A time-varying reliability topology optimization method for continuum structure
By combining the EOLE and HHT-α algorithms with an improved adjoint variable method, the problem of neglecting the cumulative effect of time-varying problems in topology optimization is solved, achieving efficient and accurate structural reliability design and ensuring the safety and reliability of the structure in dynamic environments.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- HEFEI UNIV OF TECH
- Filing Date
- 2026-02-25
- Publication Date
- 2026-06-05
Smart Images
Figure CN122154094A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of structural optimization design technology, and in particular relates to a time-varying reliability topology optimization method for continuum structures. Background Technology
[0002] Topology optimization (TO) technology, as an advanced lightweight design tool, has been widely used in aerospace, mechanical engineering, and other fields. In practical engineering, structures often bear complex time-varying dynamic loads rather than static loads; therefore, dynamic topology optimization has become a research hotspot.
[0003] However, traditional dynamic topology optimization (DTO) methods are mostly performed under deterministic scenarios, ignoring uncertainties such as material properties, manufacturing tolerances, and load fluctuations that are prevalent in reality. To address this issue, reliability-based topology optimization (RBTO) has been proposed. However, existing RBTO methods have a major drawback when dealing with dynamic problems: they typically simplify complex dynamic problems into equivalent global static problems, ignoring the time-varying correlations between different time points, especially the cumulative effect of load and response over time.
[0004] This simplification leads to a significant overestimation of the true reliability of the structure, failing to capture real physical processes such as cumulative damage or dynamic failure, thus resulting in the optimized topology not being the optimal solution under real dynamic conditions.
[0005] Therefore, integrating time-varying correlations into the topology optimization framework is a necessary requirement for achieving realistic and reliable structural design. However, this integration faces significant technical challenges, mainly including: (1) the massive computational cost brought about by fine time discretization; (2) how to handle the complex time-varying coupling effects in transient responses; and (3) efficiently solving for time-varying reliability and its complex sensitivity to design variables and random variables. Summary of the Invention
[0006] To address the aforementioned technical problems, this invention proposes a time-varying reliability topology optimization method for continuum structures. This invention addresses the shortcomings of traditional reliability-based topology optimization (RBTO), which simplifies dynamic time-varying problems to equivalent static problems and neglects the cumulative effect of load and response over time, leading to overestimation of reliability. Instead, it establishes a topology optimization framework that integrates the uncertainties of time-varying parameters. This method comprehensively considers the stochastic time-varying characteristics of material properties, load amplitude, and load direction. It utilizes the Extended Optimal Linear Estimation (EOLE) method to discretize the complex time-varying performance function into a series of statistically independent instantaneous limit state functions, and employs the Hilber-Hughes-Taylor-α (HHT-α) algorithm to solve for the transient dynamic response to ensure numerical stability. In particular, to overcome the technical bottleneck of high computational cost for time-varying reliability sensitivity, this invention proposes an improved adjoint variable method (AVM) based on an incremental integration strategy. This strategy abandons the inefficient mode of repeatedly performing direct differentiation or adjoint analysis at each time point. Instead, it uses the derivation logic of "integration to differentiation" to invert instantaneous sensitivity by utilizing historical cumulative contributions. This significantly reduces the computational dimension while retaining the ability to handle cross-time domain failure correlations, thereby achieving high-precision and high-efficiency reliability design of continuum structures under dynamic uncertain environments.
[0007] To achieve the above objectives, this invention provides a time-varying reliability topology optimization method for continuum structures, comprising: Based on continuum structure information, a time-varying stochastic parameter initial model of the continuum structure is constructed. Based on the time-varying random parameter initial model, with minimizing the structural volume fraction as the objective function, the relative density of the unit as the design variable, and the time-varying reliability index as the constraint, a time-varying reliability topology optimization mathematical model is constructed. Based on the time-varying reliability topology optimization mathematical model, the displacement, velocity, and acceleration responses of the continuum structure are obtained; Based on the displacement, velocity, and acceleration responses, an instantaneous function is constructed, and the sensitivity information of the function with respect to random variables is calculated based on the instantaneous function. Based on the sensitivity information, a global time-varying reliability index is obtained; Based on the global time-varying reliability index and the adjoint vector, the derivative relationship between the objective function, constraints and design variables is established, and the gradient information is obtained based on the derivative relationship. The continuum structure is optimized based on the gradient information.
[0008] Optionally, the initial model for the time-varying stochastic parameters of the continuum structure includes: The design domain, boundary conditions, and load application locations of the continuum structure are determined according to the actual engineering design requirements, and the design domain is discretized into a finite element mesh. Based on the finite element mesh, and combined with the material properties that follow a Gaussian random process, the time-varying load amplitude, and the load direction, the time-varying uncertain parameters are obtained. Based on the time-varying uncertain parameters, an initial model of time-varying random parameters containing time-varying characteristics is established.
