Random field data assimilation arithmetic method

Abstract

To provide an arithmetic method that takes into account the uncertainty caused by modeling errors, etc. when estimating an unknown variable such as physical properties that match observation data with data assimilation.SOLUTION: A data assimilation arithmetic method that searches for a physical property value that matches observation data in a vibrator as unknown variables is characterized by that the data assimilation arithmetic unit that comprises a sensitivity calculation unit and an optimal solution search unit, takes over the unknown variable that has been estimated to match the observation amount to a probability variable conversion unit, and performs arithmetic to convert the unknown variable into a probability density function that takes uncertainty into account.SELECTED DRAWING: Figure 1

Description

Detailed Description of the Invention [Technical field]

[0001] The present invention relates to a calculation method for estimating estimated physical property values ​​as probability density functions taking into account uncertainties caused by modeling errors and the like when estimating unknown variables, such as physical properties, that fit mainly observed data through data assimilation. [Background technology]

[0002] Data assimilation has been used to estimate the current internal stiffness of embankments from the results of measuring embankment deformation as shown in Non-Patent Document 1, and to estimate the internal state of bridges from bridge acceleration responses as shown in Non-Patent Document 2. When estimating the internal state of an object by data assimilation, the object is modeled using the finite element method, and the analytical variables used in the finite element method are set as unknown variables. The unknown variables are repeatedly updated to search for optimal unknown variables so that the error between the calculated values ​​and the measured values ​​using the finite element method is minimized.

[0003] If a sufficient number of measurement points cannot be installed in data assimilation, the unknown variables that fit the measurement results will vary. In addition, it is not easy to completely model an object using the finite element method. For example, if you want to obtain the stiffness distribution to find deteriorated areas within an object, you need to estimate variables such as the mass and damping distribution within the object in advance. However, it is difficult to obtain all variables other than stiffness, including mass and damping, in advance. In addition, if mass and damping are added as unknown variables, the number of unknown variables to be identified increases, and even more observation points are required. [Prior art documents] [Non-patent literature]

[0004] [Non-Patent Document 1] Makoto Suzuko and Kiyoshi Ishii: Estimation of spatial distribution of soil constants using extended Kalman filter, Journal of Japan Society of Civil Engineers, Vol. 406, 3-11, pp. 71-78, 1989. [Non-Patent Document 2] Hirohiro Matsuoka, Kiyoyuki Kaito, Munemasa Tokunaga, Tsutomu Watanabe, Masamichi Sogabe: Estimation of Displacement Response Based on Bridge Acceleration Response During Train Passage Using Sequential Data Assimilation, Journal of Japan Society of Civil Engineers, Proceedings of the Society of Civil Engineers, Vol. 69, No. 3, 2013. [Non-Patent Document 3] Yanev, Bojidar. Bridge Management. John Wiley & Sons., 2007. Summary of the Invention DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0005] The present invention has been made in consideration of the uncertainty of the calculation results of the internal state of an object, and has an object to provide a calculation method capable of estimating the internal state of a vibrating body by giving a range of values ​​as a probability density function.

[0006] In order to solve the above-mentioned problems and estimate the internal physical properties of the vibrating body, the present invention provides a calculation method for calculating the physical properties as a probability density function that takes into account the magnitude of the variance of the estimated value, rather than as a fixed value as in the past. By using this method, it is possible to calculate a probability density function that takes into account the statistical variance of the physical property distribution inside the vibrating body. Therefore, if the limit physical property value that satisfies the usability of the vibrating body is set as a reference value, it is possible to calculate the probability that the reference value will be exceeded.

[0007] The present invention can be used in situations where it is difficult to completely model the object or it is not possible to install a sufficient number of measurement points, and the internal physical property distribution of a vibrating body needs to be estimated with a range of values.

[0008] The random field data assimilation calculation method of the present invention is characterized by having a data assimilation calculation unit consisting of a sensitivity calculation unit that determines the sensitivity of unknown variables to the Lagrangian function and an optimal solution search unit that estimates unknown variables such as physical properties and internal states that match the observation results using the sensitivity of the unknown variables, and a random variable transformation unit that determines the statistical variation of the unknown variables from the second derivative of the Lagrangian function and determines the probability density function of the unknown variables.

