A Probability-Based Topology Optimization Method for Structural Reliability

Abstract

A probabilistic-based structural reliability topology optimization method belongs to the general technology field of advanced structures and mechanisms of aircraft. The present invention includes the following steps: for a structure with material and load uncertainties, combining probabilistic model description and reliability index definition to establish a structural topology optimization model with reliability index constraint conditions or objectives; solving the structural topology optimization model; when both the uncertainty variables and the design variables satisfy the convergence conditions, an optimal topology design meeting the requirements of the reliability index is obtained. The present invention, for a structure with material and load uncertainties, uses uncertainty description and reliability index definition under a probabilistic model framework to establish a structural topology optimization model that meets the constraint conditions of the mechanical property reliability index, and uses a functional measurement method and a single-loop strategy for a nested optimization problem sequence to solve it, obtaining an optimal topology design meeting the requirements of the reliability index.

Description

Technical Field

[0001] The present invention relates to a probability-based structural reliability topology optimization method, belonging to the general technical field of advanced structures and mechanisms of aircrafts. Background Art

[0002] Lightweight and high reliability are important goals pursued in the structural design of aerospace, aviation, and weaponry. Due to more stringent performance requirements for structural weight, stiffness, strength, dynamic performance, etc., the structural design method based on experience or specifications restricts the improvement of comprehensive performance to a certain extent. Therefore, structural optimization, especially topology optimization technology, has become an important means to support structural design innovation. Through long-term basic research and technological breakthroughs in China, significant progress has been made in the optimization design technology of such structures. However, as the structural design of various high-end equipment approaches the mechanical performance limit, the uncertainties of material properties, manufacturing errors, loads, etc. bring greater unpredictability to the performance of the final product, and even affect the reliability of the structural mechanical performance and expected functions. Although the traditional safety factor method sets a certain safety margin for considering uncertainties such as materials or loads to some extent, it is very likely that an overly conservative design will result from an unreasonably large safety factor, or the final structure achieved according to the optimization results may face failure risks due to the incomplete consideration of multiple uncertainty factors.

[0003] Western countries such as the United States attach great importance to this. For example, NASA and Sandia Laboratories regard the uncertain structural optimization as one of the challenging problems in the future, and AIAA has continuously held 18 sessions of the "Non-Deterministic Approaches Conference". As an important tool for driving design innovation in the aerospace and aviation fields, structural topology optimization design must consider these uncertainty factors. Reliability topology optimization design is to seek the optimal topology design plan for structural performance under the condition of ensuring a certain reliability requirement. Summary of the Invention

[0004] The technical problem solved by the present invention is to overcome the deficiencies of the prior art and provide a probability-based structural reliability topology optimization method. For structures with material and load uncertainties, uncertainty description and reliability index definition under a probability model framework are adopted to establish a structural topology optimization model that satisfies the constraint conditions of mechanical performance reliability indexes, and a functional measurement method and a nested optimization problem sequence single-loop strategy are used for solution to obtain the optimal topology design that meets the reliability index requirements.

[0005] The technical solution of the present invention is: a probability-based structural reliability topology optimization method, including the following steps:

[0006] For a structure with material and load uncertainties, a structural topology optimization model with reliability index constraints or objectives is established by combining probability model description and reliability index definition;

[0007] Solve the structural topology optimization model; when both the uncertainty variables and the design variables satisfy the convergence conditions, the optimal topology design that meets the reliability index requirements is obtained.

[0008] Furthermore, the structural topology optimization model is

[0009]

[0010] s.t.α j (ρ)≥0(j=1,2,...,N g ),

[0011] ρ ≤ρ e ≤1(e=1,2,...,N),

[0012] where

[0013]

[0014]

[0015] Among them, ρ represents the element density vector, u is the standardized uncertain parameter, g j (ρ,u) is the performance function of the structure, β is the reliability index, V is the total volume of the structure, ρ e is the element density, V e is the element volume, α j is the performance function value, ρ is the lower limit of the element density, N g is the number of performance functions, N is the number of elements, β j is the reliability index.

[0016] Furthermore, the method for solving the structural topology optimization model includes the following steps:

[0017] According to the structural topology optimization model, use the sequential single-loop method to continuously solve the inner and outer layer optimizations by constructing a deterministic sub-optimization problem;

[0018] Use the MMA gradient optimization algorithm to solve the sequential single-loop optimization sub-problem.

[0019] Furthermore, the deterministic sub-optimization problem is:

[0020] For k=1,2,...

