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This paper presents the state-of-the-art on different moving least square (MLS)-based dimension decomposition schemes for reliability analysis and demonstrates a modified version for high fidelity applications. The aim is to improve the performance of MLS-based dimension decomposition in terms of accuracy, number of function evaluations and computational time for large-dimensional problems. With this in view, multiple finite difference high dimension model representation (HDMR) scheme is developed. This anchored decomposition is implemented starting from an initial reference point and progressively evolving in successive iterations. Most probable point (MPP) of failure is identified in every iteration and is used as the reference point for the next decomposition until it converges. Hermite polynomials in MLS framework are used between the support points for efficient interpolation. The support points are generated sequentially using multiple sparse grids based on the Clenshaw-Curtis scheme. Once the global response surface is constructed using the support points generated in each iteration, importance sampling is employed for reliability analysis. Six different benchmark problems are solved to show its performance vis-à-vis other methods. Finally, reliability-based design of a composite plate is demonstrated, clearly showing the advantage and superiority of the proposed improvements in MLS-based multiple dimension decomposition (MDD). © 2021 World Scientific Publishing Company.

Department of Civil & Infrastructure Engineering

World University root node

International Journal of Computational Methods

Improved Moving Least Square-Based Multiple Dimension Decomposition (MDD) Technique for Structural Reliability Analysis

A novel adaptive high dimensional model representation (HDMR) is developed for stochastic finite element analysis of composite plate. Uncertainty propagating through the physical system is quantified using this technique where only the significant dimensions are retained. Influence of these dimensions is modeled by different orders which are preferred to be low for computational tractability. These dimensions are identified by a sensitivity analysis using support points in the sparse grid which are generated using multiple levels of hierarchy, joint or marginal probability distributions and dimension adaptiveness. This multi level hierarchical generation of support points fill the domain of the input variables with major emphasis on the regions of interest which may be the entire space, limit state, maxima/minima or tail ends of the probability distribution function for rare events depending upon the nature of the problem. The hybrid characteristics of the proposed dimension decomposition is introduced in this paper where the component functions are expressed using orthogonal bases and the error term which is further modeled using Gaussian random process (i.e. Kriging). The performance of the proposed Hybrid dimension adaptive HDMR (hdA-HDMR) is illustrated using stochastic finite element analysis of laminated composite plate whose material properties are considered as homogeneous non-normal random fields. © 2020 Elsevier Ltd

Fulltext

Composite Structures

Development of hybrid dimension adaptive sparse HDMR for stochastic finite element analysis of composite plate

The present work demonstrates an efficient method for reliability analysis using sequential development of the stochastic response surface. Here, orthogonal Hermite polynomials are used whose unknown coefficients are evaluated using moving least square technique. To do so, collocation points in the conventional stochastic response surface method (SRSM) are replaced by the sparse grid scheme so as to reduce the number of function evaluations. Moreover, the domain is populated sequentially by the sparse grid based on the outcome of the optimization to find out the most probable failure point. Hence, the support points are generated based on a coupled effect of the optimization for failure region and the sub-grids hierarchy. Continuous and differentiable penalty function is imposed to determine multiple failure points, if any, by repeating the optimization. Once the response surface is developed, reliability analysis is carried out using importance sampling. Five different benchmark examples are presented in this study to validate the performance of the proposed modeling. As the accuracy of the method is established, two reliability-based design examples involving nonlinear finite element (FE) analysis of plates are demonstrated. Numerical study shows the efficiency of the proposed sequential SRSM in terms of accuracy and number of time-exhaustive evaluation of the original performance function, as compared to other methods available in the literature. Based on these results, it may be concluded that the proposed method works satisfactorily for a large class of reliability-based design problems. © 2019 World Scientific Publishing Company.

