Time-Dependent Reliability Analysis Using the Total Probability Theorem

2015 ◽  
Vol 137 (3) ◽  
Author(s):  
Zissimos P. Mourelatos ◽  
Monica Majcher ◽  
Vijitashwa Pandey ◽  
Igor Baseski
Keyword(s):  
Reliability Analysis ◽  
Limit State ◽  
Random Processes ◽  
Time Dependent ◽  
Normal Space ◽  
Total Probability ◽  
Conditional Probabilities ◽  
Reliability Method ◽  
Standard Normal ◽  
Total Probability Theorem

A new reliability analysis method is proposed for time-dependent problems with explicit in time limit-state functions of input random variables and input random processes using the total probability theorem and the concept of composite limit state. The input random processes are assumed Gaussian. They are expressed in terms of standard normal variables using a spectral decomposition method. The total probability theorem is employed to calculate the time-dependent probability of failure using time-dependent conditional probabilities which are computed accurately and efficiently in the standard normal space using the first-order reliability method (FORM) and a composite limit state of linear instantaneous limit states. If the dimensionality of the total probability theorem integral is small, we can easily calculate it using Gauss quadrature numerical integration. Otherwise, simple Monte Carlo simulation (MCS) or adaptive importance sampling are used based on a Kriging metamodel of the conditional probabilities. An example from the literature on the design of a hydrokinetic turbine blade under time-dependent river flow load demonstrates all developments.

Time-Dependent Reliability Analysis Using the Total Probability Theorem

2014 ◽  
Author(s):  
Zissimos P. Mourelatos ◽  
Monica Majcher ◽  
Vijitashwa Pandey ◽  
Igor Baseski
Keyword(s):  
Reliability Analysis ◽  
Conditional Probability ◽  
Limit State ◽  
Random Variables ◽  
Random Processes ◽  
Time Dependent ◽  
Normal Space ◽  
Total Probability ◽  
Standard Normal ◽  
Total Probability Theorem

A new reliability analysis method is proposed for time-dependent problems with limit-state functions of input random variables, input random processes and explicit in time using the total probability theorem and the concept of composite limit state. The input random processes are assumed Gaussian. They are expressed in terms of standard normal variables using a spectral decomposition method. The total probability theorem is employed to calculate the time-dependent probability of failure using a time-dependent conditional probability which is computed accurately and efficiently in the standard normal space using FORM and a composite limit state of linear instantaneous limit states. If the dimensionality of the total probability theorem integral (equal to the number of input random variables) is small, we can easily calculate it using Gauss quadrature numerical integration. Otherwise, simple Monte Carlo simulation or adaptive importance sampling is used based on a pre-built Kriging metamodel of the conditional probability. An example from the literature on the design of a hydrokinetic turbine blade under time-dependent river flow load demonstrates all developments.

Most-Probable-Point-Locus Reliability Method in Standard Normal Space

1991 ◽  
Author(s):  
M. R. Khalessi ◽  
Y.-T. Wu ◽  
T. Y. Torng
Keyword(s):  
Reliability Analysis ◽  
Structural Reliability ◽  
Search Algorithm ◽  
Limit State ◽  
Iteration Procedure ◽  
Normal Space ◽  
State Function ◽  
Most Probable Point ◽  
Reliability Method ◽  
Standard Normal

Abstract This paper describes a new structural reliability analysis iteration procedure based on the concept of most probable point locus (MPPL). Using a new quadratic search algorithm, the proposed procedure examines the global behavior of the limit-state function, g, along the MPPL in the standard normal space in search of the most probable point (MPP) on the g = o surface, and identifies unusual conditions such as multiple MPPs. During the iteration procedure, the generated information is updated after each sensitivity analysis. This action helps the analyst to minimize the number of computer runs and determine the next step. By adopting two efficient convergence criteria, the proposed procedure is demonstrated to be significantly more efficient than the commonly used reliability analysis procedures, and is suitable to be integrated with existing general-purpose finite element computer programs for nondeterministic structural analysis.

