Functional gradient plate structure time domain dynamics topological optimization method and system based on principal component analysis method

Abstract

The invention discloses a functional gradient plate structure time domain dynamics topological optimization method and system based on a principal component analysis method, and belongs to the field of topological optimization design of functional gradient structures. The method is provided for improving the calculation efficiency of structural time domain dynamics topological optimization and solving the problem of synchronous time domain dynamics optimization of gradient material gradient and structural topology. The method is technically characterized by comprising the following steps: constructing an equal time domain dynamics full-order finite element model of a functional gradient plate structure on the basis of a Bezier decomposition and extraction algorithm of an NURBS (Non-Uniform Rational B-Space) function; establishing a tight and smooth functionally graded material interpolation model, and proposing a full-order time-domain dynamics multi-material topological optimization model for the problem of dynamic compliance minimization of the functionally graded plate structure; constructing an equal-time domain dynamics reduced-order finite element model of the functional gradient plate structure, cooperatively constructing a time domain adjoint equation of an optimization model, and efficiently solving sensitivity information of a target function and a constraint function; and on the basis of an MMA-GCMMA hybrid optimization algorithm, a design variable is updated, so that the optimal anti-vibration design of the functionally gradient plate structure convenient for engineering application is obtained. And the device is suitable for structural light-weight vibration reduction design of large-scale engineering equipment.

Description

Technical Field

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

[0002] With the accelerated advancement of lightweight and high-performance equipment, aircraft and underwater vehicles operate under long-term, high-vibration environments, making airborne precision instruments prone to vibration failure and load-bearing structures susceptible to fatigue damage. Functionally graded materials (FGRs), due to their excellent specific strength and stiffness, as well as the interfacial reliability resulting from continuous component transitions, have led to the widespread application of FGR plate structures in aerospace and marine equipment. Optimized design is urgently needed to fully realize their vibration resistance potential. However, temporal dynamic topology optimization of FGR structures faces two major challenges: first, the cost of iterative computation combining temporal finite element method (FEM) and temporal adjoint sensitivity analysis is high, limiting its application in large-scale engineering structures; second, the optimal topology of FGR structures depends on the gradient direction of the gradient material, and existing dynamic topology optimization methods that assume a fixed gradient direction within the plate surface fail to fully utilize the optimal dynamic performance of FGR structures. In summary, existing design technologies have failed to address the computational efficiency problem of temporal dynamic topology optimization and the problem of synchronous temporal dynamic optimization of gradient material transitions and structural topology. They lack efficient and agile dynamic design methods, making it difficult to adapt to the engineering requirements of lightweight and high-performance equipment under high-vibration environments. Summary of the Invention

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

[0004] Existing design technologies have failed to solve the computational efficiency problem of temporal dynamic topology optimization of functionally graded plate structures, as well as the problem of synchronous temporal dynamic optimization of in-plane gradient material change and structural topology. They lack efficient dynamic agile design methods and are difficult to adapt to the engineering requirements of lightweight and high-performance equipment under strong vibration environments.

[0005] To achieve the above objectives, in a first aspect, the present invention provides a method and system for time-domain dynamic topology optimization of functionally graded plate structures based on principal component analysis, comprising the following steps:

[0006] S101: Based on the Bezier decomposition and extraction algorithm of NURBS function, the stiffness, mass and damping matrices of NURBS element are calculated respectively to form a full-order finite element model of functionally graded plate structure with equal geometric time domain dynamics.

[0007] S102: Establish a compacted, smooth functional graded material interpolation model and propose a full-order time-domain dynamic multi-material topology optimization model for minimizing the dynamic compliance of functional graded plate structures.

[0008] S103: Seek the eigenmode matrix of the vibration of the functionally graded plate structure, and use it as a basis to establish a low-dimensional solution space that approximates the solution space of the above full-order model, thereby constructing a reduced-order finite element model of the functionally graded plate structure with equal geometric time-domain dynamics.

[0009] S104: A low-order time-domain dynamic multi-material topology optimization model is proposed to minimize the dynamic compliance of functionally graded plate structures. A reduced-order sensitivity time-domain adjoint equation is constructed to efficiently calculate the dynamic compliance and volume constraint of functionally graded plate structures.

[0010] sensitivity;

[0011] S105: The optimal topology design of the functionally graded plate structure is obtained by using a hybrid optimization algorithm of MMA (moving asymptote method) and GCMMA (globally convergent moving asymptote method).

