A non-probabilistic reliability topology optimization method based on aggregation function and matrix multiplication considering buckling response constraints

A non-probabilistic reliability topology optimization method is proposed based on the aggregation function and matrix multiplication.The expression of the geometric stiffness matrix is derived,the finite element linear buckling analysis is conducted,and the sensitivity solution of the linear buckling factor is achieved.For a specific problem in linear buckling topology optimization,a Heaviside projection function based on the exponential smooth growth is developed to eliminate the gray cells.The aggregation function method is used to consider the high-order eigenvalues,so as to obtain continuous sensitivity information and refined structural design.With cyclic matrix programming,a fast topology optimization method that can be used to efficiently obtain the unit assembly and sensitivity solution is conducted.To maximize the buckling load,under the constraint of the given buckling load,two types of topological optimization columns are constructed.The variable density method is used to achieve the topology optimization solution along with the moving asymptote optimization algorithm.The vertex method and the matching point method are used to carry out an uncertainty propagation analysis,and the non-probability reliability topology optimization method considering buckling responses is developed based on the transformation of non-probability reliability indices based on the characteristic distance.Finally,the differences in the structural topology optimization under different reliability degrees are illustrated by examples.

A Bayesian Updating Method for Non-Probabilistic Reliability Assessment of Structures with Performance Test Data

For structures that only the predicted bounds of uncertainties are available,this study proposes a Bayesianmethod to logically evaluate the nonprobabilistic reliability of structures based on multi-ellipsoid convex model and performance test data.According to the given interval ranges of uncertainties,we determine the initial characteristic parameters of a multi-ellipsoid convex set.Moreover,to update the plausibility of characteristic parameters,a Bayesian network for the information fusion of prior uncertainty knowledge and subsequent performance test data is constructed.Then,an updated multi-ellipsoid set with the maximum likelihood of the performance test data can be achieved.The credible non-probabilistic reliability index is calculated based on the Kriging-based surrogate model of the performance function.Several numerical examples are presented to validate the proposed Bayesian updating method.

MODIFIED SCHEME BASED ON SEMI-ANALYTIC APPROACH FOR COMPUTING NON-PROBABILISTIC RELIABILITY INDEX

A new computation scheme proposed to tackle commensurate problems is devel- oped by modifying the semi-analytic approach for minimizing computational complexity. Using the proposed scheme, the limit state equations, usually referred to as the failure surface, are obtained from transformation of an interval variable to a normalized one. In order to minimize the computational cost, two algorithms for optimizing the calculation steps have been proposed. The monotonicity of the objective function can be determined from narrowing the scope of interval variables in normalized infinite space by incorporating the algorithms into the computational scheme. Two examples are used to illustrate the operation and computational efficiency of the approach. The results of these examples show that the proposed algorithms can greatly reduce the computation complexity without sacrificing the computational accuracy. The advantage of the proposed scheme can be even more efficient for analyzing sophistic structures.

A Comprehensive Model for Structural Non-Probabilistic Reliability and the Key Algorithms

It is very difficult to know the exact boundaries of the variable domain for problems with small sample size,and the traditional convex set model is no longer applicable.In view of this,a novel reliability model was proposed on the basis of the fuzzy convex set(FCS)model.This new reliability model can account for different relations between the structural failure region and variable domain.Key computational algorithms were studied in detail.First,the optimization strategy for robust reliability is improved.Second,Monte Carlo algorithms(i.e.,uniform sampling method)for hyper-ellipsoidal convex sets were studied in detail,and errors in previous reports were corrected.Finally,the Gauss-Legendre integral algorithm was used for calculation of the integral reliability index.Three numerical examples are presented here to illustrate the rationality and feasibility of the proposed model and its corresponding algorithms.

Robust Optimization Operation of Power System Considering the Non-probabilistic Uncertainty of Parameter

Probabilistic method requires a lot of sample information to describe the probability distributions of uncertain variables and has difficulty in dealing with the optimization problem with uncertain parameters which contains unsufficient information.To solve this problem,a robust optimization operation method based on information gap decision theory(IGDT) is presented considering the non-probabilistic uncertainties of parameters.By the proposed method the maximum resistance to the disturbance of uncertain parameters is achieved and the optimization strategies with uncertain parameters are presented.Finally,numerical simulation is performed on the modified IEEE-14 bus system.Numerical results show the effectiveness of the proposed approach.

Emission Intensity in the Hydrogen Atom Calculated from a Non-Probabilistic Approach to the Electron Transitions

Quantum aspects of the Joule-Lenz law for the transmission of energy allowed us to calculate the time rate of energy transitions between the quantum states of the hydrogen atom in a fully non-probabilistic way. The calculation has been extended to all transitions between p and s states having main quantum numbers not exceeding 6. An evident similarity between the intensity pattern obtained from the Joule-Lenz law and the corresponding quantum-mechanical transition pro-babilities has been shown.

