A method and system for dynamic topology optimization of stiffened load-bearing structures under moving mass excitation.

Abstract

This invention discloses a dynamic topology optimization method and system for stiffened load-bearing structures under moving mass excitation, relating to the field of topology optimization design for load-bearing structures. The technical problem this invention addresses is that existing design techniques fail to consider the load characteristics of moving mass excitation, cannot adapt to the aforementioned practical engineering scenarios, and cannot achieve optimal control of load-bearing structure vibration, resulting in poor structural reliability under complex vibration environments. Key technical points: This invention introduces a local geometric control strategy for the cross-interference problem of stiffened elements, and constructs a dynamic topology optimization model for stiffened load-bearing structures based on the equal geometric stiffness diffusion method. An augmented Lagrangian function incorporating volume constraints and stiffened element cross-interference constraints is constructed. The sensitivity information of the dynamic topology optimization model for stiffened load-bearing structures is calculated. Design variables such as the node coordinates and cross-sectional area of ​​the stiffened elements are updated based on the moving asymptote method to seek the optimal layout design of the stiffened load-bearing structure. This invention proposes a dynamic topology optimization method for stiffened load-bearing structures under moving mass excitation, applicable to the dynamic layout design of stiffened load-bearing structures under moving mass excitation.

Description

Technical Field

[0001] This invention relates to the field of topology optimization design of load-bearing structures, and more specifically, to a method and system for dynamic topology optimization of stiffened load-bearing structures under the excitation of moving mass. Background Technology

[0002] Moving mass excitation is widely present in typical application scenarios such as bridge engineering and rail transit. Due to its spatiotemporal coupling characteristics and transient inertial effects, strong coupled vibrations exist between the bridge structure and the moving mass, which can easily lead to vibration fatigue failure of the bridge structure. The bridge structures in the aforementioned engineering context are mostly rigid load-bearing structures with lightweight, thin-walled features. Design parameters such as the position, orientation, and cross-sectional area of ​​the stiffeners have a significant impact on the dynamic performance of this type of load-bearing structure. Therefore, how to implement dynamic layout design of the stiffeners using topology optimization methods to reduce the vibration response of the structure under moving mass excitation is one of the current research hotspots and challenges in the field of rigid load-bearing structures.

[0003] Within the framework of continuum topology optimization design methods, the rib layout optimization problem is transformed into a material distribution problem by using the substrate thickness as the design variable. However, it is difficult to obtain clear rib positions, orientations, and cross-sectional dimensions. Within the framework of discrete structure topology optimization design methods, the rib layout optimization problem is transformed into a base structure size optimization problem, which can directly control the size and layout of the ribs. However, the fixed rib orientation and the dependence of the optimization results on the initial configuration of the base structure limit the design space of the optimization problem. Essentially, the rib layout optimization problem belongs to the category of discrete structure topology optimization problems. Existing time-domain dynamic topology optimization methods are limited to transient force excitation cases, while moving mass excitation has both transient force characteristics and transient inertial effects, making discrete structure topology optimization problems more complex and difficult to solve. Dynamic topology optimization of ribbed load-bearing structures under moving mass excitation directly targets the bridge design scenarios under vehicle-bridge coupling in bridge engineering and rail transit. However, existing design technologies do not consider the load characteristics of moving mass excitation, cannot adapt to the above-mentioned practical engineering scenarios, and lack clear and effective solutions and design methods. Summary of the Invention

[0004] The technical problem to be solved by this invention is:

[0005] To address the problems that existing design technologies do not consider the load characteristics of moving mass excitation, cannot adapt to the above-mentioned actual engineering scenarios, cannot achieve optimal control of the vibration of load-bearing structures, and result in poor reliability of the structure in complex vibration environments, this paper provides a dynamic topology optimization method and system for stiffened load-bearing structures under moving mass excitation.

[0006] To achieve the above objectives, in a first aspect, the present invention provides a dynamic topology optimization method for a stiffened load-bearing structure under moving mass excitation, comprising the following steps:

[0007] S101: Construct the design domain of the stiffened load-bearing structure on a uniform geometric background mesh, set boundary conditions and apply moving mass excitation, and initialize the design variables such as the node positions and cross-sectional areas of the stiffened (equivalent to a Timoshenko beam) elements, as well as the algorithm parameters: material properties of the stiffeners, Rayleigh damping coefficient, material properties and dimensional information of the uniform geometric background mesh (discrete form of the plate), mass of the moving mass excitation, initial velocity and acceleration, and optimized convergence parameters;

[0008] S102: Considering the additional mass, damping and stiffness characteristics and additional load induced by the moving mass excitation, construct the dynamic and geometric finite element equations of the stiffened load-bearing structure, and calculate the dynamic flexibility, volume constraints and local geometric control constraints to suppress the cross interference of stiffened elements of the stiffened load-bearing structure under the action of moving mass excitation.

[0009] S103: Establish a topology optimization model for minimizing the dynamic flexibility (maximizing the dynamic stiffness) of a stiffened load-bearing structure. Combine the initial design variables and algorithm parameters of the design domain to solve for the optimal dynamic topology (layout) of the stiffened load-bearing structure under the excitation of moving mass.

[0010] S104: Based on the minimum member size process principle, a stiffener unit deletion strategy and a node fusion strategy are constructed to reconstruct the dynamic optimal topology of the stiffener-type load-bearing structure, thereby obtaining a clear optimal design layout for the stiffener-type load-bearing structure.

[0011] Secondly, the present invention provides a dynamic topology optimization system for a stiffened load-bearing structure under basic moving mass excitation, comprising:

[0012] The initialization module constructs the initial layout of the stiffened load-bearing structure within a uniform geometric background mesh, initializes the design variables such as the node coordinates and cross-sectional area of ​​the stiffened elements, and pre-sets the material property parameters, volume constraints and stiffened element cross-interference constraints, HHT-α method and Newmark-β time integration scheme parameters, augmented Lagrangian optimization algorithm parameters, and stiffened element deletion and node fusion thresholds for the stiffened elements and background mesh.