[0012] Secondly, a time-domain dynamic multi-material topology optimization system for functionally graded plate structures based on principal component analysis includes:

[0013] The initialization module constructs the design domain of the functionally graded plate structure, initializes the control point density of the functionally graded plate and the design variables such as the volume fraction of strong materials attached to the plate elements, presets the property parameters and volume constraint upper limit parameters of the strong and weak phase materials of the plate structure, the algorithm parameters of the HHT-α time integration scheme, the material interpolation model parameters, and the MMA-GCMMA hybrid optimization algorithm parameters.

[0014] The preprocessing module uses isogeometric NURBS elements to discretize the design domain of the functionally graded plate structure, defines Dirichlet and Neumann boundary conditions, initial conditions, and applies time-domain dynamic loads; calculates the extraction operator, basis functions, and derivatives of the Bezier element, and uses this in conjunction with the material interpolation scheme to calculate the stiffness, mass, and damping matrices of the isogeometric NURBS element, forming an isogeometric time-domain dynamic full-order finite element model of the functionally graded plate structure;

[0015] The dynamic analysis module solves the full-order finite element model of the functionally graded plate structure in the time domain using the unconditionally stable HHT-α time integration scheme to obtain the transient image matrix; it constructs the eigenmode matrix and establishes a reduced-order finite element model of the functionally graded plate structure in the time domain based on the principal component analysis method, thereby calculating the approximate value of the dynamic compliance of the functionally graded plate structure and the volume constraint of the multiphase material.

[0016] The optimization solution module calculates the sensitivity information of the time-domain dynamic topology optimization model of the functionally graded plate structure. Based on the MMA-GCMMA hybrid optimization algorithm, it updates the design variables such as the control point density of the functionally graded plate and the volume fraction of strong materials attached to the plate elements. It then determines whether the optimization iteration results meet the convergence criteria. If not, it continues the optimization iteration until the convergence criteria are met.

[0017] This invention relates to a time-domain dynamic topology optimization method and system for functionally graded plate structures based on principal component analysis, which has the following beneficial technical effects:

[0018] This invention addresses the challenges faced by time-domain dynamic topology optimization methods for functionally graded plate structures: First, the cost of collaborative iterative calculations of time-domain finite element method and time-domain adjoint sensitivity analysis is high, which restricts its application in large-scale engineering structures; Second, the optimal topology of functionally graded structures depends on the gradient direction of the gradient material. Existing dynamic optimization methods that assume a fixed gradient direction within the plate surface cannot achieve collaborative optimization of structural topology and material distribution direction, thus failing to fully realize the optimal dynamic performance of functionally graded structures.

[0019] The method of this invention integrates the isogeometric time-domain dynamics analysis method based on Bezier extraction operator and the principal component analysis method into the structural time-domain dynamics topology optimization method. It can efficiently obtain the optimal topology of functionally graded plates with clear and smooth multiphase material boundaries, and realize the agile design of high-precision geometric expression optimization topology. This invention proposes a time-domain dynamic topology optimization method and system for functionally graded plate structures (FJPs) based on principal component analysis, using the control point density and the volume fraction of strong materials attached to the plate elements as design variables. The method includes: defining the initial design domain and algorithm parameters; constructing a full-order finite element model of the FJP structure based on the Bezier decomposition and extraction algorithm using NURBS functions; establishing a tightly supported, smooth FJP interpolation model and proposing a full-order time-domain dynamic multi-material topology optimization model for minimizing the dynamic compliance of the FJP structure; constructing a reduced-order finite element model of the FJP structure based on the same geometry, collaboratively constructing the time-domain adjoint equation of the optimization model, and efficiently solving for the sensitivity information of the objective function and constraint functions; and updating the design variables based on the MMA-GCMMA hybrid optimization algorithm to obtain the optimal vibration-resistant design of the FJP structure suitable for engineering applications.

