Topological optimization two-stage efficient solution method for structural dynamic reliability under random excitation

Abstract

The invention discloses a two-stage efficient solution method for topological optimization of structural dynamic reliability under random excitation. According to the method, firstly, deterministic topological optimization with the most unfavorable condition performance function as the constraint is carried out, and then reliability topological optimization with the reliability as the constraint is solved. A time domain explicit method and a direct probability integral method are combined, and the structural dynamic reliability is calculated efficiently and accurately. Under a sensitivity solving framework of a direct probability integral method, a time domain explicit method is further combined, an adjoint method is adopted to deduce a sensitivity formula of failure probability about design variables in a dynamic problem, and an efficient sensitivity calculation framework suitable for power reliability topological optimization is established by utilizing an importance representative point screening strategy. A sequence approximation integer programming method is used for efficiently solving the structural dynamic reliability topological optimization problem under the action of random excitation, design meeting the reliability requirement is obtained, and theoretical support is provided for design of an actual engineering structure.

Description

Technical Field

[0001] The present invention belongs to the technical field of structural topology optimization design, and relates to a two-stage efficient solution method for structural dynamic reliability topology optimization under random excitation. The method is a sensitivity analysis method developed based on the time domain explicit method, and is used to improve the solution efficiency of dynamic reliability topology optimization problems. Background Art

[0002] Topology optimization, as an advanced structural design method, can produce structural designs with excellent performance without prior experience. Existing structural topology optimization designs are mostly derived under deterministic conditions, ignoring the multi-source random uncertainties that commonly exist during the service life of engineering structures. This makes it difficult for optimized structural designs to meet safety, serviceability, and durability requirements throughout their lifecycles. Engineering structures are often subjected to dynamic forces during their service life. Due to the complexity of the service environment, these external forces are often random excitations with random uncertainties, such as earthquakes, wind, and waves. To address practical engineering applications, the impact of random excitations must be fully considered during the structural design phase. The dynamic reliability function contains a time parameter, which necessitates that dynamic reliability analysis of structures consider the time memory characteristics of the dynamic process. In this paper, the dynamic reliability problem focuses on the first-time failure reliability of responses exceeding the limit state. Since reliability assessment in dynamic reliability topology optimization requires solving the random vibration response of the structure, the computational time is excessively long. Furthermore, solving structural dynamic topology optimization problems is inherently more complex than static topology optimization problems. These two factors make dynamic reliability topology optimization a challenging problem.

[0003] Among the reliability constraint processing methods of current power reliability topology optimization, the Direct Probability Integral Method (DPIM) has the comprehensive advantages of high computational efficiency, strong universality and simple implementation. In view of this, the present invention adopts the Direct Probability Integral Method to solve the power reliability topology optimization problem. Secondly, in order to reduce the influence of the false mode problem in the power topology optimization solution, the present invention adopts the sequential approximate integer programming method of discrete variable topology optimization solution to solve the optimization problem in order to reduce the influence of this problem. Finally, since the continuum structure topology optimization problem usually has a large number of design variables, and the sensitivity analysis of the power reliability requires time-history analysis, the sensitivity analysis of the power reliability is very time-consuming. Summary of the Invention

[0004] Based on the direct probability integration method and sequential approximate integer programming, the present invention provides a two-stage efficient solution method for structural dynamic reliability topology optimization under random excitation, which is aimed at accurate and efficient reliability analysis of continuum structure dynamic reliability topology optimization, false modal problems and challenging problems with huge computational complexity.

[0005] The method of the present invention establishes a two-stage solution framework for the dynamic reliability topology optimization problem of continuum structures; uses a direct probability integration method to perform accurate and efficient dynamic reliability assessment of the structure; combines a time-domain explicit method to efficiently solve the structural response and the sensitivity of the dynamic reliability to the design variables; uses a sequential approximate integer programming method to update the design variables and obtain a discrete variable topology optimization configuration that meets the reliability requirements.

[0006] The technical solution adopted in the present invention is as follows:

[0007] A two-stage efficient solution method for structural dynamic reliability topology optimization under random excitation includes the following steps:

[0008] (1) Use the time domain explicit method to analyze the structural response of all representative points corresponding to the initial design. Including:

[0009] A two-step point selection method based on generalized F-deviation is used to partition the probability space to obtain the representative points and their assigned probabilities required by the direct probability integration method. A time-domain explicit format of the initial design is constructed to efficiently calculate the structural extreme responses corresponding to all representative points. The extreme responses corresponding to the most unfavorable situation under the initial design are obtained.

