Understanding stochastic methods: Monte Carlo simulation and polynomial chaos expansion explained with a spring-mass system

Abstract

The study of stochastic methods has gained significant attention in recent years due to the increasing need to model uncertainty in engineering systems. Structural components, material properties, and environmental conditions often exhibit inherent randomness, making it essential to incorporate uncertainty quantification techniques for reliable predictions. Advanced approaches such as Monte Carlo Simulation (MCS) and Polynomial Chaos Expansion (PCE) have demonstrated effectiveness in this context; however, their mathematical complexity can pose a steep learning curve for researchers and engineers who are new to the field.

This paper addresses this challenge by providing a clear and accessible introduction to these methods through a simple yet illustrative example: a spring-mass system. Using this elementary model, key concepts such as random variable modeling, basis function expansion, and solution convergence are explained in an intuitive manner. The simplicity of the spring-mass system enables a focused exploration of the fundamental principles underlying MCS, Intrusive Polynomial Chaos Expansion (IPCE), and Non-Intrusive Polynomial Chaos Expansion (NIPCE), without the added complexity of more sophisticated structural models.

The study evaluates the strengths and limitations of each method in terms of computational efficiency, accuracy, and ease of implementation. Through step-by-step explanations and detailed comparisons, this work serves as a foundational reference for those beginning their exploration of stochastic methods. The insights gained from this simplified model can be extended to more complex engineering problems, such as deflection and eigenfrequency analysis of beams with random elastic modulus, and heat conduction in materials with uncertain thermal conductivity. By presenting the mathematical framework in a structured and comprehensible manner, this study aims to serve as a valuable resource for researchers and students seeking to deepen their understanding of stochastic methods in engineering applications.

Keywords

Non-Intrusive polynomial chaos intrusive polynomial chaos Monte Carlo simulation

Get full access to this article

View all access options for this article.

References

Argyris

Papadrakakis

Stefanou

. Stochastic finite element analysis of shells. Comput Methods Appl Mech Eng 2002; 191: 4781–4804.

Charmpis

Papadrakakis

. Improving the computational efficiency in finite element analysis of shells with uncertain properties. Comput Methods Appl Mech Eng 2005; 194: 1447–1478.

Lagaros

Papadopoulos

. Optimum design of shell structures with random geometric, material and thickness imperfections. Int J Solids Struct 2006; 43: 6948–6964.

Papadopoulos

Charmpis

Papadrakakis

. A computationally efficient method for the buckling analysis of shells with stochastic imperfections. Comput Mech 2009; 43: 687–700.

Iman

Conover

. A distribution-free approach to inducing rank correlation among input variables. Commun Stat-Simul Comput 1982; 11: 311–334.

Stein

. Large sample properties of simulations using latin hypercube sampling. Technometrics 1987; 29: 143–151.

Ziha

. Descriptive sampling in structural safety. Struct Safety 1995; 17: 33–41.

Olsson

Sandberg

. Latin hypercube sampling for stochastic finite element analysis. J Eng Mech 2002; 128: 121–125.

Matthies

Bucher

. Finite elements for stochastic media problems. Comput Methods Appl Mech Eng 1999; 168: 3–17.

10.

Schenk

Schuëller

. Buckling analysis of cylindrical shells with random geometric imperfections. Int J Non Linear Mech 2003; 38: 1119–1132.

11.

Schenk

Schuëller

. Uncertainty assessment of large finite element systems. Berlin, Heidelberg: Springer Science & Business Media, 2005.

12.

Ghanem

Spanos

Ghanem

, et al. Stochastic finite element method: response statistics. Stoch Finit Elem: Spect Appr 1991; 101–119.

13.

Ghanem

. The nonlinear gaussian spectrum of log-normal stochastic processes and variables 1999.

14.

Ghanem

. Stochastic finite elements: a spectral approach. Cour Corpor 2003.

15.

Xiu

Karniadakis

. The wiener–askey polynomial chaos for stochastic differential equations. SIAM J Sci Comput 2002; 24: 619–644.

16.

Lucor

Karniadakis

. Adaptive generalized polynomial chaos for nonlinear random oscillators. SIAM J Sci Comput 2004; 26: 720–735.

17.

Crestaux

Le Maıtre

Martinez

. Polynomial chaos expansion for sensitivity analysis. Reliab Eng Syst Safety 2009; 94: 1161–1172.

18.

Wan

Ren

Todd

. Arbitrary polynomial chaos expansion method for uncertainty quantification and global sensitivity analysis in structural dynamics. Mech Syst Signal Process 2020; 142: 106732.

19.

Ghanem

Ghosh

. Efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition. Int J Numer Methods Eng 2007; 72: 486–504.

20.

Mulani

Kapania

Walters

. Stochastic eigenvalue problem with polynomial chaos. In: 47th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference 14th AIAA/ASME/AHS adaptive structures conference 7th, 2006, p.2068.

21.

Sarrouy

Dessombz

Sinou

. Stochastic analysis of the eigenvalue problem for mechanical systems using polynomial chaos expansion—application to a finite element rotor 2012.

22.

Elman

. Low-rank solution methods for stochastic eigenvalue problems. SIAM J Sci Comput 2019; 41: A2657–A2680.

23.

Verhoosel

Gutiérrez

Hulshoff

. Iterative solution of the random eigenvalue problem with application to spectral stochastic finite element systems. Int J Numer Methods Eng 2006; 68: 401–424.

24.

Jiang

Shi

. Combination of karhunen-loève and intrusive polynomial chaos for uncertainty quantification of thermomagnetic convection problem with stochastic boundary condition. Eng Anal Bound Elem 2024; 159: 452–465.

25.

Parekh

Verstappen

. Intrusive polynomial chaos for cfd using openfoam. In: Computational Science–ICCS 2020: 20th International conference, Amsterdam, The Netherlands, June 3–5, 2020, Proceedings, Part VII 20, 2020, pp.677–691. Springer.

26.

Xiu

. Numerical methods for stochastic computations: a spectral method approach. Princeton, New Jersey: Princeton University Press, 2010.

27.

Son

. Comparison of intrusive and nonintrusive polynomial chaos expansion-based approaches for high dimensional parametric uncertainty quantification and propagation. Comput Chem Eng 2020; 134: 106685.

28.

Son

. An efficient polynomial chaos expansion method for uncertainty quantification in dynamic systems. Appl Mech 2021; 2: 460–481.

29.

Herzog

Gilg

Paffrath

, et al. Intrusive versus non-intrusive methods for stochastic finite elements. From Nano Space: Appl Mathem Inspired Roland Bulirsch 2008; 161–174.

30.

Berveiller

Sudret

Lemaire

. Stochastic finite element: a non intrusive approach by regression. Euro J Comput Mech/Revue Eur Mécanique Numérq 2006; 15: 81–92.

31.

Branicki

Majda

. Fundamental limitations of polynomial chaos for uncertainty quantification in systems with intermittent instabilities. Commun Math Sci 2013; 11: 55–103.