Overview of eXtended IsoGeometrical FEM for non-deterministic problems

Abstract The aim of this work is to review and analyze modern advanced finite element techniques, i.e. the IsoGeometrical analysis (IGA) and the eXtended IsoGeometrical analysis (XIGA), from the viewpoint of the general interval-fuzzy-stochastic problem setting. We demonstrate the incorporation of IGA and XIGA bases into the general n-dimensional non-deterministic spectral FEM and show their application for a few problems. The first is a computational homogenization of heterogeneous materials with uncertainties in the microstructure. The second is a simple tension experiment model with large uncertainty in samples geometry. We analyze also a few applications, where the IGA basis in the parametric space by far beats the traditional polynomial chaos and the piece-wise continuous finite element basis. In these applications, the arbitrary smoothness and oscillation-free behavior of the IGA basis allows to achieve better accuracy with less computational effort as compared to standard approaches.