Structural reliability optimization design method and device based on uncertainty measurement

Abstract

The embodiment of the present invention discloses a structural reliability optimization design method and device based on uncertainty measurement, which includes: determining the performance parameters of the structure, and constructing the limit state function of the structure under cognitive uncertainty based on the uncertain variables that affect the performance parameters, wherein the uncertain variables obey a continuous or regular uncertain distribution; constructing a reliability measurement model of the structure based on uncertain testing based on the limit state function combined with the uncertain distribution or the uncertain inverse distribution; determining the structural optimization design objective function, and establishing a design optimization model with the structural performance reliability constraint and the objective function in combination with the reliability measurement model; constructing an equivalent analytical solution model of the design optimization model, and obtaining the optimal design variables in combination with the boundary conditions of the design variables. Through the above-mentioned method, the embodiment of the present invention can realize product reliability analysis under cognitive uncertainty and obtain the optimal design solution.

Description

Technical Field

[0001] The embodiments of the present invention relate to the technical field of structural reliability analysis, and in particular to a structural reliability optimization design method and device based on uncertainty measurement. Background Art

[0002] Reliability is a key quality characteristic throughout a product's life cycle and can be defined as its ability to perform its specified functions under specified conditions and within a specified timeframe. Reliability-based design optimization (RBDO) aims to find the optimal design solution for a product while ensuring that the reliability of the required performance criteria does not fall below an acceptable threshold. For example, in the reliability analysis and design of mechanical structures, various unavoidable uncertainties exist in material properties, geometric dimensions, and external loads. Sample data for these uncertainties is scarce, and the measurement space varies. Traditional reliability theory based on probability measures relies on extensive statistical data and the law of large numbers to determine the probability distribution function of the input variables. This is then used to obtain performance reliability in the probability space and optimize the design. However, when all input variables in the reliability analysis process are fuzzy or their values ​​are unclear, and sufficient historical statistical data and experimental sample data are unavailable, domain expert information with epistemic uncertainty can be used to characterize these input variables. Because the precise probability density function of the input variables cannot be obtained, directly using probabilistic reliability theory to characterize reliability under epistemic uncertainty can result in inconsistent analytical results.

[0003] To address this challenge, we need to find new quantitative theoretical tools and establish reasonable non-probabilistic reliability analysis models and optimization design methods. When sample data is insufficient and precise probability distribution characteristics are difficult to obtain, non-probabilistic theories can be used to measure epistemic uncertainty. Widely used non-probabilistic theories include fuzzy set theory, possibility theory, interval theory, convex set theory, and evidence theory. On the one hand, when insufficient sample statistical data is available to verify the accuracy of the membership function, the analysis results based on fuzzy set theory are questionable. On the other hand, due to the interval expansion problem of interval operations, reliability analysis methods based on possibility theory, interval theory, convex set theory, or evidence theory are generally overly conservative in estimating reliability indicators.

[0004] To address the shortcomings of the aforementioned non-probabilistic epistemic uncertainty quantification theories, Liu Baoding first proposed uncertainty theory in 2007 and supplemented and refined it in 2010. Because the uncertainty measure defined in the uncertainty space not only satisfies the subadditivity of epistemic uncertainty but also exhibits self-duality, uncertainty theory has been widely applied in many fields in recent years, including maintenance decision-making, optimal decision-making, uncertain finance, uncertain risk analysis, availability analysis, and residual life prediction. However, the application of uncertainty theory to structural performance reliability modeling and optimal design analysis is currently rare. Summary of the Invention

[0005] In view of the above problems, embodiments of the present invention provide a structural reliability optimization design method and apparatus based on uncertainty measurement, which overcome the above problems or at least partially solve the above problems.

[0006] According to one aspect of an embodiment of the present invention, a structural reliability optimization design method based on uncertainty measurement is provided, the method comprising: determining performance parameters of a structure, and constructing a limit state function of the structure under cognitive uncertainty based on uncertain variables affecting the performance parameters, wherein the uncertain variables obey a continuous or regular uncertain distribution; constructing a reliability measurement model of the structure based on uncertainty measurement based on the limit state function in combination with the uncertain distribution or the uncertain inverse distribution; determining a structural optimization design objective function, and establishing a design optimization model with the structural performance reliability constraints and the objective function in combination with the reliability measurement model; constructing an equivalent analytical solution model of the design optimization model, and obtaining the optimal design variables in combination with the boundary conditions of the design variables.

[0007] Optionally, the step of constructing a reliability measurement model of the structure based on uncertainty measure according to the limit state function in combination with uncertainty distribution includes: calculating the reliability of the structure based on uncertainty measure according to the limit state function through the uncertainty distribution of uncertain variables. The reliability measurement model of the structure based on uncertainty measurement is obtained:

[0008]

[0009] Where τ is the uncertainty vector τ=(τ1,τ2,…,τ n ) T,τ1,τ2,…,τ n is an uncertain variable, which obeys the continuous uncertain distribution γ1,γ2,…,γ n , the limit state function g(τ1,τ2,…,τ n ) about τ1,τ2,…,τ m Strictly monotonically increasing, and about τ m+1 ,τ m+2 ,…,τ n Strictly monotonically decreasing, is the confidence level of the failure event, and sup represents the upper bound.

[0010] Optionally, constructing the reliability measurement model of the structure based on uncertainty measure according to the limit state function in combination with uncertain inverse distribution includes: solving the reliability measurement model of the structure based on uncertainty measure according to the limit state function using uncertain inverse distribution:

[0011]

[0012] Among them, τ1, τ2,…, τ n is an uncertain variable, which obeys the regular uncertain distribution γ1,γ2,…,γ n , the failure reliability of the structure α is the solution of the equation, the reliability based on the uncertainty measure is 1-α, and the limit state function g(τ1,τ2,…,τ n ) About the uncertain variables τ1,τ2,…,τ m Strictly monotonically increasing, and with respect to the uncertain variable τ m+1 ,τ m+2 ,…,τ n Strictly monotonically decreasing.

[0013] Optionally, determining the structural optimization design objective function includes: if the structure needs to satisfy multiple mutually constrained optimization objective functions, assigning corresponding weight factors according to the importance of each objective function; and performing weighted summation on the multiple objective functions to obtain the total objective function of the structure.