[0009] Here, the Lagrangian function is a function obtained by adding the squared sum of the observation error, which is the difference between the measurement results and the estimated results from numerical analysis, and the error in prior information such as the spatial distribution of unknown variables, to the physical processes such as the equation of motion as constraints.By searching for the unknown variables that minimize the Lagrangian function, it is possible to update the unknown variables so that the observation error and prior information error are minimized while satisfying the physical processes, thereby achieving data assimilation of the unknown variables. [Brief description of the drawings]

[0010] [Figure 1] 1 is an overall calculation flow of a stochastic field data assimilation calculation method according to the present invention. [Diagram 2] This is the calculation flow of data assimilation that estimates physical properties to match the observed quantities. [Diagram 3] 13 is a calculation flow of a random variable conversion unit that converts a physical property estimate value into a probability density function and calculates the hyperprobability of a reference value according to the present invention. [Figure 4] A conceptual diagram showing the concept of uncertainty and exceedance probability of data-assimilated physical properties. [Diagram 5] 1 is a conceptual diagram of a degradation location estimation method using a random field according to the present invention; DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0011] Hereinafter, an embodiment of a random field data assimilation calculation method according to the present invention will be described with reference to the drawings.

[0012] As shown in FIG. 1, in the uncertainty evaluation method of the present invention, the sensitivity of the unknown variables to the Lagrangian function obtained in the sensitivity calculation unit is passed on to an optimum solution search unit, which searches for an optimum solution that matches the observed quantity, and the obtained unknown variables are passed on to a random variable conversion unit, which performs a calculation to convert the unknown variables from deterministic quantities to probability density functions by obtaining statistical variations around the optimum solution of the unknown variables.

[0013] The details of the data assimilation calculation performed in the sensitivity calculation section and the optimum solution section are shown in Figure 2. The purpose of this calculation is to model the structure from which observations are obtained using the finite element method, and to estimate the analysis variables (physical properties) of the finite element method as unknown variables so as to match the observations.

[0014] As shown in Figure 2, the data assimilation calculation unit models the object using the finite element method, performs finite element analysis using physical properties as analysis variables, and passes the obtained nodal displacements and velocities as state variables x to the sensitivity calculation unit. The sensitivity calculation unit in the data assimilation calculation unit sets the objective function J so as to minimize the observation error while satisfying the constraint conditions of the unknown variables, and creates the Lagrangian function L using equation 7 by adding a term multiplied by the Lagrangian multiplier to the state equation. The Lagrangian equation shown in equation 10 is derived by partially differentiating the constraint conditions of the unknown variables and the observation error with the state variable x, and the Lagrangian multiplier λ is updated in the reverse time direction to obtain it. After substituting the optimal solution λ^ of the obtained Lagrangian multiplier into the Lagrangian function, the sensitivity of the Lagrangian function is obtained using equation 13, which is partially differentiated with the unknown variable θ. The sensitivity of the Lagrangian function is passed to the optimal solution search unit, and the unknown variable θ that minimizes the Lagrangian function is searched for using the gradient descent method. A series of calculations performed in the sensitivity calculation section and the optimal solution search section are repeated a specified number of times as an inner loop, and then the process moves to an outer loop where finite element analysis is performed and the state variable x is updated. By performing the calculations of the outer loop and the inner loop within it a specified number of times as described above, the unknown variable θ that matches the observation quantity is calculated. After data assimilation is completed, the process moves to a calculation to convert the unknown variables into random variables.

[0015] The characteristic of the calculation of the sensitivity calculation part is that it partially differentiates the state variable x, which stores the nodal displacement and velocity, and the unknown variable θ, which stores the physical property values ​​and analysis variables, as independent variables when calculating the sensitivity of the Lagrangian function. First, the Lagrangian function of equation 7 is partially differentiated with respect to the state variable x, and the Lagrangian multiplier λ^ when the Lagrangian function is minimized with respect to the state variable x is found using equation 10. Next, the optimum Lagrangian multiplier λ^ is again substituted into the Lagrangian function of equation 7, and the value is partially differentiated with respect to the unknown variable θ to find the sensitivity to the unknown variable θ, and the novelty lies in the fact that only the unknown variable θ is updated.

