An nth high order perturbation-based stochastic isogeometric method and implementation for quantifying geometric uncertainty in shell structures

Abstract

This paper presents an n-th high order perturbation-based stochastic isogeometric Kirchhoff–Love shell method, formulation and implementation for modeling and quantifying geometric (thickness) uncertainty in thin shell structures. Firstly, the Non-Uniform Rational B-Splines (NURBS) is used to describe the geometry and interpolate the variables in a deterministic aspect. Then, the shell structures with geometric (thickness) uncertainty are investigated by developing an nth order perturbation-based stochastic isogeometric method. Here, we develop the shell stochastic formulations in detail (particularly, expand the random input (thickness) and IGA Kirchhoff-Love shell element based state functions analytically around their expectations via n-th order Taylor series using a small perturbation parameterε), whilst freshly providing the Matlab core codes helpful for implementation. This work includes three key novelties: 1) by increasing/utilizing the high order of NURBS basis functions, we can exactly represent shell geometries and alleviate shear locking, as well as providing more accurate deterministic solution hence enhancing stochastic response accuracy. 2) Via increasing the nth order perturbation, we overcome the inherent drawbacks of first and second-order perturbation approaches, and hence can handle uncertainty problems with some large coefficients of variation. 3) The numerical examples, including two benchmarks and one engineering application (B-pillar in automobile), simulated by the proposed formulations and direct Monte Carlo simulations (MCS) verify that thickness randomness does strongly affect the response of shell structures, such as the displacement caused by uncertainty can increase up to 35%; Moreover, the proposed formulation is effective and significantly efficient. For example, compared to MCS, only 0.014% computational time is needed to obtain the stochastic response.