[0009] Optionally, constructing a time-varying reliability topology optimization mathematical model includes: ; in, Represents the i-th design variable. This represents the total number of elements, where M, C, and K represent the mass, damping, and stiffness matrices, respectively. Represents the displacement vector. and Represents the velocity vector and acceleration vector. and V represents the initial displacement and acceleration. and These represent volume fraction, structural volume, and elemental volume, respectively, with g representing constraints. Indicates the allowed values. Indicates the probability of allowable failure. Represents random time-varying loads. For design variables, For the first There are three design variables: x is a random variable, t0 is a time point, and i is the i-th unit.
[0010] Optionally, based on the time-varying reliability topology optimization mathematical model, obtaining the displacement, velocity, and acceleration responses of the continuum structure includes: Based on the design variable distribution in the time-varying reliability topology optimization mathematical model and the stochastic process in the time-varying stochastic parameter initial model, the extended optimal linear estimation method is used to discretize the stochastic process into a finite number of independent standard normal random variables. Based on the standard normal random variables, the dynamic equilibrium equations of the structure are solved using an implicit time integration algorithm to obtain the displacement, velocity, and acceleration responses of the structure at each time node.
[0011] Optionally, the stochastic process can be discretized into a finite number of independent standard normal random variables using the extended optimal linear estimation method, including: Construct the autocorrelation coefficient matrix of the stochastic process and perform eigenvalue decomposition, then select the top few dominant eigenvalues and their corresponding eigenvectors; Based on the dominant eigenvalues and eigenvectors, the stochastic process is expanded into a linear combination of the mean function, the standard deviation function, and mutually independent standard normal random variables, thus transforming the continuous stochastic process into a deterministic function controlled by a vector of random variables.
[0012] Optionally, solving the dynamic equilibrium equations of the structure using implicit time integration algorithms includes: A modified dynamic equilibrium equation is established by introducing numerical damping parameters, and displacement and velocity are updated by combining the Newmark time integration scheme. By defining integral parameter relationships that satisfy the conditions for second-order accuracy and unconditional stability, the linear equation system is solved step by step over time to obtain the transient response data of the structure throughout the entire time history.
[0013] Optionally, calculating the gradient information of the function with respect to the random variable based on the instantaneous function includes: A Lagrangian function is constructed based on random variables of material properties, and the sensitivity expression of the performance function with respect to material properties is derived using the adjoint condition. Based on the random variable of load amplitude, a Lagrange equation is constructed using the standard normal random variable transformed by the extended optimal linear estimation, and the sensitivity of load amplitude is obtained through integral transformation by parts. The adjoint condition is constructed based on the random variable of the load direction to obtain the sensitivity of the load direction; Based on the aforementioned sensitivity expression, the sensitivity of the load amplitude, and the sensitivity of the load direction, the sensitivity of various random variables at any time point is obtained.
[0014] Optionally, based on the sensitivity information, obtaining the global time-varying reliability index includes: Based on the sensitivity information, a failure threshold is constructed, and the most likely failure point is searched in the standard normal space using the sensitivity of the function with respect to random variables. The instantaneous reliability index and instantaneous failure probability are then calculated. The correlation coefficient is calculated based on the covariance of the limit state function of adjacent time nodes, and the average crossing rate within the time interval is calculated using the correlation coefficient and the instantaneous reliability index of the time before and after the time nodes. Based on the Poisson process assumption, the average crossing rate at each moment is numerically integrated over the entire time history. The cumulative failure probability is derived by combining the instantaneous failure probability at the initial moment, and then transformed into a global time-varying reliability index through the inverse function of the standard normal distribution function.
[0015] Optionally, based on the global time-varying reliability index and the adjoint vector, a derivative relationship between the objective function, constraints, and design variables is established. Based on this derivative relationship, gradient information is obtained, including: The sensitivity of reliability indices to failure probability and the coupling term of failure probability to design variables are calculated. The sensitivity of design variables to failure probability is decomposed into instantaneous failure probability sensitivity term and pass-through rate sensitivity term. Calculate the sensitivity of the instantaneous failure probability to the design variables, which is expressed by the product of the adjoint vector and the correlation matrix of the design variables; The sensitivity of the crossing rate to the design variable is calculated, and the sensitivity is expressed by the derivative of the cumulative distribution function and probability density function of the bivariate normal distribution with respect to the design variable and the correlation coefficient. The gradient information is obtained based on the product expression and the derivative expression.
[0016] Compared with the prior art, the present invention has the following advantages and technical effects: This invention explicitly considers the time-varying uncertainties of material properties, load direction, and amplitude. A time-varying reliability design framework is developed to overcome the limitations of traditional methods that ignore time-dependent coupling effects, and to ensure the safety and reliability of the structure under dynamic operating conditions. To efficiently solve the transient dynamic response and accurately simulate time dependence, the Hilber-Hughes-Taylor-α (HHT-α) time integral scheme combined with the Extended Optimal Linear Estimation (EOLE) method is adopted. The PHI2 method is used to evaluate the first exceedance probability within a specified time interval, and the improved adjoint variable method (AVM) based on the incremental integration strategy is used to effectively derive the expressions for the required stochastic sensitivity and design sensitivity. Attached Figure Description
[0017] The accompanying drawings, which form part of this application, are used to provide a further understanding of this application. The illustrative embodiments and descriptions of this application are used to explain this application and do not constitute an undue limitation of this application. In the drawings: Figure 1 This is a flowchart illustrating the design boundary of a simply supported beam according to an embodiment of the present invention. Figure 2 This is a flowchart illustrating the design boundary of the L-shaped beam according to an embodiment of the present invention; Figure 3 This is a flowchart of a time-varying reliability topology optimization method for continuum structures according to an embodiment of the present invention. Detailed Implementation
[0018] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. This application will now be described in detail with reference to the accompanying drawings and embodiments.