[0014] Optionally, the determination of the structural optimization design objective function and the establishment of a design optimization model with the structural performance reliability constraint and objective function in combination with the reliability measurement model also include: for any of the limit state functions, obtaining the performance reliability value of the structure according to the reliability measurement model based on uncertainty measurement; and building a reliability design optimization model of the structure that meets the corresponding expected reliability value according to the total objective function and performance reliability value of the structure.

[0015] Optionally, the input variables of the structure include design variables and uncertain non-design variables, and the design variables include: deterministic design variables and uncertain design variables, and the uncertain design variables and the uncertain non-design variables are uncertain variables. The determining of the structural optimization design objective function, and establishing a design optimization model with the structural performance reliability constraint and the objective function in combination with the reliability measurement model, further includes: constructing a reliability-based structural design optimization model based on the objective function and the reliability constraint conditions determined based on the reliability measurement model in combination with the boundary conditions of the design variables in the structure:

[0016] Solve for {d,x}

[0017]

[0018] Satisfy: reliability constraints,

[0019] Boundary conditions, d Lower ≤d≤d Upper ,x Lower ≤x≤x Upper ,j=1,2,…,N g

[0020] Among them, d, x, τ are vectors composed of deterministic design variables, uncertain design variables and uncertain non-design variables, respectively, and f i (d,x,τ),ω i are the i-th objective function and the corresponding weight factor, N f is the total number of objective functions, g j (d,x,τ), are the reliability, limit state function and expected reliability value of the jth reliability constraint condition, respectively. g is the number of reliability constraints, d Lower and d Upper are the upper and lower bound vectors of the deterministic design variables, x Lower and x Upper are the upper and lower bound vectors of the uncertain design variables respectively; the reliability-based design optimization model is solved to obtain the optimal design variables that meet the reliability constraints of the structure.

[0021] Optionally, constructing an equivalent analytical solution model for the design optimization model and obtaining optimal design variables in combination with boundary conditions of design variables further includes: performing equivalent planning analysis on the reliability-based design optimization model to convert the reliability-based design optimization model into a clear equivalent planning model as described below:

[0022] Solve for {d,x}

[0023]

[0024] Satisfy: reliability constraints,

[0025]

[0026] Boundary conditions, d Lower ≤d≤d Upper ,x Lower ≤x≤x Upper ,j=1,2,…,N g ;

[0027] The clear equivalent programming model is solved to obtain the optimal design variables that meet the reliability constraints of the structure.

[0028] Based on the same inventive concept, a structural reliability optimization design device based on uncertainty measurement is provided, including: a limit state function acquisition unit, used to determine the performance parameters of the structure, and construct the limit state function of the structure under cognitive uncertainty according to the uncertain variables affecting the performance parameters, and the uncertain variables obey a continuous or regular uncertain distribution; a reliability analysis acquisition unit, used to construct a reliability measurement model of the structure based on uncertainty measurement according to the limit state function combined with the uncertain distribution or the uncertain inverse distribution; a design optimization model acquisition unit, used to determine the structural optimization design objective function, and establish a design optimization model with the structural performance reliability constraint and objective function in combination with the reliability measurement model; a design optimization solution unit, used to construct an equivalent analytical solution model of the design optimization model, and obtain the optimal design variables in combination with the boundary conditions of the design variables.

[0029] Based on the same inventive concept, an embodiment of the present invention further proposes an electronic device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the aforementioned method when executing the program.

[0030] Based on the same inventive concept, an embodiment of the present invention further proposes a computer storage medium, in which at least one executable instruction is stored, and the executable instruction enables a processor to execute the aforementioned method.

[0031] The embodiment of the present invention determines the performance parameters of the structure and constructs the limit state function of the structure under cognitive uncertainty based on the uncertain variables that affect the performance parameters, wherein the uncertain variables obey a continuous or regular uncertain distribution; constructs a reliability measurement model of the structure based on uncertainty measurement according to the limit state function in combination with the uncertain distribution or the uncertain inverse distribution; determines the structural optimization design objective function, and establishes a design optimization model with the structural performance reliability constraint and the objective function in combination with the reliability measurement model; constructs an equivalent analytical solution model of the design optimization model, and obtains the optimal design variables in combination with the boundary conditions of the design variables, thereby realizing product reliability analysis under cognitive uncertainty and obtaining the optimal design solution.

[0032] The above description is only an overview of the technical solutions of the embodiments of the present invention. In order to more clearly understand the technical means of the embodiments of the present invention, they can be implemented in accordance with the contents of the specification. In order to make the above and other purposes, features and advantages of the embodiments of the present invention more obvious and easy to understand, the specific implementation methods of the present invention are specifically listed below. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Various other advantages and benefits will become apparent to those skilled in the art upon reading the detailed description of the preferred embodiment below. The accompanying drawings are for illustration purposes only and are not to be considered as limiting the present invention. The same reference symbols are used throughout the drawings to represent the same components. In the drawings:

[0034] Figure 1 A schematic diagram of a flow chart of a structural reliability optimization design method based on uncertainty measurement provided by an embodiment of the present invention is shown;

[0035] Figure 2 A schematic diagram of a structural reliability optimization design method based on uncertainty measurement according to an embodiment of the present invention is shown;

[0036] Figure 3 An example diagram showing a reliability optimization design of a hatch door latch mechanism according to an embodiment of the present invention is shown;

[0037] Figure 4 A schematic diagram showing the reliability optimization design results of the hatch door latch mechanism according to an embodiment of the present invention is shown;

[0038] Figure 5 An example diagram showing a reliability optimization design of an automobile torque arm structure according to an embodiment of the present invention is shown;

[0039] Figure 6 A schematic diagram showing the reliability optimization design results of the automobile torque arm structure according to an embodiment of the present invention is shown;

[0040] Figure 7A schematic diagram of the structure of a structural reliability optimization design device based on uncertainty measurement provided by an embodiment of the present invention is shown;

[0041] Figure 8 A schematic diagram of an electronic device in an embodiment of the present invention is shown. DETAILED DESCRIPTION

[0042] Exemplary embodiments of the present invention will be described in more detail below with reference to the accompanying drawings. Although exemplary embodiments of the present invention are shown in the accompanying drawings, it should be understood that the present invention can be implemented in various forms and should not be limited by the embodiments set forth herein. Rather, these embodiments are provided to enable a more thorough understanding of the present invention and to fully convey the scope of the present invention to those skilled in the art.