Sequential stochastic response surface method using moving least squares-based sparse grid scheme for efficient reliability analysis

The present study aims to investigate uncertainty quantification followed by reliability analysis of structure with homogeneous non-normal random fields. In stochastic finite element formulation, these continuous fields are discretized by different methods (e.g. Karhunen-Loève Expansion) which transformed it into a set of random variables. However, this discretization often leads to large number of random variables, especially for multiple random fields. With this in view, two different meta-model based approaches are presented in this study using high dimensional model representation (HDMR) for efficient stochastic computation. First, an adaptive multiple finite difference HDMR (AMFD-HDMR) is proposed that decomposes the original performance function into summands of smaller dimensions. These subfunctions are modeled by polynomial chaos expansion (PCE) using moving least square technique which utilizes the benefits of orthogonality of the basis functions and provides adaptive interpolation between the support points. These support points are generated in a sparse grid framework based on the hierarchial tensor products of the sub-grids and the appropriate statistical properties. An iterative scheme is developed with the aim to create new support points in the desired locations such as most probable failure point and/or maxima/minima. Later, a dimension adaptive multiple finite difference HDMR (dAMFD-HDMR) is proposed utilizing sensitivity analysis to further improve the efficiency and accuracy. In the second proposal, an intermittent HDMR formulation is suggested based on the individual and mutual contributions of the significant dimensions. Once the meta-model is built, Monte Carlo simulation is performed over it, thus bypassing the time exhaustive computation of the original performance function. Numerical studies are carried out using composite plate to prove the merits of the proposed algorithms compared to other methods available in the literature. © 2018 Elsevier Ltd

Structural Safety

Dimension adaptive finite difference decomposition using multiple sparse grids for stochastic computation

Recent development of system identification using Bayesian models or stochastic filtering provides probabilistic descriptions (i.e., probability density function or statistical parameters like mean and variance) of the identified model parameters (e.g., mass, stiffness, and damping). Optimal design of passive controllers for these systems whose parameters are uncertain has remained an open problem. With this in view, the present study aims to develop numerical solution scheme for the optimal design of tuned mass damper (TMD) operating in uncertain environment. Deterministic design of TMD in these cases suffers detuning as the system parameters are random. Thus, a reliability-based design optimization (RBDO) scheme is presented in this paper for better performance of the TMD when exposed to uncertainties. To solve the RBDO problem, response surface methodology is used along with the moving least squares technique. Dual response surfaces are used for separate handling of optimization and reliability analysis. First response surface performs optimization of the design variables of TMD, while the second response surfaces are used for the estimation of the statistical properties like mean and variance to satisfy the constrained conditions. Numerical analysis is presented to show the effectiveness of the proposed algorithm for RBDO of single degree of freedom-TMD system as a proof of concept. The proposed meta-model-based algorithm can be applied for the optimal design of controller for large structures where conventional technique may face difficulty to handle both optimization and uncertainty quantification simultaneously. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.

Structural Control and Health Monitoring

Reliability-based performance optimization of TMD for vibration control of structures with uncertainty in parameters and excitation

High fidelity models for uncertainty quantification of large structures in finite element framework are computationally exhaustive. Thus, there is a constant demand for efficient algorithm that uses optimal computational cost without compromising with the quality of the end results. With this in view, present study aims to develop a sequentially evolving stochastic response surface using Hermite family of orthogonal polynomials whose support points are generated in sparse grid framework. Using the values of the original model at these support points, unknown coefficients of the stochastic response surface are optimized by moving least square technique. It helps to reduce the number of original function evaluations to determine the coefficients as compared to other deterministic or random sampling techniques. Besides sparse grid scheme for support point generation, they are also populated sequentially as the optimization progresses in every iteration. The uniqueness of the proposed scheme is its adaptability by changing the order of the polynomials and the level of the sparse grid to minimize the overall computational cost. Multiple optima present in the original response can be identified by introducing additional penalty functions whenever they are required. Once the global response surface is ready, Monte-Carlo simulation or its advance version (e.g. Latin Hypercube Sampling) is adopted to generate the probability density functions for the output variables. Numerical studies are presented to prove the efficiency and accuracy of the proposed scheme as compared to other techniques available in the literature. © 2017 The Authors.

UNCECOMP 2017 - Proceedings of the 2nd International Conference on Uncertainty Quantification in Computational Sciences and Engineering

Mls based sequential srsm in sparse grid framework for efficient uncertainty quantification

The effect of randomness in system parameters on robust design of tuned mass damper (TMD) is examined in this work. For this purpose, mean and standard deviation based robust design optimization (RDO) scheme is suggested. The performance of TMD is evaluated using the percentage reduction of the root mean square (RMS) of the output displacement. Adaptive response surface method (ARSM) is used for the optimization and for the estimation of first two moments. In this context, moving least square (MLS) based regression technique is used for better fitting of the response surface. A comparative numerical study is conducted to show the effectiveness of the proposed method to improve the reliability of the controller. © Springer India 2015.