A reliability analysis method for complex-structured equipment under consideration of measurement conditions

Abstract

The present invention provides a reliability analysis method for complex structure equipment considering measurement conditions, belonging to the field of reliability analysis. First of all, for the reliability analysis method, a finite element model of the complex structure equipment is established to obtain the structural response. Secondly, measuring points are arranged on the considered complex structure. Then, an accurate conditional probability density function expression of the reliability index considering measurement conditions is constructed. Next, the generalized quasi-Monte Carlo method is used to solve the conditional probability density function of the reliability index considering measurement conditions. Finally, the reliability analysis of the structure is completed based on the conditional probability density function. Based on the law of probability conservation and combined with the generalized quasi-Monte Carlo method, the present invention derives and solves the conditional probability density function of the mechanical probability index under measurement conditions, completes the reliability analysis of complex structure equipment, has high accuracy and good calculation efficiency, and provides a more reliable basis for the analysis of structural safety and reliability.

Description

Technical Field

[0001] The present invention belongs to the field of reliability analysis and relates to a reliability analysis method for complex structure equipment under measurement conditions. Background Art

[0002] Structural reliability estimation plays a crucial role in engineering fields, especially in the design of structures such as buildings, aerospace, bridges, machinery, and other structures that need to withstand physical stresses. Structural reliability not only involves the safety of structures but also includes serviceability and durability, which are quantitatively analyzed through a probability index - structural reliability. During the design, construction, and use processes, due to various uncertainties, the assessment of structural reliability becomes complex and necessary.

[0003] Commonly used existing structural reliability calculation methods include the Monte Carlo method, surrogate model method, first-order second-moment method, etc. For the Chinese invention patent "A Time-Varying Reliability Analysis Method for Bearings Considering the Coupling Effect of Rolling Contact Fatigue Damage and Wear" (Patent No.: 202411171631.X), this invention proposes a novel physical model for bearing reliability analysis, establishes the coupling effect of RCF damage and wear of bearings, and significantly improves the calculation efficiency while more reasonably and accurately evaluating reliability. For the Chinese invention patent "Surrogate Model Method for Turbine Casing Reliability Analysis under Evidence Uncertainty" (Patent No.: 202311359650.0), this invention can robustly and rapidly calculate the failure probability bound of the turbine casing structure under evidence uncertainty, thereby reasonably evaluating the reliability level of the turbine casing. However, these inventions all study the reliability analysis methods of structures without measurement conditions. In many engineering problems, such as building health monitoring, engineers usually arrange measurement points on structures to measure structural responses. The reliability analysis method of structures under such measurement conditions has not been studied. When quantifying the random dynamic response of a structure with random parameters, if the measurement data conditions can be considered, it is obvious that the uncertainty quantification of the random response can be further refined, providing a more reliable basis for structural safety and reliability analysis. At the same time, in practical engineering problems, the measurement errors caused by human operation and instrument errors are also random.

[0004] Therefore, under measurement conditions with random errors, how to accurately and effectively evaluate the reliability of complex structure equipment is an extremely important issue. Summary of the Invention

[0005] Aiming at the problems existing in the prior art, the present invention provides a reliability analysis method for complex structure equipment under consideration of measurement conditions. Based on the law of probability conservation, the present invention derives an exact expression of the conditional probability density function of the mechanical probability index under measurement conditions. Compared with the analysis under unconditional conditions, the reliability analysis method for complex structure equipment under measurement conditions proposed by the present invention can obtain more accurate results, thus providing a more solid basis for the safety and reliability assessment of the structure. The present invention fills the blank of the research on the reliability analysis of structure equipment when there is measurement data.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0007] A reliability analysis method for complex structure equipment under consideration of measurement conditions. The reliability analysis method first establishes a finite element model of the complex structure equipment to obtain the structural response. Secondly, measuring points are arranged on the considered complex structure. Then, an exact conditional probability density function expression of the reliability index considering measurement conditions is constructed. Then, the generalized quasi-Monte Carlo method is used to solve the conditional probability density function of the reliability index considering measurement conditions. Finally, the reliability analysis of the structure is completed based on the conditional probability density function. It includes the following steps:

[0008] The first step is to establish a finite element model of the complex structure equipment to obtain the structural response; specifically:

[0009] Step 1.1, import the existing complex structure equipment model into the three-dimensional modeling software, or model the three-dimensional model of the complex structure equipment according to the engineering drawing to obtain the finite element model of the complex structure equipment.

