Stiffened plate structure time domain dynamics topological optimization method and system based on model reduction method

Abstract

The invention discloses a stiffened plate structure time domain dynamics topological optimization method and system based on a model reduction method, and relates to the field of topological optimization design of stiffened plate structures. According to the method, efficient solving of the time domain dynamics topological optimization problem of the stiffened plate structure is achieved, and therefore the large-scale lightweight dynamics design requirement of high-performance thin-wall engineering equipment is met. According to the technical key points, a base structure introducing rib unit local geometric control constraints is constructed, the initial connectivity of rib units is ensured, and the problem of local cross interference in the rib unit optimization process is inhibited; based on an equal geometric stiffness diffusion method, a time domain dynamics full-order topological optimization model of the stiffened plate structure is established. Then, on the basis of an intrinsic orthogonal decomposition method and an incremental singular value decomposition method, a time domain dynamics reduction model of the stiffened plate structure is constructed; and meanwhile, a sensitivity analysis adjoint equation of the time domain dynamics optimization problem is cooperatively reduced, so that the time domain dynamics topological optimization problem of the stiffened plate structure is efficiently solved. And finally, on the basis of a modified penalty function method, constructing an updating iteration format of design variables, and seeking an optimal rib layout design of the stiffened plate structure. The method is used for time domain dynamics layout optimization of the stiffened plate structure.

Description

Technical Field

[0001] This invention relates to the field of topology optimization design of stiffened plate structures, and more specifically, to a time-domain dynamic topology optimization method and system for stiffened plate structures based on a model reduction method. Background Technology

[0002] Stiffened plate structures, as fundamental structural units in industrial equipment such as automobiles, ships, and aircraft, can achieve optimal stiffener layout through time-domain dynamic topology optimization design with limited material consumption, thereby ensuring optimal equipment dynamic performance. Therefore, this is of great significance for the lightweight design of engineering equipment operating in vibration environments. However, the time-domain dynamic topology optimization problem for stiffened plates requires iteratively solving strongly coupled time-domain response control equations and sensitivity analysis adjoint equations, consuming high computational costs and thus failing to meet the large-scale lightweight dynamic design requirements of the aforementioned engineering equipment. Existing technologies do not provide a technical path or effective solution for improving the efficiency of time-domain dynamic topology optimization of stiffened plate structures, or for collaboratively constructing reduced models for structural dynamics analysis and low-order models for topology optimization to achieve efficient solutions to the time-domain dynamic topology optimization problem of stiffened plate structures, fundamentally addressing the large-scale lightweight dynamic design requirements of engineering equipment. Summary of the Invention

[0003] The technical problem to be solved by this invention is:

[0004] To address the shortcomings and deficiencies of the aforementioned problems, this invention provides a time-domain dynamic topology optimization method and system for stiffened plate structures based on a model reduction approach. This method collaboratively constructs a reduced model for structural dynamics analysis and a low-order model for topology optimization, thereby achieving efficient solutions to the time-domain dynamic topology optimization problem of stiffened plate structures and meeting the large-scale lightweight dynamic design requirements of high-performance thin-walled engineering equipment.

[0005] To achieve the above objectives, in a first aspect, the present invention provides a time-domain dynamic topology optimization method for stiffened plate structures based on a model reduction method, which includes the following steps:

[0006] S101: A full-order time-domain dynamic finite element model of a stiffened plate coupled with a Timoshenko beam is established based on the equal geometric stiffness diffusion method. Subsequently, a local control constraint of the inner radius of the triangle is introduced to suppress the cross interference of the stiffener elements. The size and position information of the stiffeners are used as design variables to construct a full-order time-domain dynamic topology optimization model of the stiffened plate that considers the size and shape of the stiffeners.

[0007] S102: Based on the intrinsic orthogonal decomposition method, a reduced model of the full-order time-domain dynamic finite element model of the stiffened plate is established; then, a low-dimensional projection matrix is ​​constructed by the incremental singular value decomposition method, and the high-order matrix of the above full-order transient dynamic finite element model is projected into a low-order matrix, and then the vibration response of the stiffened plate structure is efficiently solved by the HHT-α numerical integration algorithm.

[0008] S103: Establish a low-order time-domain dynamic topology optimization model for minimizing the dynamic flexibility of stiffened plate structures, construct a reduced-order sensitivity adjoint equation to calculate dual variables, thereby achieving efficient solution of the time-domain dynamic topology optimization model of stiffened plate structures in low-rank space and obtaining the optimal topology of the time-domain dynamics of stiffened plate structures.

[0009] S104: Based on the minimum member size process principle, a stiffener unit deletion strategy and a node fusion strategy are constructed to reconstruct the time-domain dynamic optimal topology of the stiffened plate structure, thereby obtaining a clear optimal design layout for the stiffened plate structure.

[0010] Secondly, the present invention provides a time-domain dynamic topology optimization system for stiffened plate structures based on a model reduction method, comprising:

[0011] The preprocessing module constructs a base structure reflecting the initial layout of the stiffeners, establishes a stiffened plate structure coupled with a Timoshenko beam based on the equal geometric stiffness diffusion method, applies displacement boundary conditions and time-domain dynamic loads, and forms a full-order time-domain dynamic finite element model of the stiffened plate structure; at the same time, it sets the material property parameters of the stiffeners and the plate, designs the initial values ​​and ranges of the variables, the parameters of volume constraints and stiffener element cross-interference constraints, and the parameters of response calculation and optimization algorithms.

[0012] The dynamics solution module solves the full-order time-domain dynamics finite element model of the stiffened plate structure, constructs a displacement snapshot matrix, and uses this matrix to calculate the basis of the reduced model based on the incremental singular value decomposition method. Then, it establishes the time-domain dynamics reduced model of the stiffened plate structure through the eigenorthogonal decomposition method. The HHT-α numerical integration scheme is used to solve the above-mentioned time-domain dynamics reduced model of the stiffened plate structure, and calculates the dynamic compliance, volume constraints, and stiffener element cross constraints of the stiffened plate structure.

[0013] The optimization solution module, based on the strategy of differentiation followed by discretization, calculates the sensitivity information of the low-order dynamic topology optimization model of the stiffened plate structure, constructs a penalty function that considers the volume of the stiffened plate structure and the cross-interference constraint of the stiffener elements, updates the design variables such as the node coordinates and cross-sectional area of ​​the stiffener elements based on the modified penalty function 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.

[0014] The post-processing module, based on the minimum member size process principle, reconstructs the temporal dynamic optimal topology (layout) of the stiffened plate structure according to the threshold parameters (minimum size) of the cross-section, length and node fusion of the stiffening unit, thereby obtaining the optimal stiffening design layout of the stiffening plate structure that is convenient for engineering applications.