[0013] The preprocessing module projects the stiffness, mass, and damping matrices of the stiffening element onto a geometrically equivalent background mesh using the principle of energy equivalence. It then integrates the additional stiffness, mass, and damping matrices generated by the moving mass excitation and applies initial and boundary conditions to form a dynamic geometrically equivalent finite element model of the stiffening load-bearing structure.

[0014] The dynamic analysis module combines the HHT-α method and the unconditionally stable implicit Newmark-β time integration scheme to solve the dynamic and geometric finite element models of stiffened load-bearing structures, and calculates the dynamic compliance, volume constraints and cross-interference constraints of stiffened elements of the structure.

[0015] The optimization solution module constructs an augmented Lagrangian function that includes volume constraints and cross-interference constraints of stiffener elements, calculates the sensitivity information of the dynamic topology optimization model of the stiffener-type load-bearing structure, updates the design variables such as the node coordinates and cross-sectional area of ​​the stiffener elements based on the moving asymptote method, and judges whether the optimization iteration result meets the convergence criterion. If it does not meet the convergence criterion, the optimization iteration continues until the convergence criterion is met.

[0016] The post-processing module, based on the minimum member size process principle, reconstructs the optimal dynamic topology (layout) of the stiffener-type load-bearing structure according to the threshold parameters of the cross section, length and node fusion of the stiffener unit, thereby obtaining the optimal design layout of the stiffener-type load-bearing structure that is convenient for engineering applications.

[0017] This invention provides a dynamic topology optimization method for stiffened load-bearing structures under moving mass excitation, which has the following beneficial technical effects:

[0018] This invention addresses the challenges of transient dynamic topology optimization of stiffened load-bearing structures under moving mass excitation: spatiotemporal coupling characteristics and transient inertial effects. By synergistically considering the position, orientation, and cross-sectional dimensions of the stiffeners, and fully exploring the design space, this invention proposes a dynamic topology optimization method for stiffened load-bearing structures. This method achieves optimal control of the vibration of stiffened load-bearing structures, thereby improving the reliability of the structure under complex vibration environments.

[0019] This invention considers the instantaneous inertial effect of moving mass excitation and establishes an isogeometric finite element dynamic analysis model for a stiffened load-bearing structure. A local geometric control strategy is introduced to address the interference problem of stiffening elements, and a dynamic topology optimization model for the stiffened load-bearing structure is constructed based on the isogeometric stiffness diffusion method. An augmented Lagrangian function incorporating volume constraints and stiffening element interference constraints is constructed, and the sensitivity information of the dynamic topology optimization model of the stiffened load-bearing structure is calculated. The nodal coordinates and cross-sectional areas of the stiffening elements are updated based on the moving asymptote method to seek the optimal layout design of the stiffened load-bearing structure. Therefore, a dynamic topology optimization method for stiffened load-bearing structures under moving mass excitation is proposed, including: defining the initial design domain and algorithm parameters, and constructing an isogeometric finite element dynamic analysis model of the stiffened load-bearing structure; solving the above dynamic model using the HHT-α method and the unconditionally stable implicit Newmark-β time integration scheme, and calculating the dynamic compliance, volume constraints, and stiffener element cross constraints of the stiffened load-bearing structure; constructing an augmented Lagrangian function to transform the original optimization problem (containing a large number of stiffener element cross constraints) into an unconstrained optimization problem, and calculating the adjoint equation of the optimization model through a discretization-differentiation sensitivity analysis strategy to obtain sensitivity information; updating the nodal coordinates and cross-sectional areas of the stiffener elements based on the moving asymptote method; and reconstructing the dynamic optimal topology of the stiffeners based on the minimum member size process principle, thereby obtaining an optimal design of the stiffened load-bearing structure that is convenient for engineering applications.

[0020] This invention proposes a dynamic topology optimization method for stiffened load-bearing structures under moving mass excitation, which can be used to solve the dynamic layout design problem of stiffened load-bearing structures under moving mass excitation. Attached Figure Description

[0021] Figure 1 This is a flowchart of a dynamic topology optimization method for a stiffened load-bearing structure under moving mass excitation according to the present invention.

[0022] Figure 2 This is a schematic diagram of the dynamic topology optimization system for a stiffener-type load-bearing structure under the excitation of moving mass according to the present invention.

[0023] Figure 3 This is a schematic diagram of the principle of the equal geometric stiffness diffusion method.

[0024] Figure 4 (a) Schematic diagram of the arch beam problem; Schematic diagram of dynamic load calculation in Embodiment 1 of the present invention.

[0025] (b) is a schematic diagram of the design domain of Embodiment 1 of the present invention.

[0026] Figure 5The diagram shows the optimized rib layout for the arch beam problem. (a) shows the optimized rib layout in Embodiment 1 of the present invention with a moving mass velocity v=30km / h (without considering the inertia of the moving mass).

[0027] (b) Optimized rib layout for the moving mass velocity v=30km / h in Embodiment 1 of the present invention (considering the inertia of the moving mass);

[0028] (c) Optimized rib layout for the moving mass velocity v=90km / h in Embodiment 1 of the present invention (without considering the inertia of the moving mass);

[0029] (d) Optimized rib layout for the moving mass velocity v=90km / h in Embodiment 1 of the present invention (considering the inertia of the moving mass);

[0030] (e) Optimized rib layout for a moving mass velocity v=150km / h in Embodiment 1 of the present invention (without considering the inertia of the moving mass);

[0031] (f) Optimized rib layout for a moving mass velocity v=150km / h in Embodiment 1 of the present invention (considering the inertia of the moving mass).

[0032] Figure 6 (a) Schematic diagram of a square plate problem with fixed sides; Schematic diagram of dynamic load calculation in Embodiment 2 of the present invention.

[0033] (b) is a schematic diagram of the design domain of Embodiment 2 of the present invention.

