Sequential iterative analysis and design method for transient dynamic topology optimization

Abstract

The application provides a sequence iteration analysis and design method of transient dynamic topology optimization, and belongs to the field of structural optimization design. First, the optimization design domain is meshed. The method combines the modal displacement method of quasi-static response enhancement and the sequence iteration analysis and design method, performs an inverse iteration process once in each optimization step to obtain approximate modal information, forms a reduction base with the quasi-static response, and solves the original problem by order reduction, so that synchronous convergence of transient response analysis and topology optimization problem solving is realized, the calculation efficiency of finite element analysis and sensitivity analysis in the optimization design is greatly improved, and the transient dynamic topology optimization is successfully applied to three-dimensional large-scale structural optimization design. The application is a high-efficiency and easy-parallel topology optimization method, and is suitable for structural optimization design which needs to consider transient dynamic response.

Description

Technical Field

[0001] This invention belongs to the field of structural optimization design and relates to a highly efficient and easily parallelizable transient dynamic topology optimization method. This method is applicable to topology optimization design under both structured and unstructured meshes. Background Technology

[0002] Transient dynamic response optimization design under dynamic loads has received widespread attention in aerospace, shipbuilding, and machinery industries. Topology optimization, as a powerful design tool, has been successfully applied to structural design considering transient responses, and has yielded some high-performance, lightweight structural design schemes.

[0003] However, traditional transient dynamic topology optimization methods directly solve time-domain discrete equations. For large-scale three-dimensional structural optimization problems, the computational load is too large, making efficient optimization design difficult and increasing product design cycles. Simultaneously, solving the adjoint equations in sensitivity analysis requires computational resources similar to finite element analysis. The computational bottleneck in large-scale three-dimensional structural optimization severely restricts the application of transient dynamic topology optimization methods in industry. Kristiansen and Aage's paper "An open-source framework for large-scale transient topology optimization using PETSc" published in Structural and Multidisciplinary Optimization also illustrates the problem of excessive computational load in transient dynamic topology optimization. Some order-reduction algorithms for transient dynamics have been proposed, such as SOMMG and SOAR algorithms; however, the linear equations solved by these algorithms are dependent on each other, making high-efficiency parallelism difficult. Traditional modal displacement methods and modal acceleration methods can be used for transient dynamic analysis, but these algorithms require modal information, and the calculation results cannot meet the accuracy requirements under extreme load conditions.

[0004] This invention proposes a highly efficient and easily parallelizable transient dynamic topology optimization method based on sequential iterative analysis and design, applicable to optimization design problems considering transient responses. This method significantly reduces the computational cost of finite element analysis and sensitivity analysis in optimization design by combining the quasi-static response-enhanced modal displacement method with sequential iterative analysis and design, keeping the computational load of transient dynamic topology optimization within an acceptable range. Summary of the Invention

[0005] To address the computational burden of traditional transient dynamic topology optimization methods for large-scale problems, this invention provides a highly efficient and easily parallelizable topology optimization method. This method combines a quasi-static response-enhanced modal displacement method with sequential iterative analysis and design. In each optimization step, only one inverse iteration is performed to obtain approximate modal information, which, together with the quasi-static response, forms a reduced basis. This basis reduces the order of the original problem, achieving simultaneous convergence of transient response analysis and topology optimization problem solving. This significantly improves the computational efficiency of finite element analysis and sensitivity analysis in optimization design, successfully applying transient dynamic topology optimization to large-scale three-dimensional structural optimization design. This invention is applicable to structural optimization design that requires consideration of transient dynamic responses.

