Development of stochastic isogeometric analysis (SIGA) method for uncertainty in shape

Abstract

In this paper, a new method is proposed that extend the classical deterministic isogeometric analysis (IGA) into a probabilistic analytical framework in order to evaluate the uncertainty in shape and aim to investigate a possible extension of IGA in the field of computational stochastic mechanics. Stochastic IGA (SIGA) method for uncertainty in shape is developed by employing the geometric characteristics of the non-uniform rational basis spline and the probability characteristics of polynomial chaos expansions (PCE). The proposed method can accurately and freely evaluate problems of uncertainty in shape caused by deformation of the structural model. Additionally, we use the intrusive formulation approach to incorporate PCE into the IGA framework, and the C++ programming language to implement this analysis procedure. To verify the validity and applicability of the proposed method, two numerical examples are presented. The validity and accuracy of the results are assessed by comparing them to the results obtained by Monte Carlo simulation based on the IGA algorithm. KEYWORDSisogeometric analysis, polynomial chaos expansions, stochastic isogeometric analysis, uncertainty in shape 18

A matrix-free isogeometric Galerkin method for Karhunen-Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature

“…Stochastic methods have been proposed to quantify uncertainty due to material randomness in linear elasticity [41,31], static analysis of plates [69], vibrational analysis of shells [44], static and dynamic structural analysis of random composite structures [15] and functionally graded plates [26,42,43]. In [71] a method is proposed to quantify the effect due to uncertainty in shape. Of these, the methods proposed in [41,42,43,44] use isogeometric analysis within a spectral stochastic finite element framework [20], which is based on a KL expansion of random fields.…”

“…The methods in [15,26,69] use perturbation series of which [69] expands random fields in terms of the KLE. Standard polynomial chaos is used in [71], while the methods in [16], [31] and [56] discretize the stochastic dimensions in terms of splines. In particular, in [16] tensor product B-splines are used to expand stochastic variables, [31] proposes a spline-dimensional decomposition (SDD) and [56] proposes a spline chaos expansion, thus extending generalized polynomial chaos [70].…”

Tensor train based isogeometric analysis for PDE approximation on parameter dependent geometries

“…Moreover, the complexity of explicitly computing the matrix entries is bounded from below by the storage complexity. This bottleneck is greatly exacerbated in cases where IGA-based models are utilized within parametric studies, such as shape optimization [22,42,52,62], uncertainty quantification (UQ) for stochastic geometry deformations [24,63], or studies utilizing shape morphing techniques [64]. These parametric studies demand that multiple, often numerous, geometry configurations must be explored until an optimal shape or a statistical quantity of interest (QoI) can be estimated to sufficient accuracy.…”

Perturbation based stochastic isogeometric analysis for bending of functionally graded plates with the randomness of elastic modulus

“…Shahrokhabadi and Vahedifard (2018) combined isogeometric and generation random fields of water in the soil to model seepage in unsaturated soils. Zhang and Shibutani (2019) used polynomial chaos expansions to construct SIGA for uncertainty in shape. Ding et al (2019) considered higher-order Taylor series of functions of random variables to propose the Isogeometric generalized nth order perturbation-based stochastic method.…”