[0019] It should be noted that the steps shown in the flowchart in the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although a logical order is shown in the flowchart, in some cases the steps shown or described may be executed in a different order than that shown here.
[0020] This embodiment proposes a time-varying reliability topology optimization method for continuum structures, such as... Figure 3 As shown, the specific steps include: Based on continuum structure information, a time-varying stochastic parameter initial model of the continuum structure is constructed. Based on the time-varying random parameter initial model, with minimizing the structural volume fraction as the objective function, the relative density of the unit as the design variable, and the time-varying reliability index as the constraint, a time-varying reliability topology optimization mathematical model is constructed. Based on the time-varying reliability topology optimization mathematical model, the displacement, velocity, and acceleration responses of the continuum structure are obtained; Based on the displacement, velocity, and acceleration responses, an instantaneous function is constructed, and the sensitivity information of the function with respect to random variables is calculated based on the instantaneous function. Based on the sensitivity information, a global time-varying reliability index is obtained; Based on the global time-varying reliability index and the adjoint vector, the derivative relationship between the objective function, constraints and design variables is established, and the gradient information is obtained based on the derivative relationship. The continuum structure is optimized based on the gradient information.
[0021] Specifically, step 1: Construct an initial model of time-varying stochastic parameters for the continuum structure: This step forms the foundation of the entire technical solution, providing the necessary geometric models and parameter inputs for all subsequent mathematical modeling and physical analysis. Specifically, the design domain, boundary conditions, and load locations of the continuum structure are determined based on actual engineering design requirements, and the design domain is discretized into a finite element mesh. Based on this, the time-varying uncertainty parameters borne by the structure are defined, including: material properties following a Gaussian random process (such as the elastic modulus field), time-varying load amplitude, and load direction. Through these definitions, an initial stochastic dynamic model that reflects the true physical environment of the structure is established.
[0022] Step 2: Construct a mathematical model for Time-Varying Reliability Topology Optimization (TRBTO): This step, based on the physical model and stochastic parameters determined in Step 1, transforms them into a solvable mathematical optimization problem. Specifically, the objective function is to minimize the structural volume fraction, the relative density of the discretized elements from Step 1 is used as the design variable, and a time-varying reliability index is introduced as a constraint to ensure that the failure probability of the structure throughout the entire dynamic time history is below a preset threshold. This model establishes the optimization "objective" and "rules," and subsequent calculations are all aimed at solving for the objective function value in this model and its derivative with respect to the design variables.
[0023] Step 3: Physical field response analysis based on EOLE discretization and HHT-α method: This step builds upon the design variable distribution determined in Step 2 and the stochastic process defined in Step 1. It obtains the physical response of the structure under the current design through numerical calculations, providing data support for subsequent sensitivity analysis. Specifically, the Extended Optimal Linear Estimation (EOLE) method is first used to discretize the time-varying stochastic process in Step 1 into a finite number of independent standard normal random variables. Then, the current element design variable distribution from Step 2 is substituted into the finite element model, and the Hilber-Hughes-Taylor-α (HHT-α) implicit time integration algorithm is used to solve the structure's dynamic equilibrium equations. This step outputs the displacement, velocity, and acceleration responses of the structure at various time points, which are the direct basis for subsequent assessments of structural failure.
[0024] Step 4: Calculate the sensitivity of the function of the random variable based on the improved adjoint variable method: This step utilizes the physical response data obtained in step 3 to pre-calculate the gradient of the function with respect to random variables, aiming to provide the necessary mathematical tools for the reliability iterative solution in step 5. Specifically, instantaneous function is constructed for the random variables such as material, load amplitude, and load direction defined in step 1. To avoid the high cost of direct differentiation, an improved adjoint variable method based on incremental integration is adopted. By constructing the Lagrangian function and adjoint equation, the instantaneous sensitivity is efficiently derived from the cumulative effect using the "integral to differential" strategy. The random variable gradient information obtained in this step is a prerequisite for the FORM algorithm to find the most likely failure point in the next step.
[0025] Step 5: Time-varying reliability index evaluation based on FORM and PHI2 methods: This step comprehensively utilizes the physical response from step 3 and the gradient information calculated in step 4 to quantitatively assess the safety of the current design and determine whether it meets the constraints set in step 2. Specifically, at each time point, using the gradient information provided in step 4, the instantaneous reliability index is iteratively solved using the first-order reliability method (FORM). Based on this, the PHI2 crossover method is applied to calculate the average crossover rate of the stochastic process between adjacent time steps, and then the cumulative failure probability of the system is derived through time integration and converted into a global time-varying reliability index. The calculation results (reliability index) of this step are not only used for constraint determination but are also the core object for deriving the design sensitivity in step 6.