[0006] To achieve the above objectives, the technical solution adopted by the present invention includes the following steps:

[0007] A sequential iterative analysis and design method for transient dynamic topology optimization includes the following steps:

[0008] S1: First, mesh the optimization design domain to obtain a finite element model with N degrees of freedom and design variables x. Consider the following transient dynamics-related optimization problem:

[0009] min.J

[0010]

[0011] Where J represents the objective function; K, C, and M represent the overall stiffness matrix, overall damping matrix, and overall mass matrix, respectively. t represents time, u(t), Representing displacement, velocity, and acceleration respectively; F(t) represents the external load; u0 and These represent the initial displacement and velocity conditions, respectively; g i Let J represent the i-th constraint, and w represent the number of constraints. i There are transient dynamics-related performance indicators, such as dynamic response for a specified degree of freedom and average dynamic compliance over a certain time period.

[0012] S2: Given a random initial mode matrix of dimension N×n

[0013] S3: Calculate the overall stiffness matrix K, overall damping matrix C, and overall mass matrix M of the structure based on the current design variable x.

[0014] S4: Perform a reverse iteration on the initial mode matrix Φ obtained in step S2. Obtain the matrix And it is used to compute the modes of the sub-eigenvalue problem: (in A is the mode matrix, and Λ is the eigenvalue matrix. The updated approximate mode matrix can be obtained from this.

[0015] S5: Calculate the quasi-static response based on the external force load F(t) obtained in step S1 and the global stiffness matrix K obtained in step S3. Combined with the approximate mode matrix Φ calculated in step S4, an n+1-dimensional linear subspace is obtained, which serves as the reduced basis.

[0016] S6: Dynamic governing equations By reducing the order using the reduced basis Ψ in step S4, we obtain the following equation:

[0017]

[0018] in, y and y represent generalized acceleration, generalized velocity, and generalized displacement, respectively. From this, we can obtain:

[0019]

[0020] S7: Based on u(t) in step S6, Calculate the objective function J and constraint function g for optimization. i .

[0021] S8: Based on the optimization problem in step S1, sensitivity derivation is performed to obtain the discrete-time equation (the specific equation form varies depending on the problem, and relevant research has been conducted). Using the reduced basis Ψ from step S4 for order reduction calculation, the corresponding subproblems can be obtained, and the Lagrange multiplier λ can be calculated. The sensitivity corresponding to the design variable x from step S1 is calculated based on the Lagrange multiplier λ. The design variable x is then updated using a gradient optimization algorithm based on the sensitivity information.

[0022] S9: Repeat steps S3 to S7, where the initial mode matrix Φ of the one inverse iteration performed in step S4 is the approximate mode matrix Φ updated in the previous step S4.

[0023] S10: Determine if the optimization has converged based on the relative change in the objective function: If converged, the optimization ends, and the optimization result is obtained based on the design variable x. If not converged, proceed to steps S8 and S9.

[0024] Furthermore, the convergence condition in step S10 is: the relative change of the objective function J is less than the convergence criterion, and all constraints are satisfied. The convergence criterion refers to: the relative change of the objective function J is less than 10. -5 .

[0025] The beneficial effects of this invention are as follows:

[0026] To address the problem of excessive computational resource consumption in traditional transient dynamic topology optimization methods for large-scale problems, this invention provides a highly efficient and easily parallelizable topology optimization method. This method combines the quasi-static response-enhanced modal displacement method with sequential iterative analysis and design methods. In each optimization step, it performs an inverse iteration to update modal information, forming a reduced basis with the quasi-static response, and reduces the order of the original problem for solution, achieving simultaneous convergence of transient response analysis and topology optimization problem solving. This method significantly improves the computational efficiency of finite element analysis and sensitivity analysis in optimization design, successfully applying transient dynamic topology optimization to three-dimensional large-scale structural optimization design. This invention is applicable to structural optimization design that requires consideration of transient dynamics. Attached Figure Description

[0027] Figure 1 This is a schematic diagram illustrating the loading of the four simply supported boundary conditions in a specific implementation case.