[0043] Figure 1 FIG. 1 shows a flow chart of a structural reliability optimization design method based on uncertainty measurement provided by an embodiment of the present invention. Figure 1 As shown in Figure 2, the structural reliability optimization design method based on uncertainty measurement is applied to servers, including:

[0044] Step S11: determining performance parameters of the structure, and constructing a limit state function of the structure under epistemic uncertainty based on uncertain variables that affect the performance parameters, wherein the uncertain variables obey a continuous or regular uncertainty distribution.

[0045] For the uncertain variables that affect the structural performance parameters, the performance limit state of the structure under a certain failure mode is generally described by the limit state function of the uncertain variables. The limit state function involves n-dimensional uncertain variables x1, x2, ..., x n , including material properties, geometric dimensions, and external load conditions. The epistemic uncertainty problem can be defined as the existence of epistemic uncertainty in all uncertain variables, which can be expressed as an uncertainty vector τ = (τ1, τ2, ..., τ n ) T In uncertain space All input variables are uniformly represented within. Uncertain variables follow continuous or regular uncertainty distributions, such as normal uncertainty distribution, zigzag uncertainty distribution, and linear uncertainty distribution, and are not specifically limited here. The embodiments of the present invention do not impose specific limitations on the structure; it can be a mechanism that can perform a certain function or a single component structure, such as a hatch lock mechanism or a torque arm structure.

[0046] In uncertainty theory, in order to reasonably describe the epistemic uncertainty information and address the problem that non-probabilistic measures such as possibility measures do not satisfy the self-duality, a class of uncertainty measures with subadditive properties that satisfy the self-duality property is adopted. ).

[0047] Definition of uncertainty measure: If Γ is a non-empty set, it is a σ-algebra in Γ, where each element Λ is an event. Then if the set function If the following four axioms are satisfied, it is defined as an uncertainty measure:

[0048] Axiom 1 (Normality) For the set Γ,

[0049] Axiom 2 (Duality) For any event Λ,

[0050] Axiom 3 (Subadditivity) For any countable sequence of events Λ1, Λ2, ..., it satisfies:

[0051]

[0052] Axiom 4 (Product Axiom) If is a series of uncertain spaces, Λ k For any For an event selected from , k = 1, 2, ..., then the product of uncertainty measures is still an uncertainty measure and satisfies the following equation:

[0053]

[0054] Definition of uncertainty space: If Γ is a non-empty set, is a σ-algebra in Γ, is an uncertain measure, then the triple consisting of the above elements An uncertain space.

[0055] Definition of uncertain variables: The uncertain variable ξ is a variable from the uncertain space A measurable function to the set of real numbers, that is, {ξ∈B} is an event on any real Borel set B.

[0056] Assume ξ1,ξ2,…,ξ n is a set of uncertain variables, f is a real-valued measurable function, then ξ=f(ξ1,ξ2,…,ξ n ) is also an uncertain variable, that is, for any τ∈Γ, ξ(τ) can be expressed as follows:

[0057] ξ(τ)=f(ξ1(τ),ξ2(τ),…,ξ n(τ))

[0058] Satisfying Theorem 1 (independence): uncertain variables ξ1,ξ2,…,ξ n Mutual independence exists if and only if the following equations hold:

[0059]

[0060] Among them, B1, B2, …, B n is any Bore set on the set of real numbers.

[0061] Definition of uncertainty distribution: If ξ is an uncertain variable, then the uncertainty distribution Φ(x) of ξ with respect to any real number x can be defined as follows:

[0062]

[0063] In general, if the uncertainty distribution Φ(x) is a continuous function that increases monotonically with respect to x and satisfies 0<Φ(x)<1, lim x→-∞ Φ(x)=0,lim x→∞ Φ(x)=1, then the uncertainty distribution is defined as the canonical uncertainty distribution function.

[0064] Example (normal uncertainty distribution): If the uncertain variable ξ has the following normal uncertainty distribution, it is defined as a normal uncertain variable:

[0065]

[0066] It is expressed as Where m and σ are the uncertainty expectation and uncertainty variance, respectively. In particular, if m = 0 and σ = 1 are satisfied in the normal uncertainty distribution, it is called the standard normal uncertainty distribution, which is expressed as

[0067] Satisfy Theorem 2: Let ξ1,ξ2,…,ξ n is a set of independent uncertain variables, each with a continuous uncertainty distribution function Φ1, Φ2, ..., Φ n If f(ξ1,ξ2,…,ξ n ) is about ξ1,ξ2,…,ξ m Monotonically increasing, about ξ m+1 ,ξ m+2 ,…,ξ n If the function ξ(ξ1,ξ2,…,ξ n ) can be expressed as:

[0068]

[0069] In addition, assuming Φ1, Φ2, …, Φ n is a regular uncertain distribution function, then the uncertain distribution f(ξ1,ξ2,…,ξ n ) can be expressed as follows:

[0070]

[0071] Definition of uncertain programming: If ξ is an uncertain vector, x is a decision vector, and f(x,ξ) is an uncertain objective function. i (x,ξ)≤0,i=1,2,…,l does not necessarily hold. Decision makers naturally hope that the uncertain constraints are within the confidence levels α1,α2,…,α l The following relationship is satisfied:

[0072]

[0073] Because it is impossible to directly minimize the uncertain variable f(x,ξ), the decision maker can minimize the expectation of f(x,ξ) In order to obtain the minimum optimal solution with uncertain expected target value under a set of chance constraints, the uncertain programming model can be defined as follows:

[0074]

[0075] The characteristic of this chance-constrained uncertain programming model is that the objective function and constraints are composed of uncertain variables rather than random variables.

[0076] In the embodiment of the present invention, the performance reliability analysis is performed based on the “stress-strength interference” model, where the generalized stress S(τ1,τ2,…,τ n ,ω) includes load, temperature and corrosion, etc., and the corresponding generalized strength R(τ1,τ2,…,τ n ,ω) includes fatigue strength, heat resistance and corrosion resistance, etc., then the uncertainty limit state function (ULSF) can be defined as follows:

[0077] Z(τ)=g(τ1,τ2,…,τ n )=R(τ1,τ2,…,τ n )-S(τ1,τ2,…,τ n )

[0078] The reliability of structural performance can be measured by uncertainty The confidence level of a quantitatively reliable event {Z(τ)>0}, namely, Uncertain Measure Based Reliability (UMBR), is defined as follows:

[0079]

[0080] in, Unlike probabilistic reliability, insufficient sample data on the input variables prevents accurate probability distribution functions from being obtained and structural reliability estimates from being performed. This embodiment of the present invention proposes a reliability measurement model based on uncertainty measures in an uncertain space to address the shortcomings of probabilistic reliability measurement models.

