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Uncertainty quantification of spatially uncorrelated loads with a reduced-order stochastic isogeometric method

Authors:
Chensen Ding at Peking University
Chensen Ding
Kumar K. Tamma at University of Minnesota
Kumar K. Tamma
Haojie Lian at Taiyuan University of Technology
Haojie Lian
Yanjun Ding
Yanjun Ding
Yanjun Ding
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Abstract and Figures

This work models spatially uncorrelated (independent) load uncertainty and develops a reduced-order Monte Carlo stochastic isogeometric method to quantify the effect of the load uncertainty on the structural response of thin shells and solid structures. The approach is tested on two demonstrative applications of uncertainty, namely, spatially uncorrelated loading, with (1) Scordelis–Lo Roof shell structure, and (2) a 3D wind turbine blade. This work has three novelties. Firstly, the research models spatially uncorrelated (independent) load uncertainties (including both their magnitude and/or direction) using stochastic analysis. Secondly, the paper advances a reduced-order Monte Carlo stochastic isogeometric method to quantify the spatially uncorrelated load uncertainty. It inherits the merits of isogeometric analysis, which enables the precise representation of geometry and alleviates shell shear locking, thereby reducing the model’s uncertainties. Moreover, the method retains the generality and accuracy of classical Monte Carlo simulation (MCS), with significant efficiency gains. The demonstrative results suggest that there is a cost, which is 3% of the time used by the standard MCS. Furthermore, a significant observation is made from the conducted numerical tests. It is noticed that the standard deviation of the output (i.e., displacement) is strongly influenced when the load uncertainty is spatially uncorrelated. Namely, the standard derivation (SD) of the output is roughly 10 times smaller than the SD for correlated load uncertainties. Nonetheless, the expected values remain consistent between the two cases.
Structure subjected to independent distributed loads (Fn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ F_{n} $$\end{document}) that are modelled as having uncertainties in magnitude (αn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha_{n} $$\end{document}) and/or direction (θn,γn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \theta_{n} ,\,\gamma_{n} $$\end{document})
Structure subjected to independent distributed loads (Fn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ F_{n} $$\end{document}) that are modelled as having uncertainties in magnitude (αn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha_{n} $$\end{document}) and/or direction (θn,γn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \theta_{n} ,\,\gamma_{n} $$\end{document})
… 
a Direct MCS that conducts M times forward evaluation of the full order system; b proposed approach composed of an offline phase solving the full isogeometric m samples to obtain the reduced system, and an online phase conducting the reduced system M samples
a Direct MCS that conducts M times forward evaluation of the full order system; b proposed approach composed of an offline phase solving the full isogeometric m samples to obtain the reduced system, and an online phase conducting the reduced system M samples
… 
Schematic illustration of shell kinematics and spaces in IGA
Schematic illustration of shell kinematics and spaces in IGA
… 
The ith uncertainty load Fiun(αj,θk,γl)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbf{F}}_{i}^{un} (\alpha_{j} ,\theta_{k} ,\gamma_{l} ) $$\end{document} and its ingredients in three directions
The ith uncertainty load Fiun(αj,θk,γl)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbf{F}}_{i}^{un} (\alpha_{j} ,\theta_{k} ,\gamma_{l} ) $$\end{document} and its ingredients in three directions
… 
Scordelis–Lo Roof: a definition; b subjected to spatially uncorrelated loads (Fn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ F_{n} $$\end{document}) with only magnitude uncertainty (αn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha_{n} $$\end{document}); c subjected to spatially uncorrelated loads with only direction uncertainty (θn,γn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \theta_{n} ,\,\gamma_{n} $$\end{document}); and d subjected to spatially uncorrelated loads with both magnitude and direction uncertainty (αn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha_{n} $$\end{document},θn,γn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \theta_{n} ,\,\gamma_{n} $$\end{document})
+13
Scordelis–Lo Roof: a definition; b subjected to spatially uncorrelated loads (Fn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ F_{n} $$\end{document}) with only magnitude uncertainty (αn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha_{n} $$\end{document}); c subjected to spatially uncorrelated loads with only direction uncertainty (θn,γn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \theta_{n} ,\,\gamma_{n} $$\end{document}); and d subjected to spatially uncorrelated loads with both magnitude and direction uncertainty (αn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha_{n} $$\end{document},θn,γn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \theta_{n} ,\,\gamma_{n} $$\end{document})
… 
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Computational Mechanics (2021) 67:1255–1271
https://doi.org/10.1007/s00466-020-01944-9
ORIGINAL PAPER
Uncertainty quantification of spatially uncorrelated loads
with a reduced-order stochastic isogeometric method
Chensen Ding1,2 ·Kumar K. Tamma3·Haojie Lian1·Yanjun Ding4·Timothy J. Dodwell2,5 ·
Stéphane P. A. Bordas1,6,7
Received: 12 February 2020 / Accepted: 3 November 2020 / Published online: 24 March 2021
© Springer-Verlag GmbH Germany, part of Springer Nature 2021
Abstract
This work models spatially uncorrelated (independent) load uncertainty and develops a reduced-order Monte Carlo stochastic
isogeometric method to quantify the effect of the load uncertainty on the structural response of thin shells and solid structures.