[0021] Find ρ

[0022]

[0023] Subject to g j (ρ, u (k) ) ≥ 0 (j = 1, 2,..., N g )

[0024] ρ ≤ ρ e ≤ 1 (e = 1, 2,..., N)

[0025] Where ρ represents the element density vector, u is the standardized uncertain parameter, g j (ρ, u) is the performance function of the structure, V is the total volume of the structure, ρ e is the element density, V e is the element volume, α j is the performance function value, ρ is the lower limit of the element density, N g is the number of performance functions, and N is the number of elements.

[0026] Furthermore, the MPP point u in the deterministic sub-optimization problem at the k-th step (k) is calculated using the trust-region iteration algorithm:

[0027]

[0028] length = ||(1 - λ k )u (k) + λ k u C ||2 / β j

[0029]

[0030] Where u is the standardized uncertain parameter, β j is the reliability index, is the gradient of the performance function, λ k is the adjustment parameter, u (k) is the iteration point at the k-th step, and u C is the temporary iteration point.

[0031] Furthermore, after obtaining the optimal topology design that meets the requirements of the reliability index, calculate the reliability index and the failure probability, and check and verify the final results; if the check and verification are successful, end; otherwise, re-solve the structure topology optimization model.

[0032] Furthermore, the method for verifying and checking the final result is the first-order second-moment method and the response surface method.

[0033] A probability-based structural reliability topology optimization system includes:

[0034] A modeling module, for a structure with material and load uncertainties, combines probability model description and reliability index definition to establish a structural topology optimization model with reliability index constraints or objectives;

[0035] A design module, solves the structural topology optimization model; when both the uncertainty variables and the design variables meet the convergence conditions, an optimal topology design that meets the reliability index requirements is obtained;

[0036] The structural topology optimization model is

[0037]

[0038] s.t.α j (ρ)≥0(j=1,2,...,N g ),

[0039] ρ ≤ρ e ≤1(e=1,2,...,N),

[0040] where

[0041]

[0042]

[0043] Among them, ρ represents the element density vector, u is the standardized uncertain parameter, g j (ρ,u) is the performance function of the structure, β is the reliability index, V is the total volume of the structure, ρ e is the element density, V e is the element volume, α j is the performance function value, ρ is the lower limit of the element density, N g is the number of performance functions, N is the number of elements, β j is the reliability index;

[0044] The method for solving the structural topology optimization model includes the following steps:

[0045] According to the structural topology optimization model, using the sequential single-loop method, by constructing a deterministic sub-optimization problem, the inner and outer layer optimizations are solved continuously;

[0046] Solve the sequential single-loop sub-optimization problem using the MMA gradient optimization algorithm;

[0047] The deterministic sub-optimization problem is:

[0048] For k = 1, 2,...

[0049] find ρ

[0050]

[0051] subject to g j (ρ, u (k) ) ≥ 0 (j = 1, 2,..., N g )

[0052] ρ ≤ ρ e ≤ 1 (e = 1, 2,..., N)

[0053] where ρ represents the element density vector, u is the standardized uncertain parameter, g j (ρ, u) is the performance function of the structure, V is the total volume of the structure, ρ e is the element density, V e is the element volume, α j is the performance function value, ρ is the lower limit of the element density, N g is the number of performance functions, N is the number of elements;

[0054] The MPP point u in the k-th step deterministic sub-optimization problem (k) is calculated using the trust region iteration algorithm:

[0055]

[0056] length = ||(1 - λ k )u (k) + λ k u C ||2 / β j

[0057]

[0058] where u is the standardized uncertain parameter, β j is the reliability index, is the gradient of the performance function, λ k is the adjustment parameter, u (k) is the k-th step iteration point, u C is the temporary iteration point;

[0059] After obtaining the optimal topological design that meets the requirements of the reliability index, calculate the reliability index and the failure probability, and check and verify the final result; if the check and verification are successful, end; otherwise, re-solve the structural topology optimization model.

[0060] The method for checking and verifying the final result is the first-order second-moment method and the response surface method.

[0061] A computer-readable storage medium stores a computer program, and when the computer program is executed by a processor, the steps of the method for probabilistic-based structural reliability topology optimization are implemented.

[0062] A probabilistic-based structural reliability topology optimization device includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, the steps of the method for probabilistic-based structural reliability topology optimization are implemented.

[0063] The advantages of the present invention compared with the prior art are as follows:

[0064] (1) The optimization result has higher reliability. Compared with the traditional design based on the safety factor method, the method proposed in this patent can give a structural topology that strictly meets the reliability requirements and is lighter in weight under the conditions of considering the uncertainties of loads and material properties through optimization design.