[0034] Figure 7 The diagram shows the optimized rib layout for the problem of a square plate with fixed sides. (a) shows the optimized rib layout in Embodiment 2 of the present invention with a moving mass velocity v=150km / h (without considering the inertia of the moving mass).

[0035] (b) Optimized rib layout for a moving mass velocity v=150km / h in Embodiment 2 of the present invention (considering the inertia of the moving mass). Detailed Implementation

[0036] This invention addresses the challenges of transient dynamic topology optimization for stiffened load-bearing structures, specifically the combination of transient force characteristics and transient inertial effects caused by moving mass excitation. It proposes a dynamic topology optimization method for stiffened load-bearing structures under moving mass excitation, achieving dynamic topology optimization (i.e., layout optimization) for these structures.

[0037] Combined with appendix Figures 1-7This paper elaborates on the dynamic topology optimization method for a stiffened load-bearing structure under moving mass excitation, so that the advantages and features of this invention can be more easily understood by those skilled in the art, thereby making a clearer and more explicit definition of the scope of protection of this invention.

[0038] In a first aspect, the present invention provides a dynamic topology optimization method for a stiffened load-bearing structure under moving mass excitation, the flowchart of which is shown below. Figure 1 As shown, the main steps include:

[0039] S101: Construct the design domain of the stiffened load-bearing structure on a uniform geometric background mesh, set boundary conditions and apply moving mass excitation, and initialize the design variables such as the node positions and cross-sectional areas of the stiffened (equivalent to a Timoshenko beam) elements, as well as the algorithm parameters: material properties of the stiffeners, Rayleigh damping coefficient, material properties and dimensional information of the uniform geometric background mesh (discrete form of the plate), mass of the moving mass excitation, initial velocity and acceleration, and optimized convergence parameters.

[0040] In this invention, such as Figure 2 As shown, to ensure sufficient smoothness of the sensitivity field and the establishment of plate-stiffener coupling, the plate is discretized using geometrically degenerate shell elements, and the stiffeners are discretized using Timoshenko beam elements. Simultaneously, the initial layout is generated through a base structure model to ensure connectivity between the stiffeners.

[0041] S102: Considering the additional mass, damping and stiffness characteristics and additional loads induced by the moving mass excitation, construct the dynamic and geometric finite element equations of the stiffened load-bearing structure, and calculate the dynamic flexibility, volume constraints and local geometric control constraints to suppress the cross interference of stiffened elements of the stiffened load-bearing structure under the action of moving mass excitation.

[0042] In this invention, step S102 may further include:

[0043] Step S102-1: Calculate the additional mass, damping and stiffness characteristics and additional load induced by the moving mass excitation, and construct the dynamic and geometric finite element equations of the stiffened load-bearing structure.

[0044] By considering the displacement and force compatibility conditions at the contact point between the moving mass and the uniformly geometric background mesh, the moving mass... The additional mass matrix induced at time step t Additional damping matrix With additional stiffness matrix for:

[0045]

[0046] In the formula: , and Let be the displacement, velocity, and acceleration of the moving mass at time step t, respectively. The shape function matrix of the geometric background mesh at the contact point is... yes Set the first column to 0.

[0047] The dynamic and geometric finite element equations of a stiffened load-bearing structure under the excitation of moving mass are as follows:

[0048]

[0049]

[0050]

[0051] In the formula: , , and These are the additional mass, damping, and [other factors] induced by the moving mass excitation at time step t.

[0052] The influence of stiffness characteristics on the overall mass, damping, stiffness, and external load matrix. The transformation matrix is ​​the same geometric background mesh. Let be the external load matrix equivalent to the gravity of the moving mass directed towards the control point. , and These represent the displacement, velocity, and acceleration of the control points on the geometric background mesh at time step t. The total number of time steps for the quality incentive effect.

[0053] Based on the principle of energy equivalence, the stiffness, mass, and damping matrix of the stiffening element are diffused to a geometrically uniform background mesh, thus increasing the global stiffness of the stiffened load-bearing structure. ,quality With damping matrix Represented as:

[0054]

[0055] In the formula: This represents the total number of stiffening elements. , and These are the global stiffness, mass, and damping matrices for a mesh with an equal geometric background, respectively. , and These are the stiffness, mass, and damping diffusion matrices of the stiffener on a mesh with a uniform geometric background, respectively. , The Rayleigh damping parameters are for the background mesh and the stiffeners with uniform geometry.

[0056] Step S102-2: Construct the HHT-α method and the unconditionally stable implicit Newmark-β time integration scheme to solve the above equations, and calculate the dynamic flexibility, volume constraints, and local geometric control constraints to suppress cross-interference of stiffener elements in the stiffener-type load-bearing structure under the excitation of moving mass.

[0057] By combining the HHT-α method and the unconditionally stable implicit Newmark-β time integration scheme, the dynamic and geometric finite element model of the stiffened load-bearing structure is solved. The solution format is as follows:

[0058]

[0059]

[0060]

[0061] In the formula: Let be the residual of the HHT-α method integration scheme at time step t. , and The algorithm parameters for the above integration scheme are: Increment the time step.

[0062] By applying a minimum inscribed circle radius constraint within any triangle in the base structure (the initial layout of the stiffened load-bearing structure) to suppress the overlapping of stiffened elements and avoid the unreasonable stiffened layout caused by such geometric interference, we have:

[0063]

[0064]

[0065] In the formula: To control the local geometric constraints (i.e., minimum inscribed circle radius constraints) of the cross interference of stiffener elements, , The first The perimeter and area of ​​the triangle, For the first The radius of the inscribed circle of a triangle. The threshold value is the minimum inscribed circle radius. This represents the total number of triangles within the base structure.

[0066] In this invention, step S103 may further include:

[0067] Step S103-1: Establish a topology optimization model for the problem of minimizing the dynamic flexibility (maximizing the dynamic stiffness) of a stiffened load-bearing structure, solve the adjoint sensitivity analysis equation based on discretization followed by differentiation, and calculate the sensitivity of the dynamic flexibility, volume constraints, and local geometric control constraints of the cross-interference of stiffened elements in the stiffened load-bearing structure.

