the selected quantity for each integration point of the selected element set. For an element \(e\) and integration point \(g\), this reference value is denoted by

\[ \begin{equation} q^\mathrm{ref}_{e,g} = q_{e,g}(t_0) \end{equation} \]

where \(t_0\) is the time at the beginning of the step.

The reference value is important because it defines the direction in which the prescribed termination value has to be reached. The termination criterion is therefore not based on the sign of the prescribed value \(q^\mathrm{t}\). Instead, it is based on whether the current value approaches \(q^\mathrm{t}\) from below or from above.

This distinction is essential for unloading paths. For example, if the vertical stress at the beginning of an unloading step is \(\sigma_{22} = -500~\mathrm{kPa}\) and the termination value is \(\sigma_{22}^\mathrm{t} = -100~\mathrm{kPa}\), the criterion must be checked as

\[ \begin{equation} \sigma_{22} \geq -100~\mathrm{kPa} \end{equation} \]

although the prescribed termination value is negative.

Termination criterion

Let \(q^{i}_{e,g}\) be the current value of the selected quantity at increment \(i\). The termination criterion is evaluated by comparing \(q^{i}_{e,g}\) with the prescribed value \(q^\mathrm{t}\), using the reference value \(q^\mathrm{ref}_{e,g}\) to determine the crossing direction.

For each integration point, a numerical tolerance is defined as

\[ \begin{equation} \eta_{e,g}^{i} = 100\,\epsilon_\mathrm{mach} \max \left( 1, \left| q^\mathrm{ref}_{e,g} \right|, \left| q^{i}_{e,g} \right|, \left| q^\mathrm{t} \right| \right) \end{equation} \]

where \(\epsilon_\mathrm{mach}\) is the machine precision.

The criterion is fulfilled at integration point \(g\) of element \(e\) if

\[ \begin{equation} \mathrm{fulfilled}_{e,g}^{i} = \begin{cases} q^{i}_{e,g} \geq q^\mathrm{t} - \eta_{e,g}^{i} & \mathrm{if} \quad q^\mathrm{ref}_{e,g} < q^\mathrm{t} - \eta_{e,g}^{i} \\ q^{i}_{e,g} \leq q^\mathrm{t} + \eta_{e,g}^{i} & \mathrm{if} \quad q^\mathrm{ref}_{e,g} > q^\mathrm{t} + \eta_{e,g}^{i} \\ \left| q^{i}_{e,g} - q^\mathrm{t} \right| \leq \eta_{e,g}^{i} & \mathrm{otherwise} \end{cases} \end{equation} \]

The first case applies if the prescribed termination value has to be reached from below. The second case applies if the prescribed termination value has to be reached from above. The third case covers the situation where the reference value is already equal to the prescribed termination value within numerical tolerance.

The step termination criterion is fulfilled if the criterion is fulfilled at any integration point of any element in the selected element set \(\mathcal{E}_\mathrm{t}\):

\[ \begin{equation} \mathrm{fulfilled}^{i} = \exists\, e \in \mathcal{E}_\mathrm{t}, \, g \in \mathcal{G}_e : \mathrm{fulfilled}_{e,g}^{i} \end{equation} \]

where \(\mathcal{G}_e\) denotes the set of integration points of element \(e\).

Example: loading and unloading

Consider a displacement-controlled compression step in which the axial strain \(\varepsilon_{22}\) should stop at \(-1~\%\). If the strain at the beginning of the step is approximately zero, the reference and termination values are

\[ \begin{equation} \varepsilon_{22}^\mathrm{ref} \approx 0, \qquad \varepsilon_{22}^\mathrm{t} = -0.01 \end{equation} \]

Since \(\varepsilon_{22}^\mathrm{ref} > \varepsilon_{22}^\mathrm{t}\), the criterion is fulfilled once

\[ \begin{equation} \varepsilon_{22}^{i} \leq -0.01 + \eta^{i} \end{equation} \]

During a subsequent unloading step, the vertical stress may increase from \(\sigma_{22} = -500~\mathrm{kPa}\) towards \(\sigma_{22} = -100~\mathrm{kPa}\). The corresponding reference and termination values are

\[ \begin{equation} \sigma_{22}^\mathrm{ref} = -500~\mathrm{kPa}, \qquad \sigma_{22}^\mathrm{t} = -100~\mathrm{kPa} \end{equation} \]

Since \(\sigma_{22}^\mathrm{ref} < \sigma_{22}^\mathrm{t}\), the criterion is fulfilled once

\[ \begin{equation} \sigma_{22}^{i} \geq -100~\mathrm{kPa} - \eta^{i} \end{equation} \]

Thus, the unloading step does not terminate immediately at \(\sigma_{22} = -500~\mathrm{kPa}\), although \(-500 < -100\). The comparison direction follows from the reference state at the beginning of the step.

The sign of the prescribed termination value does not determine the comparison direction. The direction is determined from the reference value stored at the beginning of the step and the prescribed termination value.