Review: Recent Developments in the Non-Probabilistic Finite Element Analysis

Generally, the finite element analysis of a structure is completed under deterministic inputs.However,uncertainties corresponding to geometrical dimensions,material properties, boundary conditions cannot be neglected in engineering applications. The probabilistic methods are the most popular techniques to handle these uncertain parameters but subjective results could be obtained if insufficient information is unavailable. Non-probabilistic methods can be alternatively employed,which has led to the procedures for nonprobabilistic finite element analysis. Each non-probabilistic finite element analysis method consists of two individual parts,including the core algorithm and pre-processing procedure. In this context,three types of algorithms and two typical pre-processing procedures as well as their effectiveness are described in detail,based on which novel hybrid algorithms can be conceived for the specific problems and the future work in this research field can be fostered.

Application of non-probabilistic reliability on aerostat capsule

Aerostat capsule is small sample data,so designing reliability is very difficult to be obtained accurately by conventional probabilistic reliability method. Based on the interval non-probabilistic reliability theory,an instability mathematics model of envelope structure is studied,and the calculation formula of interval reliability index is put forward. Through the mechanical experiments of three capsule structures,the experimental results of the interval reliability are obtained. By comparing the theoretical and measured values,it is found that the theoretical reliability index is more conservative. Non-probabilistic reliability method can reflect the reliability degree of the capsule body under different loading conditions,which can provide some guidance for engineering application.

Efficient Computational Method for the Non-Probabilistic Reliability of Linear Structural Systems

The non-probabilistic reliability in higher dimensional situations cannot be calcu- lated efficiently using traditional methods, which either require a large amount of calculation or cause significant error. In this study, an efficient computational method is proposed for the cal- culation of non-probabilistic reliability based on the volume ratio theory, specificMly for linear structural systems. The common expression for non-probabilistic reliability is obtained through formula derivation with the amount of computation considerably reduced. The compatibility be- tween non-probabilistic and probabilistic safety measures is demonstrated through the Monte Carlo simulation. The high efficiency of the presented method is verified by several numerical examples.

An Efficient Strategy for Non-probabilistic Reliability-Based Multi-material Topology Optimization with Evidence Theory

It is essential to consider the effects of incomplete measurement,inaccurate information and inadequate cognition on structural topology optimization.For the multi-material structural topology optimization with non-probability uncertainty,the multi-material interpolation model is represented by the ordered rational approximation of mat erial properties(ordered RAMP).Combined with structural compliance minimization,the multi-material topology optimization with reliability constraints is established.The corresponding non-probability uncertainties are described by the evidence theory,and the uniformity processing method is introduced to convert the evidence variables into random variables.The first-order reliability method is employed to search the most probable point under the reliability index constraint,and then the random variables are equivalent to the deterministic variables according to the geometric meaning of the reliability index and sensitivity information.Therefore,the non-probabilistic reliability-based multi-material topology optimization is transformed into the conventional deterministic optimization format,followed by the ordered RAMP method to solve the optimization problem.Finally,through numerical examples of 2D and 3D structures,the feasibility and effectiveness of the proposed method are verified to consider the geometrical dimensions and external loading uncertainties.

Theoretical analysis of non-probabilistic reliability based on interval model

The aim of this paper is to propose a theoretical approach for performing the nonprobabilistic reliability analysis of structure.Due to a great deal of uncertainties and limited measured data in engineering practice,the structural uncertain parameters were described as interval variables.The theoretical analysis model was developed by starting from the 2-D plane and 3-D space.In order to avoid the loss of probable failure points,the 2-D plane and 3-D space were respectively divided into two parts and three parts for further analysis.The study pointed out that the probable failure points only existed among extreme points and root points of the limit state function.Furthermore,the low-dimensional analytical scheme was extended to the high-dimensional case.Using the proposed approach,it is easy to find the most probable failure point and to acquire the reliability index through simple comparison directly.A number of equations used for calculating the extreme points and root points were also evaluated.This result was useful to avoid the loss of probable failure points and meaningful for optimizing searches in the research field.Finally,two kinds of examples were presented and compared with the existing computation.The good agreements show that the proposed theoretical analysis approach in the paper is correct.The efforts were conducted to improve the optimization method,to indicate the search direction and path,and to avoid only searching the local optimal solution which would result in missed probable failure points.

Non-Probabilistic Reliability Research on Multi-Variable Hydraulic Turbine Blade Model

Based on the interval mathematics and possibility theory, the variables existing in hydraulic turbine blade are described. Considering the multi-failure mode in turbine blade, multi-variable model is established to meet the actual situation. Thus, non-probabilistic reliability index is presented by comparing with the output range and the given range.

Research for Non-probabilistic Structure Reliability

The paper discusses non-probabilistic approaches for uncertainty treatment in structure reliability analysis. Based on rough set theory, the uncertain parameters of structures are expressed by rough variables, the structure reliability index is computed by rough function and metric. This new methodology for structural reliability is proved to be valid and efficient using theory analysis and examples of practical application.

