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

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.