190
J. Hochhalter et al.
Essentially, the uncertainty in g ∗
s is similar to the prior, which indicates that the test
used in the calibration did not provide significant insight on g ∗
s . Because g ∗
s is related
to saturation hardness, the above might indicate that the applied strain loading of
the experiment was too small to see saturation; repeating the test to higher strains
might help identify g ∗
s . It is worth noting that, if a deterministic calibration was
performed, a single value of g ∗
s would result, without knowing that the parameter
was essentially unidentifiable by the test. Apart from gathering additional data to
aid identification, a more informative prior could have been used to regularize
the inverse problem. However, this was beyond the scope of this work, and the
uncertainty in the calibrated g ∗
s was accepted as uniform over the given bounds.
6.2 Using Global-Local Calibration
The global-local calibration was set up using a finite element model with the same
geometry as the simulated experiment. In order to remove the effect of erroneous
boundary conditions, the same boundary conditions were applied as the simulated
experiment.
The model was coarsened to decrease the model evaluation time. The coarse
mesh contained 25,500 quadratic tetrahedral elements and 45,700 nodes. The time
discretization of the model was also decreased by a factor of 2 compared to the
simulated experiment. Model evaluation of the coarsened finite element model took
approximately 9 min on 40 cores of a Dual socket 20 core 2.40 GHz Intel Gold
6148 Skylake Processor. Because a non-deterministic calibration akin to the one in
Sect. 6.1 would take at least 156 days, a deterministic optimization was pursued
instead. This means that a single deterministic set of calibration parameters is
obtained, without an idea of how certain that calibration is.
The error metric for the optimization was the weighted sum of the error norms of
the global (homogenized stress) and local (displacement) fields. The weighting was
performed in the fashion of [26] which weights the errors at each scale based on the
resolution of the measurement technique. In this case, the magnitude of the added
measurement noise was used. After normalization of the two fields, equal weight
was placed on global and local measurements.
The optimization was performed via Nelder-Mead simplex method [11] with the
initial guesses of the parameters at 105% of their true values. The optimization
took about 50 h to complete. The resulting optimal parameters are shown in
Table 3. Example comparisons of the global and local response from the simulated
experiment and optimal parameters are shown in Figs. 14 and 15. The global
response is close to the simulated data with a slight under prediction. Presumably
the bias seen in the global response was compensated by a more accurate local field,
i.e., a balance of the global and local errors would be found.
From Figs. 14 and 15, it can be seen that overall the calibration was somewhat
successful in matching the global and local behavior of the model; however, it is
difficult to place confidence in the calibration without a measure of uncertainty.
J. Hochhalter et al.
Essentially, the uncertainty in g ∗
s is similar to the prior, which indicates that the test
used in the calibration did not provide significant insight on g ∗
s . Because g ∗
s is related
to saturation hardness, the above might indicate that the applied strain loading of
the experiment was too small to see saturation; repeating the test to higher strains
might help identify g ∗
s . It is worth noting that, if a deterministic calibration was
performed, a single value of g ∗
s would result, without knowing that the parameter
was essentially unidentifiable by the test. Apart from gathering additional data to
aid identification, a more informative prior could have been used to regularize
the inverse problem. However, this was beyond the scope of this work, and the
uncertainty in the calibrated g ∗
s was accepted as uniform over the given bounds.
6.2 Using Global-Local Calibration
The global-local calibration was set up using a finite element model with the same
geometry as the simulated experiment. In order to remove the effect of erroneous
boundary conditions, the same boundary conditions were applied as the simulated
experiment.
The model was coarsened to decrease the model evaluation time. The coarse
mesh contained 25,500 quadratic tetrahedral elements and 45,700 nodes. The time
discretization of the model was also decreased by a factor of 2 compared to the
simulated experiment. Model evaluation of the coarsened finite element model took
approximately 9 min on 40 cores of a Dual socket 20 core 2.40 GHz Intel Gold
6148 Skylake Processor. Because a non-deterministic calibration akin to the one in
Sect. 6.1 would take at least 156 days, a deterministic optimization was pursued
instead. This means that a single deterministic set of calibration parameters is
obtained, without an idea of how certain that calibration is.
The error metric for the optimization was the weighted sum of the error norms of
the global (homogenized stress) and local (displacement) fields. The weighting was
performed in the fashion of [26] which weights the errors at each scale based on the
resolution of the measurement technique. In this case, the magnitude of the added
measurement noise was used. After normalization of the two fields, equal weight
was placed on global and local measurements.
The optimization was performed via Nelder-Mead simplex method [11] with the
initial guesses of the parameters at 105% of their true values. The optimization
took about 50 h to complete. The resulting optimal parameters are shown in
Table 3. Example comparisons of the global and local response from the simulated
experiment and optimal parameters are shown in Figs. 14 and 15. The global
response is close to the simulated data with a slight under prediction. Presumably
the bias seen in the global response was compensated by a more accurate local field,
i.e., a balance of the global and local errors would be found.
From Figs. 14 and 15, it can be seen that overall the calibration was somewhat
successful in matching the global and local behavior of the model; however, it is
difficult to place confidence in the calibration without a measure of uncertainty.