[0015] This invention relates to a time-domain dynamic topology optimization method and system for stiffened plate structures based on model reduction, which has the following beneficial technical effects:

[0016] This invention addresses the challenges of temporal dynamic topology optimization for stiffened plate structures: the high computational cost caused by large-scale spatiotemporal coupling iterations and sensitivity analysis in temporal dynamics problems. By introducing a model reduction method, extracting the "eigenmodes" that best represent the main characteristics of the vibration system, and the adjoint equations for the synergistic reduced-order temporal dynamic topology optimization sensitivity analysis, this invention achieves efficient solution of the temporal dynamic topology optimization problem in low-rank space, significantly improving the design efficiency of temporal dynamic layout optimization for stiffened plate structures.

[0017] This invention constructs a base structure incorporating local geometric control constraints on stiffener elements to ensure initial connectivity and suppress local cross-interference during the stiffener element optimization process. Based on the equal geometric stiffness diffusion method, a full-order time-domain dynamic topology optimization model for the stiffened plate structure is established. Subsequently, based on the intrinsic orthogonal decomposition method and incremental singular value decomposition method, a reduced time-domain dynamic model for the stiffened plate structure is constructed. Simultaneously, sensitivity analysis of the adjoint equations of the reduced-order time-domain dynamic optimization problem is performed, thereby achieving efficient solution to the time-domain dynamic topology optimization problem of the stiffened plate structure. Finally, based on the modified penalty function method, an iterative update format for the design variables is constructed to seek the optimal stiffener layout design for the stiffened plate structure. Therefore, this invention proposes a time-domain dynamic topology optimization method and system for stiffened plate structures based on model reduction, which includes: defining the base structure of the stiffened plate structure; constructing a full-order time-domain dynamic finite element model of the stiffened plate structure based on the stiffness diffusion method; establishing a time-domain dynamic reduced model and a reduced-order sensitivity analysis adjoint equation of the stiffened plate structure based on the intrinsic orthogonal decomposition method and the incremental singular value decomposition method; using the time integration scheme of the HHT-α method to solve the above equations collaboratively, calculating the dynamic compliance, volume constraints, local constraints of stiffener element cross-interference, and sensitivity information of the stiffened plate structure; constructing a penalty function containing the volume of the stiffened plate structure and the local constraints of stiffener element cross-interference; updating the design variables such as the node coordinates and cross-sectional area of ​​the stiffener elements based on the modified penalty function method; and reconstructing the dynamic optimal topology (layout) of the truss structure according to the minimum member size process principle, thereby obtaining the optimal design layout of the stiffeners in the stiffened plate structure that is convenient for engineering applications.

[0018] This invention proposes a time-domain dynamic topology optimization method and system for stiffened plate structures based on model reduction, which can be used to efficiently solve the time-domain dynamic layout optimization problem of stiffened plate structures under transient loads.

[0019] Attached Figures and Tables

[0020] Figure 1 This is a flowchart of a time-domain dynamic topology optimization method for stiffened plate structures based on a model reduction method, according to the present invention.

[0021] Figure 2 This is a schematic diagram of a time-domain dynamic topology optimization system for stiffened plate structures based on a model reduction method according to the present invention.

[0022] Figure 3 (a) is a schematic diagram of a cantilever beam problem, and (b) is a schematic diagram of the design domain of Embodiment 1 of the present invention.

[0023] (b) Schematic diagram of half-wave sinusoidal load in Embodiment 1 of the present invention;

[0024] (c) The initial design base structure of Embodiment 1 of the present invention.

[0025] Figure 4 This is a schematic diagram of the truss layout optimization results for the cantilever beam problem. (a) shows the optimized rib layout (load application time) of the full-order model in Embodiment 1 of the present invention. and );

[0026] (b) Optimized rib layout (load application time) for the reduced model of Embodiment 1 of the present invention and );

[0027] (c) is the optimization iteration history of Embodiment 1 of the present invention.

[0028] Figure 5 Comparison of optimization performance (load application time) between the full-order model and the reduced model in Embodiment 1 of the invention. and ).

[0029] Figure 6 The vertical displacement at the load application point (load application time) in Embodiment 1 of the present invention. ). Detailed Implementation

[0030] This invention addresses the challenges of temporal dynamic topology optimization for stiffened plate structures, specifically the high computational costs resulting from large-scale spatiotemporal coupling iterations and sensitivity analysis in discrete structure temporal dynamics problems. It proposes a method and system for temporal dynamic topology optimization of stiffened plate structures based on a model reduction approach, enabling efficient temporal dynamic topology optimization design (i.e., layout optimization design) of stiffened plate structures.

[0031] Combined with appendix Figures 1-6 Appendix 1 details the time-domain dynamic topology optimization method and system for stiffened plate structures based on the model reduction method described in this invention, 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.

[0032] In a first aspect, the present invention provides a time-domain dynamic topology optimization method for stiffened plate structures based on a model reduction method, the flowchart of which is as follows: Figure 1 As shown, the main steps include:

[0033] S101: A full-order time-domain dynamic finite element model of a stiffened plate coupled with an isogeometric degenerate shell and a Timoshenko beam is established based on the isogeometric stiffness diffusion method. Subsequently, a local control constraint of the inner radius of the triangle is introduced to suppress the cross interference of the stiffener elements. The size and position information of the stiffeners are used as design variables to construct a full-order time-domain dynamic topology optimization model of the stiffened plate that considers the size and shape of the stiffeners.

[0034] This invention proposes a modeling method for stiffened plate foundation structures that considers local geometric constraints. This method ensures the connectivity of stiffening elements in the initial configuration and effectively suppresses cross-interference during optimization. Boundary conditions and time-domain dynamic loads are applied to the stiffening elements.

[0035] Design variables and initial values ​​are assigned to the elements, and algorithm parameters for solving time-domain dynamics and topology optimization are set.

[0036] In this invention, step S101 may further include:

[0037] Step S101-1: Based on the principle of energy equivalence, the stiffness, mass, and damping matrices of the Timoshenko beam (stiffener) element are projected onto the geometric background mesh (a geometrically degenerate shell discrete by finite element) to form a full-order time-domain dynamic finite element model of the stiffened plate structure.

[0038] Based on the equivalence principle of deformation energy, kinetic energy, and damping energy dissipation, the stiffness, mass, and damping matrices of the stiffening element are diffused into the element stiffness, mass, and damping matrices of the flat plate structure through the NURBS basis functions of the geometrically degenerate shell element, expressed as follows:

[0039] , ,

[0040] , ,

[0041] In the formula: , and These are the displacement, velocity, and acceleration matrices of stiffener element i, respectively. , and To represent the displacement, velocity, and acceleration matrices of the control points corresponding to the geometrically degenerate shell element i, For the corresponding geometrically degenerate shell elements, (The rest of the text appears to be a list of NURBS basis functions.) , and Let be the stiffness, mass, and damping matrices of stiffener element i, respectively. , and These are the stiffness diffusion matrix, mass diffusion matrix, and damping diffusion matrix of the stiffener element, respectively.