[0028] Figure 2 This is a loading curve showing the load changing over time in a specific implementation case;

[0029] Figure 3 Here is a flowchart of the transient dynamics optimization algorithm;

[0030] Figure 4 This refers to the optimization results obtained from specific implementation cases. Detailed Implementation

[0031] Structural response optimization under impact loading is a common transient dynamic optimization problem that has attracted widespread attention in practical engineering. This paper considers an optimization scheme under impact loading that achieves a lightweight and high-performance result, significantly reducing the average dynamic compliance over a period of time. The specific embodiments of this invention are described in detail below with reference to the technical solution and accompanying drawings.

[0032] Figure 1 The diagram describes a loading schematic of a four-point simply supported boundary condition, where a = 2m, b = 1.6m, and c = 0.8m. Figure 2 The loading curves as a function of time are presented. Figure 3 This is a flowchart of the implementation process of this method.

[0033] S1: Based on the needs of the actual engineering problem, a suitable optimization design domain is given, and the optimization design domain is meshed. Optimization is performed on 1 / 4 of the design domain based on symmetry. The degrees of freedom of the finite element model are 15,630,363. This implementation scheme selects the average dynamic compliance over a period of time as the objective of the optimization problem. This implementation case uses the variable density method for optimization, adopts the RAMP interpolation model, and selects a penalty factor of 6. Considering the 30% volume fraction constraint, the specific optimization formula is as follows:

[0034]

[0035] Where T is the total time considered in the transient dynamics problem, V represents the volume of the optimization result, V0 represents the volume of the design domain, and x e This represents the e-th design variable.

[0036] S2: Given a random initial mode matrix of dimension 15,630,363×5.

[0037] S3: Calculate the overall structural stiffness matrix K, overall damping matrix C, and overall mass matrix M based on the current design variable x.

[0038] S4: Perform a reverse iteration on the initial mode matrix Φ obtained in step S2. Obtain the matrix And it is used to compute the modes of the sub-eigenvalue problem: (in A is the mode matrix, and Λ is the eigenvalue matrix. The updated approximate mode matrix can be obtained from this.

[0039] S5: Calculate the quasi-static response based on the external force load F(t) obtained in step S1 and the global stiffness matrix K obtained in step S3. Combined with the approximate mode matrix Φ calculated in step S4, a 6-dimensional linear subspace is obtained, which serves as the reduced basis.

[0040] S6: Dynamic governing equations By reducing the order using the reduced basis Ψ in step S4, the following equation is obtained.

[0041]

[0042] in y and y represent generalized acceleration, generalized velocity, and generalized displacement, respectively, from which we can obtain

[0043]

[0044] S7: Based on u(t) in step S6, Calculate the objective function J and constraint function g for optimization. i .

[0045] S8: Based on the optimization problem in step S1, derive the sensitivity and obtain the time-domain discrete equation.

[0046]

[0047] Using the reduced basis Ψ order reduction calculation in step S4, the subproblem can be obtained:

[0048]

[0049] The Lagrange multipliers required to solve for the sensitivity can be obtained:

[0050] λ(t)≈Ψy(t).

[0051] according to Approximate sensitivity analysis results were obtained. Based on the sensitivity information, the design variable x was updated using the MMA optimization algorithm.

[0052] S9: Repeat steps S3 to S7, where the initial mode matrix Φ of the one inverse iteration in step S4 is changed to the approximate mode matrix Φ updated in the previous step S4.

[0053] S10: Determine if the optimization has converged: If converged, the optimization ends, and the optimization result is obtained based on the design variable x. If it has not converged, proceed to steps S8 and S9.

[0054] Furthermore, the convergence condition for step S10 is: the relative change in the objective function J is less than 10. -5 And the volume constraint condition is satisfied.

[0055] Figure 4 The optimization results of this embodiment are presented. The optimization converged after 45 iterations, with a total time of approximately 37.2 hours. It can be seen that the computational cost of topology optimization for transient dynamic structures with more than 10 million degrees of freedom is controlled within an acceptable range, which demonstrates the efficiency of this invention.