[0068] With the optimization objective of minimizing dynamic flexibility (maximizing dynamic stiffness), and taking the nodal coordinates and cross-sectional area of ​​the stiffening element as design variables, considering the amount of stiffening material and the cross-interference constraints of the stiffening elements, a dynamic topology optimization model of the stiffened load-bearing structure based on the equal geometric stiffness diffusion method is established, which yields:

[0069]

[0070] In the formula: For the dynamic flexibility of the stiffened load-bearing structure (abbreviated as...) ), Due to volume constraints, The design variables attached to the i-th stiffener element, where and Let be the coordinates of node i and node i+1. Let be the cross-sectional area of ​​the i-th rib element. Let be the length of the i-th stiffener element. This represents the maximum permissible volume usage for a stiffened load-bearing structure. and The range of x-coordinate variation for the nodes of the stiffener element. and The range of y-coordinate variation for the rib element node depends on the background mesh size.

[0071] Dynamic flexibility of ribbed load-bearing structure The sensitivity to design variables is:

[0072]

[0073] Accompanying variables The iteration format over time is:

[0074] for

[0075]

[0076]

[0077]

[0078] for

[0079]

[0080] In the formula: for The amount, , For the accompanying variables corresponding to the residual form of the implicit Newmark-β time integration scheme.

[0081] In addition, residual The sensitivity for the design variables is:

[0082] for

[0083]

[0084] for

[0085]

[0086] Sensitivity of volume constraints That is, for , and The sensitivity is expressed as:

[0087]

[0088]

[0089] Similarly, volume constraints can be calculated for... and Sensitivity.

[0090] Sensitivity of local geometric control constraints for cross-interference of stiffener elements for:

[0091]

[0092] Step S103-2: The original problem (including local geometric control constraints and volume constraints of cross interference of stiffener elements) is processed into a series of unconstrained problems using the augmented Lagrangian method. The design variables are updated using the moving asymptote method until the convergence criterion is met or the maximum number of iterations is reached.

[0093] Using the augmented Lagrangian method, a penalty term is constructed for the cross-interference constraint between volume constraints and stiffener elements. This forms the unconstrained subproblem at step k. When there are enough rib units, the penalty term applies. It may be much larger than the original problem's objective function. This normalizes the penalty term to avoid singularities in the optimization problem. Therefore, the Lagrange day problem can be expressed as:

[0094]

[0095]

[0096]

[0097] In the formula: Let the objective function be the unconstrained subproblem. For the Lagrange multipliers of the Lagrange day problem, The squared penalty factor for the Lagrange day problem. The inequality constraint between suppressing cross interference of stiffener elements and stiffener volume usage is transformed into an equality constraint.

[0098] The update format for the squared penalty factor is as follows:

[0099]

[0100] In the formula: Update parameters for the penalty factor, upper bound Used to suppress numerical instability problems in algorithms.

[0101] The update format for Lagrange multipliers is:

[0102]

[0103] The sensitivity of the objective function of the Lagrange day problem is:

[0104]

[0105] Based on the above sensitivity information, the design variables are updated using the moving asymptote method until the convergence criterion is met or the maximum number of iterations is reached.

[0106] S104: Based on the minimum member size process principle, a stiffener unit deletion strategy and a node fusion strategy are constructed to reconstruct the dynamic optimal topology of the stiffener-type load-bearing structure, thereby obtaining a clear optimal design layout for the stiffener-type load-bearing structure.

[0107] If the optimized rib cross-sectional area Or the length of the reinforcing bar Then, the corresponding stiffening elements are deleted to obtain a clear design layout for the final stiffened load-bearing structure. Additionally, the distance between adjacent nodes... When this happens, node fusion is performed. Generally, the following is chosen: (Maximum cross-sectional area of ​​the stiffeners in the optimal layout) If the maximum length of the stiffeners in the optimal layout is determined, then the corresponding stiffener units are deleted to obtain a clear design layout for the final stiffener-type load-bearing structure.

[0108] Secondly, the present invention provides a dynamic topology optimization system for a stiffened load-bearing structure under the excitation of moving mass, such as... Figure 3 As shown, it includes:

[0109] The initialization module constructs the initial layout of the stiffened load-bearing structure within a uniform geometric background mesh, initializes the design variables such as the node coordinates and cross-sectional area of ​​the stiffened elements, and pre-sets the material property parameters, volume constraints and stiffened element cross-interference constraints, HHT-α method and Newmark-β time integration scheme parameters, augmented Lagrangian optimization algorithm parameters, and stiffened element deletion and node fusion thresholds for the stiffened elements and background mesh.

[0110] The preprocessing module projects the stiffness, mass, and damping matrices of the stiffening element onto a geometrically equivalent background mesh using the principle of energy equivalence. It then integrates the additional stiffness, mass, and damping matrices generated by the moving mass excitation and applies initial and boundary conditions to form a dynamic geometrically equivalent finite element model of the stiffening load-bearing structure.

[0111] The dynamic analysis module combines the HHT-α method and the unconditionally stable implicit Newmark-β time integration scheme to solve the dynamic and geometric finite element models of stiffened load-bearing structures, and calculates the dynamic compliance, volume constraints and cross-interference constraints of stiffened elements of the structure.

[0112] The optimization solution module constructs an augmented Lagrangian function that includes volume constraints and cross-interference constraints of stiffener elements, calculates the sensitivity information of the dynamic topology optimization model of the stiffener-type load-bearing structure, updates the design variables such as the node coordinates and cross-sectional area of ​​the stiffener elements based on the moving asymptote method, and judges whether the optimization iteration result meets the convergence criterion. If it does not meet the convergence criterion, the optimization iteration continues until the convergence criterion is met.