[0010] (2) Establish a deterministic topology optimization formula for discrete variables of continuum structures with structural volume as the objective function and the most unfavorable extreme response as the constraint function.

[0011] (3) Solve the deterministic topology optimization problem in (2) based on the sequential approximate integer programming method. Including:

[0012] Sensitivity analysis of the worst-case performance function; updating of design variables for discrete variable topology optimization using sequential approximate integer programming methods.

[0013] (4) The time domain explicit method is used to calculate the structural extreme responses corresponding to all representative points of the current design and the direct probability integration method is used to estimate the failure probability of the current design under random excitation. When the failure probability is greater than zero, the optimization enters the solution of the dynamic reliability topology optimization problem.

[0014] (5) Use direct probability integration method to conduct structural dynamic reliability sensitivity analysis. Including:

[0015] The time-domain explicit method is used to efficiently solve the structural extreme responses corresponding to all representative points, and the direct probability integration method is used to calculate the reliability of the current design. The representative points that play a key role in the reliability sensitivity calculation are selected according to the importance representative point screening strategy to reduce the amount of calculation. The reliability sensitivity is efficiently solved using the adjoint method and the time-domain explicit method.

[0016] (6) Based on the calculated sensitivity analysis, the reliability topology optimization problem is solved using the sequential approximate integer programming algorithm, and the structural topology design that meets the design requirements is iteratively obtained.

[0017] The beneficial effects of the present invention are:

[0018] The present invention establishes an efficient two-stage solution method for the reliability topology optimization of continuum structures. The method uses the direct probability integration method for reliability assessment and the sequential approximate integer programming method to solve the discrete variable topology optimization problem. By combining the time domain explicit method, the speed of the direct probability integration method for reliability and sensitivity calculation is accelerated. At the same time, the use of the importance representative point screening technology greatly reduces the number of representative points used for sensitivity analysis, so that the dynamic reliability topology optimization can be solved efficiently. It provides an efficient solution technology for the structural dynamic reliability topology optimization design and can significantly improve the safety of the structure under random excitation. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 It is a schematic diagram of the process of the present invention.

[0020] Figure 2 This is the full triangular unit load excitation diagram required for solving the time domain explicit method coefficient matrix in the present invention.

[0021] Figure 3 Schematic diagram of the cantilever beam structure provided for an example of the present invention.

[0022] Figure 4 A representative excitation time history curve of the cantilever beam structure provided in the example of the present invention.

[0023] Figure 5 The optimized configuration diagram of the cantilever beam at four cutoff frequencies provided for the example of the present invention.

[0024] Figure 6 Iteration history diagram of the cantilever beam at four cutoff frequencies provided for the example of the present invention. DETAILED DESCRIPTION

[0025] The specific embodiments of the present invention are described in detail below in conjunction with the technical solutions and drawings.

[0026] Example 1

[0027] A two-stage efficient solution method for structural dynamic reliability topology optimization under random excitation, such as Figure 1 As shown. Includes:

[0028] Step (1): Use the time domain explicit method to analyze the structural response of all representative points corresponding to the initial design.

[0029] For a linear system, by discretizing in the time domain, we can get t i The equation of motion of the system at this moment is

[0030]

[0031] Where M is the overall mass matrix, C is the overall damping matrix, and K is the overall stiffness matrix. is the acceleration vector, is the velocity vector, U is the displacement vector, F is the random excitation, and L is the positioning matrix used to locate the position of the random excitation. In order to simplify the writing, the above equations are i Indicates the form of a subscript.

[0032] When using the Newmark-β method to solve the motion equation in the formula, the basic expression of the velocity and displacement corresponding to each time step is

[0033]

[0034] Where Δt is the time step, and γ and β are two basic parameters in the Newmark-β method that determine its stability and accuracy. In this chapter, we set γ = 0.5 and β = 0.25, which makes the Newmark-β method unconditionally stable.

