Truss structure nonlinear buckling stability optimization design method based on finite mass point method and automatic differentiation

Abstract

The invention discloses a truss structure nonlinear buckling stability optimization design method based on a finite mass point method and automatic differentiation, and the method comprises the steps: constructing a parameterized truss model, and taking the cross-sectional area and / or node coordinates of a rod piece in the parameterized truss model as design variables; constructing an optimization objective function for determining the nonlinear buckling stability performance of the structure as an objective; and on the basis of the parameterized truss model, constructing a corresponding finite mass point method analysis model, and performing iterative optimization on the optimization objective function in the process of simulating nonlinear static response to buckling in the finite mass point method analysis model in combination with automatic differential software, so as to output a final design variable as the optimal truss structure design. The method provided by the invention can effectively solve the problem of buckling stability optimization of the truss structure which becomes abnormally complex due to geometric nonlinearity and path dependence characteristics.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of the intersection of engineering structure optimization design and computational mechanics, and particularly relates to a truss structure nonlinear buckling stability optimization design method based on a finite particle method and automatic differentiation. BACKGROUND

[0002] The theory and method of structure optimization design have developed for more than a hundred years, and the core goal thereof is to obtain a structure form with the highest material utilization efficiency and the optimal mechanical performance through scientific calculation. Among numerous structure types, a truss structure is widely applied due to its high specific strength and specific stiffness. However, the bearing capacity of the truss structure is often controlled by the stability thereof, especially the nonlinear buckling behavior, which makes the buckling stability optimization a key difficulty in the design thereof.

[0003] At present, most of the mainstream structure optimization methods are based on the finite element method. The finite element method is very effective in processing linear elasticity, small deformation and static optimization problems. However, in processing the nonlinear buckling stability optimization of the truss structure, the finite element method faces significant challenges: The nonlinear buckling of a structure is a typical path-dependent process, which involves the tracking of an equilibrium path and even the crossing of a bifurcation point. The traditional finite element method is based on the global stiffness matrix and the iterative solution (such as the Newton-Raphson method), and in the vicinity of a critical point, the stiffness matrix is often singular or ill-conditioned, which leads to the failure of the solution, so that the structure response analysis in the optimization process cannot be reliably completed.

[0004] Gradient-based optimization algorithms need to calculate the sensitivity (gradient) of the objective function (such as the buckling load) and the constraint condition to the design variable (such as the bar section and the node coordinate). For complex problems such as nonlinear buckling, it is extremely tedious and prone to error to analytically derive the sensitivity formula, and the traditional finite difference method has an inherent contradiction between calculation accuracy and efficiency, and in the above unstable analysis, the difference result is unreliable.

[0005] Automatic differentiation is an accurate and efficient gradient calculation technology, which automatically calculates the derivative by decomposing the basic operations of a computer program using the chain rule. By combining automatic differentiation with the finite particle method, the gradient of the entire nonlinear analysis process to the design parameter can be automatically and accurately obtained in theory.

[0006] Patent document CN111709096A discloses a special-shaped fin structure design method for strengthening natural convection heat exchange. First, the physical model of the natural convection heat exchange problem is determined. According to the actual fin structure design problem based on natural convection heat exchange, the analysis domain and geometric and physical conditions are extracted, and a physical model suitable for topology optimization is established. Then, a finite body analysis model of the base structure is established. Then, an optimization model of the special-shaped fin structure is established. The topology optimization method using variable density method structure description is used to minimize the uniform temperature objective function at the heat source, establish an optimization model driven by the physical field, and determine the optimization direction each time. Then, the iteration optimization of the special-shaped fin structure is carried out to obtain the optimized special-shaped fin structure. Finally, the optimized special-shaped fin is smoothed and rounded.

[0007] Patent document CN113032974A discloses a hexahedral element finite particle method, including the following steps: first, initializing the model parameters and updating the particle displacement, then solving the particle pure deformation of the hexahedral element inverse motion, and then calculating the particle internal force, and finally bringing it into the central difference motion formula to solve the next step particle displacement. SUMMARY

[0008] The purpose of the present application is to provide a truss structure nonlinear buckling stability optimization design method based on finite particle method and automatic differentiation, which is used to solve the truss structure buckling stability optimization problem which becomes extremely complex due to geometric nonlinearity and path-dependent characteristics.