[0020] This invention proposes a time-domain dynamic topology optimization method and system for functionally graded plate (FGP) structures based on principal component analysis (PCA). This method significantly improves the dynamic optimization efficiency of FGP structures, efficiently achieving synergistic optimization of the topology along the gradient material transition direction within the plate surface and maximizing the optimal vibration resistance design of the FGP. The method integrates isogeometric time-domain dynamic analysis based on Bezier extraction operators and PCA into the structural time-domain dynamic topology optimization method. This efficiently obtains the optimal topology of FGPs with clear and smooth multiphase material boundaries, enabling agile design of the optimized topology with high-precision geometric expression. This invention is a highly efficient dynamic agile design method, adaptable to the engineering requirements of lightweight and high-performance equipment under strong vibration environments. It significantly improves the dynamic optimization efficiency of FGP structures, maximizes the optimal vibration resistance performance of FGP structures, and achieves agile design of the optimized topology with high-precision geometric expression, making it particularly suitable for the lightweight vibration reduction design of large-scale engineering equipment. Attached Figure Description

[0021] Figure 1 This is a flowchart of a time-domain dynamic topology optimization method for functionally graded plate structures based on principal component analysis, according to the present invention.

[0022] Figure 2 This is a schematic diagram illustrating the calculation principle of the stiffness matrix and mass matrix of the functionally graded plate solid element based on the Bezier extraction operator in this invention.

[0023] Figure 3 This is a schematic diagram of a time-domain dynamic topology optimization system for a functionally graded plate structure based on principal component analysis, according to the present invention.

[0024] Figure 4 This is a problem involving a square plate with fixed corners and functional gradients under a single-point load.

[0025] (a) is the structural design domain in Embodiment 1 of the present invention;

[0026] (b) is the half-wave cosine load in Embodiment 1 of the present invention.

[0027] Figure 5 This is a schematic diagram of the topology optimization results for a functionally graded square plate with fixed corners under a single-point load.

[0028] (a) The load application time in Embodiment 1 of the present invention The topology optimization results;

[0029] (b) The load application time in Embodiment 1 of the present invention The topology optimization results;

[0030] (c) The load application time in Embodiment 1 of the present invention The topology optimization results.

[0031] Figure 6 This represents the normal displacement of the center of a square plate with fixed corners and functional gradient under a single-point load.

[0032] (a) The full-order model and the reduced-order model in Embodiment 1 of the present invention (corresponding to) Figure 5 .a's optimized topology) in Comparison of the normal displacement of the center of the square plate below;

[0033] (b) The full-order model and the reduced-order model in Embodiment 1 of the present invention (corresponding to) Figure 5 .a's optimized topology) in Comparison of the normal displacement of the center of the square plate;

[0034] (c) The full-order model and the reduced-order model in Embodiment 1 of the present invention (corresponding to) Figure 5 .c optimized topology) in Comparison of the normal displacement of the center of the square plate.

[0035] Figure 7 This is to optimize the calculation time and accuracy of the descending order model with different load application times in Embodiment 1 of the present invention.

[0036] Figure 8 For the problem of a functionally graded square plate with four fixed sides under multi-point load, (a) the structural design domain in Embodiment 2 of the present invention;

[0037] (b) Half-wave sinusoidal load in Embodiment 2 of the present invention.

[0038] Figure 9 This is a schematic diagram of the topology optimization results for a four-sided fixed functionally graded square plate under multi-point loading.

[0039] (a) The load application time in Embodiment 2 of the present invention The topology optimization results;

[0040] (b) The load application time in Embodiment 2 of the present invention The topology optimization results;

[0041] (c) The load application time in Embodiment 2 of the present invention The topology optimization results.

[0042] Figure 10 This represents the normal displacement of the center of a four-sided fixed functionally graded square plate under multi-point loading.

[0043] (a) The full-order model and the reduced-order model in Embodiment 1 of the present invention (corresponding to) Figure 9 .a's optimized topology) in Comparison of the normal displacement of the center of the square plate below;

[0044] (b) The full-order model and the reduced-order model in Embodiment 1 of the present invention (corresponding to) Figure 9 .a's optimized topology) in Comparison of the normal displacement of the center of the square plate;

[0045] (c) The full-order model and the reduced-order model in Embodiment 1 of the present invention (corresponding to) Figure 9 .c optimized topology) in Comparison of the normal displacement of the center of the square plate.

[0046] Figure 11 This is to optimize the calculation time and accuracy of the descending order model with different load application times in Embodiment 1 of the present invention. Detailed Implementation

[0047] The challenges of topology optimization in the time domain dynamics of functionally graded plate structures (FJGPS) include computational efficiency and the dependence of the FJGPS topology on the in-plane gradient direction of the gradient material. This invention proposes a time domain dynamics topology optimization method and system for FJGPS based on principal component analysis, achieving optimal vibration-resistant design for FJGPS structures.