[0035] Using the above two formulas, we can derive

[0036]

[0037] The coefficients in the above two equations are

[0038]

[0039] Using equation (3) and equation (2), after simple derivation and arrangement, we can get the time domain explicit method response recursive expression:

[0040] V i =TV i-1 +QF i (5)

[0041] in,

[0042]

[0043] When the initial condition of the system is 0, that is, when V0 is a zero vector, the recursive formula (5) can be simplified to

[0044]

[0045] Where the coefficient matrix A1, A2, ···, A in front of the random excitation at each moment is i It is only related to the structural parameters. Therefore, if these coefficient matrices can be calculated in advance, the response at any time can be obtained by linear superposition. In order to obtain these coefficient matrices, the structure can be applied as follows Figure 2 The deterministic unit load excitation shown above can obtain the entire coefficient matrix after completing a complete structural time history analysis.

[0046] Step (2): Establish a deterministic topology optimization formula for a continuum structure with discrete variables, taking the structural volume as the objective function and the most unfavorable extreme response as the constraint function.

[0047] To reduce the computational complexity of the reliability topology optimization solution, the original problem was divided into two phases. In the first phase, the optimization problem is volume minimization under the constraint of the worst-case extreme response. If the optimized design in the first phase fails, the optimization enters the second phase, which is dynamic reliability topology optimization.

[0048] The deterministic topology optimization in the first stage is formulated as

[0049]

[0050] Among them, ρ e = {0,1} represents the e-th discrete design variable, ρ = [ρ1,ρ2,…,ρ M ] is an N-dimensional design variable vector, N is the number of units, Θ = [θ1,θ2,…,θ n ] represents the n-dimensional input random vector, V e represents the unit volume, V represents the total volume of the structure, is the permissible failure probability, P f Represents the failure probability of the power system. The constraint term in the above formula is the extreme value response Y under the most unfavorable conditions. ext (ρ k ,θ wc ,T) constraint, where the extreme value response Y ext (ρ k ,θ wc ,T) refers to the representative point θ under the most unfavorable conditions wc The maximum value of the response at all times, the penultimate constraint in the above equation is the trust region constraint, and the parameter r kIt represents the trust region radius of the kth subproblem, and its physical meaning is the number of design variables that can be changed under the current design.

[0051] Step (3): Solve the deterministic topology optimization problem in step (2) based on the sequential approximate integer programming method. This includes:

[0052] The regular dual algorithm in the sequential approximate integer programming method is used to update the design variables of discrete variable topology optimization, so that the design variables are updated in the direction of the optimal solution of the optimization problem.

[0053] Step (4): Use the time domain explicit method to calculate the structural extreme responses corresponding to all representative points of the current design and use direct probability integration.

[0054] Under a certain topological design ρ, assuming that the dynamic response function of interest is g(ρ, Θ, t), the failure probability based on the first exceedance failure criterion is defined as

[0055] P f (T)=1-Pr{g(ρ,Θ,t)∈Ω s |0≤t≤T} (9)

[0056] Among them, Ω s is the safety domain, and T is the given life cycle. According to the definition in the above formula, once the structural dynamic response exceeds the safety boundary at a certain moment in the time interval [0, T], the structure will fail.

[181] Therefore, if the extreme value of the structural response in [0, T] exceeds the threshold, the structure will fail. Therefore, the first exceedance problem can be solved by constructing extreme value events.

[0057] The extreme value of the structural response g(ρ,Θ,t) in the time interval [0,T] can be expressed as

[0058]

[0059] At a given threshold of the response extreme value of interest Then, the time-varying function can be written as

[0060]

[0061] The probability of first-time failure of the structure derived from the probability density integral theory can be expressed as

[0062]

[0063] The representative points are selected by the two-step point selection method, and the failure probability is calculated by the direct probability integration method.

[0064]

[0065] Among them, θ q and P q are the representative points and assigned probabilities with the same meaning as in the previous chapter, and Nq is the number of representative points.

[0066] The structural responses corresponding to all the representative points above can be efficiently solved using the time domain explicit method.

[0067] Step (5): Use direct probability integration method to perform structural dynamic reliability sensitivity analysis.

[0068] In the framework of direct probability integration method, the sensitivity of power system failure probability to design variables can be expressed as

[0069]

[0070] Where σ DPIM The smoothing parameter is calculated by the weighted kernel density estimation method. From the above formula, we can see that the key to solving the failure probability sensitivity is to solve the sensitivity of the extreme value response.