[0065] (2) It has high optimization efficiency. The method proposed in this patent adopts the functional measurement method, the sequential single-loop method, and the trust-region iteration algorithm, effectively realizing the decoupling of the original nested optimization problem, avoiding problems such as poor convergence and huge computational amount in two-layer loop optimization, and having higher optimization efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] Figure 1 Overall design concept;

[0067] Figure 2 Schematic diagram of the reliability index. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0068] In order to better understand the above technical solutions, the technical solutions of the present application will be described in detail below with reference to the drawings and specific embodiments. It should be understood that the specific features in the embodiments of the present application are detailed descriptions of the technical solutions of the present application, rather than limitations on the technical solutions of the present application. Without conflict, the technical features in the embodiments of the present application and the embodiments can be combined with each other.

[0069] The following further details the method for probabilistic-based structural reliability topology optimization provided by the embodiments of the present application with reference to the accompanying drawings of the specification. The specific implementation manners may include (such asFigure 1 , 2 as shown

[0070] In the solution provided in the embodiments of the present application, the following steps are specifically included:

[0071] For a structure with material and load uncertainties, a structural topology optimization model with reliability index constraints or objectives is established by combining probability model description and reliability index definition;

[0072] In the design of the uncertainty structural topology optimization algorithm, first, based on the functional measure method, the original reliability index constraint is transformed into a functional measure constraint to obtain a more robust structural topology optimization model. Furthermore, the two-layer nested optimization problem is decoupled, a sequential single-loop topology optimization model is constructed, and an iterative update algorithm is constructed in combination with the trust region method;

[0073] Transformation and decoupling of the original nested optimization problem

[0074] The two-layer nested optimization problem can be summarized as the following optimization formulation:

[0075]

[0076] s.t.β[g j (ρ,u)≥0]≥ β j (j = 1,2,...,N g ),

[0077] ρ ≤ρ e ≤1(e = 1,2,...,N),

[0078] where

[0079]

[0080] s.t.g j (ρ,u)=0,

[0081] where ρ represents the element density vector, u is the standardized uncertain parameter, g j (ρ,u) is the structural performance function, β is the reliability index, and its meaning is "the shortest distance from the origin to the limit state surface in the standard uncertain space", as Figure 2 shown

[0082] The reliability index constraint is transformed into its more robust equivalent form by using the performance measure approach PMA. The performance measure approach does not directly compare the reliability index in the original optimization problem with its target lower limit value. Instead, it finds the point that minimizes the most probable failure function value within the allowable distribution range of random uncertainties, which can also be called the MPP point, and requires that the minimum function value be non - negative, that is Using this method can significantly improve the convergence of solving the optimization problem. The original optimization problem is equivalent to:

[0083]

[0084] s.t.α j (ρ)≥0(j=1,2,...,N g ),

[0085] ρ ≤ρ e ≤1(e=1,2,...,N),

[0086] where

[0087]

[0088]

[0089] Using the sequential single - loop method, by constructing a series of approximate deterministic sub - optimization problems, the inner and outer layer optimizations are solved continuously. The sequential deterministic sub - optimization problem is

[0090] For k=1,2,...

[0091] findρ

[0092]

[0093] subject to g j (ρ,u (k) )≥0(j=1,2,...,N g )

[0094] ρ ≤ρ e ≤1(e=1,2,...,N)

[0095] Among them, the MPP point u in the k - th step deterministic sub - optimization problem (k) is calculated using the trust - region iteration algorithm, that is

[0096]

[0097] length=||(1 - λ k )u (k) +λk u C ||2 / β j

[0098]

[0099] The MMA gradient optimization algorithm is used to solve the sequence single-loop optimization sub-problem. After the uncertainty variables and design variables both meet the convergence conditions, the optimal topology design that meets the requirements of the reliability index is obtained. The basic idea of the overall design is as Figure 1 shown;

[0100] For the calculation of the sensitivity of uncertainty variables, the adjoint method and the finite difference method are respectively used according to different problems; for the calculation of the sensitivity of design variables, the adjoint method is used for calculation;

[0101] After obtaining the optimal design through the uncertainty structure optimization algorithm, the improved first-order second-moment method and the response surface method are used to calculate the reliability index and the failure probability, and the final results are checked and verified;

[0102] The improved first-order second-moment method (or HL-RF method) is widely used to iteratively solve the reliability index, that is, to solve the following nonlinear optimization problem