The approach is tested on two demonstrative applications of uncertainty, namely, spatially uncorrelated loading, with (1)
Scordelis–Lo Roof shell structure, and (2) a 3D wind turbine blade. This work has three novelties. Firstly, the research models
spatially uncorrelated (independent) load uncertainties (including both their magnitude and/or direction) using stochastic
analysis. Secondly, the paper advances a reduced-order Monte Carlo stochastic isogeometric method to quantify the spatially
uncorrelated load uncertainty. It inherits the merits of isogeometric analysis, which enables the precise representation of
geometry and alleviates shell shear locking, thereby reducing the model’s uncertainties. Moreover, the method retains the
generality and accuracy of classical Monte Carlo simulation (MCS), with significant efficiency gains. The demonstrative
results suggest that there is a cost, which is 3% of the time used by the standard MCS. Furthermore, a significant observation
is made from the conducted numerical tests. It is noticed that the standard deviation of the output (i.e., displacement) is
strongly influenced when the load uncertainty is spatially uncorrelated. Namely, the standard derivation (SD) of the output is
roughly 10 times smaller than the SD for correlated load uncertainties. Nonetheless, the expected values remain consistent
between the two cases.
Keywords Uncertainty quantification ·Spatially uncorrelated load uncertainties ·Stochastic isogeometric analysis ·
Reduced-order models ·Monte Carlo simulation
BStéphane P. A. Bordas
stephane.bordas@alum.northwestern.edu
1Institute of Computational Engineering, University of
Luxembourg, 4364 Esch-Sur-Alzette, Luxembourg
2Institute of Data Science and Artificial Intelligence,
University of Exeter, Exeter EX4 4PY, UK
3Department of Mechanical Engineering, University of
Minnesota-Twin Cities, Minneapolis, MN 55455, USA
4Department of Forensic Science, School of Basic Medical
Sciences, Central South University, Changsha 410013,
People’s Republic of China
5The Alan Turing Institute, London NW1 2DB, UK
6Department of Medical Research, China Medical University
Hospital, China Medical University, Taichung, Taiwan
7Institute of Mechanics and Advanced Materials, School of
Engineering, Cardiff University, Cardiff, UK
1 Introduction
Uncertainty and stochasticity are ubiquitous in natural and
engineering systems and strongly impact their behaviour
(e.g., reliability [1]); therefore, computational stochastic
analyses are becoming increasingly important [2,3]. Gener-
ally, the procedures for stochastic analysis include two ingre-
dients [4]. The first one comprises of numerical techniques,
which are coupled with the stochastic theory. Examples
include the finite element method (FEM) [5–7], smoothed
finite element method (SFEM) [8,9], boundary element
method (BEM) [10], and finite difference method (FDM)
[11], which have been used for stochastic simulation. Recent
studies have also used isogeometric analysis for stochas-
tic analysis Noh [12], Li [13–15], and Ding [16–19]. The
second ingredient is a stochastic algorithm that models uncer-
tainty. Widely used algorithms include (a) stochastic spectral
approaches [20,21,41], (b) perturbation-based techniques
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... To examine how these uncertainties might affect acoustic system responses, a number of uncertainty analysis techniques have been proposed, such as Monte Carlo simulation (MCs) [11][12][13], stochastic spectral methods [14,15], and perturbation based methods [16][17][18][19][20]. Among them, the MCs has been the most versatile and simplest one, but its accuracy is heavily dependent on sampling size, which can increase computational costs [21][22][23]. Therefore, the MCs has limited practical applicability and is mainly used for validating other uncertainty analysis methods [24,25]. ...
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