[0010] Step 1.2, based on the finite element model of the complex structure equipment established in Step 1.1, determine the corresponding material properties, constraints and loads of the complex structure equipment according to the actual problem.

[0011] Step 1.3, according to the actual requirements, determine the uncertain random variables existing in the complex structure equipment , and the probability density function of the variables . When there are multiple uncertain variables, is a vector, and each component corresponds to a different uncertain variable.

[0012] Based on the above steps, the finite element model of the complex structure equipment, the load constraints and the distribution of the uncertain variables are obtained. When a given value is given, it can be substituted into the finite element model of the complex structure equipment to calculate the response of the complex structure equipment at moment .

[0013] Step 2: Arrange measurement points on the complex-structured equipment under consideration. Specifically:

[0014] In step 2.1, arrange measurement points at the key nodes of the complex-structured equipment, conduct measurements, and obtain the actual measurement values , and the measurement values accumulate over time. The key node positions can be determined based on experience or historical data judgment.

[0015] During the measurement process, there are always inevitable random measurement errors. Determine the measurement noise according to the range of the selected measuring instrument. The measurement noise is set as a stationary Gaussian process, denoted by , and its probability density is .

[0016] After the measured value and the measurement noise are known, the measurement equation is obtained:

[0017] (1)

[0018] where is the measurement function of the complex-structured equipment, which describes the relationship between the measurement value and the structural response under the condition of no measurement noise; represents the simplified writing of ; represents the time value.

[0019] Step 3: Construct an expression for the exact conditional probability density function of the reliability index considering the measurement conditions. Specifically:

[0020] In step 3.1, define the mechanical reliability index to be analyzed. This mechanical reliability index is a function of the response , and thus is also a function of the uncertainty random variable . Therefore, is expressed as .

[0021] In step 3.2, solve for the conditional probability density function of when the known measurement data condition is satisfied. is expressed as:

[0022] (2)

[0023] where is the joint probability density of ; is the measured value Probability density.

[0024] According to the law of probability conservation, the joint probability density of the system composed of the finite element model of the complex structure equipment and the measurement conditions is written as:

[0025] (3)

[0026] where is the Dirac function, whose function value is equal to zero at points other than zero, and the integral over the entire domain is equal to 1; represents the joint probability density function of all time-step measurement noises, for the definition of which, see Step 1.3, represents the value of the corresponding joint probability density function when all time-step measurement noises are exactly ;

[0027] By integrating over the random variable and the measurement noise the marginal probability density is obtained, as shown in Equation (4):

[0028] (4)

[0029] Then, integrating over gives:

[0030] (5)

[0031] At this time, taking and as the numerator and denominator respectively, substituting them into the probability density function expression (2), the conditional probability density function of is obtained under the condition of the known measurement data .

[0032] Fourthly, adopt the generalized quasi-Monte Carlo method to calculate the conditional probability density function of the reliability index considering the measurement conditions. Specifically:

[0033] Step 4.1, use the quasi-Monte Carlo method to solve this high-dimensional integral.

[0034] First, use the generalized low-discrepancy sampling method to sample the random variable in the considered complex structure equipment. The sampling follows the probability distribution of samples are obtained, and the corresponding point-set weights are obtained.

[0035] The obtained sample points are substituted into the finite element model of the complex structure equipment in Step 1.1, and the corresponding is calculated. Then, the probability density function of the measured value is calculated using the generalized quasi-Monte Carlo calculation format shown in formula (6): :

[0036] (6)

[0037] Step 4.2, is a high-dimensional integral containing non-smooth Dirac functions and cannot be directly calculated. It is necessary to first approximate the Dirac function using a Gaussian function and then solve it using the generalized quasi-Monte Carlo method. At this time, formula (4) is rewritten as:

[0038] (7)

[0039] where, when using the Gaussian function approximation, represents the smoothing factor.

[0040] Step 4.3, substitute the obtained and into formula (2). At this time, the solution is completed.