[0081] Step S12: constructing a reliability measurement model of the structure based on uncertainty measure according to the limit state function combined with uncertainty distribution or uncertain inverse distribution.

[0082] Assume that the limit state function Z(τ)=g(τ1,τ2,…,τ n ) obeys an uncertain distribution Φ z(τ) (z), where τ1,τ2,…,τ n are mutually independent uncertain variables, then the reliability of the failure event {Z(τ)≤0} can be calculated as:

[0083]

[0084] However, in actual engineering structural problems, it is often difficult for designers to obtain the uncertainty distribution Φ of the uncertain limit state function. z(τ) (z), the above formula is difficult to solve and apply. In order to solve this problem and improve the practicality of the unified reliability measurement model under uncertainty space established by the present invention, the following method for solving the UMBR reliability under different conditions will be proposed.

[0085] In step S12, optionally, first, for any of the limit state functions, a reliability measurement model of the structure is constructed according to the limit state function combined with uncertain variable characteristics, and then the reliability of the structure based on the uncertainty measure is calculated.

[0086] In the embodiment of the present invention, if the limit state function Z(τ)=g(τ1,τ2,…,τ n ) are mutually independent uncertain variables τ1,τ2,…,τ n Obeying the continuous uncertain distribution γ1,γ2,…,γ n When the limit state function is about τ1,τ2,…,τ m Strictly monotonically increasing, and about τ m+1 ,τ m+2 ,…,τn Strictly monotonically decreasing, the reliability measurement model of the structure is constructed based on the limit state function combined with the characteristics of uncertain variables, and the reliability of the structure based on uncertainty measurement is calculated

[0087]

[0088] Where τ is the uncertainty vector τ=(τ1,τ2,…,τ n ) T ,τ1,τ2,…,τ n are mutually independent uncertain variables, obeying the uncertain distribution γ1,γ2,…,γ n , the limit state function g(τ1,τ2,…,τ n ) about τ1,τ2,…,τ m Strictly monotonically increasing, and about τ m+1 ,τ m+2 ,…,τ n Strictly monotonically decreasing, is the confidence level of the failure event, and sup represents the upper bound.

[0089] Considering the difficulty in calculating the above formula due to the complexity of the limit state function, the embodiment of the present invention also proposes the following solution to UMBR based on the Inverse Distribution-Based Method (IDBM). If the performance limit state function Z(τ) = g(τ1,τ2,…,τ n ) are mutually independent uncertain variables τ1,τ2,…,τ n Obeying regular uncertainty distribution functions (Regular Uncertainty Distributions)γ1,γ2,…,γ n When the limit state function is about τ1,τ2,…,τ m Strictly monotonically increasing, and about τ m+1 ,τ m+2 ,…,τ n Strictly monotonically decreasing, the structural failure reliability is And the UMBR reliability can be calculated as According to the limit state function, the uncertainty inverse distribution is used to construct the reliability measurement model of the structure based on uncertainty measure:

[0090]

[0091] Where α is the solution of the equation.

[0092] Step S13: determining the structural optimization design objective function, and establishing a design optimization model with the structural performance reliability constraint and the objective function in combination with the reliability measurement model.

[0093] In the embodiment of the present invention, the engineering structure design problem may include N f There are mutually constrained objective functions f that need to be considered simultaneously i (d,x;τ),i=1,2,…,N f Uncertain multi-objective optimization belongs to uncertain programming, and its constraints and objective functions contain uncertain variables. In step S13, if the structure needs to satisfy multiple mutually constrained optimization objective functions, a corresponding weight factor is assigned to each objective function based on its importance; the weighted sum of the multiple objective functions is then taken to obtain the overall objective function of the structure.

[0094] Based on the reliability measurement model based on uncertainty measurement established above, UMBR reliability is introduced as a constraint condition of the reliability-based design optimization (RBDO) model, and a UMBR reliability-based design optimization model (Uncertainty Reliability-Based Design Optimization, URBDO) under cognitive uncertainty is constructed.

[0095] To address the problem of structural design optimization in the presence of epistemic uncertainty in the presence of insufficient sample data, an embodiment of the present invention proposes a structural design optimization (URBDO) model based on the UMBR reliability and the Uncertain Programming Model. Under conditions of epistemic uncertainty, the URBDO model uses uncertain variables to uniformly describe input variables and incorporates UMBR reliability requirements into the constraints of the classic RBDO model. This ensures structural safety while simultaneously seeking the optimal design for objectives such as minimum cost or minimum weight by varying design variables.

[0096] The input variables of the structural optimization design problem include design variables and uncertain non-design variables, and the design variables include: deterministic design variables and uncertain design variables, and the uncertain design variables and the uncertain non-design variables are uncertain variables. g The limit state function g j (d,x;τ),j=1,2,…,N g The reliability optimization design problem of x=(x1,x2…,x m ) T and τ=(τ1,τ2,…,τ n ) Tare vectors of deterministic design variables, uncertain design variables, and uncertain non-design variables. Design engineers tend to ensure that the UMBR reliability is higher than the target reliability. Combined with the definition of UMBR reliability given above, the corresponding reliability constraint can be expressed as:

[0097]

[0098] Structural design includes N f There are mutually constrained objective functions f that need to be considered simultaneously i (d,x;τ),i=1,2,…,N f By weighted summing each objective function, the following general URBDO model based on UMBR reliability can be established. Therefore, in step S13, a reliability-based design optimization model is first constructed based on the objective function and the reliability constraints determined based on the reliability measurement model, combined with the boundary conditions of the design variables:

[0099] Solve for {d,x}

[0100]

[0101] Satisfy: reliability constraints,

[0102] Boundary conditions, d Lower ≤d≤d Upper ,x Lower ≤x≤x Upper ,j=1,2,…,N g

[0103] Among them, d, x, τ are deterministic design variables, uncertain design variables and uncertain non-design variables respectively, f i (d,x,τ),ω i are the i-th objective function and the corresponding weight factor, and satisfy N f is the number of objective functions, g j (d,x,τ), are the reliability, limit state function and expected reliability value of the jth reliability constraint condition, respectively. g is the number of reliability constraints, d Lower and d Upper are the upper and lower bound vectors of the deterministic design variables, x Lower and x Upperare the upper and lower bound vectors of the uncertain design variables respectively. Then the reliability-based design optimization model is solved to obtain the optimal design variables that meet the reliability constraints of the structure. Since the objective function is also an uncertain variable, the objective function f is directly minimized. i (d,x;τ) is more difficult. Therefore, minimizing f i The uncertain expectation E[f i (d,x;τ)] is more reasonable.