[0056] The essence of this invention is to propose a quasi-static response-enhanced modal displacement method. This method transforms the computational burden of transient dynamics solutions into solving for quasi-static responses and modal information. By utilizing a sequential iterative analysis and design method, an inverse iterative process is performed in each optimization analysis step, reducing the computational burden of solving for modal information during optimization while ensuring simultaneous convergence of the optimization and analysis processes. The linear equations in steps S4 and S5 are independent and can be solved simultaneously, facilitating parallel computation. This method effectively overcomes the problem of excessive computational burden faced by traditional transient dynamics-related topology optimization methods, enabling stable and efficient design of lightweight, high-performance structures.

[0057] The invention modifies the objective function related to the transient response of the optimization problem described in the foregoing embodiments, changes the optimization method, or makes equivalent replacements to some of the material interpolation models, or replaces the quasi-static response enhancement modal displacement method with other modal-based order reduction algorithms (such as modal displacement method, modal acceleration method, etc.), without causing the essence of the corresponding methods and solutions to deviate from the scope of the methods and solutions of the embodiments of the present invention.

[0058] The above-described embodiments are merely illustrative of the implementation methods of the present invention, but should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the protection scope of the present invention.

Claims

1. A sequential iterative analysis and design method for transient dynamic topology optimization, characterized in that, Includes the following steps: S1: First, mesh the optimization design domain to obtain a finite element model with N degrees of freedom and design variable x; consider the following transient dynamics-related optimization problem: min.J Where J represents the objective function; K, C, and M represent the overall stiffness matrix, overall damping matrix, and overall mass matrix, respectively; t represents time, u(t), Representing displacement, velocity, and acceleration respectively; F(t) represents the external load; u0 and These represent the initial displacement and velocity conditions, respectively; g i Let J represent the i-th constraint, and w represent the number of constraints; J and g i There are transient dynamics-related performance indicators, such as dynamic response for a specified degree of freedom and average dynamic compliance over a certain time period; S2: Given a random initial mode matrix of dimension N×n S3: Calculate the overall structural stiffness matrix K, overall damping matrix C, and overall mass matrix M based on the current design variable x; S4: Perform a reverse iteration on the initial mode matrix Φ obtained in step S2. Obtain the matrix And it is used to compute the modes of the sub-eigenvalue problem: ,in A is the mode matrix, and Λ is the eigenvalue matrix; the updated approximate mode matrix can be obtained from this. S5: Calculate the quasi-static response based on the external force load F(t) obtained in step S1 and the global stiffness matrix K obtained in step S3. Combined with the approximate mode matrix Φ calculated in step S4, an n+1-dimensional linear subspace is obtained, which serves as the reduced basis. S6: Dynamic governing equations By reducing the order using the reduced basis Ψ in step S4, we obtain the following equation: in, y and y represent generalized acceleration, generalized velocity, and generalized displacement, respectively. From this, we can obtain: S7: Based on u(t) in step S6, Calculate the objective function J and constraint function g for optimization. i ; S8: Based on the optimization problem in step S1, derive the sensitivity to obtain the time-domain discrete equation. Use the reduced basis Ψ in step S4 to reduce the order and obtain the corresponding subproblems and calculate the Lagrange multiplier λ. Calculate the sensitivity corresponding to the design variable x in step S1 based on the Lagrange multiplier λ. Update the design variable x using the gradient optimization algorithm based on the sensitivity information. S9: Repeat steps S3 to S7, where the initial mode matrix Φ of the one inverse iteration performed in step S4 is the approximate mode matrix Φ updated in the previous step S4; S10: Determine whether the optimization has converged based on the relative change of the objective function: If it has converged, the optimization ends and the optimization result is obtained based on the design variable x; if it has not converged, proceed to steps S8 and S9.

2. The sequential iterative analysis and design method for transient dynamic topology optimization according to claim 1, characterized in that, The convergence condition for step S10 is: the relative change in the objective function J is less than 10. -5 And all constraints are satisfied.

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