[0048] Combined with appendix Figures 1-11 This paper elaborates on the temporal dynamic topology optimization method for functionally graded plate structures based on principal component analysis, 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.

[0049] In a first aspect, the present invention provides a time-domain dynamic topology optimization method for functionally graded plate structures based on principal component analysis, the flowchart of which is shown below. Figure 1 As shown, the main steps include:

[0050] S101: Based on the Bezier decomposition and extraction algorithm of NURBS function, the stiffness, mass and damping matrices of NURBS elements are calculated respectively to form a full-order finite element model of functionally graded plate structure in the time domain with equal geometry.

[0051] In this invention, step S101 may further include:

[0052] Step S101-1: Construct the Bezier extraction operator and Bernstein basis functions to efficiently calculate the stiffness, mass, and damping matrices of the NURBS element.

[0053] In the Bezier decomposition of NURBS curves, each internal node in the open nodal vector is repeatedly inserted into the parameter space until the number of repetitions reaches the order of the NURBS curve. Given an initial open node vector ,only and The multiplicity is All other nodes have a multiplicity of 1. Insert a new node into it. ( ), forming a new node vector Corresponding to node insertion in parameter space, the initial control point in physical space. Control updated to control point Then we have:

[0054]

[0055]

[0056] In the formula: This represents the number of initial control points in the NURBS curve.

[0057] And so on, open node vectors All nodes with a multiplicity of 1 are inserted in sequence, and each node is repeatedly inserted in sequence. Then, the NURBS curve can be decomposed into segments. A combination of continuous Bezier curves. Corresponding to the node insertion process in parameter space, in physical space, a global Bezier extraction operator is constructed to successively map the original NURBS control points to Bezier control points. Global Bezier extraction operator. The matrix representation is as follows:

[0058]

[0059]

[0060] In the formula: For the first Bezier extraction operator for node insertion This represents the total number of nodes to be inserted.

[0061] Similarly, the parameter space can be obtained according to the above formula. Global Bezier extraction operator for directions At the element level of the plate structure, operators are extracted using Bezier elements. Mapping NURBS elements to Bezier elements, we have:

[0062]

[0063] In the formula: This indicates the encoding of the Bezier unit.

[0064] Bezier curves can be represented as linear combinations of Bezier control points and Bernstein polynomials, with the Bernstein polynomials serving as basis functions of the Bezier curves, calculated using the following recursive algorithm:

[0065]

[0066] In the formula: For the first Bernstein polynomials with Bezier control points.

[0067] Similarly, the parameter space can be obtained according to the recursive algorithm above. Bernstein polynomials in direction ( (Order). Shape function matrix of Bezier element in plate structure. Represented as:

[0068]

[0069] like Figure 2 As shown, the Bezier extraction operator is used. NURBS control points Mapped to Bezier control points The calculation formula is as follows:

[0070]

[0071] In the formula: This is the diagonal matrix of the weights of the NURBS basis functions. This is the diagonal matrix of Bezier basis function weights.

[0072] For rectangular structured meshes, the stiffness of NURBS solid elements With the mass matrix The stiffness of the Bezier solid element can be used to determine this. With the mass matrix The linear transformation is obtained, that is:

[0073]

[0074]

[0075] In the formula: for The diagonal matrix of basis function weights on the NURBS cell Here is the strain-displacement matrix of the Bezier element. , These represent the equivalent elastic modulus and equivalent physical density of a functionally graded material that is a homogeneous mixture of two phases, respectively. Here is the constitutive matrix of the functionally graded plate structure. This represents the volume fraction of the strong material phase in the NURBS element.

[0076] Specifically, isogeometric Bezier solid elements and The data structure is highly similar to that of the classical finite element method, and can be directly calculated offline once using the classical finite element method. However, isogeometric NURBS solid elements need to be calculated one by one in the loop. Furthermore, the weight matrix... Extraction operator matrix of unit All of these are constant matrices, completely independent of the control point distribution and basis function type. Therefore, isogeometric analysis based on Bezier extraction operators helps improve numerical computation efficiency.