[0026] Step 6: Sensitivity analysis of time-varying reliability indices with respect to design variables: This step, based on the failure probability model obtained in step 5 and the adjoint vector calculated in step 4, establishes the derivative relationship between the objective function, constraints, and design variables, providing direction for the optimization algorithm. The reliability index and failure probability are interchangeable; the failure probability model measures the failure probability, and the adjoint variables are obtained using the adjoint variable method in step 4. Specifically, the sensitivity of the time-varying reliability index to the design variable (relative density of elements) is derived using the chain rule. This calculation process combines the adjoint analysis results from step 4, decomposing the complex time-varying failure probability derivative with respect to the design variable into a linear combination of the instantaneous failure probability derivative and the traversal rate derivative. This step outputs accurate gradient information, telling the optimizer how to modify the element density to most effectively improve structural reliability.
[0027] Step 7: Design variable updates and convergence judgment: This step receives the sensitivity information output from step 6, performs the design variable update operation, and controls the closed loop of the entire optimization process. Specifically, the calculated sensitivity is input into the Moving Asymptotic Method (MMA) optimizer to update the relative density of each element in the design domain. Then, it determines whether the optimization has converged: if the maximum rate of change of the design variables in two adjacent iterations is less than a preset threshold or the objective function tends to stabilize, convergence is determined, iteration stops, and the optimal topology configuration under the model in step 2 is output; otherwise, it returns to step 3, performs a new physical response analysis based on the updated design variables, and enters the next optimization cycle.
[0028] Furthermore, the initial model for the time-varying stochastic parameters of the continuum structure includes: The design domain, boundary conditions, and load application locations of the continuum structure are determined according to the actual engineering design requirements, and the design domain is discretized into a finite element mesh. Based on the finite element mesh, and combined with the material properties that follow a Gaussian random process, the time-varying load amplitude, and the load direction, the time-varying uncertain parameters are obtained. Based on the time-varying uncertain parameters, an initial model of time-varying random parameters containing time-varying characteristics is established.
[0029] Specifically, this step mainly includes two parts: establishing the finite element discretization model and defining the time-varying random field.
[0030] First, the design domain and mesh generation refer to defining the physical space range within which material distribution is allowed, based on the geometric dimensions of the actual engineering structure, and discretizing this continuous region into a finite number of regular elements. These elements constitute the basic design variable carrier for subsequent topology optimization. The boundary conditions refer to applying displacement constraints to nodes at specific locations in the discretized model according to the working conditions, in order to simulate the support state of the structure under real-world conditions.
[0031] Secondly, the time-varying uncertainty parameters refer to three types of key variables that follow a Gaussian random process introduced into the dynamic analysis to simulate the randomness of the real physical environment. Specifically, these include: Material property uncertainty: mainly refers to the fact that material parameters such as the elastic modulus of the structure are not constant, but random variables that follow a normal distribution, reflecting the discreteness in the material preparation process; Time-varying load amplitude: refers to the fact that the magnitude of the external load is not constant, but rather exhibits a random process that fluctuates with time, and there is a time correlation between the load amplitudes at different times, which is described by an autocorrelation function; Load direction: refers to the angle of action of the external load, which deflects randomly over time and is also modeled as a random process with statistical characteristics.
[0032] By coupling the geometric model with random parameters using the above definition, an initial stochastic dynamic model containing time-varying characteristics is established.
[0033] Furthermore, the mathematical model for time-varying reliability topology optimization includes: ; in, Represents the i-th design variable. This represents the total number of elements, where M, C, and K represent the mass, damping, and stiffness matrices, respectively. Represents the displacement vector. and Represents the velocity vector and acceleration vector. and V represents the initial displacement and acceleration. and These represent volume fraction, structural volume, and elemental volume, respectively, with g representing constraints. Indicates the allowed values. Indicates the probability of allowable failure. Represents random time-varying loads. For design variables, For the first There are three design variables: x is a random variable, t0 is a time point, and i is the i-th unit.
[0034] Furthermore, based on the aforementioned time-varying reliability topology optimization mathematical model, obtaining the displacement, velocity, and acceleration responses of the continuum structure includes: Based on the design variable distribution in the time-varying reliability topology optimization mathematical model and the stochastic process in the time-varying stochastic parameter initial model, the extended optimal linear estimation method is used to discretize the stochastic process into a finite number of independent standard normal random variables. Based on the standard normal random variables, the dynamic equilibrium equations of the structure are solved using an implicit time integration algorithm to obtain the displacement, velocity, and acceleration responses of the structure at each time node.