[0016] A detailed calculation flow for the random variable conversion of data-assimilated physical properties is shown in Figure 3. The purpose of this calculation is to evaluate the uncertainty of the data-assimilated physical property values ​​by calculating the statistical variance of the data-assimilated unknown variables (physical property values), and to convert the physical property values ​​into random variables that follow a standard normal distribution to calculate the probability that the physical property values ​​will be below the reference value, thereby evaluating the safety and usability of the structure.

[0017] As shown in Figure 3, the random variable transformation part calculates the Hessian matrix H by using Equation 35, by second-order partial differentiation of the Lagrangian function, which is the Lagrangian function L calculated by the data assimilation calculation part and the unknown variable θ^ and the Lagrangian multiplier λ^ substituted when the Lagrangian function L is minimized. The inverse matrix H of this Hessian matrix is -1 is equal to the covariance matrix of the unknown variables. -1 The magnitude of the diagonal components of can be used to evaluate the statistical variability of the unknown variable θ.

[0018] Using the covariance matrix, we calculate the probability density function around the optimal unknown variable θ^ with mean θ^ and variance H -1 Since it can be expressed as a normal distribution by Equation 31, the unknown variable θ can be converted to a standard normal distribution statistic Θ with mean Θ^ and variance 1 by applying the variable transformation of Equation 37. By comparing the value of the standard normal distribution statistic Θ with the standard normal distribution table, the probability that the unknown variable exceeds the reference value can be obtained. Therefore, by storing the reference value of the physical quantity in the unknown variable θ and transforming it into the unknown variable Θ of the standard normal distribution by Equation 37, the probability that the physical quantity exceeds the reference value can be evaluated.

[0019] The features of the random variable transformation section are that it finds the covariance matrix of the unknown variables by the inverse Hessian matrix H obtained by second-order differentiation of the Lagrangian function L, and that it uses the Hessian matrix H to transform the unknown variable θ into an unknown variable Θ of a standard normal distribution, thereby evaluating the probability that the physical quantity exceeds a reference value.

[0020] When it is desired to evaluate whether the displacement and velocity of a vibrating body subjected to an external force are below a reference value, the perturbation amount δθ of the unknown variable is defined as the difference between the normalized unknown variable and the unknown variable estimated value θ^ when the exceedance probability is at a significant level, as shown in Equation 55. By substituting the perturbation amount δθ of the unknown variable into the time evolution equation of the uncertainty of the state variable x shown in Equation 57 after setting the uncertainty δx(0)=0 at the initial time of the state variable x, the uncertainty δx(t) of the state variable x at each time is obtained. By determining whether the value obtained by adding the absolute value of the uncertainty δx(t) to the absolute value of the state variable x is below a reference value, it is possible to determine whether the probability that the displacement and velocity of the vibrating body will exceed the reference value is below a significant level.

[0021] When it is desired to evaluate the statistical variation of the displacement and velocity of the vibrating body due to the statistical variation of the unknown variables, the perturbation amount δθ is defined as the diagonal element of the inverse Hessian matrix as shown in Equation 57, and the uncertainty δx(t) of the state variable x is similarly calculated.

[0022] Figure 4 shows the definition of the uncertainty of the unknown variable θ obtained by data assimilation, as well as a conceptual diagram of the probability field representation of the uncertain unknown variable and the probability of exceeding the reference value.

[0023] In the proposed method according to the present invention, the physical properties (unknown variables) are calculated as random variables with a range of values. Figure 5 is a conceptual diagram summarizing the calculation flow from determining the reference value of the unknown variable to calculating the probability of each location falling below the reference value, and the calculated results.

[0024] An example in which the calculation of uncertainty evaluation of unknown variables accompanying data assimilation according to the present invention is applied to an elastic vibrator will be described below.