[0042] Combining the matrices of the assembled stiffening elements and the geometrically degenerate shell elements, the full-order time-domain dynamic equations of the stiffened plate structure are as follows:

[0043]

[0044]

[0045] In the formula: , and These are the global stiffness matrix, mass matrix, and damping matrix of the stiffened plate structure, respectively. , and These represent the global stiffness matrix, mass matrix, and damping matrix of the plate, respectively. and Let be the acceleration, velocity, and displacement matrices of the control point on the flat plate at time step t. It is the load array at time step t. This represents the total time step of the dynamic event.

[0046] Step S101-2: Construct a stiffened plate base structure that introduces a local geometric control strategy (triangle inner radius constraint) to ensure the connectivity between stiffening elements in the initial configuration and to suppress the cross-stacking problem of stiffening elements during optimization iteration, thereby establishing a full-order time-domain dynamic topology optimization model of the stiffened plate structure.

[0047] By applying a minimum inscribed circle radius constraint within any triangle in the base structure to suppress the overlapping of stiffener elements and avoid the unreasonable stiffener layout caused by such geometric interference, we have:

[0048] ,

[0049] In the formula: To control the local geometric constraints of 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 basic structure.

[0050] With the optimization objective of minimizing dynamic flexibility (maximizing dynamic stiffness), the nodal coordinates and cross-sectional area of ​​the stiffening element are used as design variables.

[0051] Considering the amount of stiffener material and the cross-interference constraints of stiffener elements, a full-order time-domain dynamic topology optimization model for stiffened plate structures is established:

[0052]

[0053]

[0054]

[0055]

[0056]

[0057]

[0058] In the formula: To improve the dynamic flexibility of stiffened plate structures, Due to volume constraints, The design variables attached to the i-th stiffener element, where and Let i be the coordinates of node i and node i+1. Let be the cross-sectional area of ​​the i-th stiffener element. Let be the length of the i-th stiffener element. This represents the total number of stiffening elements. This represents the maximum permissible volume of reinforcing bars. 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.

[0059] Step S102: Based on the intrinsic orthogonal decomposition method, a reduced model of the full-order time-domain dynamic finite element model of the stiffened plate is established; then, a low-dimensional projection matrix is ​​constructed by the incremental singular value decomposition method, and the high-order matrix of the above full-order transient dynamic finite element model is projected into a low-order matrix, and then the vibration response of the stiffened plate structure is efficiently solved by the HHT-α numerical integration algorithm.

[0060] In this invention, step S102 may further include:

[0061] Step S102-1: Based on the intrinsic orthogonal decomposition method, a reduced model of the full-order time-domain dynamic finite element model of the stiffened plate is established. The solution formula is constructed by the HHT-α numerical integration algorithm. Then, the low-rank solution space of the reduced model is used to best and uniformly approximate the high-dimensional solution space of the full-order model.

[0062] The displacement solutions of the full-order time-domain dynamic model of the stiffened plate structure are obtained by using the intrinsic orthogonal decomposition method. Approximate decomposition into:

[0063]

[0064] In the formula: for The best uniform approximation in the low-rank solution space It serves as a basis (low-dimensional projection matrix) for the low-rank solution space. for The column number (representing the number of "eigenmodes") To define the degrees of freedom of the full-order time-domain dynamic finite element model of the stiffened plate, satisfying . as base The coefficient vector at the t-th time step.

[0065] Substituting the above equation into the dynamic equation Then, a reduced time-domain dynamic model of the stiffened plate structure can be obtained:

[0066]

[0067]

[0068] In the formula: , and To reduce the global mass, damping, and stiffness matrices of the model, To reduce the load array of the model.

[0069] Combining the numerical integration scheme of the HHT-α method, the time-domain reduced dynamic model of the stiffened plate structure is solved, with the iterative format as follows:

[0070]

[0071]

[0072] In the formula: , and The algorithm parameters for the HHT-α method integration scheme are... Increment the time step.

[0073] Step S102-2: Obtain the displacement field snapshot matrix by calculating the full-order time-domain dynamic finite element model of the stiffened plate structure, construct an update algorithm for the low-rank solution space projection matrix, and project the high-order matrix of the above full-order time-domain dynamic finite element model into a low-order matrix to achieve efficient solution of the time-domain dynamic model of the stiffened plate structure.

[0074] Constructing a snapshot matrix ( (The solution of the full-order dynamic finite element model at the i-th time step) is used to find the basis. The full-order model is converted into a reduced-order model to reduce the storage of the snapshot matrix. To address the significant storage costs incurred, a low-dimensional projection matrix update algorithm based on incremental singular value decomposition is proposed. An orthogonal basis for the optimal uniform approximation subspace is extracted from the solution space of the full-order finite element model. (Low-dimensional projection matrix), snapshot matrices at time step i and time step (i+1). , They are respectively:

[0075] ,

[0076]

[0077] In the formula: SVD represents the singular value decomposition operation of a matrix. , and They are respectively The left singular matrix, diagonal matrix, and right singular matrix.

[0078] The singular value decomposition is as follows:

[0079]

[0080] In the formula: , and They are respectively The left singular matrix, diagonal matrix, and right singular matrix.

[0081] if In Orthogonal projection of a low-rank space with a basis ,satisfy ( (where the error limit is the basis update), then at the (i+1)th time step Updated to:

[0082]

[0083] In the formula: , and These are the left singular matrix, the diagonal matrix, and the right singular matrix in the above equation, respectively.

[0084] if In The orthogonal projection of a low-rank space with basis satisfies Then at the (i+1)th time step Updated to:

[0085]

[0086] In the formula: , and Let these be the left singular matrix, the diagonal matrix, and the right singular matrix in the above equation, respectively. and Projection deviation The unit vector and magnitude.

[0087] In summary, traversing the snapshot matrix By looking at all columns, you can get the results for all time periods. The singular value decomposition is as follows:

[0088]

[0089] In the formula: , and Snapshot matrices The left singular matrix, diagonal matrix, and right singular matrix, for of There are ordered singular values, among which This forms the basis for the time-domain dynamic reduction model of stiffened plate structures. .

[0090] To balance computational efficiency and accuracy, discarding The nonnegative singular values ​​are close to zero. A sufficiently small number of bases are chosen such that the ratio of the total energy of the stiffened plate structure in low-rank space to the energy in full-rank space is... Satisfy the following formula:

[0091]

[0092] In the formula: r is the truncation number, representing The optimal value.