[0103]

[0104] s.t. g(u) = 0,

[0105]

[0106] The HL-RF iterative algorithm first selects an iterative initial point, and then determines the next iterative point according to the iterative format until the convergence condition is met. Let u (k) be the point obtained in the k-th iteration in the standard space, and let the distance from the (k + 1)-th iterative point to the origin be β (k+1) , and the iterative direction α (k+1) is the unit vector of the negative gradient direction at the point u (k) on the limit state surface, that is

[0107]

[0108] Then there is

[0109] u (k+1) = β (k+1) α (k+1)

[0110] After determining the iterative direction, considering that the critical failure point u * must satisfy the limit state equation g(u) = 0, then u * the distance β from the coordinate origin to(k+1) It can be obtained by performing a one-dimensional search on this non-linear equation. To avoid one-dimensional search and reduce the computational amount, generally, the limit state equation is linearly approximated at the u (k) point, and let u k+1 satisfy this approximate equation, and we can obtain

[0111]

[0112] Thus

[0113]

[0114] Therefore, the following iterative algorithm can be constructed:

[0115]

[0116] This explicit expression is the HL-RF iteration formula of the improved first-order second-moment method. According to this iteration formula, the optimal solution of the optimized column layout can be found, and the β value corresponding to the optimal solution is the structural reliability evaluation index.

[0117] Based on the same inventive concept as Figure 1 the present invention also provides a probability-based structural reliability topology optimization system, including:

[0118] A modeling module, for a structure with material and load uncertainties, combined with probability model description and reliability index definition, to establish a structural topology optimization model with reliability index constraint conditions or objectives;

[0119] A design module, to solve the structural topology optimization model; when both the uncertainty variables and the design variables satisfy the convergence conditions, the optimal topology design meeting the reliability index requirements is obtained.

[0120] This application provides a computer-readable storage medium, and the computer-readable storage medium stores computer instructions. When the computer instructions run on a computer, the computer is made to execute Figure 1 the method described above.

[0121] Those skilled in the art should understand that the embodiments of the present application can be provided as methods, systems, or computer program products. Therefore, the present application can take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present application can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk memories and optical memories, etc.) containing computer-usable program codes.

[0122] This application is described with reference to the flowcharts and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the present application. It should be understood that each flow and / or block in the flowchart and / or block diagram, and the combination of flows and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to the processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to generate a machine, such that the instructions executed by the processor of the computer or other programmable data processing device generate means for implementing the functions specified in one or more of the flows Figure 1 one or more of the flows and / or blocks Figure 1 or in one or more of the blocks.

[0123] These computer program instructions can also be stored in a computer-readable memory that can direct a computer or other programmable data processing device to operate in a specific manner, such that the instructions stored in the computer-readable memory produce a manufacture including instruction means for implementing the functions specified in one or more of the flows Figure 1 one or more of the flows and / or blocks Figure 1 or in one or more of the blocks.

[0124] These computer program instructions can also be loaded onto a computer or other programmable data processing device, such that a series of operation steps are executed on the computer or other programmable device to generate a computer-implemented process, and thus the instructions executed on the computer or other programmable device provide steps for implementing the functions specified in one or more of the flows Figure 1 one or more of the flows and / or blocks Figure 1 or in one or more of the blocks.

[0125] Obviously, those skilled in the art can make various modifications and variations to this application without departing from the spirit and scope of this application. Thus, if these modifications and variations of this application fall within the scope of the claims of this application and their equivalent technologies, this application is also intended to include these modifications and variations.

[0126] The content not described in detail in this specification of the present invention belongs to the well-known technology of those skilled in the art.

Claims

1. A probability-based topological optimization method for structural reliability, characterized in that It includes the following steps: For a structure with material and load uncertainties, a structural topology optimization model with reliability index constraints or objectives is established by combining probability model description and reliability index definition; Solve the structural topology optimization model; when both the uncertainty variables and the design variables meet the convergence conditions, obtain the optimal topology design that meets the reliability index requirements; The method for solving the structural topology optimization model includes the following steps: According to the structural topology optimization model, use the sequential single-loop method to continuously solve the inner and outer layer optimizations by constructing a deterministic sub-optimization problem; Use the MMA gradient optimization algorithm to solve the sequential single-loop optimization sub-problem; The deterministic sub-optimization problem is: For k = 1, 2,... find ρ subject to g j (ρ,u (k) )≥0 ρ ≤ ρ e ≤ 1 Among them, ρ represents the element density vector, u is the standardized uncertain parameter, and g j (ρ, u) is the performance function of the structure, V is the total volume of the structure, and ρ e is the element density, V e is the element volume, α j is the performance function value, ρ is the lower limit of the element density, N g is the number of performance functions, N is the number of elements, j = 1, 2,..., N g , e = 1, 2,..., N; Determine the MPP point u in the k-th step deterministic sub-optimization problem (k) Use the trust region iteration algorithm for calculation: length = ||(1 - λ k )u (k) + λ k u C ||² / β j where \(u\) is the standardized uncertain parameter, β j is the reliability index, is the gradient of the performance function, \(\lambda\) k is the adjustment parameter, \(u\) (k) is the \(k\)-th iteration point, \(u\) C is the temporary iteration point.