[0077] The mass matrix of the NURBS element within the design domain Stiffness matrix With damping matrix They are represented as follows:

[0078]

[0079] In the formula: and For Rayleigh damping parameters, , These are the volume and stiffness interpolation functions, respectively. Control points The density at that location satisfies , , Physical coordinates , The total number of directional control points.

[0080] Step S101-2: Combine the hybrid material model of two-phase functionally graded materials to construct a time-domain dynamic full-order finite element model of functionally graded plate structure based on geometric analysis such as Bezier.

[0081] The upper and lower bounds of the equivalent elastic modulus of functionally graded materials are calculated using the Hashin-Shtrikman model, and the average of the two is taken as the equivalent elastic modulus. An approximation of is then:

[0082]

[0083] In the formula: The upper bound of the equivalent elastic modulus of the corresponding strong material as the coating phase is defined. The lower bound of the equivalent elastic modulus corresponding to the weak material as the coating phase. , These are the elastic moduli of the strong and weak material phases, respectively. denoted as the volume fraction of the strong material phase at any point in the structure.

[0084] The equivalent density at any point in a uniformly mixed two-phase functionally graded structure for:

[0085]

[0086] In the formula: , These are the physical densities of the strong and weak material phases, respectively.

[0087] Based on the above material properties, the full-order finite element equations for the isogeometric time-domain dynamics of the functionally graded plate structure are as follows:

[0088]

[0089]

[0090] In the formula: , and These represent the global stiffness matrix, mass matrix, and damping matrix of the functionally graded plate structure. , and The first Payload array at each time step The acceleration, velocity, and displacement matrices of the NURBS control points under the influence of the NURBS. This represents the total number of steps in the dynamic events.

[0091] Using the unconditionally stable HHT-α time integration scheme, the full-order finite element equations of the isogeometric transient dynamics of the above functionally graded plate structure are solved, resulting in:

[0092]

[0093]

[0094] In the formula: The residual matrix is... , and For HHT-α algorithm parameters, Increment the time step.

[0095] Update the displacement and velocity matrix using Newmark form:

[0096]

[0097] S102: Establish a compacted, smooth functionally graded material interpolation model, and propose a full-order time-domain dynamic multi-material topology optimization model for minimizing the dynamic compliance of functionally graded plate structures.

[0098] Smoothing control point density using the Shepard function Construct control point density distribution function by combining NURBS basis functions To significantly reduce the presence of intermediate densities, a threshold projection Heaviside threshold projection function is introduced. Pushing the control point density towards 0 or 1, we have:

[0099]

[0100] In the formula: For sharpness parameters, This is the threshold parameter.

[0101] To suppress the numerical singularity problem in low-density regions, the Ersatz parameter is introduced. To construct a material interpolation model for a functionally graded plate structure: using the volume interpolation function and the stiffness interpolation function, we have:

[0102]

[0103]

[0104] In the formula: , These are adjustable parameters.

[0105] To reduce the computational cost of subsequent material interpolation models, the above formula only calculates the material interpolation model at the Gaussian integration point located at the center point of the NURBS element, i.e. and Additionally, regarding design variables... It characterizes the continuous and smooth gradient of functionally graded materials without the need to establish an additional material interpolation model.

[0106] In summary, to control point density and the volume fraction of strong materials on the unit Using the volumetric quantities of strong and weak phase materials as design variables and minimizing dynamic compliance as the optimization objective, a full-order time-domain dynamic multi-material topology optimization model for functionally graded plates is established as follows:

[0107]

[0108] In the formula: The dynamic compliance of a functionally graded plate is abbreviated as , The total number of isogeometric NURBS elements. , For design variables, , The volume constraints for strong and weak material phases are respectively abbreviated as... , , , These represent the upper volume boundaries of the strong and weak material phases, respectively. , They are plate units Upper-strength material volume fraction variable The upper and lower bounds.

[0109] S103: Seek the eigenmode matrix of the vibration of the functionally graded plate structure, and use it as a basis to establish a low-dimensional solution space that approximates the solution space of the above full-order model, thereby constructing a reduced-order finite element model of the functionally graded plate structure with equal geometric time-domain dynamics.

[0110] In this invention, step S103 may further include:

[0111] Step S103-1: Perform eigenvalue decomposition on the correlation matrix of the instantaneous image matrix of the displacement field to find the energy-optimal eigenmode matrix as the orthogonal basis of the low-dimensional solution space.