Time-Dependent Reliability Analysis of Vibratory Systems With Random Parameters

2016 ◽  
Vol 138 (3) ◽  
Author(s):  
Zissimos P. Mourelatos ◽  
Monica Majcher ◽  
Vasileios Geroulas
Keyword(s):  
Large Scale ◽  
Probability Of Failure ◽  
Time Dependent ◽  
Total Probability ◽  
Conditional Probabilities ◽  
Random Parameters ◽  
Random Vibrations ◽  
Input Parameters ◽  
Total Probability Theorem ◽  
Dependent Probability

The field of random vibrations of large-scale systems with millions of degrees-of-freedom (DOF) is of significant importance in many engineering disciplines. In this paper, we propose a method to calculate the time-dependent reliability of linear vibratory systems with random parameters excited by nonstationary Gaussian processes. The approach combines principles of random vibrations, the total probability theorem, and recent advances in time-dependent reliability using an integral equation involving the upcrossing and joint upcrossing rates. A space-filling design, such as optimal symmetric Latin hypercube (OSLH) sampling, is first used to sample the input parameter space. For each design point, the corresponding conditional time-dependent probability of failure is calculated efficiently using random vibrations principles to obtain the statistics of the output process and an efficient numerical estimation of the upcrossing and joint upcrossing rates. A time-dependent metamodel is then created between the input parameters and the output conditional probabilities allowing us to estimate the conditional probabilities for any set of input parameters. The total probability theorem is finally applied to calculate the time-dependent probability of failure. The proposed method is demonstrated using a vibratory beam example.

Reliability Analysis and Random Vibration of Nonlinear Systems Using the Adjoint Method and Projected Differentiation

2020 ◽  
Vol 143 (6) ◽  
Author(s):  
Dimitrios Papadimitriou ◽  
Zissimos P. Mourelatos ◽  
Zhen Hu
Keyword(s):  
Reliability Analysis ◽  
Equations Of Motion ◽  
Random Variables ◽  
Random Processes ◽  
Time Dependent ◽  
Vibratory System ◽  
Standard Normal ◽  
Non Gaussian ◽  
Nonlinear Equations Of Motion ◽  
Adjoint Approach

Abstract This paper proposes a new methodology for time-dependent reliability and random vibrations of nonlinear vibratory systems using a combination of a time-dependent adjoint variable (AV) method and a projected differentiation (PD) method. The proposed approach is called AV-PD. The vibratory system is excited by stationary Gaussian or non-Gaussian input random processes. A Karhunen–Loeve (KL) expansion expresses each input random process in terms of standard normal random variables. The nonlinear equations of motion (EOM) are linearized using a Taylor expansion using the first-order derivatives of the output with respect to the input KL random variables. An adjoint approach obtains the output derivatives accurately and efficiently requiring the solution of as many sets of EOM as the number of outputs of interest, independently of the number of KL random variables. The proposed PD method then computes the autocorrelation function of each output process at an additional cost of solving as many sets of EOM as the number of outputs of interest, independently of the time horizon (simulation time). A time-dependent reliability analysis is finally performed using a KL expansion of the output processes and Monte Carlo simulation (MCS). The number of solutions of the EOM scales only with the number of output random processes which is commonly much smaller than the number of input KL random variables. The efficiency and accuracy of the proposed approach is demonstrated using a four degree-of-freedom (DOF) half-car vibratory problem.