[0104] Step S14: constructing an equivalent analytical solution model of the design optimization model, and obtaining the optimal design variables in combination with the boundary conditions of the design variables.

[0105] To solve the reliability-based design optimization (URBDO) model constructed above, this embodiment of the present invention proposes an analytical equivalent programming method. Under certain conditions, the established URBDO model can be converted into a Crisp Equivalent Programming (CEP) model. This CEP model facilitates engineers to quickly obtain the optimal design solution during the initial structural design phase.

[0106] Assume that the performance limit state function corresponding to the jth constraint condition in structural optimization is g j (d,τ),j=1,2,…,N g ,in, and τ=(τ1,τ2,…,τ n ) T are vectors of deterministic design variables and uncertain variables, respectively, where the uncertain variables are independent of each other and all obey the regular uncertainty distribution γ1,γ2,…,γ n If the limit state function g j (d,τ1,τ2,…,τ n )about Strictly monotonically increasing, and about Strictly monotonically decreasing. The objective function f i (d; τ1, τ2,…, τ n ), i=1,2,…,N f about Strictly monotonically increasing, and about Strictly monotonically decreasing. Optionally, the design optimization model based on UMBR reliability is subjected to equivalent planning analytical transformation, and the design optimization model based on UMBR reliability is converted into the following clear equivalent planning CEP model:

[0107] Solve for {d,x}

[0108]

[0109] Satisfy: reliability constraints,

[0110]

[0111] Boundary conditions, d Lower ≤d≤d Upper ,x Lower ≤x≤x Upper ,j=1,2,…,N g ;

[0112] The clear equivalent planning model is then solved to obtain the optimal design variables that meet the reliability constraints of the structure. After converting the general URBDO model into an equivalent CEP model, structural engineers can use classical numerical methods or heuristic algorithms to optimize the design.

[0113] The complete structural reliability optimization design method based on uncertainty measurement of the embodiment of the present invention is as follows Figure 2 Shown, including:

[0114] Step 100: Determine the input variables of the structure.

[0115] The input variables of the structure are determined based on the design requirements. Input variables include design variables and uncertain non-design variables, while design variables can include deterministic design variables and uncertain design variables. Uncertain design variables and uncertain non-design variables are uncertain variables. Deterministic design variables refer to input variables that do not follow any distribution.

[0116] Step 101: Construct the limit state function of the structure.

[0117] According to the relationship between structural performance parameters and input variables, performance reliability analysis is performed based on the "stress-strength interference" model, and the corresponding limit state function is constructed:

[0118] Z(τ)=g(τ1,τ2,…,τ n )=R(τ1,τ2,…,τ n )-S(τ1,τ2,…,τ n )

[0119] Z(τ), g(τ1,τ2,…,τ n ) is the limit state function, S(τ1,τ2,…,τ n ) is the generalized stress of the structure, R(τ1,τ2,…,τ n ) is the corresponding generalized intensity, τ1,τ2,…,τ n is an uncertain variable.

[0120] Step 102: τ1, τ2, …, τn They are represented by continuous uncertain distributions respectively.

[0121] That is, the uncertain variables τ1, τ2,…, τ n They can obey continuous uncertain distributions respectively.

[0122] Step 103: τ1, τ2, …, τ n They are represented by regular uncertain distributions respectively.

[0123] That is, the uncertain variables τ1, τ2,…, τ n They can obey regular uncertain distribution respectively.

[0124] Step 104: Uncertainty measure based reliability UMBR.

[0125] Through uncertainty measurement Quantify the confidence of the reliable event {Z(τ)>0}, that is, the reliability based on uncertainty measure (UMBR): in, Different from probabilistic reliability, a reliability measurement model based on uncertainty measure is constructed based on the continuous uncertainty distribution of uncertain variables, or a reliability measurement model based on uncertainty measure is constructed based on the uncertainty inverse distribution.

[0126] Step 105: URBDO model with UMBR constraints.

[0127] According to the reliability constraints, the final objective function and the boundary conditions of the structural design variables, a UMBR reliability-based design optimization (URBDO) model under cognitive uncertainty is constructed to obtain the optimal design variables that meet the structural reliability constraints. At the same time, if the engineering structure design problem contains N f The multiple objective functions that need to be considered simultaneously are weighted and summed to obtain the final objective function of the structure.

[0128] Step 106: Clarify the equivalent planning CEP model.

[0129] The UMBR Reliability-Based Design Optimization (URBDO) model is converted into an equivalent planning model to obtain the corresponding Clear Equivalent Planning (CEP) model. Based on this CEP model, engineering designers can quickly obtain the optimal design solution during the initial design phase of the structure.

[0130] Step 107: Determine whether it is an optimal solution. If so, proceed to step 108, otherwise return to step 104.

[0131] It is determined whether the design variables obtained based on the clear equivalent programming model in step 106 are the optimal solution.

[0132] Step 108: Obtain the optimal design variables.

[0133] If the design variables obtained based on the clear equivalent programming model in step 106 are the optimal solutions, the optimal solutions are output to complete the reliability optimization design of the current structure.

[0134] The embodiment of the present invention takes the problem of structural reliability optimization design as the main research object, and focuses on reliability quantitative analysis based on the performance limit state. It focuses on the key point of reliability modeling and solution under uncertainty information conditions, and takes the uncertainty theory with sub-additive characteristics as the mathematical theoretical basis. It proposes a performance reliability analysis model that characterizes cognitive uncertainty through uncertain variables, and then explores the reliability design optimization problem based on the sub-additive measure reliability theory. Finally, reliability optimization design is achieved under cognitive uncertainty conditions based on small samples. The present invention provides methodological and theoretical support for reliability design and optimization analysis of actual engineering products throughout their life cycle. The following is an explanation of the hatch lock mechanism and the torque arm structure respectively.