[0025] <Method of estimating physical properties to fit measured data> We consider the problem of placing a sensor on an elastic vibrating body to measure vibrations and assimilating the data so that the physical properties are consistent with the measurement results. First, we model the vibrating body using the finite element method, and express the equation of motion of the ni degree of freedom system as the following equation using the mass matrix M, damping matrix C, and stiffness matrix K.

[0026]

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[0027] u(t) and f(t) are ni-dimensional vectors that respectively list the free node displacements and external nodal loads at time t.

[0028] Displacement and velocity vectors combined into one vector

[0029]

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[0030] By defining it as above, the following equation of state can be constructed:

[0031]

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[0032] y(t) is the observation quantity y at the ns point on the vibrating body. - (t) is the estimate, and C is the observation matrix that extracts observations from the state variables. The coefficient matrix of the state equation is

[0033]

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[0034] A and B are system matrices, F is the external force matrix, I ni is the ni-dimensional unit matrix.

[0035] The np-dimensional unknown variable vector to be estimated by data assimilation is defined as ω. For example, if you want to estimate the mass, damping, or stiffness matrix, you can simply vectorize them and assign them to the unknown variable ω. When searching for an optimal solution using the gradient method, if the unknown variables to be updated have different sizes, only the unknown variables that are highly sensitive to the objective function will be significantly updated. To prevent this phenomenon, the expected maximum value ω of the unknown variables is set as pmax and the minimum value ω pminis defined, and the unknown variables are normalized to be in the range of 0 to 1 using the following equation.

[0036]

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[0037] The gradient of the unknown variable before and after normalization satisfies the following relationship:

[0038]

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[0039] For simplicity, the time index t will be omitted from here on. In order to estimate the linear elastic body properties from the measurement data, a Lagrangian function is created using the 2ni-dimensional Lagrangian multiplier λ. Since the properties of a linear elastic body do not change with time, the unknown variable vector θ is a time invariant.

[0040]

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[0041] The unknown variable θ that minimizes the Lagrangian function L is the optimal value. Since the unknown variable θ does not change over time, its time derivative is always 0. The optimal value x^ of the state variable calculated by the finite element method always satisfies the following state equation.

[0042]

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[0043] Therefore, the second term on the right side of the adjoint equation shown in Equation 7 is always 0 when the unknown variables and state variables are at optimal values.

[0044] TIFF2025035952000010.tif6151 and can be expressed as follows using the error weighting matrix W.

[0045]

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[0046] J θ is a constraint on prior information about unknown variables. This Lagrange equation is solved by the Adjoint method. The adjoint equation obtained by differentiating the Lagrange multiplier with respect to the state variable x is

[0047]

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[0048] The dimensions of each term in the Lagrange equation are

[0049]

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[0050] Terminal condition (λ(t f )=0) and solving in the reverse time direction, the optimal value of the Lagrange multiplier λ^ is found. In the above equation, if the terminal condition is given and the Lagrange multiplier is updated in the reverse time direction, the Lagrange multiplier at the initial time becomes equal to the gradient of the Lagrange function.

[0051]

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[0052] The gradient can be found by substituting the optimal value λ̂ into the Lagrangian function and partially differentiating it with respect to the unknown variable θ.

[0053]

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[0054]

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[0055] In the inner loop, the unknown variable θ is assimilated to the observation quantity by updating the unknown variable using the gradient descent method or the like using the gradient of equation (13). After that, the finite element method is executed in the outer loop to update the state variable x. By repeating the update calculations of the outer loop and the inner loop, the unknown variable θ that fits the observation data is obtained.

[0056] The unknown variable θ is an np-dimensional vector as follows:

[0057]

number

[0058] When updating unknown variables using the gradient method, etc., it is necessary to constrain the unknown variables so that they fall within the range of 0 to 1. Therefore, the following transformation is performed to convert the constrained problem regarding the unknown variables into an unconstrained problem regarding ψ.

[0059]

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[0060] By inversely transforming the equation for finding ψ, we obtain the following equation for finding the unknown variable θ.