An improved interval model updating method via adaptive Kriging models

Interval model updating(IMU)methods have been widely used in uncertain model updating due to their low requirements for sample data.However,the surrogate model in IMU methods mostly adopts the one-time construction method.This makes the accuracy of the surrogate model highly dependent on the experience of users and affects the accuracy of IMU methods.Therefore,an improved IMU method via the adaptive Kriging models is proposed.This method transforms the objective function of the IMU problem into two deterministic global optimization problems about the upper bound and the interval diameter through universal grey numbers.These optimization problems are addressed through the adaptive Kriging models and the particle swarm optimization(PSO)method to quantify the uncertain parameters,and the IMU is accomplished.During the construction of these adaptive Kriging models,the sample space is gridded according to sensitivity information.Local sampling is then performed in key subspaces based on the maximum mean square error(MMSE)criterion.The interval division coefficient and random sampling coefficient are adaptively adjusted without human interference until the model meets accuracy requirements.The effectiveness of the proposed method is demonstrated by a numerical example of a three-degree-of-freedom mass-spring system and an experimental example of a butted cylindrical shell.The results show that the updated results of the interval model are in good agreement with the experimental results.

Combination of structural reliability and interval analysis

In engineering applications, probabilistic reliability theory appears to be presently the most important method, however, in many cases precise probabilistic reliability theory cannot be considered as adequate and credible model of the real state of actual affairs. In this paper, we developed a hybrid of probabilistic and non-probabilistic reliability theory, which describes the structural uncertain parameters as interval variables when statistical data are found insufficient. By using the interval analysis, a new method for calculating the interval of the structural reliability as well as the reliability index is introduced in this paper, and the traditional probabilistic theory is incorporated with the interval analysis. Moreover, the new method preserves the useful part of the traditional probabilistic reliability theory, but removes the restriction of its strict requirement on data acquisition. Example is presented to demonstrate the feasibility and validity of the proposed theory.

Fatigue reliability based on residual strength model with hybrid uncertain parameters

The aim of this paper is to evaluate the fatigue reliability with hybrid uncertain parameters based on a residual strength model. By solving the non-probabilistic setbased reliability problem and analyzing the reliability with randomness, the fatigue reliability with hybrid parameters can be obtained. The presented hybrid model can adequately consider all uncertainties affecting the fatigue reliability with hybrid uncertain parameters. A comparison among the presented hybrid model, non-probabilistic set-theoretic model and the conventional random model is made through two typical numerical examples. The results show that the presented hybrid model, which can ensure structural security, is effective and practical.

采用非概率方法对不同环境条件下的漂浮式风力机进行系泊损伤识别

This paper discusses the damage identification in the mooring line system of a floating wind turbine(FWT)exposed to various environmental loads.The proposed method incorporates a non-probabilistic method into artificial neural networks(ANNs).The non-probabilistic method is used to overcome the problem of uncertainties.For this purpose,the interval analysis method is used to calculate the lower and upper bounds of ANNs input data.This data contains some of the natural frequencies utilized to train two different ANNs and predict the output data which is the interval bounds of mooring line stiffness.Additionally,in order to reduce computational time and more importantly,identify damage in various conditions,the proposed method is trained using constant loads(CL)case(deterministic loads,including constant wind speed and airy wave model)and is tested using random loads(RL)case(including Kaimal wind model and JONSWAP wave theory).The superiority of this method is assessed by applying the deterministic method for damage identification.The results demonstrate that the proposed non-probabilistic method identifies the location and severity of damage more accurately compared to a deterministic one.This superiority is getting more remarkable as the difference in uncertainty levels between training and testing data is increasing.

NO-ARBITRAGE SYMMETRIES

The no-arbitrage property is widely accepted to be a centerpiece of modern financial mathematics and could be considered to be a financial law applicable to a large class of(idealized) markets.This paper addresses the following basic question:can one characterize the class of transformations that leave the law of no-arbitrage invariant?We provide a geometric formalization of this question in a non probabilistic setting of discrete time-the so-called trajectorial models.The paper then characterizes,in a local sense,the no-arbitrage symmetries and illustrates their meaning with a detailed example.Our context makes the result available to the stochastic setting as a special case.

Convex Set Theory for Reliability Assessment of Steel Beam with Bounded Uncertainty

Probabilistic reliability model established by insufficient data is inaccessible. The convex model was applied to model the uncertainties of variables. A new non-probabilistic reliability model was proposed based on the robustness of system to uncertainty. The non-probabilistic reliability model,the infinite norm model,and the probabilistic model were used to assess the reliability of a steel beam,respectively. The results show that the resistance is allowed to couple with the action effect in the non-probabilistic reliability model. Additionally,the non-probabilistic reliability model becomes the same accurate as probabilistic model with the increase of the bounded uncertain information. The model is decided by the available data and information.

Time Intervals of the Energy Emission in Quantum Systems Obtained from the Conservation Rule of the Electron Momentum

The paper presents a non-probabilistic approach to the time interval associated with the energy emission produced by the electron transition in a quantum system. The calculations were performed for the hydrogen atom and the electron particle in a one-dimensional potential box. In both cases, the rule of conservation of the electron momentum has been applied. The results, limited to the time intervals of transitions between two neighbouring quantum energy levels, occur to be much similar to those obtained earlier with the aid of the Joule-Lenz energy emission theory.