[0035] Specifically, this step follows the design variable distribution determined in step 2 and the stochastic process defined in step 1, aiming to obtain the physical response of the structure under dynamic uncertainty through numerical calculation. The execution process consists of two parts: stochastic process discretization and dynamic equation solving. First, the extended optimal linear estimation (EOLE) method is used to discretize the stochastic process. This is to enable the handling of continuously time-varying random parameters (such as time-varying load amplitude) defined in step 1 in numerical calculations. The autocorrelation coefficient matrix of the stochastic process is first constructed using the EOLE method, which approximates it as a linear combination of a finite number of standard normal random variables. And perform eigenvalue decomposition on it. Before selection... Dominant eigenvalues and its corresponding eigenvectors random process Expand to the following form: ; in, and These are the mean function and standard deviation function of the stochastic process, respectively. They are mutually independent standard normal random variables; for The correlation vector between the random variable at time step and the random vector at discrete time points. Using this formula, the originally complex continuous stochastic process is transformed into a process consisting of a vector of random variables. The deterministic function of control provides input for the subsequent deterministic finite element solution.
[0036] Secondly, the transient dynamic equilibrium equations of the structure are solved based on the HHT-α algorithm. The discretized random parameters and the current element design variables (relative density) are substituted into the finite element model to construct the global mass matrix. Damping matrix and stiffness matrix To filter out high-frequency numerical noise and ensure unconditional stability of the calculation, the Hilber-Hughes-Taylor-α (HHT-α) implicit time integration method is employed. This method introduces a numerical damping parameter. The following modified dynamic equilibrium equations were established: ; in, , , These represent the displacement, velocity, and acceleration vectors, respectively; subscripts and Indicates the first and One time step; Let be the external load vector. To solve the above equations, the displacement and velocity are updated using the Newmark time integral scheme: ; ; In the formula, The time step is [value]. To ensure the algorithm has second-order accuracy and unconditional stability, the integration parameter [value] is [value]. The following relationship must be satisfied: By solving the linear equations step by step through the above steps, the transient response data of the structure over the entire time history (including the displacement vector time history of all nodes) is finally obtained. Velocity vector time history and acceleration vector time history These data will serve as the physical basis for reliability assessment in subsequent steps.
[0037] Furthermore, the extended optimal linear estimation method is used to discretize the stochastic process into a finite number of independent standard normal random variables, including: Construct the autocorrelation coefficient matrix of the stochastic process and perform eigenvalue decomposition, then select the top few dominant eigenvalues and their corresponding eigenvectors; Based on the dominant eigenvalues and eigenvectors, the stochastic process is expanded into a linear combination of the mean function, the standard deviation function, and mutually independent standard normal random variables, thus transforming the continuous stochastic process into a deterministic function controlled by a vector of random variables.
[0038] Furthermore, solving the dynamic equilibrium equations of the structure using the implicit time integration algorithm includes: A modified dynamic equilibrium equation is established by introducing numerical damping parameters, and displacement and velocity are updated by combining the Newmark time integration scheme. By defining integral parameter relationships that satisfy the conditions for second-order accuracy and unconditional stability, the linear equation system is solved step by step over time to obtain the transient response data of the structure throughout the entire time history.
[0039] Furthermore, calculating the gradient information of the function with respect to the random variable based on the instantaneous function includes: A Lagrangian function is constructed based on random variables of material properties, and the sensitivity expression of the performance function with respect to material properties is derived using the adjoint condition. Based on the random variable of load amplitude, a Lagrange equation is constructed using the standard normal random variable transformed by the extended optimal linear estimation, and the sensitivity of load amplitude is obtained through integral transformation by parts. The adjoint condition is constructed based on the random variable of the load direction to obtain the sensitivity of the load direction; Based on the aforementioned sensitivity expression, the sensitivity of the load amplitude, and the sensitivity of the load direction, the sensitivity of various random variables at any time point is obtained.
[0040] Specifically, when considering dynamic compliance constraints, a Lagrangian function is proposed to evaluate the sensitivity of material property variables. The total dynamic compliance for this time interval is expressed as: ; The sensitivity of the random variable x is as follows: ; When the constraint is switched to a displacement constraint, the force vector F is replaced by the identity matrix I. Then, for a single time node, the sensitivity of performance g is calculated as follows: ; Since there is no correlation between the initial conditions and the structural properties, the last term in the equation becomes... Then, the accompanying conditions are as follows: ; therefore, The sensitivity is determined by the following formula: ; same, This can also be derived in the same way. Therefore, the performance function g with respect to the random variable... The sensitivity is expressed as: ; in, Indicates the corresponding time interval The adjoint vector.
[0041] When When the load amplitude is considered a random variable, it is converted to Z using EOLE, and the sensitivity of the performance function g to Z is calculated as follows: ; For calculation Lagrange equation The construction is as follows: ; The sensitivity to load amplitude is as follows: ; Where f represents It is a product of The vector is composed of l and m, where l and m represent the l-th and m-th time points, respectively. It satisfies the following condition: ; Since the initial conditions are unrelated to the load amplitude, the last term in the equation, transformed by integration by parts, becomes: ; Then, the accompanying conditions are obtained. ; therefore, The sensitivity is described as follows: ; g about Sensitivity at time points The expression is ; Similarly, when the load direction is treated as a random variable, the Lagrange equations... Its structure is as follows: ; The accompanying conditions are as follows: ; Then, calculate about in the following way Sensitivity in load direction: ; g regarding the loading direction at time points The sensitivity assessment is as follows: .