[0009] In order to achieve the purpose of the present application, the following technical scheme is provided: a truss structure nonlinear buckling stability optimization design method based on finite particle method and automatic differentiation, including the following steps: constructing a parameterized truss model, and taking the cross-sectional area of the rod in the parameterized truss model and / or the node coordinates as design variables to construct an optimization objective function for determining the nonlinear buckling stability performance of the structure as the target; Based on the parameterized truss model, a corresponding finite particle method analysis model is constructed, and the automatic differentiation software is combined to simulate the nonlinear static response of the finite particle method analysis model until the buckling occurs. In the process, the optimization objective function is iteratively optimized to output the final design variable as the best truss structure design.

[0010] The present application deeply integrates the finite particle method suitable for strong nonlinear process analysis, the automatic differentiation technology capable of realizing efficient and accurate gradient calculation, and the mathematical programming algorithm based on gradient, forming a complete, efficient and automated optimization design closed loop.

[0011] Specifically, the parameterized truss model includes the initial geometric topology, material properties, boundary conditions and applied pseudo-static load of the truss structure to be optimized.

[0012] Specifically, the optimization objective function also includes constraints, which include one or more of the following: material volume constraints, weight constraints, or rigidity constraints of the truss structure.

[0013] Specifically, the construction process of the finite point method analysis model is as follows: The truss structure is discretized into mass points and massless rods, and the dynamic response process of the truss structure under load is simulated based on the explicit time integration method.

[0014] Specifically, the basic governing equations of the finite point method analysis model are as follows: ; in, For the acceleration of a particle, These are the external forces, internal forces, and damping forces of the structure.

[0015] Specifically, the expression for the explicit time integration method is as follows: ; in, , They represent point masses respectively. t n+1 , t n and t n-1 The displacement vector at the time step. The pseudo-iteration time step, where c1 and c2 are damping-related coefficients. Indicates the internal forces of the structure. Indicates external forces on the structure. It represents the mass of a point mass.

[0016] Specifically, the optimization objective function is to minimize the buckling tendency characterization of the structure under a given load.

[0017] Specifically, the automatic differentiation software includes a backpropagation automatic differentiation function, which takes the optimization design variables as input and the values ​​of the optimization objective function and constraints as output, and automatically backpropagates to calculate the gradient.

[0018] Specifically, the automatic backpropagation gradient calculation process records the basic operation sequence of forward analysis and its derivatives, and uses the chain rule to accurately calculate the gradient of the objective function and constraints with respect to multiple design variables.

[0019] Compared with the prior art, the beneficial effects of the present invention are as follows: The introduction of automatic differentiation technology completely overcomes the infeasibility of manually deriving analytical sensitivity formulas and the shortcomings of high computational cost and low accuracy of the finite difference method, realizing automatic, accurate and efficient calculation of gradients for complex path-dependent processes. The clear, explicit computational flow of the finite point method provides ideal conditions for constructing computational graphs through automatic differentiation; while automatic differentiation unlocks the enormous potential of the finite point method in the field of optimization. The deep integration of these two methods forms a powerful tool with unique advantages in handling strongly nonlinear, path-dependent optimization problems. Attached Figure Description

[0020] Figure 1 The flowchart below shows the nonlinear buckling stability optimization design method for truss structures based on the finite mass method and automatic differentiation provided in this embodiment. Figure 2 This is a schematic diagram of the optimized columnar truss structure and the overall buckling instability deformation under load provided in this embodiment; Figure 3 This is a schematic diagram of the optimized columnar truss structure and its deformation under load provided in this embodiment; Figure 4 This is a schematic diagram of the optimized arched truss structure and elastic sudden buckling instability deformation under load provided in this embodiment; Figure 5 This is a schematic diagram of the optimized arched truss structure and its deformation under load provided in this embodiment. Detailed Implementation

[0021] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0022] like Figure 1 As shown in this embodiment, a nonlinear buckling stability optimization design method for truss structures based on the finite mass method and automatic differentiation is provided. The finite mass method reliably tracks the nonlinear buckling path of the structure, and the automatic differentiation technique automatically and accurately obtains the sensitivity of buckling performance indicators to design parameters, thereby driving the optimization algorithm to quickly find the optimal solution. The method includes: Establish a parametric truss model. Define the initial geometry, material properties, boundary conditions, and applied pseudo-static load of the truss structure to be optimized. This pseudo-static load is used to simulate the nonlinear static response of the structure until buckling occurs through the dynamic relaxation process of the finite mass method.