Time-Dependent Reliability Analysis by a Sampling Approach to Extreme Values of Stochastic Processes

2012 ◽  
Author(s):  
Zhen Hu ◽  
Xiaoping Du
Keyword(s):  
Stochastic Processes ◽  
Reliability Analysis ◽  
Extreme Values ◽  
Limit State ◽  
Time Dependent ◽  
State Function ◽  
Limit State Function ◽  
Reliability Method ◽  
Time Invariant ◽  
State Functions

Maintaining high accuracy and efficiency is a challenging issue in time-dependent reliability analysis. In this work, an accurate and efficient method is proposed for limit-state functions with the following features: The limit-state function is implicit with respect to time, and its input contains stochastic processes; the stochastic processes include only general strength and stress variables, or the limit-state function is monotonic to these stochastic processes. The new method employs random sampling approaches to estimate the distributions of the extreme values of the stochastic processes. The extreme values are then used to replace the corresponding stochastic processes, and consequently the time-dependent reliability analysis is converted into its time-invariant counterpart. The commonly used time-invariant reliability method, the First Order Reliability Method, is then applied for the time-variant reliability analysis. The results show that the proposed method significantly improves the accuracy and efficiency of time-dependent reliability analysis.

Time-Dependent System Reliability Analysis With Second-Order Reliability Method

2020 ◽  
Vol 143 (3) ◽  
Author(s):  
Hao Wu ◽  
Zhangli Hu ◽  
Xiaoping Du
Keyword(s):  
Reliability Analysis ◽  
System Reliability ◽  
Limit State ◽  
Second Order ◽  
Time Dependent ◽  
Component Reliability ◽  
Reliability Method ◽  
Series Systems ◽  
State Functions ◽  
Time Dependent System

Abstract System reliability is quantified by the probability that a system performs its intended function in a period of time without failures. System reliability can be predicted if all the limit-state functions of the components of the system are available, and such a prediction is usually time consuming. This work develops a time-dependent system reliability method that is extended from the component time-dependent reliability method using the envelope method and second-order reliability method. The proposed method is efficient and is intended for series systems with limit-state functions whose input variables include random variables and time. The component reliability is estimated by the second-order component reliability method with an improve envelope approach, which produces a component reliability index. The covariance between component responses is estimated with the first-order approximations, which are available from the second-order approximations of the component reliability analysis. Then, the joint distribution of all the component responses is approximated by a multivariate normal distribution with its mean vector being component reliability indexes and covariance being those between component responses. The proposed method is demonstrated and evaluated by three examples.

Reliability Analysis and Random Vibration of Nonlinear Systems Using the Adjoint Method and Projected Differentiation

2020 ◽  
Author(s):  
Dimitrios Papadimitriou ◽  
Zissimos P. Mourelatos ◽  
Zhen Hu
Keyword(s):  
Reliability Analysis ◽  
Equations Of Motion ◽  
Random Variables ◽  
Random Processes ◽  
Time Dependent ◽  
Vibratory System ◽  
Standard Normal ◽  
Non Gaussian ◽  
Nonlinear Equations Of Motion ◽  
Adjoint Approach

Abstract This paper proposes a new methodology for time-dependent reliability and random vibrations of nonlinear vibratory systems using a combination of a time-dependent adjoint variable (AV) method and a projected differentiation (PD) method. The proposed approach is called AV-PD. The vibratory system is excited by stationary Gaussian or non-Gaussian input random processes. A Karhunen-Loeve (KL) expansion expresses each input random process in terms of standard normal random variables. The nonlinear equations of motion (EOM) are linearized using a Taylor expansion using the first-order derivatives of the output with respect to the input KL random variables. An adjoint approach obtains the output derivatives accurately and efficiently requiring the solution of as many sets of EOM as the number of outputs of interest, independently of the number of KL random variables. The proposed PD method then computes the autocorrelation function of each output process at an additional cost of solving as many sets of EOM as the number of outputs of interest, independently of the time horizon (simulation time). A time-dependent reliability analysis is finally performed using a KL expansion of the output processes and Monte Carlo Simulation (MCS). The number of solutions of the EOM scales only with the number of output random processes which is commonly much smaller than the number of input KL random variables. The efficiency and accuracy of the proposed approach is demonstrated using a four degree-of-freedom (DOF) half-car vibratory problem.