[0113] The post-processing module, based on the minimum member size process principle, reconstructs the optimal dynamic topology (layout) of the stiffener-type load-bearing structure according to the threshold parameters of the cross section, length and node fusion of the stiffener unit, thereby obtaining the optimal design layout of the stiffener-type load-bearing structure that is convenient for engineering applications.

[0114] Example

[0115] Example 1

[0116] This invention addresses the problem of rib layout optimization in arch beam structures under moving mass excitation, further verifying the effectiveness of the method.

[0117] Step S101: Construct the design domain of the stiffened load-bearing structure on a uniform geometric background mesh, set boundary conditions and apply moving mass excitation, and initialize the design variables such as the node positions and cross-sectional areas of the stiffeners (equivalent to Temmosinco beams) and algorithm parameters: material properties of the stiffeners, Rayleigh damping coefficient, material properties and dimensional information of the uniform geometric background mesh (discrete form of the plate), mass of the moving mass excitation, initial velocity and acceleration, and optimized convergence parameters (such as...). Figure 4 (As shown).

[0118] like Figure 4 As shown, the structural dimensions of the arch beam design domain are: length ,high ,thickness The design domain is fixed at both ends, and the moving mass... The moving mass moves at a constant speed, and the moving speed is... The initial basic structure of the stiffener element is as follows: Figure 4 As shown in (b), the Young's modulus of the stiffener element. mass density The Young's modulus of the background mesh is mass density Both have a Poisson ratio of 1. The Rayleigh damping constant of the stiffener is , Number of response sampling points The background mesh size is 2.0m, the maximum volume of the stiffened load-bearing structure is 60, and the maximum number of iterations is set to 800. HHT-α algorithm parameters. Newmark-β algorithm parameters , .

[0119] S102: Considering the additional mass, damping and stiffness characteristics and additional loads induced by the moving mass excitation, construct the dynamic and geometric finite element equations of the stiffened load-bearing structure, and calculate the dynamic flexibility, volume constraints and local geometric control constraints to suppress the cross interference of stiffened elements of the stiffened load-bearing structure under the action of moving mass excitation.

[0120] Step S102-1: Calculate the additional mass, damping and stiffness characteristics and additional load induced by the moving mass excitation, and construct the dynamic and geometric finite element equations of the stiffened load-bearing structure.

[0121] By considering the displacement and force compatibility conditions at the contact point between the moving mass and the uniformly geometric background mesh, the moving mass... The additional mass matrix induced at time step t Additional damping matrix With additional stiffness matrix for:

[0122]

[0123] In the formula: , and Let be the displacement, velocity, and acceleration of the moving mass at time step t, respectively. The shape function matrix of the geometric background mesh at the contact point is... yes Set the first column to 0.

[0124] The dynamic and geometric finite element equations of a stiffened load-bearing structure under the excitation of moving mass are as follows:

[0125]

[0126]

[0127]

[0128] In the formula: , , and These represent the total mass, damping, stiffness, and external load matrices, respectively, considering the additional mass, damping, and stiffness characteristics induced by the moving mass excitation at time step t. The transformation matrix is ​​the same geometric background mesh. Let be the external load matrix equivalent to the gravity of the moving mass directed towards the control point. , and These represent the displacement, velocity, and acceleration of the control points on the geometric background mesh at time step t. The total number of time steps for the quality incentive effect.

[0129] Based on the principle of energy equivalence, the stiffness, mass, and damping matrix of the stiffening element are diffused to a geometrically uniform background mesh, thus increasing the global stiffness of the stiffened load-bearing structure. ,quality With damping matrix Represented as:

[0130]

[0131] In the formula: This represents the total number of stiffening elements. , and These are the global stiffness, mass, and damping matrices for a mesh with an equal geometric background, respectively. , and These are the stiffness, mass, and damping diffusion matrices of the stiffener on a mesh with a uniform geometric background, respectively. , The Rayleigh damping parameters are for the background mesh and the stiffeners with uniform geometry.

[0132] Step S102-2: Construct the HHT-α method and the unconditionally stable implicit Newmark-β time integration scheme to solve the above equations, and calculate the dynamic flexibility, volume constraints, and local geometric control constraints to suppress cross-interference of stiffener elements in the stiffener-type load-bearing structure under the excitation of moving mass.

[0133] By combining the HHT-α method and the unconditionally stable implicit Newmark-β time integration scheme, the dynamic and geometric finite element model of the stiffened load-bearing structure is solved. The solution format is as follows:

[0134]

[0135]

[0136]

[0137] In the formula: Let be the residual of the HHT-α method integration scheme at time step t. , and The algorithm parameters for the above integration scheme are: Increment the time step.

[0138] By applying a minimum inscribed circle radius constraint within any triangle in the base structure (the initial layout of the stiffened load-bearing structure) to suppress the overlapping of stiffened elements and avoid the unreasonable stiffened layout caused by such geometric interference, we have:

[0139]

[0140]

[0141] In the formula: To control the local geometric constraints (i.e., minimum inscribed circle radius constraints) of the cross interference of stiffener elements, , The first The perimeter and area of ​​the triangle, For the first The radius of the inscribed circle of a triangle. The threshold value is the minimum inscribed circle radius. This represents the total number of triangles within the base structure.

[0142] S103: Establish a topology optimization model for the problem of minimizing the dynamic flexibility (maximizing the dynamic stiffness) of a stiffened load-bearing structure. Combine the initial design variables and algorithm parameters of the design domain to obtain the dynamic optimal topology (layout) of the stiffened load-bearing structure under the excitation of moving mass.

[0143] Step S103-1: Establish a topology optimization model for the problem of minimizing the dynamic flexibility (maximizing the dynamic stiffness) of a stiffened load-bearing structure, solve the adjoint sensitivity analysis equation based on discretization followed by differentiation, and calculate the sensitivity of the dynamic flexibility, volume constraints, and local geometric control constraints of the cross-interference of stiffened elements in the stiffened load-bearing structure.

