Key Takeaways & Executive Findings
- •• The proposed isogeometric collocation method with adaptive moment abscissae significantly improves accuracy over classical Greville abscissae in random field discretization. • The method reduces computational cost by minimizing integral operations compared to the Galerkin method, making it suitable for complex geometries and large-scale engineering problems. • Intelligent algorithms optimize the moment parameter, enabling adaptive placement of collocation points for enhanced precision. • Numerical and engineering examples demonstrate superior accuracy and efficiency, validating the method's practical applicability in stochastic structural analysis.
Abstract
The discretization of random fields is the first and most important step in the stochastic analysis of engineering structures with spatially dependent random parameters. The essential step of discretization is solving the Fredholm integral equation to obtain the eigenvalues and eigenfunctions of the covariance functions of the random fields. The collocation method, which has fewer integral operations, is more efficient in accomplishing the task than the time-consuming Galerkin method, and it is more suitable for engineering applications with complex geometries and a large number of elements. With the help of isogeometric analysis that preserves accurate geometry in analysis, the isogeometric collocation method can efficiently achieve the results with sufficient accuracy. An adaptive moment abscissa is proposed to calculate the coordinates of the collocation points to further improve the accuracy of the collocation method. The adaptive moment abscissae led to more accurate results than the classical Greville abscissae when using the moment parameter optimized with intelligent algorithms. Numerical and engineering examples illustrate the advantages of the proposed isogeometric collocation method based on the adaptive moment abscissae over existing methods in terms of accuracy and efficiency.
1. Introduction
Realistic engineering structures exhibit stochastic characteristics because of random variations in their material properties and external loads [1]. Traditional deterministic analyses that disregard such uncertainties cannot accurately capture the actual structural responses of engineering structures [2]. Consequently, stochastic structural analysis, which considers the inherent randomness within engineering structures, is necessary for further developing structural analysis. Spatially dependent randomness in engineering structures is modelled as a random field. Representing a continuous-parameter random field using a finite set of random variables is essential for the stochastic analysis of structures with random-field uncertainties. This process, known as random field discretization, aims to achieve a balanced compromise between accuracy and computational efficiency.
The Karhunen–Loève (K–L) expansion is one of the most commonly used methods for discretizing the random field. The K–L expansion can hold both homogeneous and inhomogeneous fields, minimize the mean-square error committed by a finite-term K–L approximation, and converge the sequence of K–L approximations in mean square to the correct limit [1]. Given these advantages, the K–L expansion requires the solution of a Fredholm integral equation that cannot be solved easily. An analytical solution to the integral equation exists only when the physical domain and covariance function of the random field are simple. Numerical solutions are the only options for more complex practical engineering applications.
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Zhenyu Liu, Deshang Peng, Minglong Yang, Jin Cheng, Chan Qiu, Jianrong Tan (2025). Isogeometric Collocation Method for Random Field Discretization Based on Adaptive Moment Abscissae. Chinese Journal of Mechanical Engineering. https://doi.org/10.1186/s10033-025-01293-9
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Frequently Asked Questions
What is the main contribution of this paper?
The paper proposes an isogeometric collocation method for random field discretization that uses adaptive moment abscissae to improve accuracy over classical Greville abscissae, while maintaining computational efficiency compared to the Galerkin method.
How does the adaptive moment abscissa improve the collocation method?
The adaptive moment abscissa optimizes the placement of collocation points using intelligent algorithms, leading to more accurate eigenvalues and eigenfunctions in the Fredholm integral equation solution, thus improving the overall discretization accuracy.
Why is the collocation method preferred over the Galerkin method in this context?
The collocation method requires fewer integral operations and is more efficient for large-scale problems with complex geometries, making it more suitable for practical engineering applications.
What is the significance of the Karhunen–Loève expansion in this work?
The Karhunen–Loève expansion is used to discretize random fields, and the paper focuses on efficiently solving the associated Fredholm integral equation, which is a critical step in the expansion.
What are the practical applications of this research?
The method is applicable to stochastic analysis of engineering structures with spatially varying material properties and loads, such as in mechanical, civil, and aerospace engineering, where accurate and efficient random field discretization is essential.
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