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numgeo-ACT

Cost function

Table of contents

  1. Introduction
  2. Weighting different test types $W_{iT}$
  3. Discrepancy between experiment and simulation $\epsilon_{iT}$
    1. Oedometric compression tests $\epsilon^{it}_{oc}$
    2. Drained monotonic triaxial tests $\epsilon^{it}_{CD}$
    3. Undrained monotonic triaxial tests $\epsilon^{it}_{CU}$
    4. Undrained cyclic triaxial tests $\epsilon^{it}_{CUcyc}$

Introduction

The basis of any optimization is the existence of a (scalar) objective function (‘‘cost function’’) which has to be minimized by changing the parameters in the course of the optimization. The cost function in this work is defined as follows:

\[\epsilon = \sum_{iT}^{nT} W_{iT} \epsilon_{iT} = \sum_{iT}^{nT} W_{iT} \left( \sum_{it}^{nt} w_{iT}^{it} \epsilon_{iT}^{it} \right),\]

where $\epsilon_{iT}$ is the resulting error for laboratory test type $iT$ and $nT$ is the total number of different types of laboratory tests which form the basis for the optimization:

  • Oedometric compression tests: $iT = OC$
  • Drained monotonic triaxial tests: $iT = CD$
  • Undrained monotonic triaxial tests: $iT = CU$
  • Undrained monotonic triaxial tests: $iT = CUcyc$

$W_{iT}$ are the weighting functions, which control the calibration procedure with regard to whether the simulated material behavior should represent all test types equally well or whether the behavior in individual laboratory test types (e.g. the behavior under oedometric conditions) is more important than the others.

The resulting error $\epsilon_{iT}$ is again a weighted sum (thus an averaged value) of all errors $\epsilon_{iT}^{it}$ of all individual tests $it$ of one test type $iT$. $w_{iT}^{it}$ are the associated weightings and $nt$ the total number of all individual tests of test type $iT$. Note that the closure conditions $\sum_{iT}^{nT} W_{iT} = 1$ and $\sum_{it}^{nt} w_{iT}^{it}=1$ hold.

Weighting different test types $W_{iT}$

In the present implementation, all monotonic tests are equally weighted, i.e. $W_{OC} = W_{CD} = W_{CD} = W_{mon}/3$. Therein, $W_{mon}$ is the weighting factor applied to monotonic tests. In case only monotonic tests are considered, $W_{mon}=1$ holds.

If cyclic laboratory tests are as well included in the calibration, the default weights are $W_{mon}=W_{cyc}=1/2$.

Currently the only cyclic laboratory tests available are undrained cyclic triaxial tests, thus $W_{CUcyc}=W_{cyc}$ holds.

Weights deviating from the standard settings can be specified as follows:

weights = ACTweights.initialize()
weights["global"]["monotonic"] = 1./2.
weights["global"]["cyclic"] = 1./2.

Note that the constraint $W_{mon}+W_{cyc}=1$ holds.

Discrepancy between experiment and simulation $\epsilon_{iT}$

The calculation of $\epsilon^{it}_{iT}$ of an individual test depends on the laboratory test type. It is a natural choice to judge the quality of the optimization on the basis of the comparison of the experimental data with the simulation results. For this purpose, a comparison of the experimentally measured and simulated relationships in the stress and strain spaces is useful which takes the general form

\[\epsilon^{it}_{iT} = \sum_\alpha w_{iT}^{it,\alpha} \epsilon_{iT}^{it,\alpha}.\]

Therein, $\epsilon_{iT}^{it,\alpha}$ and $w_{iT}^{it,\alpha}$ are the individual error and the individual weight of the comparison plane $\alpha$. The comparison planes vary from test to test, e.g. axial strain - deviatoric stress ($\tilde{\varepsilon}_1-\tilde{q}$) and the axial strain - volumetric strain ($\tilde{\varepsilon}_1-\tilde{\varepsilon}_V$) planes for drained monotonic triaxial tests.