[0144] With the optimization objective of minimizing dynamic flexibility (maximizing dynamic stiffness), and taking the nodal coordinates and cross-sectional area of ​​the stiffening element as design variables, considering the amount of stiffening material and the cross-interference constraints of the stiffening elements, a dynamic topology optimization model of the stiffened load-bearing structure based on the equal geometric stiffness diffusion method is established, which yields:

[0145]

[0146] In the formula: For the dynamic flexibility of the stiffened load-bearing structure (abbreviated as...) ), Due to volume constraints, The design variables attached to the i-th stiffener element, where and Let be the coordinates of node i and node i+1. Let be the cross-sectional area of ​​the i-th rib element. Let be the length of the i-th stiffener element. This represents the maximum permissible volume usage for a stiffened load-bearing structure. and The range of x-coordinate variation for the nodes of the stiffener element. and The range of y-coordinate variation for the rib element node depends on the background mesh size.

[0147] Dynamic flexibility of ribbed load-bearing structure The sensitivity to design variables is:

[0148]

[0149] Accompanying variables The iteration format over time is:

[0150] for

[0151]

[0152]

[0153]

[0154] for

[0155]

[0156] In the formula: for The amount, , For the accompanying variables corresponding to the residual form of the implicit Newmark-β time integration scheme.

[0157] In addition, residual The sensitivity for the design variables is:

[0158] for

[0159]

[0160] for

[0161]

[0162] Sensitivity of volume constraints That is, for , and The sensitivity is expressed as:

[0163]

[0164]

[0165] Similarly, volume constraints can be calculated for... and Sensitivity.

[0166] Sensitivity of local geometric control constraints for cross-interference of stiffener elements for:

[0167]

[0168] Step S103-2: The original problem (including local geometric control constraints and volume constraints of cross interference of stiffener elements) is processed into a series of unconstrained problems using the augmented Lagrangian method. The design variables are updated using the moving asymptote method until the convergence criterion is met or the maximum number of iterations is reached.

[0169] Using the augmented Lagrangian method, a penalty term is constructed for the cross-interference constraint between volume constraints and stiffener elements. This forms the unconstrained subproblem at step k. When there are enough rib units, the penalty term applies. It may be much larger than the original problem's objective function. This normalizes the penalty term to avoid singularities in the optimization problem. Therefore, the Lagrange day problem can be expressed as:

[0170]

[0171]

[0172]

[0173] In the formula: Let the objective function be the unconstrained subproblem. Lagrange for the Lagrange Day Problem

[0174] Chengzi, The squared penalty factor for the Lagrange day problem. The inequality constraint between suppressing cross interference of stiffener elements and stiffener volume usage is transformed into an equality constraint.

[0175] The update format for the squared penalty factor is as follows:

[0176]

[0177] In the formula: Update parameters for the penalty factor, upper bound Used to suppress numerical instability problems in algorithms.

[0178] The update format for Lagrange multipliers is:

[0179]

[0180] The sensitivity of the objective function of the Lagrange day problem is:

[0181]

[0182] Based on the above sensitivity information, the design variables are updated using the moving asymptote method until the convergence criterion is met or the maximum number of iterations is reached.

[0183] S104: Based on the minimum member size process principle, a stiffener unit deletion strategy and a node fusion strategy are constructed to reconstruct the dynamic optimal topology of the stiffener-type load-bearing structure, thereby obtaining a clear optimal design layout for the stiffener-type load-bearing structure.

[0184] If the optimized rib cross-sectional area Or the length of the reinforcing bar Then, the corresponding stiffening elements are deleted to obtain a clear design layout for the final stiffened load-bearing structure. Additionally, the distance between adjacent nodes... When this happens, node fusion is performed. Generally, the following is chosen: (Maximum cross-sectional area of ​​the stiffeners in the optimal layout) If the maximum length of the stiffeners in the optimal layout is determined, then the corresponding stiffener units are deleted to obtain a clear design layout for the final stiffener-type load-bearing structure.

[0185] The specific optimization results are as follows: Figure 5 As shown.

[0186] For inertial effects that are not considered due to the excitation of moving mass (e.g.) Figure 5 As shown in (a), (c), and (e), with the increase of the moving speed, the pulse characteristics of the moving load are enhanced, guiding more material stiffener elements to move towards the area of ​​action of the moving load, thus suppressing the in-plane bending of the upper end face of the arch beam. For considering the inertial effects of the moving mass excitation (such as...), Figure 5 As shown in (b), (d), and (f), with the increase of the moving speed, the coupling effect between the moving mass and the arch beam is enhanced, which excites additional inertial forces, damping forces, and internal forces, resulting in a more complex layout of the stiffeners. Therefore, the inertial effect excited by the moving mass has a significant impact on the stiffener layout, and this phenomenon becomes more pronounced with the increase of the moving speed.

[0187] The results of Embodiment 1 of this invention show that, for the design of stiffener layout under in-plane vibration conditions, ignoring the inertial influence of moving mass excitation, i.e., performing topology optimization design of the stiffener-bearing structure according to the moving load, will lead to an unreasonable stiffener layout. For applications such as bridge engineering and rail transit, accurately calculating the load characteristics of moving mass excitation and integrating them into the dynamic topology optimization of the stiffener-bearing structure is necessary to adapt to the engineering design of these scenarios. The algorithm of this invention effectively solves the problem of stiffener dynamic layout optimization under moving mass excitation.

[0188] Example 2

[0189] Embodiment 2 of the present invention solves the problem of rib layout optimization of a square plate with four fixed sides under the excitation of moving mass, and further verifies the effectiveness of the method of the present invention.

[0190] Step S101: Construct the design domain of the stiffened load-bearing structure on a uniform geometric background mesh, set boundary conditions and apply moving mass excitation, and initialize the design variables such as the node positions and cross-sectional areas of the stiffeners (equivalent to Temmosinco beams) and algorithm parameters: material properties of the stiffeners, Rayleigh damping coefficient, material properties and dimensional information of the uniform geometric background mesh (discrete form of the plate), mass of the moving mass excitation, initial velocity and acceleration, and optimized convergence parameters (such as...). Figure 6(As shown).