[0071] Since the maximum operator is not differentiable, in order to derive the sensitivity of the failure probability, it is necessary to use a smooth function to approximate it. Here, the following condensation function is used for approximation:

[0072]

[0073] Among them, η is the aggregation parameter. The larger its value is, the closer it is to the maximum value of the structural response. However, like other aggregation functions, an excessively large aggregation parameter will increase the nonlinearity of the function, thus affecting the optimization solution. In this section, the aggregation parameter is taken as In addition, cp is the correction coefficient, which is defined as

[0074]

[0075] The adjoint method is used to derive the derivative of the extreme response with respect to the design variable. First, the Lagrangian function is constructed as shown below:

[0076]

[0077] In the formula, λ is the adjoint vector. Since the structural motion equation is valid at any time, the last term of the above formula is always zero. In this way, the value of the Lagrangian function is consistent with the extreme response. The values ​​of are the same, and the sensitivities of the two are also consistent. The derivative of the Lagrangian function with respect to the design variable can be expressed as

[0078]

[0079] When the response of interest is the displacement u at a certain degree of freedom ss (ρ, Θ, t), the partial derivative of the absolute value of the response with respect to the displacement field can be expressed as

[0080]

[0081] Among them, e s is a standard basis vector whose sth element is 1 and the rest are all 0.

[0082] Perform partial integration on the last integral term in Eq. (18) and introduce the initial condition Later you can get

[0083]

[0084] In order to speed up the calculation efficiency of sensitivity analysis, let the adjoint vector λ(t) satisfy the following adjoint equation

[0085]

[0086] Assuming τ=Tt, use the variable substitution method to replace λ(t) in the adjoint equation with Λ(τ)=λ(T-τ) and we have

[0087]

[0088] The load terms are

[0089]

[0090] When the adjoint vector Λ is solved, we can get

[0091]

[0092] The above integral expression is difficult to solve analytically, and needs to be solved by numerical integration. The trapezoidal integral formula can be used to numerically solve this integral expression, that is,

[0093]

[0094] Among them, w i The integral weight corresponding to each time integration point is 2w0=w1=···w Nt-1 =2w Nt =Δt. In addition, the derivatives of the mass matrix, stiffness matrix and damping matrix with respect to the design variables are

[0095]

[0096] Where k e and m eThey represent the stiffness matrix and mass matrix when the e-th unit is fully filled with material, p=3 is the penalty factor, and a and b are the two proportional coefficients of Rayleigh damping.

[0097] A careful comparison reveals that the adjoint equation, Equation (22), and the structural motion equation are identical in form, with only the right-hand side load term being different. Therefore, the adjoint equation can also be efficiently solved using the time domain explicit method to obtain the response Λ of the adjoint vector at each discrete time point. Nt-i .

[0098] In addition, only those representative points close to the limit state will have a significant effect on the sensitivity of the failure probability. This conclusion also applies to dynamic problems. Therefore, when performing sensitivity analysis, you can only calculate |Z(ρ,Θ,T)|≤5σ DPIM This reduction in the number of representative points required for sensitivity calculations significantly improves calculation efficiency.

[0099] Step (6): Based on the calculated sensitivity analysis, the reliability topology optimization problem is solved using the sequential approximate integer programming algorithm, and the structural topology design that meets the design requirements is iteratively obtained.

[0100] Example 2

[0101] In order to illustrate the effectiveness of the two-stage method for solving the power reliability topology optimization proposed in this invention, the following Figure 3 The dynamic reliability topology optimization design example of the cantilever beam structure shown is explained.

[0102] The geometry in this example is 200 mm × 100 mm, and the design domain is discretized into 200 × 100 elements. The material elastic modulus E is set to 70 GPa, the material density is 7800 kg / m³, and the Poisson's ratio ν is set to 0.3. The random excitation F is applied at the midpoint of the free end of the cantilever beam. The longitudinal displacement at this point is the response of interest, with a displacement response threshold of 20 mm and a failure probability threshold of 0.01. The specific calculation steps for this example are as follows:

[0103] 1. Component random excitation model. Here, the spectrum expression based on random function is used to generate non-stationary random excitation, that is,

[0104]