[0041] Fifth step, based on the conditional probability density function, complete the reliability analysis of the complex structure equipment, specifically as follows:

[0042] Step 5.1, according to the actual problem, determine the allowable threshold of when the complex structure equipment under consideration is in normal use.

[0043] Step 5.2, at this time, the reliability probability of the complex structure equipment is expressed as , and use to perform numerical integration to solve the reliability probability of the complex structure equipment, and complete the reliability analysis of the complex structure equipment.

[0044] Compared with the prior art, the present invention has the following beneficial effects:

[0045] (1) Based on the law of probability conservation, the present invention combines the generalized quasi-Monte Carlo method to deduce and solve the conditional probability density function of the mechanical probability index under measurement conditions, completes the reliability analysis of the complex structure equipment, and solves for the first time the reliability problem of the complex structure equipment in the presence of measurement conditions, filling the gap in the research on the reliability analysis of structural equipment in the presence of measurement data.

[0046] (2) Compared with the unconditional case, the results obtained by the reliability analysis method of complex structure equipment under measurement conditions given by the present invention are more accurate, providing a more reliable basis for structural safety and reliability analysis.

[0047] (3) The present invention uses the generalized quasi-Monte Carlo method to calculate high-dimensional integral problems. Compared with the traditional Monte Carlo method and quasi-Monte Carlo method, it requires fewer sample points, has higher precision, and better calculation efficiency. Description of the Drawings

[0048] Figure 1 It is a flowchart of a reliability analysis method for complex structure equipment considering measurement conditions.

[0049] Figure 2 It is a schematic diagram of the finite element model of the bridge and the measuring point positions in the embodiment. Specific Embodiments

[0050] The present invention will be described in detail below with reference to the drawings and specific embodiments. This embodiment is implemented on the premise of the technical solution of the present invention, and the reliability analysis of a certain bridge is completed, giving a detailed implementation method and specific operation process. However, the protection scope of the present invention is not limited to the following embodiments.

[0051] A reliability analysis method for complex structure equipment considering measurement conditions. In this embodiment, the complex structure equipment is a bridge, and the reliability analysis method includes the following steps:

[0052] The first step is to establish a finite element model of the bridge. Specifically:

[0053] Step 1.1: Import the existing bridge model into the three-dimensional modeling software to obtain the finite element model of the bridge.

[0054] Step 1.2: Based on the finite element model of the bridge established in Step 1.1, the material of the bridge frame is structural steel, and the elastic modulus is and the shear modulus is and the density is The thickness of the bridge deck is 0.5 m, the Poisson's ratio is 0.5, the material of the bridge deck is concrete, and the density is

[0055] Step 1.3: According to actual needs, determine the uncertain random variables existing in the bridge, and the probability density function of the variables. For this embodiment, is a 10-dimensional column vector, and each component corresponds to a different uncertain variable.

[0056] Based on the above steps, the finite element model of the bridge, the load constraints, and the distribution of uncertain variables are obtained. When given When the value is available, it can be substituted into the finite element model of the bridge to calculate the responses of points A and B on the bridge at the moment. .

[0057] Step 2: Arrange measurement points on the considered bridge. Specifically:

[0058] Step 2.1: Arrange measurement points at the key nodes of the bridge. The positions of the measurement points are as shown by the midpoints C1, C2, and C3, and measurements are taken to obtain the actual measured values Figure 2 . The measured values accumulate over time. The positions of the key nodes are determined based on empirical judgment.

[0059] Step 2.2: During the measurement process, there are always inevitable random measurement errors. According to the range of the selected measuring instrument, the measurement noise is determined. The measurement noise is assumed to be a stationary Gaussian process, denoted by , and its probability density is , where represents the covariance matrix of the measurement noise, and represents the number of iterative time steps.

[0060] Step 2.3: After the measured values and the measurement noise are known, the measurement equation is obtained:

[0061] (1)

[0062] where is the measurement function of the bridge, which describes the relationship between the measured value and the structural response under the condition of no measurement noise; represents in a simplified form; represents the time value.

[0063] Step 3: Construct an expression for the exact conditional probability density function of the reliability index considering the measurement conditions. Specifically:

[0064] Step 3.1: Define the mechanical reliability index to be analyzed. This mechanical reliability index is a function of the response , and thus is also a function of the uncertainty random variable . Therefore, is expressed as .

