A time-varying reliability analysis method and system for a functionally graded material cylinder

Abstract

The present invention relates to the field of time-varying reliability technology, and in particular to a method and system for analyzing the time-varying reliability of a functionally gradient material cylinder. The method comprises: defining random variables, random processes, and a time domain for the functionally gradient material cylinder; discretizing the random processes and time domain to obtain a number of time nodes, as well as an instantaneous functional function and independent random variables at each time node; performing dimensionality reduction processing on the instantaneous functional function based on a single-variable dimensionality reduction method, and calculating the statistical moments of the instantaneous functional function; calculating the probability density function of the instantaneous functional function based on the statistical moments, and then calculating the instantaneous failure probability; and jointly combining the instantaneous failure probabilities of all time nodes to obtain a failure probability curve, thereby completing the time-varying reliability analysis of the functionally gradient material cylinder. The present invention combines random process discretization with single-variable dimensionality reduction to calculate failure probabilities, achieving high efficiency and accuracy.

Description

Technical Field

[0001] The present invention relates to the technical field of time-varying reliability, and in particular to a time-varying reliability analysis method and system for a functionally gradient material cylinder. Background Art

[0002] Cylindrical structures are widely used in engineering structures, and studying their stability under various loads is of great significance. Cylindrical structures are often used as supporting structures in buildings such as bridges, buildings, and stadiums. They can not only withstand enormous pressure but also provide excellent stability. Cylindrical storage vessels, such as oil tanks, water towers, and gas tanks, are ideal for storing liquids and gases due to their uniform structure, effectively distributing internal and external pressure. Rockets and spacecraft often adopt a cylindrical design to reduce air resistance, increase flight stability, and improve fuel efficiency. Cylindrical structures, due to their excellent physical properties and broad application prospects, have occupied a vital position in modern engineering and technology. Furthermore, due to the increasingly stringent requirements for structural performance in extreme environments and high-speed industries, cylindrical structures manufactured from functionally graded materials have attracted widespread attention in recent years. To ensure the safety and stability of functionally graded cylindrical structures in practical applications, reliability analysis of these structures is an essential step in the research, development, and manufacturing process.

[0003] The goal of reliability analysis is to calculate the probability that a structure will achieve its intended function while accounting for various uncertainties, such as material properties, manufacturing, loading conditions, and the environment. Traditional reliability analysis methods assume that uncertainties are time-independent and can only assess instantaneous reliability. However, the performance of real-world engineering structures is also affected by time-varying or dynamic sources of uncertainty, such as time-varying loads or material degradation. Ignoring time parameters can lead to inaccurate reliability assessments. Time-varying reliability analysis is a reliability assessment method based on the entire life cycle or a specific time interval. Assessing the probability within the time interval of interest requires repeated instantaneous reliability assessments at each time point. In real-world engineering, limit states are generally difficult to assess. Furthermore, considering time, the limit state response will be a complex, time-varying random variable or process. Implementing time-varying reliability analysis requires numerous evaluations of time-varying limit state functions, which results in a significant computational burden. Therefore, achieving efficient and accurate time-varying reliability analysis is a challenge.

[0004] Existing calculation methods for time-varying reliability problems can be roughly divided into three categories. The first is the heterogeneity method, which calculates the time-varying failure probability by integrating the heterogeneity (the ratio of the probability of a limit state event occurring to the time). This method can more accurately reflect the reliability variation of a system on a time scale, but requires a specific performance function and is computationally very complex. The second is the extreme value method, which calculates the distribution parameters of the extreme values by counting the number of times the system response extreme value exceeds the limit state within a given time interval. This transforms the time-varying reliability analysis problem into a traditional time-invariant (static) reliability analysis problem, requiring a large number of simulation runs and incurring significant computational costs. The third is the response surrogate method, which replaces the actual time-varying limit state function with a surrogate model to improve efficiency. Despite advances over the past few decades, these methods still have some shortcomings and limitations. The heterogeneity method can produce inaccurate results for problems with nonlinear and multimodal performance functions. Due to model assumptions, the empirical model used to calculate the failure probability may be inaccurate in some cases. The sample-based extreme value method, due to its high computational complexity, may encounter bottlenecks in practical applications, while the surrogate-based extreme value method is still in its developmental stage in terms of ensuring model credibility. Therefore, a time-varying reliability analysis method and system for functionally graded material cylinders are proposed. Summary of the Invention

[0005] The purpose of the present invention is to provide a time-varying reliability analysis method and system for functionally graded material cylinders, which combines random process discretization and single variable dimensionality reduction method to calculate failure probability with high efficiency and accuracy.