[0105] Among them, ω k =k(Δω),k=1,2,···,N f is the discrete frequency, Δω is the frequency step, N f =(ω u -ω l ) / Δω,ω uis the upper limit of the cutoff frequency, ω l is the lower limit of the cutoff frequency. In this section, the frequency step is taken as a fixed value Δω=0.5Hz, and the power spectrum density function S(ω k )=2.5N 2 ·s. In addition, R k and Q k are two standard orthogonal random variables, which are generated by the following random functions, namely

[0106] R k =cos(kΘ1)+sin(kΘ2),Q k =cos(kΘ2)+sin(kΘ1) (28)

[0107] Among them, Θ1 and Θ2 are random vectors Θ f The random variables in , and both random variables obey the uniform distribution of [-π,π]. Finally, d(t) represents the time modulation function, and the specific expression is d(t)=B(e -ψt -e -ζt ), where the parameters are ψ=0.08, ζ=0.16, and B=1000.

[0108] 2. The probability space composed of random variables is divided by the two-step point selection method based on the generalized F deviation to generate representative points and assigned probabilities. Using the generated representative points, a random excitation with a duration of 80s is generated based on formula (27). The representative random excitation time course corresponding to one representative point is as follows: Figure 4 shown.

[0109] 3. Calculate the stiffness, mass and damping matrices of the structure and combine them into an overall matrix.

[0110] 4. Construct a recursive format for the time-domain explicit method to calculate the structural response under arbitrary random excitation and find the structural response function for the most unfavorable case.

[0111] 5. Perform sensitivity analysis on the worst-case performance function, construct and solve the integer programming subproblem, i.e., Equation (8), and use the sequential approximate integer programming method to solve it, and obtain the updated design ρ k .

[0112] 6. Current design ρ k The time-domain explicit method is used to solve for the extreme responses at all representative points, and the direct probability integration method is used to evaluate the failure probability at these points. If the failure probability is greater than zero, the deterministic topology optimization process stops and the reliability topology optimization process begins. Otherwise, the iterative solution of the deterministic topology optimization problem continues.

[0113] 7. Current design ρ kUse the time-domain explicit method to solve for the extreme value response at all representative points, and use the direct probability integration method to evaluate the failure probability at this time. Select representative points that are useful for sensitivity analysis and perform sensitivity analysis.

[0114] 8. Construct the following integer programming subproblem

[0115]

[0116] 9. Solve the above optimization subproblems using sequential approximate integer programming and check whether the updated design meets the convergence criteria (when the mean of the relative change in volume fraction over five consecutive iterations is less than 0.01 and the failure probability constraint is satisfied). If so, stop the iteration. Otherwise, continue iterating until convergence is achieved.

[0117] Figure 5 and Figure 6 The optimized configurations and iteration histories corresponding to four different cutoff frequencies are shown respectively. It can be found that the proposed two-stage efficient solution method for structural dynamic reliability topology optimization under random excitation can efficiently obtain a structural optimization design that meets the dynamic reliability requirements.

Claims

1. A two-stage efficient solution method for structural dynamic reliability topology optimization under random excitation, characterized by: A two-stage solution framework for the topological optimization problem of the dynamic reliability of continuum structures is established. The direct probability integral method is used to accurately and efficiently evaluate the dynamic reliability of the structure. The time-domain explicit method is combined to efficiently solve the sensitivity of the structural response and dynamic reliability to the design variables. The sequential approximate integer programming method is used to update the design variables and obtain the discrete variable topology optimization configuration that meets the reliability requirements.

