Numerical treatment of imprecise random fields in non-linear solid mechanics

Abstract

The quantification and propagation of mixed uncertain material parameters in the context of solid mechanical finite element simulations is studied. While aleatory uncertainties appear in terms of spatial varying parameters, i.e. random fields, the epistemic character is induced by a lack of knowledge regarding the correlation length, which is therefore described by interval values. The concept and description of the resulting imprecise random fields is introduced in detail. The challenges occurring from interval valued correlation lengths are clarified. These include mainly the stochastic dimension, which can become very high under some circumstances, as well as the comparability of different correlation length scenarios with regard to the underlying truncation error of the applied Karhunen-Loève expansion.

Additionally, the computation time can increase drastically, if the straightforward and robust double loop approach is applied. Sparse stochastic collocation method and sparse polynomial chaos expansion are studied to reduce the number of required sample evaluations, i.e. the computational cost. To keep the stochastic dimension as low as possible, the random fields are described by Karhunen-Loève expansion, using a modified exponential correlation kernel, which is advantageous in terms of a fast convergence while providing an analytic solution. Still, for small correlation lengths, the investigated approaches are limited by the curse of dimensionality. Furthermore, they turn out to be not suited for non-linear material models.

As a straightforward alternative, a decoupled interpolation approach is proposed, offering a practical engineering estimate. For this purpose, the uncertain quantities only need to be propagated as a random variable and deterministically in terms of the mean values. From these results, the so-called absolutely no idea probability box (ani-p-box) can be obtained, bounding the results of the interval valued correlation length being between zero and infinity. The idea is, to interpolate the result of any arbitrary correlation length within this ani-p-box, exploiting prior knowledge about the statistical behaviour of the input random field corresponding to the correlation length.

The new approach is studied for one- and two-dimensional random fields. Furthermore, linear and non-linear finite element models are used in terms of linear-elastic or elasto-plastic material laws, the latter including linear hardening. It appears that the approach only works satisfyingly for sufficiently smooth responses but an improvement by considering also higher order statistics is motivated for future research.