[0135] The reliability analysis of the hatch lock mechanism can be simplified to the reliability analysis problem of a crank slider mechanism, such as Figure 3 As shown in Figure 2, based on the kinematic analysis of the crank slider mechanism, the horizontal distance between the fulcrum and the lock end point can be expressed as:

[0136]

[0137] Among them, the input variable α1 is the angle between the crank and the horizontal direction, and the input variables r, L1, L2, and e are size parameters, respectively. α1, r, L1, L2, and e are uncertain variables.

[0138] When the length of the performance parameter L is not less than L3 (mm), the lock can work properly, otherwise the lock cannot work properly. Due to the lack of sample data, it is assumed that all input variables ξ=(α1,r,L1,L2,e) T All obey mutually independent normal uncertainty distributions, then the following uncertain limit state function of the locking mechanism can be established:

[0139] g(α1,r,L1,L2,e)=L(α1,r,L1,L2,e)-L3

[0140] The distribution parameters of the input variables in the embodiment of the present invention are shown in Table 1. The reliability analysis method based on uncertainty measurement established by the present invention is applied to perform reliability evaluation on the locking mechanism. In order to verify the effectiveness and superiority of the method of the embodiment of the present invention, the traditional first-order reliability method (FORM) method based on probability theory is also applied to this implementation case. At this time, all input variables are regarded as obeying the normal probability distribution, as shown in Table 1. When L3∈[286,291], the reliability estimation results of the locking mechanism obtained by the probabilistic reliability method and the method of the present invention are compared. Figure 4 shown.

[0141] Table 1 Distribution type and parameters of input variables in Example 1

[0142]

[0143]

[0144] according to Figure 4 As can be seen, as L3 increases, both the UMBR reliability and the Probability Measure Based Reliability (PMBR) decrease. The overall trends of the UMBR and PMBR are consistent. This is because the reliability of the locking mechanism decreases as L3 increases. Therefore, the UMBR reliability proposed in this paper can effectively describe the changing trends of mechanism reliability under epistemic uncertainty.

[0145] It is worth noting that the reliability estimated by the uncertainty measurement method of the present invention is smaller than the result calculated based on the probability measurement. For the reliability problem under epistemic uncertainty, assuming that the uncertain variables are random variables, the use of probability measurement theory for analysis will ignore the impact of epistemic uncertainty, resulting in overly optimistic calculation results. On the one hand, when there is insufficient information related to the input variables and it is impossible to obtain a specific probability distribution function, it is appropriate to use a structural reliability measurement model based on uncertainty measurement for analysis. On the other hand, when there is sufficient statistical information related to the input variables and the actual frequency is obtained, it is more reasonable to use probability measurement theory. When statistical data of the input variables is lacking, information about the existence of epistemic uncertainty can be obtained by combining incomplete sample data and subjective experience of experts, and the reliability measurement model based on uncertainty measurement proposed by the present invention is used for analysis, which can ensure the safety of the structure / mechanism to a greater extent.

[0146] If the structure is a car torque arm structure, such as Figure 5 As shown in Figure 2, the geometrical parameters a, d1, d2 and l of the automobile torque arm structure are design variables. Other input variables include Young's modulus E, yield strength S yand external stress Q are uncertain variables. Under the conditions of strength and buckling constraints, the structural volume is required to be minimized. Its initial deterministic design optimization model can be expressed as follows:

[0147] Solve d = [a, d1, d2, l] T

[0148]

[0149] satisfy

[0150]

[0151] 5≤a≤15

[0152] 45≤d1≤55

[0153] 55≤d2≤75

[0154] 110≤l≤210

[0155] Where d is the vector of deterministic design variables, the objective function V(d) represents the volume of the torque arm, and the limit state function g1(d; Q, S y ) means Figure 5 Yield failure of section AA shown, where The limit state function g2(d; E, Q) corresponding to the second constraint condition represents the buckling failure of the two connecting rods.

[0156] Table 2 Distribution type and parameters of torque arm input variables

[0157]

[0158] Assume that due to insufficient sample data, Q, S y and E can be respectively represented by the normal uncertainty distribution γ Q (x), and γ E (x) description. Table 2 lists the distribution types and parameters of the input variables. Since the first limit state function g1(d; Q, S y ) continuous and about S y It is strictly increasing with respect to E and strictly decreasing with respect to Q. The second limit state function g2(d; E, Q) is also continuous and strictly increasing with respect to E and strictly decreasing with respect to Q. Therefore, the URBDO model of the torsion arm structure can be equivalently replaced by a clear equivalent planning model to establish the following under different expected reliability levels: The torque arm CEP model with epistemic uncertainty is as follows:

[0159] Solve d = [a, d1, d2, l] T

[0160]

[0161] satisfy

[0162]

[0163] 5≤a≤15

[0164] 45≤d1≤55

[0165] 55≤d2≤75

[0166] 110≤l≤210

[0167] After the uncertain planning problem is transformed into a CEP model, the model can be solved using MATLAB software. In order to demonstrate the rationality of the proposed method, when Q, S y and E are considered to obey normal probability distribution, that is, Q~N(5500,900 2 ), S y ~N(170,30 2 ) and E~N(2.1×10 11 ,(2.0×10 10 ) 2 ), the classic RBDO model based on probability measure and its reliability index approach (RIA) are used for reference comparison. In addition, when all input variables are regarded as constants equal to their respective expected values, the deterministic optimization design method is also performed for reference. When the expected reliability falls within the range of [0.50, 0.99], the optimization results of the objective function are compared. Figure 6 As shown in Table 3, the detailed comparison results when the expected reliability is 0.90 and 0.99 respectively are listed.

[0168] Table 3 Comparison of optimization results when the expected reliability is 0.9 and 0.99 respectively

[0169]

[0170] Depend on Figure 6 The optimization design results shown in the figure show that as the reliability requirements increase, more material is required to manufacture the torque arm. According to the comparison results, the optimal volume change trends estimated by the RBDO and URBDO models are basically similar. Figure 6As shown in Table 3, when the reliability level is in the range of [0.50, 0.98], the target value calculated by the URBDO model is lower than the target value estimated by the RBDO model. However, when the reliability level is in the range of [0.98, 0.99], the target value estimated by the URBDO model is significantly higher than the target value calculated by the RBDO model. This is because in the RBDO model, uncertain variables with epistemic uncertainty are also treated as random variables, and the significant impact of epistemic uncertainty is ignored by the RBDO model. When epistemic uncertainty information is present, the URBDO model based on uncertainty measurement can more accurately handle structural optimization design problems under epistemic uncertainty. As reliability requirements increase, the performance difference between the two optimization models increases, indicating that structural design is more sensitive to the impact of epistemic uncertainty. Therefore, the URBDO model can measure and reflect the significant impact of epistemic uncertainty on the optimization design of engineering structures. In addition, design variables a and d2 have a significant impact on the torque arm optimization results, and structural engineers should pay special attention to these two design variables during the design process.