[0112] Solving the full-order finite element model Construct the instantaneous image matrix .

[0113] In order to construct a low-dimensional displacement solution space, we seek solutions that are similar to high-dimensional displacement solutions. Orthogonal basis satisfying the least squares best approximation The above optimization problem can be transformed into the following eigenvalue problem:

[0114]

[0115]

[0116] In the formula: for and The positive eigenvalues ​​are arranged in descending order. and These are the corresponding feature vectors, typically .

[0117] Retain the intrinsic modes that contribute significantly to structural vibration energy. Select a sufficiently small number This ensures that the ratio of the vibrational energy of the intrinsic modes preserved in the low-dimensional space to the vibrational energy of all intrinsic modes satisfies the following cutoff criterion:

[0118]

[0119] In the formula: This represents the optimal number of intrinsic modes.

[0120] Therefore, the truncated eigenmode matrix Then the displacement field approximation Represented as:

[0121]

[0122] In the formula: for The coefficient matrix, abbreviated as .

[0123] Step S103-2: Based on principal component analysis, establish a time-domain dynamic reduced-order model of the functionally graded plate structure to improve the computational efficiency of the instantaneous displacement field.

[0124] Based on principal component analysis, Substituting into the full-order finite element equation Thus, a reduced-order time-domain dynamic model of the functionally graded plate structure can be obtained, which yields:

[0125]

[0126]

[0127] The dimension of the solution space of the above time-domain dynamical reduced-order model This significantly reduces the solution size, and its initial conditions are:

[0128]

[0129] In the formula: These represent the initial displacements for the full-order finite element equations. Combining the initial conditions described above, applying the aforementioned HHT-α time integration scheme to the reduced-order model yields the displacements at each time step. , back to Thus, an approximate solution for the displacement field can be obtained.

[0130] S104: A low-order time-domain dynamic multi-material topology optimization model is proposed to minimize the dynamic compliance of functionally graded plate structures. A reduced-order sensitivity time-domain adjoint equation is constructed to efficiently calculate the sensitivity of dynamic compliance and volume constraints of functionally graded plate structures.

[0131] Based on the time-domain dynamics reduced-order model of the functionally graded plate structure in S103-2, a low-order time-domain dynamics multi-material topology optimization model for the functionally graded plate structure is proposed, which yields:

[0132]

[0133] In the formula: for An approximation of, denoted as .

[0134] A sensitivity analysis strategy of first discretizing and then differentiating is used to calculate the dual variables, thereby obtaining an approximate value of dynamic compliance for the design variables. , Sensitivity.

[0135] Dynamic compliance for design variables The sensitivity, expressed in components, is as follows:

[0136]

[0137] In the formula: To the residual of the reduced order The corresponding dual variable.

[0138] Constructing a reduced-order sensitivity time-domain adjoint equation to calculate the dual variable The formula is as follows:

[0139] Then:

[0140]

[0141]

[0142] Then:

[0143]

[0144] In the formula, , These are the dual variables corresponding to the reduced-order Newmark difference scheme.

[0145] Similarly, the approximate value of dynamic compliance For the volume fraction of strong material on the plate unit The sensitivity is:

[0146]

[0147] Finally, using the chain rule, the volume constraints of strong and weak material phases are applied to the design variables. , The sensitivity, expressed in component form, is as follows:

[0148]

[0149]

[0150] S105: The optimal topology design of the functionally graded plate structure is obtained by using a hybrid optimization algorithm of MMA (moving asymptote method) and GCMMA (globally convergent moving asymptote method).

[0151] The initial iteration phase of optimization uses the MMA algorithm, while the GCMMA algorithm is switched when the objective function approaches a local optimum. Once the objective function satisfies the following oscillation condition, MMA will switch to GCMMA, with the switching criterion being:

[0152]

[0153] In the formula: For the first The approximate dynamic compliance value obtained through step-by-step optimization iteration. For the middle A tiny positive number between.

[0154] In summary, the method of this invention integrates the isogeometric temporal dynamics analysis method based on Bezier extraction operator and the principal component analysis method into the structural temporal dynamics topology optimization method. It can efficiently obtain the optimal topology of functionally graded plates with clear and smooth multiphase material boundaries, and realize the agile design of high-precision geometric expression optimization topology.