2. A probability-based structural reliability topology optimization method according to claim 1, characterized in that: The structural topology optimization model is s.t.α j (ρ)≥0, ρ ≤ ρ e ≤ 1, where where ρ represents the element density vector, u is the normalized uncertain parameter, and g j (ρ, u) is the performance function of the structure, β is the reliability index, V is the total volume of the structure, and ρ e is the element density, V e is the element volume, α j is the performance function value, ρ is the lower limit of the element density, N g is the number of performance functions, N is the number of elements, β j is the reliability index, j = 1, 2,..., N g , e = 1, 2,..., N.

3. A probability-based structural reliability topology optimization method according to claim 1, characterized in that: After obtaining the optimal topology design that meets the reliability index requirements, calculate the reliability index and the failure probability, and check and verify the final result; if the check and verification are successful, end; Otherwise, re-solve the structural topology optimization model.

4. A probability-based structural reliability topology optimization method according to claim 3, characterized in that: The method for checking and verifying the final result is the first-order second-moment method and the response surface method.

5. A probability-based structural reliability topology optimization system, characterized in that, It includes: A modeling module, which, for a structure with material and load uncertainties, establishes a structural topology optimization model with reliability index constraints or objectives by combining probability model description and reliability index definition; A design module, which solves the structural topology optimization model; when both the uncertainty variables and the design variables meet the convergence conditions, obtains the optimal topology design that meets the reliability index requirements; The structural topology optimization model is s.t.α j (ρ)≥0, ρ ≤ ρ e ≤ 1, where Among them, ρ represents the element density vector, u is the standardized uncertain parameter, and g j (ρ, u) is the performance function of the structure, β is the reliability index, V is the total volume of the structure, and ρ e is the element density, V e is the element volume, α j is the performance function value, ρ is the lower limit of the element density, N g is the number of performance functions, N is the number of elements, β j is the reliability index, j = 1, 2,..., N g , e = 1, 2,..., N; The method for solving the structural topology optimization model includes the following steps: According to the structural topology optimization model, use the sequential single-loop method to continuously solve the inner and outer layer optimizations by constructing a deterministic sub-optimization problem; Use the MMA gradient optimization algorithm to solve the sequential single-loop optimization sub-problem; The deterministic sub-optimization problem is: For k = 1, 2,... find ρ subject to g j (ρ,u (k) )≥0 ρ ≤ ρ e ≤ 1 where ρ represents the element density vector, u is the normalized uncertain parameter, and g j (ρ, u) is the performance function of the structure, V is the total volume of the structure, and ρ e is the element density, V e is the element volume, α j is the performance function value, ρ is the lower limit of the element density, N g is the number of performance functions, N is the number of elements, j = 1, 2,..., N g , e = 1, 2,..., N; Determine the MPP point u in the k-th step deterministic sub-optimization problem (k) Use the trust region iteration algorithm for calculation: u C = - β j ▽G u(k) / ||▽G u(k) ||² length=||(1-λ k )u (k) +λ k u C ||2 / β j where \(u\) is the standardized uncertain parameter, β j is the reliability index, is the gradient of the performance function, \(\lambda\) k is the adjustment parameter, \(u\) (k) is the \(k\)-th iteration point, \(u\) C is the temporary iteration point; After obtaining the optimal topology design that meets the reliability index requirements, calculate the reliability index and the failure probability, and check and verify the final result; if the check and verification are successful, end; otherwise, re-solve the structural topology optimization model; The method for checking and verifying the final result is the first-order second-moment method and the response surface method.

6. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by a processor, it implements the steps of the method according to any one of claims 1 to 4.

7. A probability-based structural reliability topology optimization device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that: When the processor executes the computer program, it implements the steps of the method according to any one of claims 1 to 4.

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