[0171] When sample data of input variables in reliability analysis and design optimization problems are lacking, that is, when epistemic uncertainty exists, the reliability measurement model based on uncertainty measurement and its URBDO model can obtain the optimal design solution in the initial design stage of the product and provide theoretical guidance for engineering designers.

[0172] In summary, the structural reliability optimization design method based on uncertainty measurement in the embodiment of the present invention determines the performance parameters of the structure, and constructs the limit state function of the structure under cognitive uncertainty based on the uncertain variables affecting the performance parameters, and the uncertain variables obey a continuous or regular uncertain distribution; constructs a reliability measurement model of the structure based on uncertainty measurement according to the limit state function combined with the uncertain distribution or the uncertain inverse distribution; determines the structural optimization design objective function, and establishes a design optimization model with the structural performance reliability constraint and the objective function in combination with the reliability measurement model; constructs an equivalent analytical solution model of the design optimization model, and obtains the optimal design variables in combination with the boundary conditions of the design variables, which can realize product reliability analysis and obtain the optimal design scheme under cognitive uncertainty.

[0173] The foregoing description is of specific embodiments of the present invention. In some cases, the actions or steps described in the embodiments of the present invention may be performed in an order different from that shown in the embodiments and still achieve the desired results. In addition, the processes depicted in the accompanying drawings do not necessarily require the specific order or sequential order shown to achieve the desired results. In some embodiments, multitasking and parallel processing are also possible or may be advantageous.

[0174] Based on the same concept, the embodiment of the present invention also provides a structural reliability optimization design device based on uncertainty measurement. Applied to the server. Figure 7 As shown, the structural reliability optimization design device based on uncertainty measurement includes: a limit state function acquisition unit, a reliability analysis acquisition unit, a design optimization model acquisition unit and a design optimization solution unit.

[0175] a limit state function acquisition unit, configured to determine performance parameters of a structure and construct a limit state function of the structure under epistemic uncertainty based on uncertain variables affecting the performance parameters, wherein the uncertain variables obey a continuous or regular uncertainty distribution;

[0176] A reliability analysis acquisition unit, configured to construct a reliability measurement model of the structure based on uncertainty measure according to the limit state function combined with uncertainty distribution or uncertainty inverse distribution;

[0177] A design optimization model acquisition unit is used to determine the structural optimization design objective function and establish a design optimization model with the structural performance reliability constraint and the objective function in combination with the reliability measurement model;

[0178] The design optimization solution unit is used to construct an equivalent analytical solution model of the design optimization model and obtain the optimal design variables in combination with the boundary conditions of the design variables.

[0179] For the convenience of description, the above devices are described as being divided into various modules according to their functions. Of course, when implementing the embodiments of the present invention, the functions of each module can be implemented in the same or multiple software and / or hardware.

[0180] The apparatus of the above embodiment is applied to the corresponding method of the above embodiment and has the beneficial effects of the corresponding method embodiment, which will not be described in detail here.

[0181] Based on the same inventive concept, an embodiment of the present invention further provides an electronic device, which includes a memory, a processor, and a computer program stored in the memory and runnable on the processor, wherein when the processor executes the program, the method described in any one of the above embodiments is implemented.

[0182] An embodiment of the present invention provides a non-volatile computer storage medium, wherein the computer storage medium stores at least one executable instruction, and the computer executable instruction can execute the method described in any one of the above embodiments.

[0183] Figure 8805. A more specific schematic diagram of the hardware structure of an electronic device provided in this embodiment is shown. The device may include: a processor 801, a memory 802, an input / output interface 803, a communication interface 804, and a bus 805. The processor 801, the memory 802, the input / output interface 803, and the communication interface 804 are communicatively connected to each other within the device via the bus 805.

[0184] The processor 801 can be implemented as a general-purpose CPU (Central Processing Unit), a microprocessor, an application-specific integrated circuit (ASIC), or one or more integrated circuits, and is used to execute relevant programs to implement the technical solutions provided by the method embodiments of the present invention.

[0185] The memory 802 can be implemented in the form of ROM (Read Only Memory), RAM (Random Access Memory), static storage devices, dynamic storage devices, etc. The memory 802 can store an operating system and other application programs. When the technical solutions provided by the method embodiments of the present invention are implemented through software or hardware, the relevant program codes are stored in the memory 802 and called and executed by the processor 801.

[0186] The input / output interface 803 is used to connect to input / output modules to implement information input and output. The input / output modules can be configured as components in the device (not shown in the figure) or can be externally connected to the device to provide corresponding functions. Input devices may include a keyboard, mouse, touch screen, microphone, various sensors, etc., and output devices may include a display, speaker, vibrator, indicator light, etc.

[0187] The communication interface 804 is used to connect to a communication module (not shown) to enable communication between the device and other devices. The communication module can communicate via a wired method (such as USB, network cable, etc.) or a wireless method (such as mobile network, WIFI, Bluetooth, etc.).

[0188] The bus 805 comprises a pathway for transmitting information between the various components of the device (eg, the processor 801 , the memory 802 , the input / output interface 803 , and the communication interface 804 ).

[0189] It should be noted that although the above device only shows the processor 801, memory 802, input / output interface 803, communication interface 804, and bus 805, in a specific implementation, the device may also include other components necessary for normal operation. In addition, those skilled in the art will understand that the above device may only include the components necessary to implement the embodiments of the present invention, and does not necessarily include all the components shown in the figure.

[0190] Those skilled in the art should understand that the discussion of any of the above embodiments is merely illustrative and is not intended to imply that the scope of the present disclosure is limited to these examples. Based on the concept of the present disclosure, the technical features in the above embodiments or different embodiments may be combined, the steps may be implemented in any order, and there are many other variations of the different aspects of the embodiments of the present invention as described above, which are not provided in detail for the sake of simplicity.