[0191] like Figure 6 As shown, the design domain dimensions of the four-sided fixed square plate structure are: length ,thickness Moving mass Movement speed The ribs and the background mesh have the same material properties: Young's modulus. mass density Both have a Poisson ratio of 1. The Rayleigh damping constant of the stiffener is , Number of response sampling points The background mesh size is 2.5m, the maximum volume of the stiffened load-bearing structure is 60, and the maximum number of iterations is set to 800. HHT-α algorithm parameters. Newmark-β algorithm parameters , .

[0192] Step S102 is the same as in Example 1.

[0193] Step S103 is the same as in Example 1.

[0194] Step S104 is the same as in Example 1.

[0195] The specific optimization results are as follows: Figure 7 As shown.

[0196] Compared to a rib layout that does not consider the inertial effects of moving mass excitation (e.g.) Figure 7 (a) shows that the inertia of the moving mass excitation causes additional inertial forces, damping forces, and internal forces, resulting in an increase in the number of ribs near the moving mass's trajectory (e.g., ...). Figure 7 (b) shown), thereby suppressing out-of-plane vibration of the thin plate caused by the additional load.

[0197] The results of Embodiment 2 of this invention show that, for the design of stiffener layout under out-of-plane vibration conditions, ignoring the inertial influence of moving mass excitation, i.e., performing topology optimization design of the stiffener-bearing structure according to the moving load, will lead to an unreasonable stiffener layout. For applications such as bridge engineering and rail transit, accurately calculating the load characteristics of moving mass excitation and integrating them into the dynamic topology optimization of the stiffener-bearing structure is necessary to adapt to the engineering design of these scenarios. The algorithm of this invention effectively solves the problem of stiffener dynamic layout optimization under moving mass excitation.

[0198] The present invention has been described in detail above with reference to specific embodiments and exemplary examples. However, these descriptions should not be construed as limiting the present invention. Those skilled in the art will understand that various equivalent substitutions, modifications, or improvements can be made to the technical solutions and embodiments of the present invention without departing from the spirit and scope of the present invention, and all such modifications and improvements fall within the scope of the present invention.

Claims

1. A method for dynamic topology optimization of a truss structure under moving mass excitation, characterized in that, Includes the following steps: S101: Construct the design domain of the stiffened load-bearing structure on a uniform geometric background mesh, set boundary conditions and apply moving mass excitation, and initialize design variables such as node positions and cross-sectional areas of the stiffened elements, as well as algorithm parameters: material properties of the stiffeners, Rayleigh damping coefficient, material properties and size information of the uniform geometric background mesh, mass of the moving mass excitation, initial velocity and acceleration, and optimized convergence parameters; S102: Considering the additional mass, damping and stiffness characteristics and additional load induced by the moving mass excitation, construct the dynamic and geometric finite element equations of the stiffened load-bearing structure, and calculate the dynamic flexibility, volume constraints and local geometric control constraints to suppress the cross interference of stiffened elements of the stiffened load-bearing structure under the action of moving mass excitation. S103: Establish a topology optimization model for minimizing the dynamic flexibility (i.e. maximizing the dynamic stiffness) of a stiffened load-bearing structure. Combine the initial design variables and algorithm parameters of the design domain to solve for the optimal dynamic topology layout of the stiffened load-bearing structure under the excitation of moving mass. S104: Based on the minimum member size process principle, a stiffener unit deletion strategy and a node fusion strategy are constructed to reconstruct the dynamic optimal topology of the stiffener-type load-bearing structure, thereby obtaining a clear optimal design layout for the stiffener-type load-bearing structure. Step S102 includes: Step S102-1: Calculate the additional mass, damping and stiffness characteristics and additional load induced by the moving mass excitation, and construct the dynamic and geometric finite element equations of the stiffened load-bearing structure. Step S102-2: Construct the HHT-α method and the unconditionally stable implicit Newmark-β time integration scheme to solve the above equations, and calculate the dynamic flexibility, volume constraints and local geometric control constraints to suppress cross interference of stiffener elements of the rigid load-bearing structure under the excitation of moving mass. Step S102-1 includes: The moving mass The additional mass matrix induced at time step t The additional damping matrix The additional stiffness matrix is In the formula: , and Let be the displacement, velocity, and acceleration of the moving mass at time step t, respectively. The shape function matrix of the geometric background mesh at the contact point is... yes Set the first column to 0; The dynamic and geometric finite element equations of a stiffened load-bearing structure under the excitation of moving mass are as follows: In the formula: , , and These represent the total mass, damping, stiffness, and external load matrices, respectively, considering the additional mass, damping, and stiffness characteristics induced by the moving mass excitation at time step t. The transformation matrix is ​​the same geometric background mesh. Let be the external load matrix equivalent to the gravity of the moving mass directed towards the control point. , and These represent the displacement, velocity, and acceleration of the control points on the geometric background mesh at time step t. The total number of time steps for the motion quality incentive effect; Based on the principle of energy equivalence, the stiffness, mass, and damping matrix of the stiffening element are diffused to a geometrically uniform background mesh, thus increasing the global stiffness of the stiffened load-bearing structure. ,quality With damping matrix Represented as: In the formula: This represents the total number of stiffening elements. , and These are the global stiffness, mass, and damping matrices for a mesh with an equal geometric background, respectively. , and These are the stiffness, mass, and damping diffusion matrices of the stiffener element on a mesh with a uniform geometric background, respectively. , The Rayleigh damping parameters are for the background mesh and the stiffeners with uniform geometry.