[0061]

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[0062] From equation 17, we can see that the range of θ is limited to 0 to 1 regardless of the value of ψ. Therefore, when updating the unknown variables, it is advisable to replace θ with ψ. The differential value of the Lagrangian function J with respect to the unknown variables θ and ψ satisfies the following relationship,

[0063]

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[0064] After calculating the differential value with respect to θ in Equation 13, it is advisable to substitute this when converting the variable to ψ.

[0065] <How to set constraints on unknown variables> Constraint term J on unknown variables θ In the finite element method, the unknown variable numbers that exist at positions adjacent to the p-th component of the unknown variable are defined as a subscript set P p The index set P p The average value of each type of unknown variable is calculated using the formula below, and the difference between the value of the unknown variable of element e is minimized to smooth the unknown variables between adjacent elements. Here, the p-th component of the unknown variable θ is included in the set P b The np-dimensional defining function, where the b-th component is 1 when the element belongs to the np-dimensional ...

[0066]

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[0067] Using this unknown variable vector, set P p The mean value of the unknown variables belonging to p The average value of the unknown variable adjacent to can be calculated by the following formula:

[0068]

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[0069] The deviation of the unknown variable from the mean value is calculated using the following formula:

[0070]

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[0071] The sample variance of the unknown variable in the neighboring elements around the p-th component of the unknown variable is

[0072]

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[0073] The variance of the adjacent unknown variables is defined as a constraint term using the following equation:

[0074]

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[0075] By differentiating the constraint terms with respect to the unknown variables to obtain the sensitivities, the set P around the mean value obtained at each update process of the unknown variables is obtained. p It is possible to minimize the sample variance for each

[0076]

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[0077] <Relationship between the variance of unknown variables and the Lagrangian function> First, consider the relationship between the posterior probability of the unknown variable θ and the Lagrangian function. To do this, we calculate the posterior probability of the unknown variable θ when the observable D is known as prior information. The posterior probability density of the observable and the unknown variable can be inverted using Bayes' theorem as follows:

[0078]

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[0079] Since the probability density Pr(D) of the observed data, which is a definite value, is always a constant value, the posterior probability of the unknown variable θ is proportional to the product of the probability density of the unknown variable and the posterior probability of the observed data.

[0080]

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[0081] If we take prior information about the unknown variables as probability densities for the unknown variables,

[0082]

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[0083] The posterior probability density of the observation is equal to the probability density of the observation error, and therefore follows a multivariate normal distribution with the covariance Σ of the observation error as shown in the following equation.

[0084]

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[0085] ns is the number of observation points. From the above, the posterior probability of the unknown variable is

[0086]

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[0087] Therefore, the log-likelihood is equal to the Lagrangian function.

[0088]

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[0089] The Lagrangian function assumes that the observation error follows a normal distribution, and substituting this assumption into Bayes' theorem in equation 25 leads to the proposition that the unknown variable θ in the object of analysis also follows a normal distribution. When this normal distribution assumption holds, the posterior probability density of the unknown variable θ can be expressed as a probability density function of a normal distribution with the optimal value θ^ of the unknown variable determined from the observation amount and prior information as the mean value. Therefore, if the inverse matrix of the covariance matrix of the unknown variable θ is defined as the np-dimensional precision matrix H, the posterior probability density of the unknown variable can be expressed by the following equation.

[0090]

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[0091] Substituting the posterior probability density of the unknown variables (Equation 31) into the Lagrangian function (Equation 30), we obtain

[0092]

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[0093] If the unknown variable is fixed as θ = θ^, the first and second terms on the right hand side become constants, so only the third term on the right hand side needs to be treated as a variable. Therefore, the precision matrix H is equal to the Hessian matrix obtained by second-order partial differentiation of the Lagrangian function with respect to the unknown variable.

[0094]

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[0095] When θ = θ^, the first derivative of the Lagrangian function with respect to the unknown variable is 0, so Equation 32 can be approximated by Taylor expansion of the Lagrangian function up to the second-order term with respect to the unknown variable θ.

[0096]

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[0097] In the above equation, we can confirm that the second partial derivative of the Lagrangian function L is equal to the Hessian matrix.