[0042] Furthermore, based on the sensitivity information, obtaining the global time-varying reliability index includes: Based on the sensitivity information, a failure threshold is constructed, and the most likely failure point is searched in the standard normal space using the sensitivity of the function with respect to random variables. The instantaneous reliability index and instantaneous failure probability are then calculated. The correlation coefficient is calculated based on the covariance of the limit state function of adjacent time nodes, and the average crossing rate within the time interval is calculated using the correlation coefficient and the instantaneous reliability index of the time before and after the time nodes. Based on the Poisson process assumption, the average crossing rate at each moment is numerically integrated over the entire time history. The cumulative failure probability is derived by combining the instantaneous failure probability at the initial moment, and then transformed into a global time-varying reliability index through the inverse function of the standard normal distribution function.
[0043] Furthermore, based on the global time-varying reliability index and the adjoint vector, a derivative relationship between the objective function, constraints, and design variables is established. Based on this derivative relationship, gradient information is obtained, including: The sensitivity of reliability indices to failure probability and the coupling term of failure probability to design variables are calculated. The sensitivity of design variables to failure probability is decomposed into instantaneous failure probability sensitivity term and pass-through rate sensitivity term. Calculate the sensitivity of the instantaneous failure probability to the design variables, which is expressed by the product of the adjoint vector and the correlation matrix of the design variables; The sensitivity of the crossing rate to the design variable is calculated, and the sensitivity is expressed by the derivative of the cumulative distribution function and probability density function of the bivariate normal distribution with respect to the design variable and the correlation coefficient. The gradient information is obtained based on the product expression and the derivative expression.
[0044] Specifically, this step follows the transient physical response data of the structure obtained in step 3 and directly utilizes the sensitivity information of the function with respect to random variables calculated in step 4. The aim is to quantitatively assess the safety of the current topology configuration across the entire time domain, providing a basis for subsequent judgments on whether constraints are met, and also providing a foundational model for the design sensitivity analysis in step 6. The specific execution process is as follows: First, the instantaneous limit state function is constructed and the instantaneous reliability index is calculated. Based on the structure obtained in step 3, at any time point... displacement response Combined with a preset failure threshold (such as the maximum allowable displacement) Construct the instantaneous limit state function To calculate the instantaneous failure probability at that moment, a first-order reliability method (FORM) is employed. In this process, the sensitivity of the function to the random variable, obtained in step 4 beforehand, is utilized. As the iterative gradient, the most likely failure point is searched in the standard normal space to calculate the instantaneous reliability index at that moment. .but Instantaneous failure probability at time 1 It can be represented as: ; in, The cumulative distribution function (CDF) of the standard normal distribution is used. Secondly, the average crossing rate of the stochastic process is calculated based on the PHI2 method. Since the structural response exhibits time autocorrelation, simply adding instantaneous probabilities would introduce errors; therefore, the PHI2 crossing method is used to calculate the rate at which the stochastic process enters the failure domain from the safe domain. For adjacent time nodes... and The correlation coefficient is calculated using the covariance of the limit state function. Using the correlation coefficient and the instantaneous reliability index at the preceding and following times, the time interval is calculated. Average crossing rate within The calculation formula is as follows: ; in, The correlation coefficient is The binary standard normal cumulative distribution function is used. Finally, the cumulative failure probability is evaluated and a global time-varying reliability index is output. Based on the Poisson process assumption, the crossing events at each time period are considered to be independent, and the average crossing rate at each time point calculated above is used as the average crossing rate over the entire time history. Numerical integration is performed on the above, combined with the initial time. From the instantaneous failure probability, the cumulative failure probability of the structure in the entire time domain is derived. : ; To facilitate its treatment as a constraint in mathematical optimization models, the cumulative failure probability is transformed into a global time-varying reliability index of the system using the inverse function of the standard normal distribution function. : ; If this indicator Greater than or equal to the target reliability index preset in step 2 If the design meets the reliability constraints, then the current design is deemed to satisfy the reliability constraints; otherwise, step 6 is required to calculate the sensitivity to guide the design update.
[0045] The sensitivity of the reliability index to design variables can be calculated as the product of its sensitivity to the failure probability and the coupling term between the failure probability and the design variables, as follows: ; Subsequently, the sensitivity of design variables to failure probability can be derived as follows: ; in and It can be calculated as follows: ; ; Therefore, the sensitivity of the crossing rate to design variables can be derived as follows: ; and They are expressed as follows: ; ; ; in, and These are the cumulative distribution function (CDF) and probability density function (PDF) of the standard univariate normal distribution, respectively. and These are the CDF and PDF of the binary normal distribution, respectively.