2. The two-stage efficient solution method for structural dynamic reliability topology optimization under random excitation according to claim 1 is characterized in that The steps are as follows: (1) Analyze the structural response of all representative points corresponding to the initial design using the time domain explicit method; including: A two-step point selection method based on generalized F-deviation is used to partition the probability space, obtaining the representative points and their assigned probabilities required by the direct probability integration method. A time-domain explicit format for the initial design is constructed to efficiently calculate the structural extreme responses corresponding to all representative points. The extreme responses corresponding to the most unfavorable situation under the initial design are obtained. (2) Establish a deterministic topology optimization formula for discrete variables of continuum structures with the structural volume as the objective function and the most unfavorable extreme response as the constraint function; (3) Solve the deterministic topology optimization problem in (2) based on the sequential approximate integer programming method; including: Sensitivity analysis of the worst-case performance function; updating design variables for discrete variable topology optimization using sequential approximate integer programming; (4) Using the time domain explicit method to calculate the structural extreme responses corresponding to all representative points of the current design and using the direct probability integration method to estimate the failure probability of the current design under random excitation, when the failure probability is greater than zero, the optimization enters the solution of the dynamic reliability topology optimization problem; (5) Using direct probability integration method to perform structural dynamic reliability sensitivity analysis; including: The time-domain explicit method is used to efficiently solve the structural extreme responses corresponding to all representative points, and the direct probability integration method is used to calculate the reliability of the current design. The representative points that play a key role in the reliability sensitivity calculation are selected based on the importance representative point screening strategy to reduce the amount of calculation. The reliability sensitivity is efficiently solved using the adjoint method and the time-domain explicit method. (6) Based on the calculated sensitivity analysis, the reliability topology optimization problem is solved using the sequential approximate integer programming algorithm, and the structural topology design that meets the design requirements is iteratively obtained.

3. The two-stage efficient solution method for structural dynamic reliability topology optimization under random excitation according to claim 1 is characterized in that: Step (1): Analyze the structural response of all representative points corresponding to the initial design using the time domain explicit method; including: For a linear system, by discretizing in the time domain, we can get t i The equation of motion of the system at this moment is Where M is the overall mass matrix, C is the overall damping matrix, and K is the overall stiffness matrix. is the acceleration vector, is the velocity vector, U is the displacement vector, F is the random excitation, and L is the positioning matrix used to locate the position of the random excitation. i Indicates the form of a subscript; When using the Newmark-β method to solve the motion equation in the formula, the basic expression of the velocity and displacement corresponding to each time step is Where Δt is the time step, γ and β are two basic parameters in the Newmark-β method, which determine the stability and accuracy of the method. When γ = 0.5 and β = 0.25, the Newmark-β method is unconditionally stable. Using the above two formulas, we can derive The coefficients in the above two equations are Using equation (3) and equation (2), we can get the time domain explicit method response recursive expression: V i =TV i-1 +QF i (5) in, When the initial condition of the system is 0, that is, when V0 is a zero vector, the recursive formula (5) can be simplified to Where the coefficient matrix A1, A2, ···, A in front of the random excitation at each moment is i It is only related to the structural parameters.

4. The two-stage efficient solution method for structural dynamic reliability topology optimization under random excitation according to claim 1 is characterized in that: Step (2): Establish a deterministic topology optimization formula for a continuum structure with discrete variables, taking the structural volume as the objective function and the most unfavorable extreme response as the constraint function; the details are as follows: The original problem is divided into two stages. In the first stage, the optimization problem is the volume minimization problem under the constraint of the most unfavorable extreme response. When the optimization design in the first stage fails, the optimization enters the second stage, which is the dynamic reliability topology optimization. The deterministic topology optimization in the first stage is formulated as Among them, ρ e = {0,1} represents the e-th discrete design variable, ρ = [ρ1,ρ2,…,ρ M ] is an N-dimensional design variable vector, N is the number of units, Θ = [θ1,θ2,…,θ n ] represents the n-dimensional input random vector, V e represents the unit volume, V represents the total volume of the structure, P is the allowable failure probability, P f represents the failure probability of the power system; the constraint term in the above formula is the extreme response Y under the most unfavorable conditions ext (ρ k ,θ wc ,T) constraint, where the extreme value response Y ext (ρ k ,θ wc ,T) refers to the representative point θ under the most unfavorable conditions wc The maximum value of the response at all times, the penultimate constraint in the above equation is the trust region constraint, and the parameter r k represents the trust region radius of the k-th subproblem.

5. The two-stage efficient solution method for structural dynamic reliability topology optimization under random excitation according to claim 1 is characterized in that: Step (3): Solve the deterministic topology optimization problem in step (2) based on the sequential approximate integer programming method; including: using the regular dual algorithm in the sequential approximate integer programming method to update the design variables of the discrete variable topology optimization, so that the design variables of the optimization problem are updated in the direction of the optimal solution.