[0098] <Method of evaluating uncertainty in estimated physical properties> By partially differentiating the Lagrangian function L with respect to the unknown variable θ, the precision matrix H, which represents the degree of statistical variation of the unknown variable around the optimal value Θ^, is obtained. The method for evaluating the uncertainty of the unknown variable using this precision matrix is ​​summarized below. By partially differentiating the Lagrangian equation in equation 13 with respect to the unknown variable θ, the precision matrix is ​​calculated as follows.

[0099]

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[0100] The variable arrangement and dimensions used in the second partial derivative of the Lagrangian function are as follows:

[0101]

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[0102] Unknown variable θ has mean θ^ and variance H -1 When the unknown variable θ follows a normal distribution,

[0103]

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[0104] and substituting it into Equation 31, the posterior probability density of the unknown variable is normalized to a standard normal distribution with a mean of 0 and a variance of 1, as shown in Figure 4. The inverse matrix and root of the precision matrix can be calculated by calculating the inverse and root of the eigenvalue obtained by performing eigenvalue decomposition on the precision matrix and converting it to an orthogonal coordinate system, and then multiplying by the eigenvectors on both sides to return the coordinate axes to the base.

[0105]

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[0106] The exceedance probability of the standard normal distribution can be uniquely determined from the standard normal distribution table once the value of Θ is determined. Due to this property, for example, after substituting the reference value into the unknown variable θ, the variable is converted to the standard normal distribution variable Θ using Equation 37, and the distribution of Θ within the vibrating body is obtained, and the exceedance probability where the physical property value does not satisfy the reference value can be expressed as a contour diagram as shown in Figure 5 together with the data assimilation results.

[0107] By determining the distribution of the probability of exceeding the standard value within the body, it is possible to identify areas where the probability of exceeding a certain level is greater than or equal to a certain level as deteriorated areas. In addition, by prioritizing visual inspection of areas with a high probability of exceeding the standard, it is possible to contribute to the formulation of an efficient inspection plan.

[0108] <Calculation of sensitivity to equation of state of unknown variables> Assuming a case where the distribution of element stiffness ratios in a vibrating body is estimated by data assimilation, we summarize the formulation for calculating the sensitivity of unknown variables. The element stiffnesses in the healthy state and after damage changes in the current state are K e 0 , Ke The element stiffness ratio α and the unknown variable ω are related to each other to define α(ω e The element stiffness ratio is expressed as the ratio of the element stiffness in the healthy state to the element stiffness in the current state, as shown in the following equation.

[0109]

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[0110] e is the element index, and ne is the number of elements. Since it is assumed that the unknown variable ω in the vibrating body always follows a normal distribution, it is necessary to find some variable that follows a normal distribution and is related to the element stiffness ratio α.

[0111] Element stiffness ratio α(ω e ) corresponds to the rate of decline in element stiffness from soundness. The unknown variable θ must follow a normal distribution within the object to be analyzed, but since stiffness degradation progresses rapidly, it is difficult to imagine that the stiffness degradation ratio α follows a normal distribution. On the other hand, many components are used in bridges, and the distribution within the object of the value obtained by dividing the service life of the components by their useful life is expected to be close to a normal distribution. If we assume that the element stiffness ratio α is proportional to the soundness, the relationship between the element stiffness ratio α and the unknown variable ω is proportional to the relationship when the component soundness is defined as α and the service life divided by the component useful life as ω, according to Non-Patent Document 3. Therefore, the element stiffness ratio α is defined to be a function with a shape similar to the component deterioration curve shape described in Non-Patent Document 3.

[0112]

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[0113] γ is the shape coefficient of the component deterioration curve, and generally has a value of around 2; the larger γ is, the faster the component deterioration rate will be. The function shape of Equation 40 was determined on the assumption that the value obtained by dividing the component's service life by its useful life follows a normal distribution, but if the target structure or component is different, there is a possibility that it will deviate from the normal distribution. In that case, it will be necessary to separately find an unknown variable that follows a normal distribution and formulate the relationship with the element stiffness.