[0023] Define the buckling stability optimization problem. Determine the optimization design variables, including the cross-sectional area of ​​members and / or node coordinates; determine the optimization objective function with the nonlinear buckling stability performance of the structure as the objective, which is to minimize the buckling tendency of the structure under a given load (e.g., determined by monitoring the structural displacement response or a sharp increase in overall flexibility); and simultaneously determine the relevant constraints, such as the total material volume constraint.

[0024] A differentiable finite mass method (FMT) analysis model is constructed. Within a programming framework supporting automatic differentiation (such as PyTorch or JAX), a FMT model of the truss structure is built. This model discretizes the structure into mass points and massless elements, and simulates the dynamic response of the structure under load using an explicit time integration method (central difference method) until static equilibrium is reached or buckling instability occurs. The entire calculation process, including internal force calculation and time integration, is implemented within the automatic differentiation framework, forming a complete and differentiable computational graph.

[0025] Automatic differentiation calculation of sensitivity. Based on the analysis results of step 03, the values ​​of the objective function and constraints are calculated. Subsequently, using the backpropagation automatic differentiation function of the programming framework, with the optimization design variables as input and the values ​​of the objective function and constraints as output, the gradient is automatically calculated through backpropagation. This process records the basic operation sequence and its derivatives of the forward analysis, and uses the chain rule to accurately calculate the gradient of the objective function and constraints with respect to multiple design variables, serving as the sensitivity information required for optimization iteration.

[0026] Gradient-based optimization iteration is employed. Mathematical programming algorithms based on gradient information (such as moving asymptote algorithms or sequential quadratic programming algorithms) are used to obtain the sensitivity information required for optimization iteration, driving the iterative update of design variables until the convergence condition is met. Ultimately, a truss structure design with optimal nonlinear buckling stability under given constraints is obtained.

[0027] The basic governing equations of the above finite point method analysis model are as follows: ; in, For the acceleration of a particle, These are the external forces, internal forces, and damping forces of the structure.

[0028] Using the explicit central difference method for time integration, the displacement update formula can be expressed as: ; in, , They represent point masses respectively. t n+1 , t n and t n-1 The displacement vector at the time step. Where c1 is the iteration time step, and c2 are coefficients related to damping. Indicates the internal forces of the structure. This indicates the external force on the structure.

[0029] In this example, the automatic differentiation technique is not simply applied; rather, it differentiates the entire forward dynamic analysis process of the finite mass method, which includes hundreds to thousands of time steps. The automatic differentiation system automatically tracks and records the entire calculation graph from the input of design variables (such as the cross-sectional area A of the member), to the calculation of internal forces and displacement updates at each time step, until the final output of buckling tendency indicators (such as maximum displacement). After the forward calculation is completed, automatic differentiation in reverse mode is performed, starting from the objective function value and traversing backward along the calculation graph, recursively calculating the gradient of the objective function with respect to each intermediate variable (such as internal forces, displacements) and even the initial design variables using the chain rule. This process can obtain the accurate gradients of all design variables in one go, and its computational complexity is constant compared to the forward calculation, making it efficient and error-free.

[0030] To better illustrate the effect of the method provided in this embodiment, a vertical truss and an arched truss are used as examples.

[0031] like Figure 2 As shown, this is the vertical truss provided in this example, where... Figure 2 (a) in the figure represents an undeformed vertical truss with fixed bottom nodes.

[0032] Without loss of generality, the parameters below are dimensionless values. The truss height is 7.0, the width is 0.5, and the top vertex is subjected to a vertically downward load of P1 = 5.0. The top left node is subjected to a horizontal load to the right of P2 = 0.01P1. The initial cross-sectional area of ​​all components is A = 1.0, and the material elastic modulus is E = 1000.