[0155] Secondly, this invention provides a time-domain dynamic topology optimization system for functionally graded plate structures based on principal component analysis, such as... Figure 3 As shown, it includes:

[0156] The initialization module constructs the design domain of the functionally graded plate structure, initializes the control point density of the functionally graded plate and the design variables such as the volume fraction of strong materials attached to the plate elements, presets the property parameters and volume constraint upper limit parameters of the strong and weak phase materials of the plate structure, the algorithm parameters of the HHT-α time integration scheme, the material interpolation model parameters, and the MMA-GCMMA hybrid optimization algorithm parameters.

[0157] The preprocessing module uses isogeometric NURBS elements to discretize the design domain of the functionally graded plate structure, defines Dirichlet and Neumann boundary conditions, initial conditions, and applies time-domain dynamic loads; calculates the extraction operator, basis functions, and derivatives of the Bezier element, and uses this in conjunction with the material interpolation scheme to calculate the stiffness, mass, and damping matrices of the isogeometric NURBS element, forming an isogeometric time-domain dynamic full-order finite element model of the functionally graded plate structure;

[0158] The dynamic analysis module solves the full-order finite element model of the functionally graded plate structure in the time domain using the unconditionally stable HHT-α time integration scheme to obtain the transient image matrix; it constructs the eigenmode matrix and establishes a reduced-order finite element model of the functionally graded plate structure in the time domain based on the principal component analysis method, thereby calculating the approximate value of the dynamic compliance of the functionally graded plate structure and the volume constraint of the multiphase material.

[0159] The optimization solution module calculates the sensitivity information of the time-domain dynamic topology optimization model of the functionally graded plate structure. Based on the MMA-GCMMA hybrid optimization algorithm, it updates the design variables such as the control point density of the functionally graded plate and the volume fraction of strong materials attached to the plate elements. It then determines whether the optimization iteration results meet the convergence criteria. If not, it continues the optimization iteration until the convergence criteria are met.

[0160] Example

[0161] Example 1

[0162] This invention addresses the temporal dynamic topology optimization problem of a functionally graded square plate with fixed corners under single-point load, further verifying the effectiveness of the method.

[0163] Step S101: Based on the Bezier decomposition and extraction algorithm of NURBS function, calculate the stiffness, mass and damping matrices of NURBS element respectively to form a full-order finite element model of functionally graded plate structure in the time domain with equal geometry.

[0164] like Figure 4 As shown, the functionally graded square plate structure has the following dimensions: length is... ,thickness The design domain is fixed at all four corners, with a concentrated half-wave cosine load applied at the center, and the load amplitude is... Load application time Elastic modulus of strong material phase The elastic modulus of the weak material phase The mass density of the strong material phase is The mass density of the weak material phase is Total number of steps in the dynamics time The upper limit of the volume of the strong material phase is Upper limit of volume of weak material phase , This represents the total volume of the functionally graded plate. The parameters of the stiffness interpolation function are... , The equivalent Rayleigh damping constant of the functionally graded plate is: , HHT-α algorithm parameters , , .

[0165] Step S101-1: Construct the Bezier extraction operator and Bernstein basis functions to efficiently calculate the stiffness, mass, and damping matrices of the NURBS element.

[0166] In the Bezier decomposition of NURBS curves, each internal node in the open-node vector is repeatedly inserted into the parameter space until the number of repetitions reaches the order of the NURBS curve. Given an initial open node vector ,only and The multiplicity is All other nodes have a multiplicity of 1. Insert a new node into it. ( ), forming a new node vector Corresponding to node insertion in parameter space, the initial control point in physical space. Control updated to control point Then we have:

[0167]

[0168]

[0169] In the formula: This represents the number of initial control points in the NURBS curve.

[0170] And so on, open node vectors All nodes with a multiplicity of 1 are inserted in sequence, and each node is repeatedly inserted in sequence. Then, the NURBS curve can be decomposed into segments. A combination of continuous Bezier curves. Corresponding to the node insertion process in parameter space, in physical space, a global Bezier extraction operator is constructed to successively map the original NURBS control points to Bezier control points. Global Bezier extraction operator. The matrix representation is as follows:

[0171]

[0172]

[0173] In the formula: For the first Bezier extraction operator for node insertion This represents the total number of nodes to be inserted.