[0114] In order to obtain the sensitivity of the unknown variables of the components to the state equation, we first organize the relationships between the variables. First, in order to explicitly express the superposition of element e, we use the coupling matrix L e and the element nodal displacement vector u e and the nodal displacement vector U.

[0115]

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[0116] connection matrix L e is a matrix in which the i-th component of the j-th row is 1 when the j-th component of element e corresponds to the i-th component of the global nodal displacement vector, and the other j-th row components are 0.

[0117] The overall composite matrix K is obtained by superposing this element stiffness matrix with the coupling matrix.

[0118]

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[0119] An indicator function χ is defined to associate an element number e with one unknown variable number p.

[0120]

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[0121] The index of the unknown variable is converted from the element quantity e to the unknown variable number p by multiplying it by the indicator function χ and taking a weighted average in the element direction.

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[0123] Using the element stiffness ratio and indicator function, the first and second derivatives of the stiffness matrix are calculated by Equations 45 and 46, respectively, and the second derivative of the state equation is calculated from the second derivative of the stiffness matrix K by Equation 47.

[0124]

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[0125]

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[0126]

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[0127] The second derivative of term B, which has the stiffness of the state equation as a variable, is calculated for each unknown variable number, and each component of the matrix is ​​calculated by applying it to equation 35, which leads to a reduction in the calculation memory.

[0128] <Random variable transformation when there are multiple optimal values ​​for unknown variables> So far, we have summarized the methods of random variable transformation when the optimum value of the unknown variable is determined to be one. However, when the shape of the vibrating body itself cannot be completely reproduced by numerical analysis and there are multiple candidates for the numerical model shape itself, or when the parameters of the Lagrangian function cannot be uniquely determined, there will be multiple optimal solution candidates for the unknown variable. In this article, we summarize the methods of calculating the probability density function and the reference value excess probability of the unknown variable when the numerical model and analysis conditions are uncertain and there are multiple solution candidates.

[0129] Calculate the mean and variance (inverse matrix of precision matrix) of the probability density function when there are nκ optimal solution candidates for the unknown variable θ. Define the κth optimal solution as θ^(κ) and the precision matrix as H(κ), and the variables before transformation in the standard normal distribution are

[0130]

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[0131] Therefore, when the sum of nκ normal distribution probability density functions is expressed as a characteristic function, the first and second terms on the right-hand side can be divided into the mean and variance terms of the normal distribution, as follows:

[0132]

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[0133] Here, i is a complex number, t is a variable vector of the characteristic function, and E[·] is the averaging process. The variance term in the second term on the right-hand side is the characteristic function of the np-dimensional standard normal distribution with Θ as a variable, so

[0134]

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[0135] O np and I np are np-dimensional zero and unit matrices, respectively. By substituting the characteristic function of Equation 50 into Equation 49 and dividing both sides by nκ, the mean value and variance of the probability density function that combines multiple normal distributions can be obtained in the first and second terms on the right-hand side of the characteristic function as shown in the following equation.

[0136]

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[0137] From this characteristic function, the mean vector θ^ of the normal distribution obtained by averaging nκ probability density functions that follow normal distributions is obtained. ave and the covariance matrix (precision matrix H ave The inverse matrix of

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[0139] Therefore, the conditional posterior probability of the unknown variable θ on the probability density function of the synthesized normal distribution is given by the following equation.

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[0141] From the above, if we transform the unknown variables into standard normal distributions in the synthesized probability space,

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[0143] and the standard normal variable Θ, which is a composite of multiple normal distributions ave can be derived, and the corresponding exceedance probability can be calculated from a normal distribution table. [Industrial Applicability]

[0144] Consider a situation where we need to check whether the probability of the displacement or velocity of a vibrating body exceeding a standard value is below a significant level in order to ensure safety and comfort for public use. The unknown variable θ is set as the distribution of physical properties within the vibrating body. The normalized unknown variable Θ when the exceedance probability is at a significant level is found from a standard normal distribution table, and after variable conversion to the unknown variable θ, the difference from the estimated value θ^ is found and defined as the perturbation amount δθ.

