Uncertainty quantification and its impact on the analysis/design have remained an active area of research in science and engineering. It is not only a better option to deal with the actual field conditions, but it is unavoidable in many cases, as uncertainty is inevitable in practical problems. Thus, the interests of this research group are focused on the study of uncertainty propagation and its quantification using novel numerical techniques. It includes Reliability Analysis, Reliability-Based Design Optimization (RBDO), Robust Design Optimization (RDO), Stochastic Finite Element Analysis (SFEA) and Surrogate Modelling. The scope of these methods involves multi-disciplinary applications in a random environment. The traditional techniques/algorithms often suffer the curse of dimensionality, escalated computational cost and inaccuracies. New schemes are developed for reliability analysis and uncertainty propagation to address these issues in a non-intrusive framework for better computational efficiency and accuracy. Thus, different algorithms are proposed using orthogonality and multi-level dimension decompositions. The following are the salient contributions from this research group
Sequential Stochastic Response Surface Method (Seq. SRSM) is developed using polynomial chaos expansion (PCE) for uncertainty quantification and reliability analysis. It forms the orthogonal bases where the MLS determines the unknown coefficients. It helps in addressing ill-conditioning (especially for large dimensional problems) for better accuracy.
Besides the above improvements, orthogonal sub-functions using analysis of variance (ANOVA) are also incorporated to enhance the surrogate modelling for uncertainty quantification. It uses multiple sub-functions for high fidelity representations of the original performance function. The method is known as Adaptive Multiple Finite Difference High Dimensional Model Representation (AMFD-HDMR). The overall contribution of each HDMR leads to an adaptive high-fidelity mixed-order approximation of the original performance function for large dimensional problems.
Simultaneously, another version (i.e. dAMFD-HDMR, the term ‘d’ represents dimension adaptive nature of the algorithm) is developed to curtail the undesired contributions of the orthogonal sub-functions in the higher-order dimension decomposition. It results in a novel sparse dimension decomposition formulation based on the significant dimensions, proving effective in evaluating uncertainty propagation in spatial randomness.
Many surrogate methods in uncertainty quantification suffer unique computational challenges while solving large dimensional problems such as matrix inversions to determine unknown coefficients and/or evaluating distance due to inefficient structuring/hierarchy. Both of them result in exponential computational cost (sometimes out of reach). Thus, an improved technique is developed using the Gaussian error modelling within the adaptive dimension decomposition framework. This Hybrid Dimension Adaptive HDMR (Hda-HDMR) is based on a mixed approach using algebraic formulation and weight parameters defined for error correlation analysis.
Additionally, the research group has developed an intelligent design of experiment (DoE) strategy using sparse grids and dimension decomposition. A tensor product based on the hierarchy modifying Smolyak’s algorithm is formulated. It employs the optimization process to generate experimental points in the region of interest adaptively. Three key features of the proposed DoE generation scheme are multi-level generation, distribution and dimension adaptive nature.
Different multi-disciplinary problems of reliability analysis, robust based design optimizations, uncertainty quantification, and FE model updating are solved using these techniques/algorithms. They include non-linear structural systems, laminate composites, Gaussian and non-Gaussian correlated random variables/fields, among many others.
