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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Approximate analytical solutions for a class of nonlinear stochastic differential equations
An approximate analytical solution is derived for a certain class of stochastic differential equations with constant diffusion, but nonlinear drift coefficients. Specifically, a closed form expression is derived for the response process transition probability density function (PDF) based on the concept of the Wiener path integral and on a Cauchy–Schwarz inequality treatment. This is done in conjunction with formulating and solving an error minimisation problem by relying on the associated Fokker–Planck equation operator. The developed technique, which requires minimal computational cost for the determination of the response process PDF, exhibits satisfactory accuracy and is capable of capturing the salient features of the PDF as demonstrated by comparisons with pertinent Monte Carlo simulation data. In addition to the mathematical merit of the approximate analytical solution, the derived PDF can be used also as a benchmark for assessing the accuracy of alternative, more computationally demanding, numerical solution techniques. Several examples are provided for assessing the reliability of the proposed approximation.
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- Award ID(s):
- 1748537
- NSF-PAR ID:
- 10097863
- Date Published:
- Journal Name:
- European Journal of Applied Mathematics
- ISSN:
- 0956-7925
- Page Range / eLocation ID:
- 1 to 17
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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These files are supplementary data for this publication:
Uhl JH & Leyk S (2022). "Assessing the relationship between morphology and mapping accuracy of built-up areas derived from global human settlement data (https://doi.org/10.1080/15481603.2022.2131192).
Each geopackage (GPKG) file contains a set of point locations (in EPSG:3857) attributed with focal accuracy metrics of the GHS-BUILT-R2018A epochs 1975 and 2014, calculated within different levels of spatial support (i.e., focal window size) and for different analytical units (i.e., 30m grid cells, and 3x3 grid cell blocks). Moreover, each location is attributed with focal landscape metrics of built-up areas calculated in the same focal windows using the software Fragstats. These landscape metrics are calculated based on both, GHS built-up areas and reference built-up areas. Reference built-up areas were derived from the Multi-temporal building footprint database for 33 U.S. counties (MTBF-33). These datasets can be used for spatially explicit predictive modeling of the GHS-BUILT R2018A data accuracy using landscape metrics as predictor variables.
File nomenclature:
lsm_ref_accuracy_sample_2014_1000.gpkg : landscape metrics calculated from the reference built-up areas, for the epoch 2014, using a quadratic focal window of 1,000m x 1,000m.
lsm_ghs_accuracy_sample_1975_10000.gpkg : landscape metrics calculated from the ghs built-up areas, for the epoch 1975, using a quadratic focal window of 10,000m x 10,000m.
Data processing: Johannes H. Uhl, University of Colorado Boulder (USA), 2020-2022.
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/S0956792518000530