Note that for the planes scaled stress and strain measures are used instead of the actual values. This is necessary to ensure that the variables used for the error calculation - despite their different value ranges and units (e.g. $\varepsilon \in [0-0.3]$ and $q \in [0-500]$ kPa for drained monotonic triaxial tests) - have a comparable influence on the optimization. Therefore, both the experimental ($exp$) and numerical ($sim$) data are made dimensionless using the following relations:

\[\tilde{(\bullet)}^{\sqcup} = \frac{(\bullet)^{\sqcup}}{\max(\max((\bullet)^{exp}),\max((\bullet)^{sim}))} ~~~\text{with:}~~~\sqcup=\{exp,sim\}\]

This normalization transfers the experimental and numerical results into representations which have their origin in $(0,0$) and range to $(1,1)$ or $(-1,1)$.

The cost functions $\epsilon^{it}_{iT}$ driving the optimization of the parameters are thus built by accounting for the discrepancy between the numerical predictions and the measured (experimental) data. Traditionally, this discrepancy is often quantified using a sum-of-square based cost function. However, in the present case this is not always possible. One problem is a potentially different number of data points on the experimental curve compared to the numerical curve. Admittedly, this circumstance could be remedied by linear interpolation - provided there are enough data points to keep the introduced error low. However, in many cases there is no unique relationship between stress and strain - the material behavior is path dependent. An example is the oedometric compression test with loading, unloading and reloading. Thus, the use of alternative objective functions is advised to measure the similarity between the experimental data and the simulation results.

Oedometric compression tests $\epsilon^{it}_{oc}$

The cost function for oedometric compression tests $\epsilon^{it}_{oc}$ is evaluated in the axial stress - axial strain ($\tilde{\sigma}_1-\tilde{\varepsilon}_1$) plane.

Drained monotonic triaxial tests $\epsilon^{it}_{CD}$

For the the drained monotonic triaxial tests, both the axial strain - deviatoric stress ($\tilde{\varepsilon}_1-\tilde{q}$) and the axial strain - volumetric strain ($\tilde{\varepsilon}_1-\tilde{\varepsilon}_V$) planes are used:

\[\epsilon^{it}_{CD}(\tilde{q},\tilde{\varepsilon}_1,\tilde{\varepsilon}_V) = w_{CD}^{\varepsilon q} \epsilon^{it,\varepsilon q}_{CD}(\tilde{\varepsilon}_1,\tilde{q}) + w_{CD}^{\varepsilon \varepsilon}\epsilon^{it, \varepsilon \varepsilon}_{CD}(\tilde{\varepsilon}_1,\tilde{\varepsilon}_V).\]

Therein, $w_{dmt}^{\varepsilon q}$ and $w_{dmt}^{\varepsilon \varepsilon}$ are weights controlling whether the focus during calibration should be on achieving the best possible reproducibility of the response in the $\varepsilon_1-q$ plane, $\varepsilon_1-\varepsilon_V$ plane, or a combination of both. The default values of the weights are $w_{CD}^{\varepsilon q} = 2/3$ and $w_{CD}^{\varepsilon \varepsilon}=1/3$. The stronger weighting of the ($\tilde{\varepsilon}_1-\tilde{q}$) plane takes into account the stronger uncertainty in the volumetric strain measurement. If desired, these values can be changed:

weights = ACTweights.initialize()
weights["triaxCD"]["eps1-q"] = 1./2.
weights["triaxCD"]["eps1-epsV"] = 1./2.

Note that the constraint $\sum_\alpha w_{CD}^{\alpha}=1$ with $\alpha=(\varepsilon q,\varepsilon \varepsilon)$ must be respected.