[0006] To achieve the above object, the present invention provides the following solutions:

[0007] A time-varying reliability analysis method for a functionally gradient material cylinder, comprising:

[0008] Defining a random variable, a random process, and a time domain of a functionally gradient material cylinder, wherein the random variable is a material parameter of the functionally gradient material cylinder, the random process is a time-varying load on the functionally gradient material cylinder, and the time domain is a preset time interval;

[0009] Discretizing the random process and the time domain to obtain a number of time nodes, as well as an instantaneous functional function and an independent random variable at each time node;

[0010] Performing dimensionality reduction processing on the instantaneous performance function based on a single variable dimensionality reduction method, and calculating a statistical moment of the instantaneous performance function;

[0011] Calculating a probability density function of the instantaneous functional function based on the statistical moment, and then calculating the instantaneous failure probability;

[0012] The instantaneous failure probabilities of all time nodes are combined to obtain a failure probability curve, thereby completing the time-varying reliability analysis of the functionally gradient material cylinder.

[0013] Optionally, the random variable is a material parameter of the functionally gradient material cylinder, the random process is a load on the functionally gradient material cylinder that changes with time, and the time domain is a preset time interval.

[0014] Optionally, after performing dimensionality reduction processing on the instantaneous performance function based on a single variable dimensionality reduction method, a number of one-dimensional integrals are obtained, and the reduced dimensionality instantaneous performance function is obtained by solving the sum of the number of one-dimensional integrals. The reduced dimensionality instantaneous performance function is:

[0015]

[0016] Among them, g(X) is the instantaneous function after dimensionality reduction, μ is the mean of the random variable, g(μ1,...,x i ,...μ n ) represents the random variable x i is a univariate function of , where n is the number of random variables.

[0017] Optionally, the method for calculating the statistical moment of the instantaneous functional function is:

[0018]

[0019] Where l represents the order of the statistical moment, represents permutation and combination calculations, and E[·] represents the moment of ·.

[0020] Optionally, the method for calculating the probability density function of the instantaneous functional function based on the statistical moment is:

[0021] max H[g(X)]=-∫ R p[g(X)]lnp[g(X)]dX

[0022] st∫ R [g(X)] j p[g(X)]dX=μ j j=0,1,…,m

[0023] Among them, max H[g(X)] is the maximum entropy of the instantaneous performance function, p[g(X)] is the probability density function of the instantaneous performance function, μ j is the jth order statistical moment, and m is the highest order of the statistical moment taken.

[0024] Optionally, the method for calculating the instantaneous failure probability is:

[0025]

[0026] Among them, P f is the instantaneous failure probability, and Z is an independent random variable.

[0027] To further achieve the above-mentioned object, the present invention also provides a time-varying reliability analysis system for a functionally gradient material cylinder, comprising: an information definition module, an information processing module, and an information analysis module;

[0028] The information definition module is used to define the dynamic information of the functionally gradient material cylinder;

[0029] The information processing module is used to discretize and reduce the dimensionality of the dynamic information and calculate the instantaneous failure probability of the functionally gradient material cylinder at each time node;

[0030] The information analysis module is used to combine the instantaneous failure probabilities of all time nodes to obtain a failure probability curve.

[0031] Optionally, the dynamic information includes random variables, random processes, and time domains, wherein the random variables are material parameters of the functionally gradient material cylinder, the random processes are loads on the functionally gradient material cylinder that change with time, and the time domain is a preset time interval.

