Towards interval analysis of large-scale uncertainty in spatially varied interval field

In this work, we investigate the large-scale interval analysis for structures with spatially varied interval uncertainty. Our motivations come from the fact that sample data of uncertainties in practical engineering problems are often limited, and the lack of an interval analysis approach for large-scale uncertain-but-bounded field problems. To quantify uncertain material properties, geometries, and external loads that may vary spatially, we employ the B-spline Interval Field Decomposition (BIFD) method to reduce the complexity and dimensionality of uncertainty modeling. Unlike the available method, we introduce the Chebyshev polynomials of the first kind to approximate the original function with interval fields, thereby establishing an interval envelope function with interval parameters that is suitable for large-scale uncertainty problems. We then suggest utilizing an optimization-based approach to find combinations that make the approximate function reach extreme values within a given interval, thereby determining the boundaries of the above interval envelope. Finally, the Chebyshev Interval Polynomials Approximation (CIPA) framework is summarized and extended to jointly apply the Finite Element Method (FEM). Several numerical examples are presented to illustrate the effectiveness of the proposed approach, in which the uncertain material properties, geometry and loading are investigated. Compared with the conventional Interval Perturbation Analysis (IPA) and Monte Carlo Simulation (MCS), the suggested approach shows good potential in balancing the accuracy and efficiency of uncertainty propagation analysis, especially for large-scale uncertainties. This work also reveals that a midpoint offset phenomenon of the IPA when dealing with large-scale uncertainties is one of the reasons for its inaccuracy in interval analysis.