Study On Application Of Dimensionality Reduction And Visualization Technology In Multi-dimensional Implicit Reliability Problems

With the development of science and technology,product structure is becoming more and more complex,and immediately the reliability of product is required more and more highly.The limit state function which is based on the finite element or the nonlinear differential equation usually doesn't have analytical expression in many practical engineering,so called the implicit limit state function.Besides,there are many random variables involved in the complex structure.For the multi-dimensional implicit reliability problems,the classical reliability methods are not adequate procedure for estimating reliability,such as the First Order Reliability Method,the Second Order Reliability Method,and so on.The numerical simulation methods have been proposed to solve these problems,such as Monte Carlo,Importance Sampling,Truncated Importance Sampling,Line Sampling and so on.However,these algorithms need a great deal of computation time.Therefore,the existing reliability assessment methods cannot meet engineering requirements gradually.How to efficiently solve these special problems is becoming more and more importance.Dimensionality reduction and visualization technology is an improved numerical simulation method,which is achieved by means of a polar transformation for reducing the dimensionality of the multi-dimensional structure reliability to only two independent dimensions.And the safe and failure classes of samples are clearly distinguishable by a line in the plane of polar representation.It is important to find the design point for dimensionality reduction and visualization technology.During the process of finding design point,it needs to estimate the gradient vector of the limit state function,which is carried out by finite differences for multi-dimensional implicit structure reliability.To reduce the truncation error,it is necessary to impose a judicious value of the perturbation.In order to solve this problem,an original technique is suggested in this paper called 'projection gradient with error control'.Based on the theory of structure reliability,this paper is to solve two multi-dimensional implicit structure reliability problems by means of dimensionality reduction and visualization technology.The first one is about reliability research for the nonlinear vibration of gear transmission,the limit function of which is resulting from the finite element method.The second one is about reliability research for bear clearance,the limit function of which is resulting from the nonlinear differential equation.The examples demonstrate the simplicity and versatility of dimensionality reduction and visualization technology,which is not affected by dimensionality or nonlinear.This paper provides new solving thoughts and structure theory for multi-dimensional implicit structure reliability problems.