[0032] Optionally, the information processing module includes: a discretization unit, a dimension reduction unit, and a calculation unit;

[0033] The discretization unit is used to discretize the random process and the time domain to obtain a number of time nodes, as well as the instantaneous functional function and independent random variables of each time node;

[0034] The dimension reduction unit is used to perform dimension reduction processing on the instantaneous functional function to obtain a one-dimensional function;

[0035] The calculation unit is configured to calculate a statistical moment of the instantaneous functional function based on the one-dimensional function; and calculate a probability density function of the instantaneous functional function based on the statistical moment, thereby calculating an instantaneous failure probability.

[0036] The beneficial effects of the present invention are:

[0037] To address the computationally intensive and highly nonlinear nature of time-varying reliability analysis for functionally graded cylindrical structures, this paper proposes a time-varying reliability analysis method based on dimensionality reduction. By discretizing the random process, the time-varying functional function is converted into multiple instantaneous functional functions at different moments. Since random processes require additional equivalent random variables to represent them, to avoid the significant computational cost of solving the reliability problem, a single-variable dimensionality reduction method is used to calculate the probability density curves of all instantaneous functions and obtain the failure probability at each moment. This method significantly improves efficiency over traditional methods and also maintains high computational accuracy for nonlinear and multimodal functional functions. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0039] Figure 1 This is a flow chart of a time-varying reliability analysis method for a functionally graded material cylinder according to an embodiment of the present invention;

[0040] Figure 2 This is a schematic diagram of a functionally graded cylindrical structure according to an embodiment of the present invention;

[0041] Figure 3 4 is a comparison chart of failure probability curves of the method according to an embodiment of the present invention and MCS. DETAILED DESCRIPTION

[0042] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0043] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0044] This embodiment provides a time-varying reliability analysis method for a functionally gradient material cylinder. Figure 1 Shown, including:

[0045] Define the random variables, random processes, and time domain of the functionally gradient material cylinder, where the random variables are the material parameters of the functionally gradient material cylinder, the random processes are the time-varying loads on the functionally gradient material cylinder, and the time domain is a preset time interval.

[0046] Discretize the random process and time domain to obtain several time nodes, as well as the instantaneous functional function and independent random variables of each time node;

[0047] The instantaneous functional function is reduced in dimension based on the single variable dimensionality reduction method, and the statistical moment of the instantaneous functional function is calculated;

[0048] Calculate the probability density function of the instantaneous functional function based on the statistical moment, and then calculate the instantaneous failure probability;

[0049] The instantaneous failure probabilities of all time nodes are combined to obtain the failure probability curve and complete the time-varying reliability analysis of the functionally graded material cylinder.

[0050] Specifically, by discretizing the random process, the time-varying performance function is converted into multiple instantaneous performance functions at different moments. Since random processes require additional equivalent random variables to represent them, and to avoid the significant computational cost of reliability solutions, a single-variable dimensionality reduction method is used to calculate the probability density curves of all instantaneous functions and obtain the failure probability at each moment. This method not only significantly improves efficiency over traditional methods but also maintains high computational accuracy for nonlinear and multimodal performance functions.

[0051] The following is Figure 2 Taking the functionally gradient material cylindrical structure shown in FIG. 1 as an example, a time-varying reliability analysis method for a functionally gradient material cylinder proposed in this embodiment is described in detail, including the following steps:

[0052] Step 1: Set random variables;

[0053] The material parameters of functionally graded materials are used as random variables X{x i}, including the elastic modulus, Poisson's ratio, density, etc. The form of material parameters is:

[0054] E=E1V1+E2V2

[0055] ν=ν1V1+ν2V2

[0056] ρ=ρ1V1+ρ2V2

[0057] Where E, ν, ρ, and V are Young's modulus, Poisson's ratio, density, and volume fraction, respectively. Subscripts 1 and 2 denote the two materials, respectively.

[0058] Step 2: Set boundary conditions, including random process and time domain;

[0059] The time-varying load is regarded as the random process Y(t), and the distribution form and autocorrelation function in the time domain T are given.

[0060] The functionally graded material cylindrical structure is subjected to a lateral force Y(t) that varies with time. The distribution of the random parameters is shown in Table 1.

