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

2020 ◽  
Vol 18 (01) ◽  
pp. 2050024
Author(s):  
Amit Kumar Rathi ◽  
Arunasis Chakraborty
Keyword(s):  
Reliability Analysis ◽  
Reference Point ◽  
Structural Reliability ◽  
Least Square ◽  
Computational Time ◽  
Benchmark Problems ◽  
Moving Least Square ◽  
Support Points ◽  
Multiple Dimension ◽  
Dimension Decomposition

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).

Asymptotic distribution method for structural reliability analysis in high dimensions

2005 ◽  
Vol 461 (2062) ◽  
pp. 3141-3158 ◽  
Author(s):  
Sondipon Adhikari
Keyword(s):  
Reliability Analysis ◽  
Asymptotic Distribution ◽  
Structural Reliability ◽  
Random Variables ◽  
Computational Time ◽  
High Dimensions ◽  
Engineering Structures ◽  
Distribution Method ◽  
Complex Engineering ◽  
Asymptotic Formulation

In the reliability analysis of safety critical complex engineering structures, a very large number of the system parameters can be considered as random variables. The difficulty in computing the failure probability using the classical first- and second-order reliability methods (FORM and SORM) increases rapidly with the number of variables or ‘dimension’. There are mainly two reasons behind this. The first is the increase in computational time with the increase in the number of random variables. In principle, this problem can be handled with superior computational tools. The second reason, which is perhaps more fundamental, is that there are some conceptual difficulties typically associated with high dimensions. This means that even when one manages to carry out the necessary computations, the application of existing FORM and SORM may still lead to incorrect results in high dimensions. This paper is aimed at addressing this issue. Based on the asymptotic distribution of quadratic form in Gaussian random variables, two formulations for the case when the number of random variables n →∞ is provided. The first is called ‘strict asymptotic formulation’ and the second is called ‘weak asymptotic formulation’. Both approximations result in simple closed-form expressions for the probability of failure of an engineering structure. The proposed asymptotic approximations are compared with existing approximations and Monte Carlo simulations using numerical examples.

scholarly journals Quadric SFDI for Laplacian Discretisation in Lagrangian Meshless Methods

2020 ◽  
Vol 19 (3) ◽  
pp. 362-380
Author(s):  
Shiqiang Yan ◽  
Q. W. Ma ◽  
Jinghua Wang
Keyword(s):  
Finite Difference ◽  
Least Square Method ◽  
Least Square ◽  
Particle Methods ◽  
Computational Time ◽  
Initial State ◽  
Patch Tests ◽  
Moving Least Square ◽  
Particle Hydrodynamics ◽  
Convergent Rate

Abstract In the Lagrangian meshless (particle) methods, such as the smoothed particle hydrodynamics (SPH), moving particle semi-implicit (MPS) method and meshless local Petrov-Galerkin method based on Rankine source solution (MLPG_R), the Laplacian discretisation is often required in order to solve the governing equations and/or estimate physical quantities (such as the viscous stresses). In some meshless applications, the Laplacians are also needed as stabilisation operators to enhance the pressure calculation. The particles in the Lagrangian methods move following the material velocity, yielding a disordered (random) particle distribution even though they may be distributed uniformly in the initial state. Different schemes have been developed for a direct estimation of second derivatives using finite difference, kernel integrations and weighted/moving least square method. Some of the schemes suffer from a poor convergent rate. Some have a better convergent rate but require inversions of high order matrices, yielding high computational costs. This paper presents a quadric semi-analytical finite-difference interpolation (QSFDI) scheme, which can achieve the same degree of the convergent rate as the best schemes available to date but requires the inversion of significant lower-order matrices, i.e. 3 × 3 for 3D cases, compared with 6 × 6 or 10 × 10 in the schemes with the best convergent rate. Systematic patch tests have been carried out for either estimating the Laplacian of given functions or solving Poisson’s equations. The convergence, accuracy and robustness of the present schemes are compared with the existing schemes. It will show that the present scheme requires considerably less computational time to achieve the same accuracy as the best schemes available in literatures, particularly for estimating the Laplacian of given functions.

Approximations for Safety Optimization of Large Systems

2000 ◽  
Author(s):  
R. J. Yang ◽  
L. Gu ◽  
L. Liaw ◽  
C. Gearhart ◽  
C. H. Tho ◽  
...  
Keyword(s):  
Latin Hypercube Sampling ◽  
Response Surfaces ◽  
Least Square ◽  
Multivariate Adaptive Regression Splines ◽  
Benchmark Problems ◽  
Moving Least Square ◽  
Adaptive Regression ◽  
Large Systems ◽  
Optimal Latin Hypercube Sampling ◽  
Adaptive Regression Splines

Abstract This paper presents four approximation methods for the construction of safety related functions. These methods are: Enhanced Multivariate Adaptive Regression Splines, Stepwise Regression, Artificial Neural Network, and the Moving Least Square. The optimal Latin Hypercube Sampling method is used to distribute the sampling points uniformly over the entire design space. Four benchmark problems used in crash and occupant simulation are employed to investigate the accuracy of the approximate or surrogate models. An occupant safety optimization problem is solved using these four response surfaces. Based on numerical results, a best, applicable approximation strategy for safety optimization is proposed in the end.

STRUCTURAL RELIABILITY ANALYSIS BASED ON P-BOXES

2020 ◽  
Vol 92 (6) ◽  
pp. 51-58
Author(s):  
S.A. SOLOVYEV ◽  
Keyword(s):  
Probability Distribution ◽  
Reliability Analysis ◽  
Structural Reliability ◽  
Epistemic Uncertainty ◽  
Random Variable ◽  
Strength Criterion ◽  
Structural Elements ◽  
Structural Element ◽  
Limit States ◽  
Distribution Shape

The article describes a method for reliability (probability of non-failure) analysis of structural elements based on p-boxes. An algorithm for constructing two p-blocks is shown. First p-box is used in the absence of information about the probability distribution shape of a random variable. Second p-box is used for a certain probability distribution function but with inaccurate (interval) function parameters. The algorithm for reliability analysis is presented on a numerical example of the reliability analysis for a flexural wooden beam by wood strength criterion. The result of the reliability analysis is an interval of the non-failure probability boundaries. Recommendations are given for narrowing the reliability boundaries which can reduce epistemic uncertainty. On the basis of the proposed approach, particular methods for reliability analysis for any structural elements can be developed. Design equations are given for a comprehensive assessment of the structural element reliability as a system taking into account all the criteria of limit states.

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