[0093] Therefore, the optimal basis for the time-domain dynamic reduction model of the stiffened plate structure is:

[0094]

[0095] In summary, the basis of the reduced model is obtained using incremental singular value decomposition. There is no need to additionally store snapshot matrices for all times. It only needs to store the basis from the previous time step and a small diagonal matrix, and update the basis at the current time step in real time, effectively reducing the basis time. The computational cost.

[0096] S103: Establish a low-order time-domain dynamic topology optimization model for minimizing the dynamic flexibility of stiffened plate structures, construct a reduced-order sensitivity adjoint equation to calculate dual variables, thereby achieving efficient solution of the time-domain dynamic topology optimization model of stiffened plate structures in low-rank space and obtaining the optimal topology of the time-domain dynamics of stiffened plate structures.

[0097] In this invention, step S103 may further include:

[0098] Step S103-1: Establish a low-order time-domain dynamic topology optimization model for minimizing the dynamic flexibility of stiffened plate structures, construct a reduced-order sensitivity adjoint equation, and achieve efficient solution of dual variables and sensitivity calculation.

[0099] use Flexibility The lower-order approximation is:

[0100]

[0101] The full-order time-domain dynamic topology optimization model of the stiffened plate structure is reduced to the following low-order time-domain dynamic topology optimization model, expressed as:

[0102]

[0103] Based on the time-domain problem sensitivity analysis strategy of differentiation followed by discretization, the low-order dynamic flexibility estimation of the original problem is performed. For design variables The sensitivity is:

[0104]

[0105] In the formula: To reduce the Lagrange multipliers of the model, the corresponding sensitivity adjoint equation is:

[0106]

[0107] make and The low-order sensitivity adjoint equation above is transformed into:

[0108]

[0109]

[0110] In the formula: and For the transformed Lagrange multipliers The initial "displacement" and "velocity".

[0111] Comparison of time-domain dynamic reduction models of stiffened plate structures The transformed adjoint equation has the same expression form, and therefore can be solved using the same HHT-α numerical integration scheme.

[0112] Step S103-2: Construct a modified penalty function method to process the local geometric control constraints and volume constraints of the stiffener element, so as to achieve efficient updating of design variables of the low-order dynamic topology optimization model.

[0113] Construct a penalty function that includes the volume of the stiffened plate structure and the cross-interference constraints of the stiffener elements, and introduce it into... In the middle, there are:

[0114]

[0115] In the formula: Let be the augmented function of the original problem. This is a piecewise function relating to the intersection of stiffener elements and volume constraints.

[0116] Regarding variables The piecewise function is:

[0117]

[0118] In the formula: , , , , for The penalty function is defined as follows: when the constraints of rib element intersection and volume are satisfied, it uses an exponential function to apply a small penalty to the constraints; when the constraints deviate from the above conditions, it uses a quadratic function to apply the penalty, and the greater the deviation, the more significant the penalty.

[0119] After introducing the modified penalty function, the objective function and constraints of the optimization problem can be written as follows:

[0120]

[0121] Solving the above optimization problem yields the optimal solution for the time-domain dynamic topology optimization of the stiffened plate structure.

[0122] S104: Based on the minimum member size process principle, a stiffener unit deletion strategy and a node fusion strategy are constructed to reconstruct the time-domain dynamic optimal topology of the stiffened plate structure, thereby obtaining a clear optimal design layout for the stiffened plate structure.

[0123] If the optimized cross-sectional area or Then, delete the corresponding stiffening elements to obtain a final design layout with a clear stiffening 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 stiffener is determined (in the optimal layout), then the corresponding stiffener unit is deleted to obtain the final design layout with a clear stiffener structure.

[0124] Secondly, this invention provides a time-domain dynamic topology optimization system for stiffened plate structures based on a base model reduction method, such as... Figure 2 As shown, it includes:

[0125] The preprocessing module constructs a base structure reflecting the initial layout of the stiffeners, establishes a stiffened plate structure coupled with a Timoshenko beam based on the equal geometric stiffness diffusion method, applies displacement boundary conditions and time-domain dynamic loads, and forms a full-order time-domain dynamic finite element model of the stiffened plate structure; at the same time, it sets the material property parameters of the stiffeners and the plate, designs the initial values ​​and ranges of the variables, the parameters of volume constraints and stiffener element cross-interference constraints, and the parameters of response calculation and optimization algorithms.

[0126] The dynamics solution module solves the full-order time-domain dynamics finite element model of the stiffened plate structure, constructs a displacement snapshot matrix, and uses this matrix to calculate the basis of the reduced model based on the incremental singular value decomposition method. Then, it establishes the time-domain dynamics reduced model of the stiffened plate structure through the eigenorthogonal decomposition method. The HHT-α numerical integration scheme is used to solve the above-mentioned time-domain dynamics reduced model of the stiffened plate structure, and calculates the dynamic compliance, volume constraints, and stiffener element cross constraints of the stiffened plate structure.

[0127] The optimization solution module, based on the strategy of differentiation followed by discretization, calculates the sensitivity information of the low-order dynamic topology optimization model of the stiffened plate structure, constructs a penalty function that considers the volume of the stiffened plate structure and the cross-interference constraint of the stiffener elements, updates the design variables such as the node coordinates and cross-sectional area of ​​the stiffener elements based on the modified penalty function 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.

[0128] The post-processing module, based on the minimum member size process principle, reconstructs the temporal dynamic optimal topology (layout) of the stiffened plate structure according to the threshold parameters (minimum size) of the cross-section, length and node fusion of the stiffening unit, thereby obtaining the optimal stiffening design layout of the stiffening plate structure that is convenient for engineering applications.

[0129] Example

[0130] Example 1

[0131] The embodiments of the present invention solve the problem of stiffening plate layout optimization under half-wave sinusoidal load, and further verify the effectiveness of the method of the present invention.

[0132] Step S101: Establish a full-order time-domain dynamic finite element model of a stiffened plate coupled with a Timoshenko beam based on the equal geometric stiffness diffusion method; subsequently, introduce local control constraints on the inner radius of triangles to suppress cross-interference of stiffener elements, and use the size and position information of the stiffeners as design variables to construct a full-order time-domain dynamic topology optimization model of the stiffened plate considering the size and shape of the stiffeners (e.g., Figure 3 (As shown).

[0133] like Figure 3 As shown, the structural dimensions of the cantilever beam design domain are: length ,high ,thickness Boundary conditions and time-domain loads: The left end of the design domain is completely fixed, and an amplitude is applied at the midpoint of the free surface on the right side. A half-wave sinusoidal load, considering the load duration and Two working conditions. Young's modulus of the stiffening element. mass density The Young's modulus of the flat plate is mass density Both have a Poisson ratio of 1. The Rayleigh damping constants of the stiffeners and the plate are: , Number of response sampling points Reduce the error limit of model basis updates. To ensure the unconditional stability of the vibration response iteration process, the HHT-α algorithm parameters... , , .