[0061] Table 1

[0062]

[0063] The autocorrelation function of T(t) is expressed as:

[0064] ρ(t1,t2)=exp[-(t2-t1) 2 ]

[0065] The random process is expressed as:

[0066]

[0067] Where p is the number of dominant eigenvalues and is less than N t ,λ h and Φ h are the hth eigenvalue and eigenvector of the covariance matrix, respectively, and the (k, l)th element is σ y (t k )σ y (t l )σ y (t k ,t l ). ρ Y (t) is given by [σ Y (t)σ Y (t1)σ Y (t,t1),...,σ Y (t)σ Y (t n )σ Y (t,t n )] T Compute the time-varying covariance vector Z. h is the hth independent standard normal random variable.

[0068] Step 3: Discretize the random process and time domain to obtain several time nodes, as well as the instantaneous functional function and independent random variables of each time node;

[0069] In the time interval [t0,t e ] uniform selection within N t A moment in time, N t The number of N determines the accuracy of the discretization of the random process.t The larger the value, the more accurately the dependencies between different instantaneous functional functions are captured.

[0070] According to the random process obtained by discretization, the dynamic calculation function g(x, Y(t), t) of the functionally graded material cylinder is transformed into g={g(x, Z, t), t k ∈[t0,t e ],k=1,...,N t}, where t k The instantaneous performance function at g(x, Z, t) is correlated with the performance functions at other moments. The dependencies between different instantaneous performance functions can be implicitly captured in the discretization of Y(t). Although Y(t) is represented by a set of independent random variables Z at each instant in time, the coefficients of Z are extracted from the covariance matrix, whose elements contain information about Y(t) at different instants in time. The coefficients of Z remain correlated across the discrete g(x, Z, t).

[0071] The dynamic response of the functionally graded material cylinder, i.e., the functional function, is calculated based on the solution of the dynamic control equation. In this embodiment, the dynamic control equation is solved using the differential orthogonal method and the Newmark method. The dynamic equation can be expressed as:

[0072]

[0073] Where σ, ρ and u are stress, density and displacement respectively, and the relationship between strain and displacement is expressed as:

[0074]

[0075] Where ε is the strain, the physical equation can be expressed as

[0076]

[0077] Substituting the physical equation and strain-displacement relationship into the control equation, we get

[0078]

[0079] The initial boundary conditions are expressed as follows:

[0080]

[0081] The stress-based performance function is defined as

[0082]

[0083] Where r represents the radial axis in the cylindrical coordinate system, and the displacement u at any coordinate is obtained by the differential orthogonal method and the Newmark method.

[0084] Step 4: Reduce the dimension of the instantaneous performance function based on the single variable dimensionality reduction method and calculate the statistical moment of the instantaneous performance function;

[0085] The single variable dimensionality reduction method is used to calculate the discrete g(x, Z, t). The main idea of the single variable dimensionality reduction method is to convert the multidimensional integral into several one-dimensional integrals, approximate the original function by solving the sum of these one-dimensional integrals, and obtain the instantaneous functional function after dimensionality reduction.

[0086] The approximation of the instantaneous performance function g(X) after dimensionality reduction takes the following form:

[0087]

[0088] Where μ is the mean of the random variable g(μ1,...,x i ,...μ n ) represents the random variable x i The calculation formula of the statistical moment of the instantaneous function after dimension reduction can be expressed as:

[0089]

[0090] Where l represents the order of the statistical moment, represents the random variable x i The probability density function of . The integral formula is calculated by Gaussian integral, so the integral expression for moment is:

[0091]

[0092] Step 5: Calculate the probability density function of the instantaneous functional function based on the statistical moment;

[0093] The maximum entropy method is used to calculate the probability density function (PDF) of the performance function. According to the information theory principle, entropy is defined as the expression of the probability density function:

[0094] H=-∫p[g(X)]lnp[g(X)]dX

[0095] Under different constraints, the expression of maximizing entropy can be used to obtain the distribution expression of minimizing prior information. For continuous variables, the expression of the maximum entropy method is:

[0096] max H[g(X)]=-∫ R p[g(X)]lnp[g(X)]dX

[0097] st∫ R [g(X)] j p[g(X)]dX=μ j j=0,1,…,m

[0098] This optimization problem can be solved with the help of Lagrange function and Lagrange multiplier, namely:

[0099]

[0100] Among them, there are (N+1) Lagrange multipliers λ=[λ0,λ1,...,λ n ],φ n (x) takes the form of φ n [g(X)]=[g(X)] n p[g(X)] is the function of the maximum entropy probability density required, μ j is the jth-order statistical moment, m is the highest order of the statistical moment, usually 4th order. The constraint problem can be solved iteratively using the standard Newton method.

