Predicting the process of porosity-based ductile damage in polycrystalline metallic materials
is an essential practical topic. Ductile damage and its precursors are represented by extreme
values in stress and material state quantities, the spatial probability density function (PDF)
of which are highly non-Gaussian with strong fat tails. Traditional deterministic forecasts
utilizing sophisticated continuum-based physical models generally lack in representing the
statistics of structural evolution during material deformation. Computational tools which do
represent complex structural evolution are typically expensive. The inevitable model error and
the lack of uncertainty quantification may also induce significant forecast biases, especially
in predicting the extreme events associated with ductile damage. In this paper, a data-driven
statistical reduced-order modeling framework is developed to provide a probabilistic forecast
of the deformation process of a polycrystal aggregate leading to porosity-based ductile damage
with uncertainty quantification. The framework starts with computing the time evolution of
the leading few moments of specific state variables from the spatiotemporal solution of full-
field polycrystal simulations. Then a sparse model identification algorithm based on causation
entropy, including essential physical constraints, is utilized to discover the governing equations
of these moments. An approximate solution of the time evolution of the PDF is obtained from
the predicted moments exploiting the maximum entropy principle. Numerical experiments based
on polycrystal realizations of a representative body-centered cubic (BCC) tantalum illustrate a
skillful reduced-order model in characterizing the time evolution of the non-Gaussian PDF of the
von Mises stress and quantifying the probability of extreme events. The learning process also
reveals that the mean stress is not simply an additive forcing to drive the higher-order moments
and extreme events. Instead, it interacts with the latter in a strongly nonlinear and multiplicative
fashion. In addition, the calibrated moment equations provide a reasonably accurate forecast
when applied to the realizations outside the training data set, indicating the robustness of the
model and the skill for extrapolation. Finally, an information-based measurement is employed
to quantitatively justify that the leading four moments are sufficient to characterize the crucial
highly non-Gaussian features throughout the entire deformation history considered.
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State Estimation—The Role of Reduced Models
The exploration of complex physical or technological processes usually requires exploiting available information from different sources: (i) physical laws often represented as a family of parameter dependent partial differential equations and (ii) data provided by measurement devices or sensors. The amount of sensors is typically limited and data acquisition may be expensive and in some cases even harmful. This article reviews some recent developments for this “small-data” scenario where inversion is strongly aggravated by the typically large parametric dimension- ality. The proposed concepts may be viewed as exploring alternatives to Bayesian inversion in favor of more deterministic accuracy quantification related to the required computational complexity. We discuss optimality criteria which delineate intrinsic information limits, and highlight the role of reduced models for developing efficient computational strategies. In particular, the need to adapt the reduced models—not to a specific (possibly noisy) data set but rather to the sensor system—is a central theme. This, in turn, is facilitated by exploiting geometric perspectives based on proper stable variational formulations of the continuous model.
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- Award ID(s):
- 1720297
- NSF-PAR ID:
- 10333094
- Editor(s):
- Rebollo, Tomás C.; Donat, Rosa; Higueras, Inmaculada
- Date Published:
- Journal Name:
- SEMA SIMAI Springer series
- Volume:
- 1
- ISSN:
- 2199-3041
- Page Range / eLocation ID:
- 57-77
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1007/978-3-030-86236-7_4