[0145]

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[0146] Assume that the state equation in equation (3) is stable and the first partial differential with respect to the unknown variable is 0. In this case, the first-order perturbation with respect to the state variable x and the unknown variable θ is also 0, so

[0147]

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[0148] When we actually calculate the above equation written in terms of differential operators, we get

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[0150] By substituting the perturbation amount δθ of the unknown variable into Equation 49 with the uncertainty δx(0) = 0 at the initial time of the state variable x, and calculating the time evolution of the perturbation amount δx(t) of the state variable x(t), and adding the absolute values ​​of the state variable x and the perturbation amount δx, it is possible to check whether the probability that the displacement or speed exceeds the reference value is below the significance level.

[0151] Consider a situation where you want to evaluate the uncertainty of the displacement or velocity (state variable) of a vibrating body. In this case, the statistical variation of the unknown variable can be interpreted by replacing it with the perturbation amount δθ, so the statistical variation of the unknown variable can be calculated as the diagonal term of the inverse Hessian matrix.

[0152]

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[0153] Similarly, the uncertainty of the state variable can be evaluated by substituting the perturbation amount δθ of the unknown variable into a number and calculating the time evolution of the perturbation amount δx of the state variable x.

[0154] As described above, when the uncertainty evaluation index of the present invention is used to obtain the variance of an unknown variable, any of the items described in the claims is considered to be within the scope of the present invention. [Explanation of symbols]

[0155] 1 Analysis variable setting section 2 Data Assimilation Calculation Unit 3. Random variable transformation section 4 Calculation result display area

Claims

1. A method for calculating sensitivity of a Lagrangian function, comprising the steps of: A state equation with the state variable x, which stores the nodal displacement and velocity calculated by finite element analysis, and the unknown variable θ, which stores the physical property value or analysis variable, as variables. [0080] The Lagrangian function L is calculated by adding the constraint J on the prior information of the state variables and the unknown variables to the [0070] When creating The Lagrange function L shown in Equation 7 is partially differentiated with respect to the state variable x, and the optimal Lagrange multiplier λ^ when the Lagrange function is minimized with respect to the state variable x is obtained by solving the Lagrange equation shown in Equation 10 in the reverse time direction, and then [0010] The optimum Lagrange multiplier λ^ is again substituted into the Lagrange function shown in Equation 7, and the value is partially differentiated with respect to the unknown variable θ to obtain the sensitivity to the unknown variable. ##EQU00013## The calculation flow is as follows: The method is characterized in that, when calculating the sensitivity of the Lagrangian function L, the Lagrangian function is partially differentiated using the state variable x storing the nodal displacement and the velocity and the unknown variable θ storing the physical property value and the analysis variable as independent variables. A method for calculating the sensitivity of the Lagrangian function.

2. A method for evaluating uncertainty of unknown variables storing the physical property values ​​and analysis variables, comprising: The optimal unknown variable θ^ and the optimal Lagrangian multiplier λ^ when the Lagrangian function L calculated by the data assimilation calculation unit is minimized are substituted into the Lagrangian function L, and then the Hessian matrix H is obtained by performing a second-order partial differentiation with respect to the unknown variable θ. [Equation 35] And calculated that, The method is characterized in that the statistical variability of the unknown variables is evaluated by utilizing the property that the inverse matrix of the Hessian matrix is ​​equal to the covariance matrix of the unknown variables. A method for evaluating the uncertainty of the unknown variables.

3. A method for calculating an exceedance probability that the unknown variable, which stores the physical property value and the analysis variable, exceeds a reference value, comprising: After substituting the reference value for the unknown variable θ, Using the optimal unknown variable θ̂ and the Hessian matrix H, the unknown variable θ is converted into the standard normal distribution statistic Θ. [Equation 37] Then, The method is characterized in that the value of the statistic Θ of the standard normal distribution is compared with a standard normal distribution table to calculate the excess probability that the unknown variable exceeds the statistic Θ of the standard normal distribution. A method for calculating the exceedance probability that the unknown variable exceeds the reference value.

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