[0046] It is important to note that The sensitivity regarding the design variables is: ; and This can be deduced as: ; ; Furthermore, the specific details of variable updates and convergence judgment in step 7 are as follows: This step follows the sensitivity information calculated in step 6, aiming to use gradient guidance to find the optimal material distribution while satisfying all constraints, and to control the termination of the entire iterative process. The specific execution process is as follows: First, the design variables are updated using the Moving Asymptotic Method (MMA). Given the characteristics of topology optimization problems—numerous variables and nonlinear constraints—the MMA is used as the optimizer. This method, by introducing a moving asymptote, approximates the original complex nonlinear optimization problem at the current design point as a series of explicit convex subproblems. Specifically, in the... In this iteration, MMA approximates the original optimization problem as a subproblem of the following form for solution: ; in, Design variable (cell density); and These are the upper and lower asymptotes determined by the information from the previous iteration; as well as The coefficients are determined based on the sensitivity of the objective function and constraint function calculated in step 6. These are slack variables. The updated design variable vector is obtained by solving this convex subproblem. Secondly, a convergence check is performed. After updating the design variables, it is necessary to check whether the optimization process has converged. This invention uses the maximum rate of change of the design variables as the convergence criterion. The specific calculation formula is as follows: ; in, and They represent the first Second and third In the nth iteration The relative density values of each unit; The preset convergence tolerance is used. Finally, process control and iterative looping are implemented.
[0047] Logical judgment is made based on the calculation results of the above convergence criteria: If If the optimization process converges, the iteration stops, and the final cell density distribution is output as the optimal topology. If the convergence has not yet occurred, then we must return to step 3 and update the design variables. Substitute the finite element model back into the model, and perform transient dynamic response analysis, stochastic process discretization, and subsequent sensitivity calculations based on the new material distribution. Then, proceed to the next optimization cycle until the convergence condition is met.
[0048] The following is a detailed description of this embodiment: Example 1: In this example, the topology optimization layout of a two-dimensional simply supported beam under time-varying uncertainty conditions is studied.
[0049] like Figure 1 As shown, a fixed beam is subjected to a half-cycle cosine force at the midpoint of its upper span. Its length L is 8 meters and its width H is 2 meters. The structure contains 14,400 elements. It is assumed that random variables, including material properties, load direction, and load amplitude, follow a normal distribution. Their mean and standard deviation are shown in Table 1. The maximum load duration is 0.01 seconds. The maximum allowable dynamic compliance is 0.01. The optimization results are shown in Table 2.
[0050] The time-varying half-cycle sinusoidal load was generated by Monte Carlo simulation (MCS). Its mean and standard deviation are 1000 and 50, respectively, and the load values are expressed as follows: ; in This represents the e-th time point. Represents the initial load value. This represents the loading amplitude at the e-th time point. This represents the maximum loading time. This randomness is modeled using an autocorrelation function. The autocorrelation function describing the time correlation is analytically expressed as: ; Table 1 Table 2 Example 2: In this example, the topology optimization layout of L-shaped beams under time-varying uncertainty conditions is studied.
[0051] A test was conducted on an L-shaped beam. The design domain was divided into 10,000 quadrilateral elements, such as... Figure 2 As shown. The load is applied at point A, and the allowable displacement of point A in the Y direction is 0.08 mm. Its length H is 1 meter and its width L is 0.6 meters. Random variables are listed in Table 4. The maximum loading time is 0.005 seconds. Uncertainty parameters are shown in Table 3. Optimization results are shown in Table 4.
[0052] The time-varying half-cycle sinusoidal load was generated by Monte Carlo simulation (MCS). Its mean and standard deviation are 1000 and 50, respectively, and the load values are expressed as follows: ; in This represents the e-th time point. Represents the initial load value. This represents the loading amplitude at the e-th time point. This represents the maximum loading time. This randomness is modeled using an autocorrelation function. The autocorrelation function describing the time correlation is analytically expressed as: ; Table 3 Table 4 The above are merely preferred embodiments of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.
Claims
1. A time-varying reliability topology optimization method for continuum structures, characterized in that, include: Based on continuum structure information, a time-varying stochastic parameter initial model of the continuum structure is constructed. Based on the time-varying random parameter initial model, with minimizing the structural volume fraction as the objective function, the relative density of the unit as the design variable, and the time-varying reliability index as the constraint, a time-varying reliability topology optimization mathematical model is constructed. Based on the time-varying reliability topology optimization mathematical model, the displacement, velocity, and acceleration responses of the continuum structure are obtained; Based on the displacement, velocity, and acceleration responses, an instantaneous function is constructed, and the sensitivity information of the function with respect to random variables is calculated based on the instantaneous function. Based on the sensitivity information, a global time-varying reliability index is obtained; Based on the global time-varying reliability index and the adjoint vector, the derivative relationship between the objective function, constraints and design variables is established, and the gradient information is obtained based on the derivative relationship. The continuum structure is optimized based on the gradient information.
2. The time-varying reliability topology optimization method for continuum structures according to claim 1, characterized in that, The initial model for time-varying stochastic parameters of a continuum structure includes: The design domain, boundary conditions, and load application locations of the continuum structure are determined according to the actual engineering design requirements, and the design domain is discretized into a finite element mesh. Based on the finite element mesh, and combined with the material properties that follow a Gaussian random process, the time-varying load amplitude, and the load direction, the time-varying uncertain parameters are obtained. Based on the time-varying uncertain parameters, an initial model of time-varying random parameters containing time-varying characteristics is established.