Undrained monotonic triaxial tests $\epsilon^{it}_{CU}$

For the the undraineddrained monotonic triaxial tests, three comparision planes are implemented:

  • the axial strain - deviatoric stress ($\tilde{\varepsilon}_1-\tilde{q}$) plane,
  • the mean effective stress - deviatoric stress ($\tilde{p}-\tilde{q}$) plane
  • and the normalized mean effective stress - stress ratio ($\tilde{p}_n-\tilde{\eta}$). The normalized mean effective stress is defined as $p_n=p/p_0$ and the stress ratio as $\eta=p/q$.

The overall error is calculated as follows:

\[\epsilon^{it}_{CU}(\tilde{q},\tilde{\varepsilon}_1,\tilde{\varepsilon}_V) = w_{CU}^{\varepsilon q} \epsilon^{it,\varepsilon q}_{CU}(\tilde{\varepsilon}_1,\tilde{q}) + w_{CU}^{pq}\epsilon^{it, pq}_{CU}(\tilde{p},\tilde{q}) + w_{CU}^{p\eta}\epsilon^{it, p \eta}_{CU}(\tilde{p}_n,\tilde{\eta}).\]

Therein, $w_{CU}^{\alpha}$ and are weights controlling whether the focus during calibration should be on achieving the best possible reproducibility of the response in the $\varepsilon_1-q$ plane, the $p-q$ plane, the $p_n-\eta$ plane, or a combination of those. The default values of the weights are $w_{CU}^{\varepsilon q} = 1/3$ and $w_{CU}^{p \eta}=2/3$. The stronger weighting of the ($\tilde{\varepsilon}_1-\tilde{q}$) plane takes into account the stronger uncertainty in the volumetric strain measurement. If desired, these values can be changed:

weights = ACTweights.initialize()
weights["triaxCU"]["eps1-q"] = 1./3.
weights["triaxCU"]["p-q"] = 1./3.
weights["triaxCU"]["p-eta"] = 1./3.

Note that the constraint $\sum_\alpha w_{CU}^{\alpha}=1$ with $\alpha=(\varepsilon q, p q, p \eta)$ must be respected.

Undrained cyclic triaxial tests $\epsilon^{it}_{CUcyc}$

For the the undraineddrained cyclic triaxial tests, three comparision planes are implemented:

  • the mean effective stress - deviatoric stress ($\tilde{p}-\tilde{q}$) plane
  • the cycle - accumulated pore water pressure ($\tilde{p}^{acc}_{w}-\tilde{N}$) plane
  • the cycle - accumulated axial strain ($\tilde{\varepsilon}^{acc}_{ax}-\tilde{N}$) plane

The overall error is calculated as follows:

\[\epsilon^{it}_{CUcyc}(\tilde{q},\tilde{\varepsilon}_{ax},\tilde{p}_w) = w_{CUcyc}^{pq}\epsilon^{it, pq}_{CUcyc}(\tilde{p},\tilde{q}) + w_{CUcyc}^{p^wN-acc}\epsilon^{it, p^wN-acc}_{CUcyc}(\tilde{N},\tilde{p}^w) + w_{CUcyc}^{\varepsilon N-acc}\epsilon^{it, \varepsilon N-acc}_{CUcyc}(\tilde{N},\tilde{\varepsilon}^{acc}_{ax}).\]

Therein, $w_{CUcyc}^{\alpha}$ and are weights controlling whether the focus during calibration should be on achieving the best possible reproducibility of the response in the different planes. The default values of the weights are $w_{CUcyc}^{pq} = 1/2$ and $w_{CUcyc}^{p^wN-acc}=1/2$. If desired, these values can be changed:

weights = ACTweights.initialize()
weights["triaxCUcyc"]["p-q"] = 1./3.
weights["triaxCUcyc"]["N-pwacc"] = 1./3.
weights["triaxCUcyc"]["N-eps1acc"] = 1./3.

Note that the constraint $\sum_\alpha w_{CUcyc}^{\alpha}=1$ with $\alpha=(pq, p^wN-acc, \varepsilon N-acc)$ must be respected.