[0033] like Figure 2 Figure (b) shows the deformation diagram of the truss structure. It is clear that the structure has undergone significant overall buckling instability, and due to the occurrence of buckling instability, most of the nodes of the structure have undergone significant large displacements.

[0034] Considering that large geometric deformation is a typical characteristic of the aforementioned overall buckling instability, the physical quantity that measures the overall resistance to deformation of the structure, namely compliance P, will be used. T Minimize d as the objective function, and use the cross-sectional area of ​​the members and the horizontal coordinates of all nodes except support and load nodes as optimization design variables to construct the following optimization model for structural optimization design: ; ; , ; Where V is the total volume of the structural materials. Take the maximum material usage limit as That is, the maximum allowable volume is the same as the initial material volume of the structure; The lower and upper limits of the allowable cross-sectional area of ​​component i are respectively taken as: , The node coordinates are respectively The lower and upper limits are respectively set as follows: , .

[0035] A finite mass method model for structural nonlinear analysis was established using the PyTorch programming framework, which supports automatic differentiation. After obtaining the structural response under load, the sensitivity information of the objective function to the design variables was obtained efficiently and accurately through automatic differentiation. Then, the MMA algorithm was used for iterative optimization design, with a maximum number of iterations set to 50, and finally the optimized structure was obtained.

[0036] Figure 3 Figure (a) shows the optimized structural configuration. It is evident that the optimized configuration has a wider lower section and relatively larger cross-sectional areas for the members on both sides, effectively improving the structure's overall buckling resistance. The objective function of the optimized configuration is less than 1% of that of the unoptimized structure, demonstrating a significant reduction in deformation.

[0037] Figure 3 Figure (b) shows the deformation of the optimized structure under load. It can be seen that the deformation is very small and can be considered to remain within the linear elastic range. This embodiment demonstrates the effectiveness of the proposed method in handling nonlinear optimization of structures involving overall buckling instability.

[0038] Figure 4(a) shows an arched truss structure to be optimized, with fixed bottom nodes on both sides. Without loss of generality, dimensionless values ​​are used for the parameters below. The horizontal span of the truss arch is 68, the height is 16, the module unit height is 0.5, and the upper nodes are subjected to a vertically downward load of P = 4.0. The initial cross-sectional area of ​​all components is A = 1.0, and the material elastic modulus is E = 2000.

[0039] Figure 4 Figure (b) shows the deformation of the arch structure under the aforementioned loads. During the loading process, the upward arch shape suddenly jumps to a downward arch shape, clearly indicating a significant abrupt instability. To avoid this abrupt instability, the method of this invention is used for optimization design.

[0040] Considering that large geometric deformation remains a typical characteristic of the aforementioned abrupt bouncing phenomenon, the physical quantity used to measure the overall deformation resistance of the structure, namely the overall flexibility P, will still be used here. T Minimize d as the objective function, and use the cross-sectional area of ​​the members and the coordinates of all nodes except support and load nodes as optimization design variables to construct the following optimization model for structural optimization design: ; ; , ; Where V is the total volume of the structural materials. Take the maximum material usage limit as That is, the maximum allowable volume is the same as the initial material volume of the structure; The lower and upper limits of the allowable cross-sectional area of ​​component i are respectively taken as: , The node coordinates are respectively The lower and upper limits are respectively set as follows: , .

[0041] A finite mass method model for structural nonlinear analysis was established using the PyTorch programming framework, which supports automatic differentiation. After obtaining the structural response under load, the sensitivity information of the objective function to the design variables was obtained efficiently and accurately through automatic differentiation. Then, the moving asymptote algorithm was used for iterative optimization design, with a maximum number of iterations set to 50, and finally the optimized structure was obtained.

[0042] Figure 4Figure (a) shows the optimized structural configuration. It can be seen that the optimized truss arch structure is wider overall, with relatively larger cross-sectional areas in the upper middle members near the supports on both sides, effectively improving the overall stability of the structure. The objective function of the optimized configuration is reduced by approximately 95% compared to the unoptimized structure, demonstrating a significant reduction in deformation.