[0101] Step 6: Calculate the instantaneous failure probability;

[0102] Substitute the performance function probability density function (PDF) into the following formula:

[0103]

[0104] The failure probability of the instantaneous limit state function can be obtained.

[0105] Step 7: Combine the instantaneous failure probabilities of all time nodes to obtain the failure probability curve and complete the time-varying reliability analysis of the functionally graded material cylinder.

[0106] In order to further optimize the above technical solution, the present invention also provides a time-varying reliability analysis system for a functionally gradient material cylinder, comprising: an information definition module, an information processing module, and an information analysis module;

[0107] Information definition module, used to define the dynamic information of the functionally gradient material cylinder;

[0108] An information processing module, configured to discretize and reduce the dimensionality of the dynamic information and calculate the instantaneous failure probability of the functionally gradient material cylinder at each time node;

[0109] The information analysis module is used to combine the instantaneous failure probabilities of all time nodes and obtain the failure probability curve.

[0110] Specifically, the dynamic information includes random variables, random processes, and time domains, wherein the random variables are material parameters of the functionally gradient material cylinder, the random processes are loads on the functionally gradient material cylinder that change with time, and the time domain is a preset time interval.

[0111] Specifically, the information processing module includes: a discretization unit, a dimensionality reduction unit, and a calculation unit;

[0112] The discretization unit is used to discretize the random process and the time domain to obtain a number of time nodes, as well as the instantaneous functional function and independent random variables of each time node;

[0113] The dimensionality reduction unit is used to perform dimensionality reduction processing on the instantaneous functional function to obtain a one-dimensional function; the calculation unit is used to calculate the statistical moment of the instantaneous functional function based on the one-dimensional function; and calculate the probability density function of the instantaneous functional function based on the statistical moment, and then calculate the instantaneous failure probability.

[0114] Comparing the method proposed in this embodiment with the results of MCS, Table 2 shows the error of failure probability. Figure 3 The comparison of failure probability curves is given. It can be seen that the maximum error with the MCS result is 3.76%, indicating that this method has high accuracy in solving the time-varying reliability analysis problem of functionally graded material cylinder dynamics. In terms of computational efficiency, the number of times MCS calls the functional function is 21×10 6 times, while the number of function calls of this method is only 21×313=6573 times, which greatly reduces the calculation cost.

[0115] Table 2

[0116]

[0117] The embodiments described above are merely descriptions of preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Without departing from the spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by persons skilled in the art should fall within the scope of protection defined by the claims of the present invention.

