Conference Agenda

This study has been performed within the research project MuScaBlaDes "Multi scale failure analysis with polymorphic uncertainties for optimal design of rotor blades", which is part of the DFG Priority Programme (SPP 1886) "Polymorphic Uncertainty Modelling for the Numerical Design of Structures" started in 2016.

The modeling of real engineering structures is a tough challenge and always accompanied by uncertainties. Geometry, material and all boundary conditions should be quantified as accurately as possible. The quality of the numerical prediction of the system behavior and of desired system outcomes depends on the underlying model. Real measurements on the structure provide the possibility to assess and to verify the numerics. In general, discrepancies exist between the predicted and the measured values. Within the data assimilation framework, it is possible to consider both for the estimation of the system state. Additionally, the estimation of unknown parameters at once can be achieved in nonlinear problems by using the ensemble Kalman filter (EnKF).

In this contribution, the EnKF is extended by parameters, which influence the system state and which are subject to aleatory or epistemic uncertainty. These parameters have to be quantified by suitable uncertainty models first, and then integrated into the numerical simulation. Stochastic, interval and fuzzy variables are used leading to a multilayered uncertainty space and a nested numerical simulation in which the EnKF is embedded. Besides an academic example, the practical applicability is demonstrated on real engineering structures with synthetic and also with real measurement data.

Composite materials, such as (reinforced) concrete, which are designed by combining different constituents to obtain materials with beneficial properties for specific applications, are involved in many current research topics. The combination of different materials yields heterogeneities. These must be taken into account in the numerical simulation in order to obtain realistic results. Traditionally, the FE2 method based on the concept of numerical homogenization is used to obtain the macro-structural constitutive response at each integration point through a nested finite element analysis, whereby the meso-structural behavior is characterized by representative volume elements (RVE).

The main drawback of this method is the large computational effort because the representative volume elements, which are usually very complex, need to be evaluated at every material point. An approach to reduce the computational effort is the concept of decoupled numerical homogenization. Therefore, a database representing the macroscopic material behavior is derived by solving the boundary value problem of the considered RVE for different applied boundary conditions. Subsequently, the approach of data-driven computational mechanics is utilized to receive an approximate solution for the boundary value problem on the macroscale with direct reference to stress-strain data obtained from mesoscale evaluations. In order to receive accurate results by data-driven analyses, a sufficient data set density with respect to the present problem is essential.

With respect to the definition of the concrete mesostructure, aleatoric uncertainties are introduced by natural variability especially in the material behavior. Additional epistemic uncertainties are caused by manufacturing tolerances and an insufficient amount of measurement data. A combined consideration is realized by polymorphic uncertainty models. The acquisition of data sets consisting of uncertain macroscopic stress-strain states leads to a large number of required evaluations of the considered RVEs and correspondingly high computational effort, which is addressed by incorporating surrogate models for uncertainty quantification. The large number of uncertainty propagations that must be performed for data set generation is the main challenge in creating the surrogates. Accordingly, overhead and training time caused by surrogate creation need to be as low as possible in order to avoid impracticably high computational cost. In this contribution, a polynomial chaos assisted data set acquisition approach enabling the efficient consideration of polymorphic uncertainty is presented and applied in the context of data-driven computational homogenization.

We will explore sensitivity analysis for mechanical engineering problems in presence of polymorphic uncertainty. Polymorphic uncertainty quantification allows for the incorporation of different sources of uncertainty, such as epistemic and aleatory, which have varying levels of complexity and dependence.

A measurement of distances between the most common polymorphic uncertainties will be at the core of the computation of sensitivity indices.

We will discuss how sensitivity analysis can aid in understanding the effects of input uncertainties on system performance and inform further polymorphic uncertainty quantification analysis. Additionally, we will cover methods for efficiently computing sensitivity measures for high-dimensional systems based on tensor surrogates.

The method of influence functions is a well-known engineering tool in structural analysis to investigate the consequences of load variations on deflections and stress resultants. Based on its strong relationship with adjoint sensitivity analysis [1], the traditional method of influence functions can be generalized as an engineering tool for sensitivity analysis [2]. The aim of our contribution is to give insights into these methodical extensions and to demonstrate their added value.

The traditional influence function approach can be seen as work balance based on Betti’s theorem. In our contribution we show how that work expression can be extended for sensitivity analysis with respect to various parameters. We discuss the significance of the resulting mechanically interpretable sensitivity analysis and its limitations. In that regard, we also specify how the graphical analysis procedure, for which the traditional influence function technique is well-known, can be generalized. The intention is to use those “sensitivity maps” to identify the positions of extreme influences and the individual effects of the partitions to the final sensitivity and its spatial distribution. In this way, structural analysis models of real-world engineering problems can be systematically explored and important model parameters to be considered in uncertainty quantification can be identified. Hence, our method has the potential to provide valuable support for preliminary investigations of structural models as a basis of polymorphic uncertainty analysis.

References

[1] A. Belegundu, Interpreting Adjoint Equations in Structural Optimization, Journal of Structural Engineering 112 (1986) 1971–1976. https://doi.org/10.1061/(ASCE)0733-9445(1986)112:8(1971).

[2] M. Fußeder, R. Wüchner, K.-U. Bletzinger, Towards a computational engineering tool for structural sensitivity analysis based on the method of influence functions, Engineering Structures 265 (2022) 114402. https://doi.org/10.1016/j.engstruct.2022.114402.