Time-Dependent System Reliability Analysis With Second Order Reliability Method

2020 ◽  
Author(s):  
Hao Wu ◽  
Xiaoping Du
Keyword(s):  
Reliability Analysis ◽  
System Reliability ◽  
Limit State ◽  
Second Order ◽  
Time Dependent ◽  
Component Reliability ◽  
Reliability Method ◽  
Series Systems ◽  
State Functions ◽  
Time Dependent System

Abstract System reliability is quantified by the probability that a system performs its intended function in a period of time without failure. System reliability can be predicted if all the limit-state functions of the components of the system are available, and such a prediction is usually time consuming. This work develops a time-dependent system reliability method that is extended from the component time-dependent reliability method that uses the envelop method and second order reliability method. The proposed method is efficient and is intended for series systems with limit-state functions whose input variables include random variables and time. The component reliability is estimated by the existing second order component reliability method, which produces component reliability indexes. The covariance between components responses are estimated with the first order approximations, which are available from the second order approximations of the component reliability analysis. Then the joint probability of all the component responses is approximated by a multivariate normal distribution with its mean vector being component reliability indexes and covariance being those between component responses. The proposed method is demonstrated and evaluated by three examples.

Time-Dependent Reliability Analysis of Vibratory Systems With Random Parameters

2015 ◽  
Author(s):  
Zissimos P. Mourelatos ◽  
Monica Majcher ◽  
Vasileios Geroulas
Keyword(s):  
Large Scale ◽  
Probability Of Failure ◽  
Time Dependent ◽  
Total Probability ◽  
Conditional Probabilities ◽  
Random Parameters ◽  
Random Vibrations ◽  
Input Parameters ◽  
Total Probability Theorem ◽  
Dependent Probability

The field of random vibrations of large-scale systems with millions of degrees of freedom is of significant importance in many engineering disciplines. In this paper, we propose a method to calculate the time-dependent reliability of linear vibratory systems with random parameters excited by non-stationary Gaussian processes. The approach combines principles of random vibrations, the total probability theorem and recent advances in time-dependent reliability using an integral equation involving the up-crossing and joint up-crossing rates. A space-filling design, such as optimal symmetric Latin hypercube sampling, is first used to sample the input parameter space. For each design point, the corresponding conditional time-dependent probability of failure is calculated efficiently using random vibrations principles to obtain the statistics of the output process and an efficient numerical estimation of the up-crossing and joint up-crossing rates. A time-dependent metamodel is then created between the input parameters and the output conditional probabilities allowing us to estimate the conditional probabilities for any set of input parameters. The total probability theorem is finally applied to calculate the time-dependent probability of failure. The proposed method is demonstrated using a vibratory beam example.

Reliability Analysis for Multidisciplinary Systems Involving Stationary Stochastic Processes

2015 ◽  
Author(s):  
Zhifu Zhu ◽  
Zhen Hu ◽  
Xiaoping Du
Keyword(s):  
Stochastic Processes ◽  
Reliability Analysis ◽  
Limit State ◽  
Time Dependent ◽  
State Function ◽  
Limit State Function ◽  
Current Time ◽  
Reliability Method ◽  
Stationary Stochastic Processes ◽  
Multidisciplinary System

The response of a component in a multidisciplinary system is affected by not only the discipline to which it belongs, but also by other disciplines of the system. If any components are subject to time-dependent uncertainties, responses of all the components and the system are also time dependent. Thus, time-dependent multidisciplinary reliability analysis is required. To extend the current time-dependent reliability analysis for a single component, this work develops a time-dependent multidisciplinary reliability method for components in a multidisciplinary system under stationary stochastic processes. The method modifies the First and Second Order Reliability Methods (FORM and SORM) so that the Multidisciplinary Analysis (MDA) is incorporated while approximating the limit-state function of the component under consideration. Then Monte Carlo simulation is used to calculate the reliability without calling the original limit-state function. Two examples are used to demonstrate and evaluate the proposed method.

Sign in / Sign up

Export Citation Format

Share Document