2. The dynamic topology optimization method for a stiffened load-bearing structure under moving mass excitation as described in claim 1, characterized in that, Step S102-2 includes: By combining the HHT-α method and the unconditionally stable implicit Newmark-β time integration scheme, the dynamic and geometric finite element model of the stiffened load-bearing structure is solved. The solution format is as follows: In the formula: Let be the residual of the HHT-α method integration scheme at time step t. , and The algorithm parameters for the above integration scheme are: Increment the time step; Applying a minimum inscribed circle radius constraint to any triangle in the initial layout of the stiffener-type load-bearing structure to suppress the cross-stabilization of stiffener elements and avoid the unreasonable stiffener layout caused by such geometric interference, we have: In the formula: To control the local geometric constraints of cross interference between stiffener elements, the minimum inscribed circle radius constraint is represented. , The first The perimeter and area of ​​the triangle, For the first The radius of the inscribed circle of a triangle. The threshold value is the minimum inscribed circle radius. This represents the total number of triangles within the basic structure.

3. A dynamic topology optimization method for a stiffened load-bearing structure under moving mass excitation as described in claim 1 or 2, characterized in that, Step S103 includes: Step S103-1: Establish a topology optimization model for the problem of minimizing the dynamic flexibility of a stiffened load-bearing structure, i.e. maximizing its dynamic stiffness. Solve the adjoint sensitivity analysis equation based on discretization followed by differentiation, and calculate the sensitivity of the dynamic flexibility, volume constraints, and local geometric control constraints of the cross-interference of stiffened elements in the stiffened load-bearing structure. Step S103-2: The original problem is treated as a series of unconstrained problems using the augmented Lagrangian method. The design variables are updated using the moving asymptote method until the convergence criterion is met or the maximum number of iterations is reached. The original problem includes local geometric control constraints and volume constraints of cross interference of stiffener elements.

4. The dynamic topology optimization method for a stiffened load-bearing structure under moving mass excitation as described in claim 3, characterized in that, Step S103-1 includes: With the optimization objective of minimizing dynamic flexibility (i.e. maximizing dynamic stiffness), and taking the nodal coordinates and cross-sectional area of ​​the stiffening element as design variables, considering the amount of stiffening material and the cross-interference constraints of the stiffening elements, a dynamic topology optimization model of the stiffened load-bearing structure based on the equal geometric stiffness diffusion method is established, resulting in: In the formula: Dynamic flexibility of the stiffened load-bearing structure , Due to volume constraints, The design variables attached to the i-th stiffener element, where and Let be the coordinates of node i and node i+1. Let be the cross-sectional area of ​​the i-th rib element. Let be the length of the i-th stiffener element. This represents the maximum permissible volume usage for a stiffened load-bearing structure. and The range of x-coordinate variation for the nodes of the stiffener element. and The range of y-direction coordinates for the stiffener element nodes is given, and this range depends on the background mesh size. Dynamic flexibility of ribbed load-bearing structure The sensitivity to design variables is: Accompanying variables The iteration format over time is: for for In the formula: for The amount, , These are the accompanying variables corresponding to the residual form of the implicit Newmark-β time integration scheme; residual The sensitivity for the design variables is: for for 。 5. The dynamic topology optimization method for a stiffened load-bearing structure under moving mass excitation as described in claim 3, characterized in that, Step S103-2 includes: Using the augmented Lagrangian method, a penalty term is constructed for the cross-interference constraint between volume constraints and stiffener elements. This forms the unconstrained subproblem at step k. When there are enough rib units, the penalty term applies. Much larger than the original problem's objective function This normalizes the penalty term to avoid singularities in the optimization problem; the Lagrange day problem is expressed as: In the formula: Let the objective function be the unconstrained subproblem. For the Lagrange multipliers of the Lagrange day problem, The squared penalty factor for the Lagrange day problem. The inequality constraint between suppressing cross interference of stiffener elements and stiffener volume usage is transformed into an equality constraint. The update format for the squared penalty factor is as follows: In the formula: Update parameters for the penalty factor, upper bound Used to suppress numerical instability problems in algorithms; The update format for Lagrange multipliers is: The sensitivity of the objective function of the Lagrange day problem is: Based on the above sensitivity information, the design variables are updated using the moving asymptote method until the convergence criterion is met or the maximum number of iterations is reached.

6. A system for dynamic topology optimization of a stiffened load-bearing structure under moving mass excitation, the system being implemented based on the method described in any one of claims 1 to 5, characterized in that... Includes the following program modules: The initialization module constructs the initial layout of the stiffened load-bearing structure within a uniform geometric background mesh, initializes the node coordinates and cross-sectional area of ​​the stiffened elements, and pre-sets the material property parameters, volume constraints, and cross-interference constraints of the stiffened elements and the background mesh, as well as the parameters of the HHT-α method and the Newmark-β time integration scheme, the parameters of the augmented Lagrangian optimization algorithm, and the thresholds for stiffened element deletion and node fusion. The preprocessing module projects the stiffness, mass, and damping matrices of the stiffening element onto a geometrically equivalent background mesh using the principle of energy equivalence. It then integrates the additional stiffness, mass, and damping matrices generated by the moving mass excitation and applies initial and boundary conditions to form a dynamic geometrically equivalent finite element model of the stiffening load-bearing structure. The dynamic analysis module combines the HHT-α method and the unconditionally stable implicit Newmark-β time integration scheme to solve the dynamic and geometric finite element models of stiffened load-bearing structures, and calculates the dynamic compliance, volume constraints and cross-interference constraints of stiffened elements of the structure. The optimization solution module constructs an augmented Lagrangian function that includes volume constraints and cross-interference constraints of stiffener elements, calculates the sensitivity information of the dynamic topology optimization model of the stiffener-type load-bearing structure, updates the node coordinates and cross-sectional area of ​​the stiffener elements based on the moving asymptote method, and determines whether the optimization iteration results meet the convergence criteria. If not, the optimization iteration continues until the convergence criteria are met. The post-processing module, based on the minimum member size process principle, reconstructs the optimal dynamic topology layout of the stiffener-type load-bearing structure according to the threshold parameters of the cross-section, length and node fusion of the stiffener unit, thereby obtaining the optimal design layout of the stiffener-type load-bearing structure that is convenient for engineering applications.

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