[0134] Step S101-1: Based on the principle of energy equivalence, the stiffness, mass, and damping matrices of the Timoshenko beam (stiffener) element are projected onto the geometric background mesh (a geometrically degenerate shell discrete by finite element) to form a full-order time-domain dynamic finite element model of the stiffened plate structure.

[0135] Based on the equivalence principle of deformation energy, kinetic energy, and damping energy dissipation, the stiffness, mass, and damping matrices of the stiffening element are diffused into the element stiffness, mass, and damping matrices of the flat plate structure through the NURBS basis functions of the geometrically degenerate shell element, expressed as follows:

[0136] , ,

[0137] , ,

[0138] In the formula: , and These are the displacement, velocity, and acceleration matrices of stiffener element i, respectively. , and To represent the displacement, velocity, and acceleration matrices of the control points corresponding to the geometrically degenerate shell element i, For the corresponding geometrically degenerate shell elements, (The rest of the text appears to be a list of NURBS basis functions.) , and Let be the stiffness, mass, and damping matrices of stiffener element i, respectively. , and These are the stiffness diffusion matrix, mass diffusion matrix, and damping diffusion matrix of the stiffener element, respectively.

[0139] Combining the matrices of the assembled stiffening elements and the geometrically degenerate shell elements, the full-order time-domain dynamic equations of the stiffened plate structure are as follows:

[0140]

[0141] ,

[0142] In the formula: , and These are the global stiffness matrix, mass matrix, and damping matrix of the stiffened plate structure, respectively. , and These represent the global stiffness matrix, mass matrix, and damping matrix of the plate, respectively. and Let be the acceleration, velocity, and displacement matrices of the control point on the flat plate at time step t. It is the load array at time step t. This represents the total time step of the dynamic event.

[0143] Step S101-2: Construct a stiffened plate base structure that introduces a local geometric control strategy (triangle inner radius constraint) to ensure the connectivity between stiffening elements in the initial configuration and to suppress the cross-stacking problem of stiffening elements during optimization iteration, thereby establishing a full-order time-domain dynamic topology optimization model of the stiffened plate structure.

[0144] By applying a minimum inscribed circle radius constraint within any triangle in the base structure to suppress the overlapping of stiffener elements and avoid the unreasonable stiffener layout caused by such geometric interference, we have:

[0145] ,

[0146] In the formula: To control the local geometric constraints of 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 basic structure.

[0147] With the optimization objective of minimizing dynamic flexibility (maximizing dynamic stiffness), and the nodal coordinates and cross-sectional area of ​​the stiffening elements as design variables, a full-order time-domain dynamic topology optimization model for the stiffened plate structure is established, considering the amount of stiffening material and the cross-interference constraints of the stiffening elements.

[0148]

[0149]

[0150]

[0151]

[0152]

[0153]

[0154] In the formula: To improve the dynamic flexibility of stiffened plate structures, Due to volume constraints, The design variables attached to the i-th stiffener element, where and Let i 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 total number of stiffening elements. This represents the maximum permissible volume of reinforcing bars. 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.

[0155] S102: Based on the intrinsic orthogonal decomposition method, a reduced model of the full-order time-domain dynamic finite element model of the stiffened plate is established; then, a low-dimensional projection matrix is ​​constructed by the incremental singular value decomposition method, and the high-order matrix of the above full-order transient dynamic finite element model is projected into a low-order matrix, and then the vibration response of the stiffened plate structure is efficiently solved by the HHT-α numerical integration algorithm.

[0156] Step S102-1: Based on the intrinsic orthogonal decomposition method, establish a reduced model of the full-order time-domain dynamic finite element model of the stiffened plate, construct its solution formula through the HHT-α numerical integration algorithm, and then use the low-rank solution space of the reduced model to best and uniformly approximate the high-dimensional solution space of the full-order model.

[0157] The displacement solutions of the full-order time-domain dynamic model of the stiffened plate structure are obtained by using the intrinsic orthogonal decomposition method. Approximate decomposition into:

[0158]

[0159] In the formula: for The best uniform approximation in the low-rank solution space It serves as a basis (low-dimensional projection matrix) for the low-rank solution space. for The column number (representing the number of "eigenmodes") To define the degrees of freedom of the full-order time-domain dynamic finite element model of the stiffened plate, satisfying . as base The coefficient vector at the t-th time step.

[0160] Substituting the above equation into... Then, a reduced time-domain dynamic model of the stiffened plate structure can be obtained:

[0161]

[0162]

[0163] In the formula: , and To reduce the global mass, damping, and stiffness matrices of the model, To reduce the load array of the model.

[0164] Using the HHT-α numerical integration scheme, the time-domain reduced dynamic model of the stiffened plate structure is solved, and the solution format is as follows:

[0165]

[0166]

[0167] In the formula: , and The algorithm parameters for the HHT-α method integration scheme are... Increment the time step.

[0168] Step S102-2: Obtain the displacement field snapshot matrix by calculating the full-order time-domain dynamic finite element model of the stiffened plate structure, construct an update algorithm for the low-rank solution space projection matrix, and project the high-order matrix of the above full-order time-domain dynamic finite element model into a low-order matrix to achieve efficient solution of the time-domain dynamic model of the stiffened plate structure.

[0169] Constructing a snapshot matrix ( (The solution of the full-order dynamic finite element model at the i-th time step) is used to find the basis. The full-order model is converted into a reduced-order model to reduce the storage of the snapshot matrix. To address the significant storage costs incurred, a low-dimensional projection matrix update algorithm based on incremental singular value decomposition is proposed. An orthogonal basis for the optimal uniform approximation subspace is extracted from the solution space of the full-order finite element model. (Low-dimensional projection matrix), snapshot matrices at time step i and time step (i+1). , They are respectively:

[0170] ,

[0171]

[0172] In the formula: SVD represents the singular value decomposition operation of a matrix. , and They are respectively The left singular matrix, diagonal matrix, and right singular matrix.

[0173] The singular value decomposition is as follows:

[0174]

[0175] In the formula: , and They are respectively The left singular matrix, diagonal matrix, and right singular matrix.

[0176] if In Orthogonal projection of a low-rank space with a basis ,satisfy ( (where the projection error limit is), then at the (i+1)th time step Updated to:

[0177]

[0178] In the formula: , and These are the left singular matrix, the diagonal matrix, and the right singular matrix in the above equation, respectively.