[0065] Step 3.2: Solve for the conditional probability density function of when the known measurement data condition is available. , is expressed as:

[0066] (2)

[0067] wherein, is the joint probability density; is the measured value of the probability density.

[0068] According to the law of probability conservation, the joint probability density of the system composed of the bridge finite element model and the measurement conditions is written as:

[0069] (3)

[0070] wherein, is the Dirac function, the function value is equal to zero at points other than the zero point, and the integral over the entire domain is equal to 1. represents the joint probability density function of all time-step measurement noises, the definition of which is shown in detail in Step 1.3, represents the value of the corresponding joint probability density function when all time-step measurement noises are exactly ;

[0071] By integrating the random variable and the measurement noise , the marginal probability density is obtained, as shown in formula (4):

[0072] (4)

[0073] Integrating again gives:

[0074] (5)

[0075] At this time, taking and as the numerator and denominator and substituting them into the probability density function expression (2), the conditional probability density function of under the condition of the known measurement data is obtained.

[0076] Fourthly, the generalized quasi-Monte Carlo method is adopted to calculate the conditional probability density function of the reliability index considering the measurement conditions. Specifically:

[0077] Step 4.1, use the quasi-Monte Carlo method to solve this high-dimensional integral.

[0078] First, the generalized low-discrepancy sampling method is used to sample the random variables in the bridge under consideration . The sampling follows the probability distribution, and sample points are obtained , and the corresponding point set weights are obtained.

[0079] The obtained sample points are substituted into the bridge finite element model in Step 1.1, and the corresponding is calculated. Then, the probability density function of the measured value is calculated using the generalized quasi-Monte Carlo calculation format shown in formula (6):

[0080] (6)

[0081] Step 4.2, is a high-dimensional integral containing a non-smooth Dirac function and cannot be calculated directly. It is necessary to first approximate the Dirac function using a Gaussian function and then solve it using the generalized quasi-Monte Carlo method. At this time, formula (4) is rewritten as:

[0082] (7)

[0083] where, when using the Gaussian function approximation, represents the smoothing factor.

[0084] Step 4.3, substitute the obtained and into formula (2). At this time the solution is completed.

[0085] Fifth, based on the conditional probability density function, the reliability analysis of the bridge is completed as follows:

[0086] Step 5.1, according to the actual problem, determine the allowable threshold of when the bridge under consideration is in normal use.

[0087] Step 5.2, at this time, the reliability probability of the bridge is expressed as . Use to perform numerical integration to solve it to obtain the reliability probability of the bridge, and complete the reliability analysis of the bridge.

[0088] The above-described embodiments merely represent the implementation modes of the present invention, but should not be construed as limiting the scope of the patent for the present invention. It should be noted that for those skilled in the art, without departing from the concept of the present invention, several modifications and improvements can still be made, and these all fall within the protection scope of the present invention.

