Spline Dimensional Decomposition with Interpolation-based Optimal Knot Selection for Stochastic Dynamic Analysis

2025-05-19

AI Summary

Forward uncertainty quantification for dynamic systems faces challenges in modeling nonsmooth or locally oscillatory responses, while conventional spline-based dimension decomposition (SDD) suffers from computationally expensive global optimization for optimal node selection. Method: This paper proposes a gradient-driven interpolation-type SDD framework featuring a novel piecewise maximum-gradient localization strategy, integrated with input–output profile interpolation to bypass global optimization. Contribution/Results: Validated on 401 finite-element simulations, the method reduces the relative variance error of the first natural frequency distribution to 2.89%, outperforming uniform nodes (12.31%), random nodes (15.27%), and Gaussian process regression (5.32%). It achieves high-accuracy second-order statistical characterization and reliability assessment using only O(10²) simulations, markedly enhancing computational efficiency and robustness over existing approaches.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Forward uncertainty quantification in dynamic systems is challenging due to non-smooth or locally oscillating nonlinear behaviors. Spline dimensional decomposition (SDD) effectively addresses such nonlinearity by partitioning input coordinates via knot placement, yet its accuracy is highly sensitive to the location of internal knots. Optimizing knots through sequential quadratic programming can be effective, yet the optimization process becomes computationally intense. We propose a computationally efficient, interpolation-based method for optimal knot selection in SDD. The method involves three steps: (1) interpolating input-output profiles, (2) defining subinterval-based reference regions, and (3) selecting optimal knot locations at maximum gradient points within each region. The resulting knot vector is then applied to SDD for accurate approximation of non-smooth and locally oscillating responses. A modal analysis of a lower control arm demonstrates that SDD with the proposed knot selection achieves higher accuracy than SDD with uniformly or randomly spaced knots, and also a Gaussian process surrogate model. The proposed SDD exhibits the lowest relative variance error (2.89%), compared to SDD with uniformly spaced knots (12.310%), randomly spaced knots (15.274%), and Gaussian process (5.319%) in the first natural frequency distribution. All surrogate models are constructed using the same 401 simulation datasets, and the relative errors are evaluated against a 2000-sample Monte Carlo simulation. The scalability and applicability of proposed method are demonstrated through stochastic and reliability analyses of mathematical functions (N=1, 3) and a lower control arm system (N=10). The results confirm that both second-moment statistics and reliability estimates can be accurately achieved with only a few hundred function evaluations or finite element simulations.

Problem

• Optimizing knot selection in Spline Dimensional Decomposition for accuracy
• Reducing computational cost in stochastic dynamic system analysis
• Improving approximation of non-smooth and oscillating system responses

Innovation

• Interpolation-based optimal knot selection method
• Subinterval-based reference regions for knot placement
• Optimal knots at maximum gradient points