3. The time-varying reliability topology optimization method for continuum structures according to claim 1, characterized in that, The construction of a time-varying reliability topology optimization mathematical model includes: ； in, Represents the i-th design variable. This represents the total number of elements, where M, C, and K represent the mass, damping, and stiffness matrices, respectively. Represents the displacement vector. and Represents the velocity vector and acceleration vector. and V represents the initial displacement and acceleration. and These represent volume fraction, structural volume, and elemental volume, respectively, with g representing constraints. Indicates the allowed values. Indicates the probability of allowable failure. Represents random time-varying loads. For design variables, For the first There are three design variables: x is a random variable, t0 is a time point, and i is the i-th unit.
4. The time-varying reliability topology optimization method for continuum structures according to claim 1, characterized in that, Based on the aforementioned time-varying reliability topology optimization mathematical model, the displacement, velocity, and acceleration responses of the continuum structure are obtained, including: Based on the design variable distribution in the time-varying reliability topology optimization mathematical model and the stochastic process in the time-varying stochastic parameter initial model, the extended optimal linear estimation method is used to discretize the stochastic process into a finite number of independent standard normal random variables. Based on the standard normal random variables, the dynamic equilibrium equations of the structure are solved using an implicit time integration algorithm to obtain the displacement, velocity, and acceleration responses of the structure at each time node.
5. The time-varying reliability topology optimization method for continuum structures according to claim 4, characterized in that, The extended optimal linear estimation method is used to discretize the stochastic process into a finite number of independent standard normal random variables, including: Construct the autocorrelation coefficient matrix of the stochastic process and perform eigenvalue decomposition, then select the top few dominant eigenvalues and their corresponding eigenvectors; Based on the dominant eigenvalues and eigenvectors, the stochastic process is expanded into a linear combination of the mean function, the standard deviation function, and mutually independent standard normal random variables, thus transforming the continuous stochastic process into a deterministic function controlled by a vector of random variables.
6. The time-varying reliability topology optimization method for continuum structures according to claim 4, characterized in that, Solving the dynamic equilibrium equations of a structure using implicit time integration algorithms includes: A modified dynamic equilibrium equation is established by introducing numerical damping parameters, and displacement and velocity are updated by combining the Newmark time integration scheme. By defining integral parameter relationships that satisfy the conditions for second-order accuracy and unconditional stability, the linear equation system is solved step by step over time to obtain the transient response data of the structure throughout the entire time history.
7. The time-varying reliability topology optimization method for continuum structures according to claim 1, characterized in that, The calculation of the gradient information of the function with respect to the random variable based on the instantaneous function includes: A Lagrangian function is constructed based on random variables of material properties, and the sensitivity expression of the performance function with respect to material properties is derived using the adjoint condition. Based on the random variable of load amplitude, a Lagrange equation is constructed using the standard normal random variable transformed by the extended optimal linear estimation, and the sensitivity of load amplitude is obtained through integral transformation by parts. The adjoint condition is constructed based on the random variable of the load direction to obtain the sensitivity of the load direction; Based on the aforementioned sensitivity expression, the sensitivity of the load amplitude, and the sensitivity of the load direction, the sensitivity of various random variables at any time point is obtained.
8. The time-varying reliability topology optimization method for continuum structures according to claim 1, characterized in that, Based on the sensitivity information, the global time-varying reliability index is obtained as follows: Based on the sensitivity information, a failure threshold is constructed, and the most likely failure point is searched in the standard normal space using the sensitivity of the function with respect to random variables. The instantaneous reliability index and instantaneous failure probability are then calculated. The correlation coefficient is calculated based on the covariance of the limit state function of adjacent time nodes, and the average crossing rate within the time interval is calculated using the correlation coefficient and the instantaneous reliability index of the time before and after the time nodes. Based on the Poisson process assumption, the average crossing rate at each moment is numerically integrated over the entire time history. The cumulative failure probability is derived by combining the instantaneous failure probability at the initial moment, and then transformed into a global time-varying reliability index through the inverse function of the standard normal distribution function.
9. The time-varying reliability topology optimization method for continuum structures according to claim 1, characterized in that, Based on the global time-varying reliability index and the adjoint vector, a derivative relationship between the objective function, constraints, and design variables is established. Based on this derivative relationship, gradient information is obtained, including: The sensitivity of reliability indices to failure probability and the coupling term of failure probability to design variables are calculated. The sensitivity of design variables to failure probability is decomposed into instantaneous failure probability sensitivity term and pass-through rate sensitivity term. Calculate the sensitivity of the instantaneous failure probability to the design variables, which is expressed by the product of the adjoint vector and the correlation matrix of the design variables; The sensitivity of the crossing rate to the design variable is calculated, and the sensitivity is expressed by the derivative of the cumulative distribution function and probability density function of the bivariate normal distribution with respect to the design variable and the correlation coefficient. The gradient information is obtained based on the product expression and the derivative expression.