[0191] This application is intended to cover all such substitutions, modifications, and variations that fall within the broad scope of all embodiments. Therefore, any omissions, modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the embodiments of the present invention should be included in the scope of protection of this disclosure.

Claims

1. A structural reliability optimization design method based on uncertainty measurement, characterized in that: The method comprises: determining performance parameters of a structure and constructing a limit state function of the structure under epistemic uncertainty based on uncertain variables affecting the performance parameters, wherein the uncertain variables obey a continuous or regular uncertainty distribution; Constructing a reliability measurement model of the structure based on uncertainty measure according to the limit state function combined with uncertainty distribution or uncertainty inverse distribution; The step of constructing a reliability measurement model of the structure based on uncertainty measurement according to the limit state function and uncertainty distribution includes: According to the limit state function, the reliability of the structure based on uncertainty measure is calculated through the uncertainty distribution of uncertain variables , we get the reliability measurement model of the structure based on uncertainty measure: in, is an uncertain vector , is an uncertain variable that obeys a continuous uncertain distribution , limit state function about Strictly monotonically increasing, and about Strictly monotonically decreasing, is the confidence level of the failure event, Indicates taking the upper bound; The step of constructing a reliability measurement model of the structure based on uncertainty measure according to the limit state function and the uncertain inverse distribution includes: According to the limit state function, the uncertainty inverse distribution is used to solve the reliability measurement model of the structure based on uncertainty measure: in, is an uncertain variable, obeying a regular uncertain distribution , the failure reliability of the structure , is the solution of the equation, and the reliability based on the uncertainty measure is , limit state function About uncertain variables Strictly monotonically increasing, and about uncertain variables Strictly monotonically decreasing; Determining a structural optimization design objective function, and establishing a design optimization model having the structural performance reliability constraints and the objective function in combination with the reliability measurement model; An equivalent analytical solution model of the design optimization model is constructed, and the optimal design variables are obtained by combining the boundary conditions of the design variables.

2. The method according to claim 1, characterized in that Determining the structural optimization design objective function includes: If the structure needs to satisfy multiple mutually constrained optimization objective functions, then a corresponding weight factor is assigned according to the importance of each objective function; A weighted sum is performed on the multiple objective functions to obtain the overall objective function of the structure.

3. The method according to claim 1, characterized in that The determining of the structural optimization design objective function and establishing a design optimization model having the structural performance reliability constraint and the objective function in combination with the reliability measurement model further includes: For any of the limit state functions, obtaining a performance reliability value of the structure according to the reliability measurement model based on uncertainty measure; According to the overall objective function and performance reliability value of the structure, a reliability design optimization model of the structure that meets the corresponding expected reliability value is established.

4. The method according to claim 1, wherein The input variables of the structure include design variables and uncertain non-design variables, the design variables include: deterministic design variables and uncertain design variables, the uncertain design variables and the uncertain non-design variables are uncertain variables, the structural optimization design objective function is determined, and a design optimization model with the structural performance reliability constraint and the objective function is established in combination with the reliability measurement model, further comprising: According to the objective function and the reliability constraints determined based on the reliability measurement model, combined with the boundary conditions of the design variables in the structure, a reliability-based design optimization model is constructed: Solution min Satisfy: reliability constraints, Boundary conditions, in, are vectors consisting of deterministic design variables, uncertain design variables and uncertain non-design variables, respectively. , Respectively objective functions and corresponding weight factors, is the total number of objective functions, 、 、 Respectively reliability constraints, limit state functions, and expected reliability values, is the number of reliability constraints, and are the upper and lower bound vectors of the deterministic design variables, and are the upper and lower bound vectors of the uncertain design variables respectively; The reliability-based design optimization model is solved to obtain optimal design variables that meet the reliability constraints of the structure.

5. The method according to claim 4, characterized in that The equivalent analytical solution model of the design optimization model is constructed, and the optimal design variables are obtained in combination with the boundary conditions of the design variables, including: Perform equivalent planning analysis on the reliability-based design optimization model to convert the reliability-based design optimization model into the following clear equivalent planning model: Solution Satisfy: reliability constraints, Boundary conditions, ; The clear equivalent programming model is solved to obtain the optimal design variables that meet the reliability constraints of the structure.

6. A structural reliability optimization design device based on uncertainty measurement, characterized by: The device comprises: a limit state function acquisition unit, configured to determine performance parameters of a structure and construct a limit state function of the structure under epistemic uncertainty based on uncertain variables affecting the performance parameters, wherein the uncertain variables obey a continuous or regular uncertainty distribution; A reliability analysis acquisition unit, configured to construct a reliability measurement model of the structure based on uncertainty measure according to the limit state function combined with uncertainty distribution or uncertainty inverse distribution; The step of constructing a reliability measurement model of the structure based on uncertainty measurement according to the limit state function and uncertainty distribution includes: According to the limit state function, the reliability of the structure based on uncertainty measure is calculated through the uncertainty distribution of uncertain variables , we get the reliability measurement model of the structure based on uncertainty measure: in, is an uncertain vector , is an uncertain variable, obeying a continuous uncertain distribution , limit state function about Strictly monotonically increasing, and about Strictly monotonically decreasing, is the confidence level of the failure event, Indicates taking the upper bound; The step of constructing a reliability measurement model of the structure based on uncertainty measurement according to the limit state function and the uncertain inverse distribution includes: According to the limit state function, the uncertainty inverse distribution is used to solve the reliability measurement model of the structure based on uncertainty measure: in, is an uncertain variable, obeying a regular uncertain distribution , the failure reliability of the structure , is the solution of the equation, and the reliability based on the uncertainty measure is , limit state function About uncertain variables Strictly monotonically increasing, and about uncertain variables Strictly monotonically decreasing; A design optimization model acquisition unit is used to determine the structural optimization design objective function and establish a design optimization model with the structural performance reliability constraint and the objective function in combination with the reliability measurement model; The design optimization solution unit is used to construct an equivalent analytical solution model of the design optimization model and obtain the optimal design variables in combination with the boundary conditions of the design variables.

7. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When the processor executes the program, the method according to any one of claims 1 to 5 is implemented.

8. A computer storage medium, characterized in that: The storage medium stores at least one executable instruction, and the executable instruction enables the processor to execute the method according to any one of claims 1 to 5.

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