Claims

1. A time-varying reliability analysis method for a functionally graded material cylinder, characterized in that: include: Defining a random variable, a random process, and a time domain of a functionally gradient material cylinder, wherein the random variable is a material parameter of the functionally gradient material cylinder, the random process is a time-varying load on the functionally gradient material cylinder, and the time domain is a preset time interval; Discretizing the random process and the time domain to obtain a number of time nodes, as well as an instantaneous functional function and an independent random variable at each time node; Performing dimensionality reduction processing on the instantaneous performance function based on a single variable dimensionality reduction method, and calculating a statistical moment of the instantaneous performance function; Calculating a probability density function of the instantaneous functional function based on the statistical moment, and then calculating the instantaneous failure probability; The instantaneous failure probabilities of all time nodes are combined to obtain a failure probability curve, thereby completing the time-varying reliability analysis of the functionally gradient material cylinder; The material parameters are: E=E1V1+E2V2 ν=ν1V1+ν2V2 ρ=ρ1V1+ρ2V2 Where E, ν, ρ, and V are Young's modulus, Poisson's ratio, density, and volume fraction, respectively, and subscripts 1 and 2 represent two materials, respectively; The random process is: Where p is the number of dominant eigenvalues and is less than N t ,λ h and Φ h are the hth eigenvalue and eigenvector of the covariance matrix, respectively, and the (k, l)th element is σ y (t k )σ y (t l )σ y (t k ,t l ), ρ Y (t) is given by [σ Y (t)σ Y (t1)σ Y (t,t1),...,σ Y (t)σ Y (t n )σ Y (t,t n )] T Calculate the time-varying covariance vector, Z h is the hth independent standard normal random variable; Discretize the random process and time domain to obtain several time nodes, as well as the instantaneous functional function and independent random variables of each time node, including: In the time interval [t0,t e ] uniform selection within N t The time instants are discretized and the dynamic calculation function g(x, Y(t), t) of the functional gradient material cylinder is transformed into g={g(x, Z, t), t k ∈[t0,t e ],k=1,...,N t }, where t k The instantaneous performance function at , i.e. g(x, Z, t), is related to the performance functions at other moments; The dynamic response of the functionally graded material cylinder, i.e., the functional function, is calculated based on the solution of the dynamic control equation. The dynamic control equation is solved using the differential orthogonal method and the Newmark method. The dynamic equation is expressed as: Among them, σ, ρ and u are stress, density and displacement respectively, and the relationship between strain and displacement is expressed as: Where ε is the strain, and the physical equation can be expressed as: Substituting the physical equation and strain-displacement relationship into the control equation, we get: The initial boundary conditions are expressed as: The stress-based performance function is defined as: Where r represents the radial axis in the cylindrical coordinate system, and the displacement u at any coordinate is obtained by the differential orthogonal method and the Newmark method; After the instantaneous functional function is reduced in dimension based on the single variable dimensionality reduction method, a number of one-dimensional integrals are obtained, and the reduced instantaneous functional function is obtained by solving the sum of the several one-dimensional integrals. The reduced instantaneous functional function is: Among them, g(X) is the instantaneous function after dimensionality reduction, μ is the mean of the random variable, g(μ1,...,x i ,...μ n ) represents the random variable x i A univariate function of , n represents the number of random variables; The method for calculating the statistical moment of the instantaneous functional function is: Among them, l represents the order of statistical moment, represents permutation and combination calculations, and E[·] represents moment calculation of ·; The method for calculating the probability density function of the instantaneous functional function based on the statistical moment is: Among them, max H[g(X)] is the maximum entropy of the instantaneous performance function, p[g(X)] is the probability density function of the instantaneous performance function, μ j is the jth order statistical moment, and m is the highest order of the statistical moment taken; The method for calculating the instantaneous failure probability is: Among them, P f is the instantaneous failure probability, and Z is an independent random variable.

2. A time-varying reliability analysis system for a functionally gradient material cylinder, used to implement the time-varying reliability analysis method for a functionally gradient material cylinder according to claim 1, characterized in that: include: Information definition module, information processing module, information analysis module; The information definition module is used to define the dynamic information of the functionally gradient material cylinder; The information processing module is used to discretize and reduce the dimensionality of the dynamic information and calculate the instantaneous failure probability of the functionally gradient material cylinder at each time node; The information analysis module is used to combine the instantaneous failure probabilities of all time nodes to obtain a failure probability curve.

3. The time-varying reliability analysis system for a functionally gradient material cylinder according to claim 2, characterized in that: The dynamic information includes random variables, random processes, and a time domain, wherein the random variables are material parameters of the functionally gradient material cylinder, the random processes are loads on the functionally gradient material cylinder that change with time, and the time domain is a preset time interval.

4. The time-varying reliability analysis system for a functionally gradient material cylinder according to claim 3, characterized in that: The information processing module includes: a discretization unit, a dimension reduction unit, and a calculation unit; The discretization unit is used to discretize the random process and the time domain to obtain a number of time nodes, as well as the instantaneous functional function and independent random variables of each time node; The dimension reduction unit is used to perform dimension reduction processing on the instantaneous functional function to obtain a one-dimensional function; The calculation unit is configured to calculate a statistical moment of the instantaneous functional function based on the one-dimensional function; and calculate a probability density function of the instantaneous functional function based on the statistical moment, thereby calculating an instantaneous failure probability.

Images