[0179] if In The orthogonal projection of a low-rank space with basis satisfies Then at the (i+1)th time step

[0180] of Updated to:

[0181]

[0182] In the formula: , and Let these be the left singular matrix, the diagonal matrix, and the right singular matrix in the above equation, respectively. and Projection deviation The unit vector and magnitude.

[0183] In summary, traversing the snapshot matrix By looking at all columns, you can get the results for all time periods. The singular value decomposition is as follows:

[0184]

[0185] In the formula: , and Snapshot matrices The left singular matrix, diagonal matrix, and right singular matrix, for of There are ordered singular values, among which This forms the basis for the time-domain dynamic reduction model of stiffened plate structures. .

[0186] To balance computational efficiency and accuracy, discarding The nonnegative singular values ​​are close to zero. A sufficiently small number of bases are chosen such that the ratio of the total energy of the stiffened plate structure in low-rank space to the energy in full-rank space is... Satisfy the following formula:

[0187]

[0188] In the formula: r is the truncation number, representing The optimal value.

[0189] Therefore, the optimal basis for the time-domain dynamic reduction model of the stiffened plate structure is:

[0190]

[0191] In summary, the basis of the reduced model is obtained using incremental singular value decomposition. There is no need to additionally store snapshot matrices for all times. It only needs to store the basis from the previous time step and a small diagonal matrix, and update the basis at the current time step in real time, effectively reducing the basis time. The computational cost.

[0192] S103: Establish a low-order time-domain dynamic topology optimization model for minimizing the dynamic flexibility of stiffened plate structures, construct a reduced-order sensitivity adjoint equation to calculate dual variables, thereby achieving efficient solution of the time-domain dynamic topology optimization model of stiffened plate structures in low-rank space and obtaining the optimal topology of the time-domain dynamics of stiffened plate structures.

[0193] Step S103-1: Establish a low-order time-domain dynamic topology optimization model for minimizing the dynamic flexibility of stiffened plate structures, construct a reduced-order sensitivity adjoint equation, and achieve efficient solution of dual variables and sensitivity calculation.

[0194] use Flexibility The lower-order approximation is:

[0195]

[0196] The full-order time-domain dynamic topology optimization model of the stiffened plate structure is reduced to the following low-order time-domain dynamic topology optimization model, expressed as:

[0197]

[0198] Based on the time-domain problem sensitivity analysis strategy of differentiation followed by discretization, the low-order dynamic flexibility estimation of the original problem is performed. For design variables The sensitivity is:

[0199]

[0200] In the formula: To reduce the Lagrange multipliers in the model, the corresponding low-order sensitivity adjoint equation is:

[0201]

[0202] make and The above sensitivity adjoint equation is transformed into:

[0203]

[0204]

[0205] In the formula: and For the transformed Lagrange multipliers The initial "displacement" and "velocity".

[0206] Comparison of time-domain dynamic reduction models of stiffened plate structures The transformed adjoint equation has the same expression form, and therefore can be solved using the same HHT-α numerical integration scheme.

[0207] Step S103-2: Construct a modified penalty function method to process the local geometric control constraints and volume constraints of the stiffener element, so as to achieve efficient updating of design variables of the low-order dynamic topology optimization model.

[0208] Construct a penalty function that includes the volume of the stiffened plate structure and the cross-interference constraints of the stiffener elements, and introduce it into... In the middle, there are:

[0209]

[0210] In the formula: Let be the augmented function of the original problem. This is a piecewise function relating to the intersection of stiffener elements and volume constraints.

[0211] Regarding variables The piecewise function is:

[0212]

[0213] In the formula: , , , , for The penalty function is defined as follows: when the constraints of rib element intersection and volume are satisfied, it uses an exponential function to apply a small penalty to the constraints; when the constraints deviate from the above conditions, it uses a quadratic function to apply the penalty, and the greater the deviation, the more significant the penalty.

[0214] After introducing the modified penalty function, the objective function and constraints of the optimization problem can be written as follows:

[0215]

[0216] Solving the above optimization problem yields the optimal solution for the time-domain dynamic topology optimization of the stiffened plate structure.

[0217] S104: Based on the minimum member size process principle, a stiffener unit deletion strategy and a node fusion strategy are constructed to reconstruct the time-domain dynamic optimal topology of the stiffened plate structure, thereby obtaining a clear optimal design layout for the stiffened plate structure.

[0218] If the optimized cross-sectional area or Then, delete the corresponding stiffening elements to obtain a final design layout with a clear stiffening 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 stiffener is determined (in the optimal layout), then the corresponding stiffener unit is deleted to obtain the final design layout with a clear stiffener structure.

[0219] The specific optimization results are as follows: Figure 4 and Figure 5As shown.

[0220] like Figure 4 As shown in 4.(a) and 4.(b), the full-order and reduced models ( This produces almost identical optimal stiffened plate structures. Furthermore, as the loading time decreases, the stiffener distribution tends to converge towards the load-bearing end region to counteract the weakening effect of structural inertial loads on the structural dynamic stiffness. For example... Figure 4 (c) and Figure 5 As shown, the optimal dynamic compliance of the reduced model converges almost asymptotically to the optimization objective value of the full-order model. However, the optimization computation time of the reduced model is much lower than that of the full-order model, thus verifying the efficiency of the reduced model for the time-domain dynamic topology optimization problem of stiffened plate structures. The influence of the number of basis columns ("number of eigenmodes") of the reduced model on its optimization performance (load application time) in Embodiment 1 of the invention is also discussed. Using the full-order model as a reference, we analyze the orthogonal basis. Number of columns (number of intrinsic modes) The impact on optimization cost and accuracy. Compared with the full-order model, as the intrinsic mode order N increases... b With the increase in [something], the computation time for topology optimization relatively increased, the speedup ratio decreased from 26.95 to 6.84, and the relative error of optimal dynamic flexibility (optimization target value) decreased from 24.6% to 1.2%. In fact, At that time, the vertical displacement of the reduced model at the load application point is very close to that of the full-order model (e.g. Figure 6 As shown in the figure, therefore, to balance accuracy and sufficient acceleration, N is recommended. b The value range is 4 to 8.

[0221] The results of Embodiment 1 of the present invention show that the introduction of intrinsic orthogonal decomposition and incremental singular value decomposition significantly reduces the topology optimization time of stiffened plate structures under time-domain dynamic loads, effectively solves the technical contradiction between optimization accuracy and optimization cost, and provides an efficient engineering optimization method for the design of stiffener dynamic layout of complex large-scale high-performance thin-walled structures, thereby verifying the superiority and effectiveness of the method of the present invention.

[0222] Table 1

[0223]

[0224] Table 1 shows the effect of the number of basis columns ("the number of intrinsic modes") of the reduced model on its optimization performance (load application time) in Embodiment 1 of the invention.

