Professor Dr. Katja Ickstadt, Dr. Swetlana Herbrandt
TU Dortmund University, Faculty Statistic, Chair Mathematical Statistics with Applications in Biometrics
DFG SPP 1886: subproject 12
Funding phase 1: 01.01.2017 - 31.12.2019
The fundamental objective was and is to characterize and analyze the influence of polymorphic uncertain data on the deterministic modeling of multi-physical behavior of saturated soils. For this purpose, we have chosen a consolidation initial value problem as a benchmark in order to be able to test and compare both existing and self-developed methods.
A two-phase model (solid body and fluid, both materially incompressible and linearly elastic) was prepared within the framework of the Theory of Porous Media (TPM), so that the otherwise deterministic FEM calculations can be calculated automatically with probabilistic and stochastic approaches (in FEAP). Furthermore, we have developed a user friendly tool (in R) for the generation of random fields with different properties. The tool allows to model aleatoric uncertainties like soil properties for several soil layers and epistemic uncertainties such as the numberof layers, their positions and geometries.
For the practical realization between R and FEAP, a shell script has been created that controls the automated organization and processing of the individual batches of random fields and makes the outputs of the TPM available and sorted in a suitable format for the following analyses. By applying the Monte Carlo approach we have analyzed the development of fluid pore pressure over time as well as the settlements due to own weight and variing top surface loads. Furthermore the combined impact of a variation in correlation lengths and the natural variability of properties was investigated by a fuzzy-probabilistic approach.
Alongside these global uncertainty quantification approaches, another focus was on the development of the variational sensitivity analysis for the underlying physical model (TPM), which can be assigned to the local uncertainty quantification methods. The theoretical derivation of the model equations as well as the implementation in FEAP led to an efficient algorithm, which provides extensive information about the considered problem with only one deterministic calculation. For appropriate evaluation and assessment, different sensitivity measures have been developed, which provide information about the influence of individual parameters (global impact analysis) but also the spatial distribution of a parameter (local impact analysis).
The advantages and disadvantages of the different methods were worked out on the basis of comprehensive calculations and evaluations. In parallel, the DEIM method was prepared for the TPM in order to reduce the computational effort for analyses that require a high number of model runs.
Funding phase 2: 01.01.2020 - 31.12.2022
Based on the first project phase, the characterization and analysis of the influence of polymorphic uncertain data on the deterministic modeling of the multi-physical behavior of saturated soil will be investigated in the second project phase. As before, the physical behaviour will be modelled based on the theory of porous media (TPM). In the first phase of the project, it was successful to use the information obtained from the variational sensitivity analysis (VSA) to increase the efficiency of probabilistic methods. For this purpose, the VSA was used on the one hand as a method to increase prior knowledge and to reduce the input parameter space of a subsequent Monte Carlo analysis, and on the other hand it served as additional tangential information to support Bayesian models and the Bayesian sensitivity analysis derived from them.
In addition, probabilistic analyses can ideally be supplemented by fuzzy arithmetic in order to also take into account expert knowledge of parameters in the form of fuzzy quantities.
Consequently, it is planned to combine the variational and Bayesian sensitivity approaches in the second funding phase. This is expected to lead to both more efficient and more precise uncertainty quantification. In the variational/analytical (variolytic) predefined probabilistic sensitivity analysis, the tangential information of the variational sensitivity analysis is incorporated into the Bayesian sensitivity approach to improve the model used to perform the global sensitivity indices.
In a second step, the fuzzy theory is integrated into the variolytic predefined probabilistic sensitivity analysis in order to capture the individual input variables more flexibly.
For the next step towards real problems, the formulation of the TPM model will be extended. Motivated by problems such as hydraulic ground failure and tectonic disorders, erosion processes, plasticity and non-linear strains will be considered.
Due to the model extension the computational effort increases, so that further steps towards model reduction have to be worked out and investigated. In order to reduce the number of TPM evaluations, the Kriging approach is used as a metamodel for the description of TPM field quantities depending on different initial value settings. This metamodel has the advantage of a much faster evaluation with acceptable accuracy. This allows the uncertainty quantification for unknown parameter ranges even for more complex and non-linear models. Furthermore, the POD-DEIM method will be applied and extended as a model reduction method to accelerate the calculations.
Finally, the calculation program should be able to generate DIN-compliant decision aids for the engineer. A suitable user interface, sufficient flexibility of the program, efficiency and decision making as well as interpretable results are essential for this. To facilitate the decision making process, we will provide probabilities of worst case scenarios which can be derived from the a posteriori distribution of the target variable. In addition, we will use model-based optimization (MBO) to identify parameter combinations (scenarios) that lead to unfavorable target variables. The analysis of these parameter combinations can, for example, help to recommend positions for further on-site soil sampling. It should be possible for the user to transfer any problems to the program without much effort. Representative benchmark examples are used to evaluate the architecture of the program and, if necessary, to optimize the workflow and algorithms.