[0043] Figure 4 Figure (b) shows the deformation of the optimized structure under load. It can be seen that the structure did not experience any sudden bouncing instability, and the overall deformation was very small, which can be considered to be within the linear elastic range. This embodiment demonstrates the effectiveness of the proposed method in handling nonlinear optimization of structures including sudden bouncing instability.

[0044] Furthermore, the terms "upper," "lower," "inner," "outer," "front," and "rear" are used for descriptive purposes only and should not be construed as indicating or implying relative importance. Unless otherwise specifically stated, the relative steps, numerical expressions, and values ​​of the components and steps set forth in these embodiments do not limit the scope of the invention.

[0045] Of course, the above description is only a specific embodiment of the present invention and is not intended to limit the scope of the present invention. All equivalent changes or modifications made to the structure, features and principles described in the claims of the present invention should be included in the scope of the claims of the present invention.

[0046] Finally, it should be noted that the above-described embodiments are merely specific implementations of the present invention, used to illustrate the technical solutions of the present invention, and not to limit it. The scope of protection of the present invention is not limited thereto. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that any person skilled in the art can still modify or easily conceive of changes to the technical solutions described in the foregoing embodiments within the technical scope disclosed in the present invention, or make equivalent substitutions for some of the technical features; and these modifications, changes, or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention, and should all be covered within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for nonlinear buckling stability optimization design of a truss structure based on a finite particle method and automatic differentiation, characterized by, The method comprises: constructing a parameterized truss model, and taking the cross-sectional area of a rod and / or the coordinates of a node in the parameterized truss model as a design variable to construct an optimization objective function for determining a nonlinear buckling stability performance of a structure as a target; based on the parameterized truss model, constructing a corresponding finite particle method analysis model, and combining automatic differentiation software to iteratively optimize the optimization objective function in the process of simulating the nonlinear static response of the finite particle method analysis model until buckling occurs, to output the final design variable as the optimal truss structure design. 2.The finite particle method and automatic differentiation based optimization design method for nonlinear buckling stability of a truss structure according to claim 1, wherein, The parameterized truss model includes the initial geometric topology, material properties, boundary conditions, and applied pseudo-static loads of the truss structure to be optimized. 3.The finite particle method and automatic differentiation based optimization design method for nonlinear buckling stability of a truss structure according to claim 1, wherein, The optimization objective function further includes a constraint condition, and the constraint condition includes one or more of a material volume constraint, a weight constraint, or a rigidity constraint of the truss structure. 4.The finite particle method and automatic differentiation based optimization design method for nonlinear buckling stability of a truss structure according to claim 1, wherein, The construction process of the finite particle method analysis model is as follows: discretize the truss structure into particles and massless rod elements, and simulate the dynamic response process of the truss structure under the action of the load based on an explicit time integration method.

5. The finite particle method and automatic differentiation based optimal design method for nonlinear buckling stability of a truss structure according to claim 4, wherein, The expression of the explicit time integration method is as follows: ; where, , respectively represent the mass of the particle t n+1 , t n and t n-1 the displacement vector of the particle at the time step, the pseudo-iterative time step size, and c1 and c2 are damping related coefficients, represents the internal force of the structure, represents the external force of the structure, represents the mass of the particle. 6.The finite particle method and automatic differentiation based optimization design method for nonlinear buckling stability of a truss structure according to claim 1, wherein, The optimization objective function is to minimize the buckling tendency of the structure under a given load. 7.The finite particle method and automatic differentiation based optimization design method for nonlinear buckling stability of a truss structure according to claim 1, wherein, The automatic differentiation software includes a reverse mode automatic differentiation function, which takes the optimization design variable as input, and the value of the optimization objective function and the constraint condition as output, and automatically backpropagates to calculate the gradient. 8.The finite particle method and automatic differentiation based optimization design method for nonlinear buckling stability of a truss structure according to claim 7, wherein, The process of automatically backpropagating to calculate the gradient is to record the basic operation sequence of the forward analysis and its derivative, and accurately calculate the gradient of the objective function and the constraint condition with respect to multiple design variables by using the chain rule.

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