[0225] 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 time-domain dynamic topology optimization method for stiffened plate structures based on model reduction, characterized in that, Includes the following steps: S101: A full-order time-domain dynamic finite element model of a stiffened plate coupled with an isogeometric degenerate shell and a Timoshenko beam is established based on the isogeometric stiffness diffusion method. Subsequently, a local control constraint of the inner radius of the triangle is introduced to suppress the cross interference of the stiffener elements. The size and position information of the stiffeners are used as design variables to construct a full-order time-domain dynamic topology optimization model of the stiffened plate that considers the size and shape of the stiffeners. S102: A reduced model of the full-order time-domain dynamic finite element model of stiffened plate is established based on the intrinsic orthogonal decomposition method. Subsequently, a low-dimensional projection matrix is ​​constructed using the incremental singular value decomposition method, and the high-order matrix of the above full-order transient dynamic finite element model is projected into a low-order matrix. Then, the vibration response of the stiffened plate structure is efficiently solved using the HHT-α numerical integration algorithm. S103: Establish a low-order time-domain dynamic topology optimization model for minimizing the dynamic flexibility of stiffened plate structures, construct a reduced-order sensitivity adjoint equation to calculate dual variables, thereby achieving efficient solution of the time-domain dynamic topology optimization model of stiffened plate structures in low-rank space and obtaining the optimal topology of the time-domain dynamics of stiffened plate structures. S104: Based on the minimum member size process principle, a stiffener unit deletion strategy and a node fusion strategy are constructed to reconstruct the time-domain dynamic optimal topology of the stiffened plate structure, thereby obtaining a clear optimal design layout for the stiffened plate structure.

2. The time-domain dynamic topology optimization method for stiffened plate structures based on model reduction method according to claim 1, characterized in that, Step S101 includes: Step S101-1: Based on the principle of energy equivalence, the stiffness, mass and damping matrices of Timoshenko beam elements or stiffener elements are projected onto the geometrically background mesh, i.e., the geometrically degenerate shell discretized by the finite element method, to form a full-order time-domain dynamic finite element model of the stiffened plate structure. Step S101-2: Construct a stiffened plate base structure that introduces a local geometric control strategy (triangle inner radius constraint) to ensure the connectivity between stiffening elements in the initial configuration and to suppress the cross-stacking problem of stiffening elements during optimization iteration, thereby establishing a full-order time-domain dynamic topology optimization model of the stiffened plate structure.

3. The time-domain dynamic topology optimization method for stiffened plate structures based on model reduction method according to claim 2, characterized in that, Step S101-1 includes: Based on the equivalence principle of deformation energy, kinetic energy, and damping energy dissipation, the stiffness, mass, and damping matrices of the stiffening element are diffused into the element stiffness, mass, and damping matrices of the flat plate structure through the NURBS basis functions of the geometrically degenerate shell element, expressed as follows: In the formula: , and These are the displacement, velocity, and acceleration matrices of rib element i, respectively. , and To represent the displacement, velocity, and acceleration matrices of the control points corresponding to the geometrically degenerate shell element i, For the corresponding geometrically degenerate shell elements, (The rest of the text appears to be a list of NURBS basis functions.) , and Let be the stiffness, mass, and damping matrices of stiffener element i, respectively. , and These are the stiffness diffusion matrix, mass diffusion matrix, and damping diffusion matrix of the stiffener element, respectively. Combining the matrices of the assembled stiffening elements and the geometrically degenerate shell elements, the full-order time-domain dynamic equations of the stiffened plate structure are as follows: In the formula: , and These represent the global stiffness matrix, mass matrix, and damping matrix of the stiffened plate structure. , and These are the global stiffness matrix, mass matrix, and damping matrix of the plate, respectively. and Let be the acceleration, velocity, and displacement matrices of the control point on the flat plate at time step t. It is the load array at time step t. This represents the total time step of the dynamic event.

4. The time-domain dynamic topology optimization method for stiffened plate structures based on model reduction method according to claim 2, characterized in that, Step S101-2 includes: By applying a minimum inscribed circle radius constraint within any triangle in the base structure to suppress the overlapping 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 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. The total number of triangles within the base structure; With the optimization objective of minimizing dynamic flexibility (i.e. maximizing dynamic stiffness), and taking the nodal coordinates and cross-sectional area of ​​stiffening elements as design variables, a full-order time-domain dynamic topology optimization model for stiffened plate structures is established, considering the amount of stiffening material and the cross-interference constraints of stiffening elements. In the formula: To improve the dynamic flexibility of stiffened plate structures, Due to volume constraints, The design variables attached to the i-th stiffener element, where and Let i be the coordinates of the reinforcing bar 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 total number of stiffening elements. This represents the maximum permissible volume of reinforcing bars. 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.

5. The time-domain dynamic topology optimization method for stiffened plate structures based on model reduction as described in claim 1, characterized in that, Step S102 includes: Step S102-1: Based on the intrinsic orthogonal decomposition method, establish a reduced model of the full-order time-domain dynamic finite element model of the stiffened plate, construct its solution formula through the HHT-α numerical integration algorithm, and then use the low-rank solution space of the reduced model to best and uniformly approximate the high-dimensional solution space of the full-order model. Step S102-2: Obtain the displacement field snapshot matrix by calculating the full-order time-domain dynamic finite element model of the stiffened plate structure, construct an update algorithm for the low-rank solution space projection matrix, and project the high-order matrix of the above full-order time-domain dynamic finite element model into a low-order matrix to achieve efficient solution of the time-domain dynamic model of the stiffened plate structure.

6. The time-domain dynamic topology optimization method for stiffened plate structures based on model reduction method according to claim 5, characterized in that, Step S102-1 includes: The displacement solutions of the full-order time-domain dynamic model of the stiffened plate structure are obtained by using the intrinsic orthogonal decomposition method. Approximate decomposition into: In the formula: for The best uniform approximation in the low-rank solution space It serves as a basis for the low-rank solution space, i.e., a low-dimensional projection matrix. for The column number represents the number of "eigenmodes". To define the degrees of freedom of the full-order time-domain dynamic finite element model of the stiffened plate, satisfying ; as base The coefficient vector at the t-th time step; Substituting the above equation into... Then, a reduced time-domain dynamic model of the stiffened plate structure can be obtained: In the formula: , and To reduce the global mass, damping, and stiffness matrices of the model, To reduce the load matrix of the model; Using the HHT-α numerical integration scheme, the time-domain reduced dynamic model of the stiffened plate structure is solved, and the solution format is as follows: In the formula: , and The algorithm parameters for the HHT-α method integration scheme are... Increment the time step.

