Combining Finite Elements and Random Fields to Quantify Uncertainty in a Multi‐phase Structural Analysis

Abstract

For a wide range of mechanical application fields, computational analyses are already indispensable due to their advanced impact on engineering developments. They provide a lot of information about the functioning and structural response of an arbitrary structure and the interaction with its surrounding. Nevertheless, despite preliminary investigations or the knowledge about manufacturing process of structures, assumptions of geometrical or material properties are often sources of uncertainties. These can be distinguished into aleatoric (e.g. natural scattering of properties) and epistemic (e.g. accuracy of the numerical approximation) uncertainties. Although measuring devices or process workflows are continuously improved, there always remains a residual risk. In order to provide and improve decision-making tools for end users, the priority program SPP 1886 has been installed by the DFG with the focus on polymorphic uncertainty quantification. The present subproject, which is part of the SPP 1886 (sp12), is focused on fluid saturated soil and earth structures and their multiple uncertainty sources.The very complex micro-structural consolidation mechanism of soil requires a suitable numerical description. For the strongly coupled fluid-solid response behavior, the Theory of Porous Media (TPM) will be used, a macroscopic homogenization approach based on the mixture theory restricted by the concept of volume fraction. In the framework of the Finite Element Method (FEM), arbitrary Boundary Value Problems (BVP) can be investigated. Uncertainty modelling takes place on the stochastic level by introducing random fields with a defined correlation structure for the generation of model initial material parameters. By generating many random field realizations for a FEM material parameter, we receive various FEM outputs for selected scenarios of the stated BVP. These results can be used to identify typical output scenarios, which allows to identify critical material parameter combinations which may cause instabilities.. Fig. 1: Mixture Theory: Superimposition of the homogenized solid and liquid phases on the microscopic level.To describe the complex behavior of soil, one has to incorporate the strongly coupled fluid-structure interaction on pore scale. Since a direct discretization of of the pore structure exceeds the current computational capacities for appropriate large scale applications, the Theory of Porous Media (TPM) bases on the Theory of Mixtures, which is a well-established homogenization approach for multiphase problems, is a suitable approach to account for as much details as needed to derive for a reliable model approach. For fully saturated soil, the fluid water (F) and the solid grains (S) are considered in superimposed and homogenized domains, see figure 1. To solve the set of unknowns R = {u S , λ}, wherein u S denotes the solid displacements and λ the fluid pore pressure, the formulation of mixed balance equations is used. The local forms of the mixture balance equation of mass and momentum in the referenc...

Numerical studies of earth structure assessment via the theory of porous media using fuzzy probability based random field material descriptions

“…Uncertainties can be categorized into epistemic and aleatoric uncertainties . While basic uncertainty models like intervals and fuzzy variables are used to describe epistemic uncertainties, like the lack of knowledge, the probability based random variables, and random processes/fields are associated with aleatoric uncertainties . These basic uncertainty models can be combined to cover more complex and realistic uncertainty specifications yielding polymorphic uncertainty concepts, like fuzzy probability .…”

Polymorphic Uncertainty Quantification of Computational Soil and Earth Structure Simulations via the Variational Sensitivity Analysis

“…Monte-Carlo Analyses [1], partly in combination with Random Fields (RF), allow to assess uncertainties in earth structures for diverse applications [2][3][4]. However, the high computational effort due to the large number of simulation runs required, builds a main drawback of this approach.…”