Claims

1. A reliability analysis method for complex structure equipment under measurement conditions, characterized in that: The reliability analysis method comprises the following steps: The first step is to establish a finite element model of complex structural equipment and obtain the structural response; The second step is to arrange the measuring points on the complex structure equipment under consideration and obtain the measurement equation; The third step is to construct an accurate conditional probability density function expression of the reliability index when the measurement conditions are considered; the third step includes the following steps: Step 3.1, define the mechanical reliability index G to be analyzed, which is a function of the response u(θ, t) and also a function of the uncertainty random variable θ; therefore, G is expressed as G = F(θ, t); Step 3.2, solve the conditional probability density function ρ of G = F(θ, t) when the measured value y(t) is known G|y (G|y),ρ G|y (G|y) is expressed as: Among them, ρ G,y (G,y) is the joint probability density of G and y; ρ y (y) is the probability density of the measured value y(t); According to the law of conservation of probability, the joint probability density ρ of the system composed of the finite element model of complex structural equipment and the measurement conditions G,y,θ,v (G, y, θ, v) is written as: Among them, δ(·) is the Dirac function, which is equal to zero at all points except the zero point, and the integral over the entire domain is equal to 1; ρ v (·) represents the joint probability density function of the measurement noise at all time steps, ρ θ (θ) is defined as in step 1.3, ρ v (v) represents the value of the corresponding joint probability density function when the measurement noise of all time steps is exactly v; Represents measurement functions of complex structural equipment; By G,y,θ,v The marginal probability density ρ is obtained by integrating the random variable θ and the measurement noise v. G,y (G,y), as shown in formula (4): Then integrate G to get: At this time, ρ G,y (G,y) and ρ y Substitute (y) as the numerator and denominator into the probability density function expression (2) to obtain the conditional probability density function ρ of G = F (θ, t) when the measurement data condition y (t) is known G|y (G|y); The fourth step is to use the generalized quasi-Monte Carlo method to calculate the conditional probability density function of the reliability index when the measurement conditions are considered; the fourth step includes the following steps: Step 4.1: Use the quasi-Monte Carlo method to solve ρ y (y) this high-dimensional integral; First, the generalized low-bias sampling method is used to sample the random variable θ in the complex structure equipment under consideration; the sampling follows the probability distribution of θ and obtains n sample points [θ1, θ2, …, θ n ], and obtain the corresponding point set weight w = [w1,w2,…,w n ]; The obtained n sample points [θ1,θ2,…,θ n ] is substituted into the finite element model of complex structural equipment and the corresponding Then the probability density function ρ of the measured value y(t) is calculated using the generalized quasi-Monte Carlo calculation format shown in formula (6): y (y): Step 4.2, use the Gaussian function to approximate the Dirac function, and then use the generalized quasi-Monte Carlo method to solve it. At this time, formula (4) is rewritten as: Wherein, when the Gaussian function is used for approximation, σ represents the smoothing factor; Step 4.3, the obtained ρ y (y) and ρ G,y Substitute (G, y) into formula (2), then ρ G|y (G|y) is solved; The fifth step is to complete the reliability analysis of complex structural equipment based on the conditional probability density function.

2. A reliability analysis method for complex structure equipment under measurement conditions according to claim 1, characterized in that: The first step comprises the following steps: Step 1.1, importing an existing complex structure equipment model into a 3D modeling software, or modeling the 3D model of the complex structure equipment according to an engineering drawing, to obtain a finite element model of the complex structure equipment; Step 1.2, based on the finite element model of the complex structure equipment established in step 1.1, determine the corresponding material properties, constraints and loads of the complex structure equipment according to the actual problem; Step 1.3: According to actual needs, determine the uncertain random variables θ and the probability density function ρ of the variables in the complex structure equipment. θ (θ); when there are multiple uncertainty variables, θ is a vector, and each component corresponds to a different uncertainty variable; Based on the above steps, the finite element model of the complex structural equipment, the distribution of load constraints and uncertainty variables are obtained; when the value of θ is given, it is substituted into the finite element model of the complex structural equipment to calculate the response u(θ, t) of the complex structural equipment at time t.

3. The reliability analysis method for complex structure equipment under measurement conditions according to claim 1 is characterized in that: The second step comprises the following steps: Step 2.1, arrange measuring points at key nodes of complex structural equipment, perform measurements, obtain actual measurement values ​​y(t), and accumulate the measurement values ​​over time; Step 2.2: During the measurement process, determine the test noise according to the range of the measuring instrument; the measurement noise is set to a stationary Gaussian process, represented by v(t), and its probability density is ρ v (v); Step 2.3, after knowing the measured value y(t) and the measurement noise v(t), we get the measurement equation: Among them, h(u(θ,t)) is the measurement function of complex structural equipment, which describes the relationship between the measurement value and the structural response u(θ,t) under the condition of no measurement noise; It represents the simplified expression of h(u(θ,t)); t represents the time value.

4. The reliability analysis method for complex structure equipment under measurement conditions according to claim 1, characterized in that: The fifth step comprises the following steps: Step 5.1: According to the actual problem, determine the allowable threshold value G of G for the complex structure equipment under normal use. tol ; Step 5.2, the reliability probability of complex structure equipment is expressed as P{G≤G tol |y}, use The reliability probability of complex structure equipment is obtained by numerical integration and the reliability analysis of complex structure equipment is completed.

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