6. The two-stage efficient solution method for structural dynamic reliability topology optimization under random excitation according to claim 1 is characterized in that: Step (4): Calculate the structural extreme responses corresponding to all representative points of the current design using the time domain explicit method and use direct probability integration; including: Under a certain topological design ρ, assuming that the dynamic response function of interest is g(ρ, Θ, t), the failure probability based on the first exceedance failure criterion is defined as P f (T)=1-Pr{g(ρ,Θ,t)∈Ω s |0≤t≤T}(9) Among them, Ω s is the safety domain, T is the given life cycle; according to the definition in the above formula, once the structural dynamic response exceeds the safety boundary at a certain moment in the time interval [0, T], the structure will fail [181] ; Therefore, if the extreme value of the structural response in [0, T] exceeds the threshold, the structure will fail; Therefore, the first exceedance problem can be solved by constructing extreme value events; The extreme value of the structural response g(ρ,Θ,t) in the time interval [0,T] can be expressed as At a given threshold of the response extreme value of interest Then, the time-varying function can be written as The probability of first-time failure of the structure derived from the probability density integral theory can be expressed as The representative points are selected by the two-step point selection method, and the failure probability is calculated by the direct probability integration method. Among them, θ q and P q are the representative points and assigned probabilities with the same meaning as in the previous chapter, and Nq is the number of representative points; The structural responses corresponding to all the representative points above can be efficiently solved using the time domain explicit method.

7. The two-stage efficient solution method for structural dynamic reliability topology optimization under random excitation according to claim 1 is characterized in that: Step (5): Use direct probability integration method to perform structural dynamic reliability sensitivity analysis; include: In the framework of direct probability integration method, the sensitivity of power system failure probability to design variables can be expressed as Where σ DPIM The smoothing parameter is calculated by the weighted kernel density estimation method. From the above formula, we can see that the key to solving the failure probability sensitivity is to solve the sensitivity of the extreme value response. Since the maximum operator is not differentiable, it needs to be approximated by a smooth function and a condensed function: Among them, η is the aggregation parameter. The larger its value is, the closer it is to the maximum value of the structural response. However, like other aggregation functions, an excessively large aggregation parameter will increase the nonlinearity of the function, thus affecting the optimization solution. The value of the aggregation parameter is In addition, cp is the correction coefficient, which is defined as The adjoint method is used to derive the derivatives of the extreme response with respect to the design variables; first construct the Lagrangian function shown below In the formula, λ is the adjoint vector; because the structural motion equation is valid at any time, the last term of the above formula is always zero; thus, the value of the Lagrangian function is related to the extreme response The values ​​of are the same, and the sensitivities of the two are also consistent; the derivative of the Lagrangian function with respect to the design variable can be expressed as When the response of interest is the displacement u at a certain degree of freedom s s (ρ, Θ, t), the partial derivative of the absolute value of the response with respect to the displacement field can be expressed as Among them, e s is a standard basis vector whose sth element is 1 and the rest of the elements are all 0; Perform partial integration on the last integral term in Eq. (18) and introduce the initial condition Later you can get In order to speed up the calculation efficiency of sensitivity analysis, let the adjoint vector λ(t) satisfy the following adjoint equation Assuming τ=Tt, use the variable substitution method to replace λ(t) in the adjoint equation with Λ(τ)=λ(T-τ) and we have The load terms are When the adjoint vector Λ is solved, we can get The above integral expression is difficult to solve analytically, and needs to be solved by numerical integration. The trapezoidal integral formula can be used to numerically solve this integral expression, that is, Among them, w i The integral weight corresponding to each time integration point is 2w0=w1=···w Nt-1 =2w Nt =Δt; In addition, the derivatives of the mass matrix, stiffness matrix and damping matrix with respect to the design variables are Where k e and m e They represent the stiffness matrix and mass matrix of the e-th element when it is fully filled with material, p = 3 is the penalty factor, a and b are the two proportional coefficients of Rayleigh damping; A careful comparison reveals that the adjoint equation, Equation (22), and the structural motion equation are identical in form, with only the right-hand side load term being different. Therefore, the adjoint equation can be efficiently solved using the time domain explicit method to obtain the response Λ of the adjoint vector at each discrete time point. Nt-i ; In addition, only those representative points close to the limit state will have a significant effect on the sensitivity of the failure probability. This conclusion also applies to dynamic problems. Therefore, when performing sensitivity analysis, only |Z(ρ,Θ,T)|≤5σ can be calculated. DPIM The reduction in the number of representative points required for sensitivity calculation will greatly improve the calculation efficiency.

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