7. The time-domain dynamic topology optimization method for stiffened plate structures based on model reduction method according to claim 5, characterized in that, Step S102-2 includes: Constructing a snapshot matrix ( (The solution of the full-order dynamic finite element model at the i-th time step) is used to find the basis. The full-order model is converted into a reduced model to reduce the storage of the snapshot matrix. To address the large storage costs incurred, a low-dimensional projection matrix update algorithm based on incremental singular value decomposition is proposed. Extract the orthogonal basis of the best uniform approximation subspace from the solution space of the full-order finite element model. That is, the low-dimensional projection matrix, and the snapshot matrix at the i-th time step and the (i+1)-th time step. , They are respectively: ， In the formula: SVD represents the singular value decomposition operation of a matrix. , and They are respectively The left singular matrix, diagonal matrix, and right singular matrix; The singular value decomposition is as follows: In the formula: , and They are respectively The left singular matrix, diagonal matrix, and right singular matrix; if In Orthogonal projection in a low-rank space with a basis ,satisfy ( (where the projection error limit is), then at the (i+1)th time step Updated to: In the formula: , and These are the left singular matrix, the diagonal matrix, and the right singular matrix in the above equation, respectively. if In Orthogonal projections in a low-rank space with basis satisfy Then at the (i+1)th time step Updated to: In the formula: , and Let these be the left singular matrix, the diagonal matrix, and the right singular matrix in the above equation, respectively. and Projection deviation The unit vector and magnitude; In summary, traversing the snapshot matrix By looking at all columns, you can get the results for all time periods. The singular value decomposition is as follows: In the formula: , and Snapshot matrices The left singular matrix, diagonal matrix, and right singular matrix, for of There are ordered singular values, among which This forms the basis for the time-domain dynamic reduction model of stiffened plate structures. ; To balance computational efficiency and accuracy, discard Non-negative singular values ​​close to zero are considered; a sufficiently small number of bases are selected such that the ratio of the total energy of the stiffened plate structure in low-rank space to the energy in full-rank space is... Satisfy the following formula: In the formula: r is the truncation number, representing The optimal value; Therefore, the optimal basis for the time-domain dynamic reduction model of the stiffened plate structure is: Obtaining the basis of the reduced model using incremental singular value decomposition. There is no need to additionally store snapshot matrices for all times. It only needs to store the basis from the previous time step and a small diagonal matrix, and update the basis at the current time step in real time, effectively reducing the basis. The computational cost.

8. The time-domain dynamic topology optimization method for stiffened plate structures based on model reduction method according to claim 1, characterized in that, Step S103 includes: Step S103-1: Establish a low-order time-domain dynamic topology optimization model for minimizing the dynamic flexibility of stiffened plate structures, construct a reduced-order sensitivity adjoint equation, and achieve efficient solution of dual variables and sensitivity calculation; Step S103-2: Construct a modified penalty function method to handle the local geometric control constraints and volume constraints of stiffening elements, and achieve efficient updating of design variables in the low-order dynamic topology optimization model.

9. The time-domain dynamic topology optimization method for stiffened plate structures based on model reduction method according to claim 8, characterized in that, Step S103-1 includes: use Flexibility The lower-order approximation is: The full-order time-domain dynamic topology optimization model of the stiffened plate structure is reduced to the following low-order time-domain dynamic topology optimization model: Based on the time-domain problem sensitivity analysis strategy of differentiation followed by discretization, the low-order dynamic flexibility estimation of the original problem is performed. For design variables The sensitivity is: In the formula: To reduce the Lagrange multipliers in the model, the corresponding low-order sensitivity adjoint equation is: make and The above sensitivity adjoint equation is transformed into: In the formula: and For the transformed Lagrange multipliers The initial "displacement" and "velocity"; Comparison of time-domain dynamic reduction models of stiffened plate structures The transformed adjoint equation has the same expression form, and therefore can be solved using the same HHT-α numerical integration scheme.

10. The time-domain dynamic topology optimization method for stiffened plate structures based on model reduction method according to claim 8, characterized in that, Step S103-2 includes: Construct a penalty function that includes the volume of the stiffened plate structure and the cross-interference constraints of the stiffener elements, and introduce it into... In the middle, there are: In the formula: Let be the augmented function of the original problem. This is a piecewise function relating to the intersection of stiffener elements and volume constraints; Regarding variables The piecewise function is: In the formula: , , , , for The penalty function uses an exponential function to apply a small penalty to the constraints when the rib element intersection and volume constraints are met. When the constraints deviate from the above constraints, a quadratic function is used for the penalty. The greater the deviation, the more significant the penalty. After introducing the modified penalty function, the objective function and constraints of the optimization problem can be written as follows: Solving the above optimization problem yields the optimal solution for the time-domain dynamic topology optimization of the stiffened plate structure.

11. A time-domain dynamic topology optimization system for stiffened plate structures based on a model reduction method, characterized in that, The system is implemented based on the method of any one of claims 1 to 10, and includes the following program modules: The preprocessing module constructs a base structure reflecting the initial layout of the stiffeners, establishes a stiffened plate structure coupled with a Timoshenko beam based on the equal geometric stiffness diffusion method, applies displacement boundary conditions and time-domain dynamic loads, and forms a full-order time-domain dynamic finite element model of the stiffened plate structure; at the same time, it sets the material property parameters of the stiffeners and the plate, designs the initial values ​​and ranges of the variables, the parameters of volume constraints and stiffener element cross-interference constraints, and the parameters of response calculation and optimization algorithms. The dynamics solution module solves the full-order time-domain dynamics finite element model of the stiffened plate structure, constructs a displacement snapshot matrix, and uses this matrix to calculate the basis of the reduced model based on the incremental singular value decomposition method. Then, it establishes the time-domain dynamics reduced model of the stiffened plate structure through the eigenorthogonal decomposition method. The HHT-α numerical integration scheme is used to solve the above-mentioned time-domain dynamics reduced model of the stiffened plate structure, and calculates the dynamic compliance, volume constraints, and stiffener element cross constraints of the stiffened plate structure. The optimization solution module, based on the strategy of differentiation followed by discretization, calculates the sensitivity information of the low-order dynamic topology optimization model of the stiffened plate structure, constructs a penalty function that considers the volume of the stiffened plate structure and the cross-interference constraint of the stiffener elements, updates the design variables such as the node coordinates and cross-sectional area of ​​the stiffener elements based on the modified penalty function 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. The post-processing module, based on the minimum member size process principle, reconstructs the optimal time-domain dynamic topology layout of the stiffened plate structure according to the threshold parameter of the cross-section, length and node fusion of the stiffening unit, i.e. the minimum size, so as to obtain the optimal stiffening design layout of the stiffening plate structure that is convenient for engineering applications.

Images