[0174] Similarly, the parameter space can be obtained according to the above formula. Global Bezier extraction operator for directions At the element level of the plate structure, operators are extracted using Bezier elements. Mapping NURBS elements to Bezier elements, we have:

[0175]

[0176] In the formula: This indicates the encoding of the Bezier unit.

[0177] Bezier curves can be represented as linear combinations of Bezier control points and Bernstein polynomials, with the Bernstein polynomials serving as basis functions of the Bezier curves, calculated using the following recursive algorithm:

[0178]

[0179] In the formula: For the first Bernstein polynomials with Bezier control points.

[0180] Similarly, the parameter space can be obtained according to the recursive algorithm above. Bernstein polynomials in direction ( (Order). Shape function matrix of Bezier element in plate structure. Represented as:

[0181]

[0182] Extracting operators using Bezier NURBS control points Mapped to Bezier control points The calculation formula is as follows:

[0183]

[0184] In the formula: This is the diagonal matrix of the weights of the NURBS basis functions. This is the diagonal matrix of Bezier basis function weights.

[0185] For rectangular structured meshes, the stiffness of NURBS solid elements With the mass matrix The stiffness of the Bezier solid element can be used to determine this. With the mass matrix The linear transformation is obtained, that is:

[0186]

[0187]

[0188] In the formula: for The diagonal matrix of basis function weights on the NURBS cell Here is the strain-displacement matrix of the Bezier element. , These represent the equivalent elastic modulus and equivalent physical density of a functionally graded material that is a homogeneous mixture of two phases, respectively. Here is the constitutive matrix of the functionally graded plate structure. This represents the volume fraction of the strong material phase in the NURBS element.

[0189] Specifically, isogeometric Bezier solid elements and The data structure is highly similar to that of the classical finite element method, allowing for direct offline computation using the classical finite element method. However, isogeometric NURBS solid elements require computation one by one within the loop. Furthermore, the weight matrix... Extraction operator matrix of unit All of these are constant matrices, completely independent of the control point distribution and basis function type. Therefore, isogeometric analysis based on Bezier extraction operators helps improve numerical computation efficiency.

[0190] The mass matrix of the NURBS element within the design domain Stiffness matrix With damping matrix They are represented as follows:

[0191]

[0192] In the formula: and For Rayleigh damping parameters, , These are the volume and stiffness interpolation functions, respectively. Control points The density at that location satisfies , , Physical coordinates , The total number of directional control points.

[0193] Step S101-2: Combine the hybrid material model of two-phase functionally graded materials to construct a time-domain dynamic full-order finite element model of functionally graded plate structure based on geometric analysis such as Bezier.

[0194] The upper and lower bounds of the equivalent elastic modulus of functionally graded materials are calculated using the Hashin-Shtrikman model, and the average of the two is taken as the equivalent elastic modulus. An approximation of is then:

[0195]

[0196] In the formula: The upper bound of the equivalent elastic modulus of the corresponding strong material as the coating phase is defined. The lower bound of the equivalent elastic modulus corresponding to the weak material as the coating phase. , These are the elastic moduli of the strong and weak material phases, respectively. denoted as the volume fraction of the strong material phase at any point in the structure.

[0197] The equivalent density at any point in a uniformly mixed two-phase functionally graded structure for:

[0198]

[0199] In the formula: , These are the physical densities of the strong and weak material phases, respectively.

[0200] Based on the above material properties, the full-order finite element equations for the isogeometric time-domain dynamics of the functionally graded plate structure are as follows:

[0201]

[0202]

[0203] In the formula: , and These represent the global stiffness matrix, mass matrix, and damping matrix of the functionally graded plate structure. , and The first Payload array at each time step The acceleration, velocity, and displacement matrices of the NURBS control points under the influence of the NURBS. This represents the total number of steps in the dynamic events.

[0204] Using the unconditionally stable HHT-α time integration scheme, the full-order finite element equations of the isogeometric transient dynamics of the above-mentioned functionally graded plate structure are solved.

[0205] S102: Establish a compacted, smooth functionally graded material interpolation model, and propose a full-order time-domain dynamic multi-material topology optimization model for minimizing the dynamic compliance of functionally graded plate structures.

[0206] Smoothing control point density using the Shepard function Construct control point density distribution function by combining NURBS basis functions To significantly reduce the presence of intermediate densities, a threshold projection Heaviside threshold projection function is introduced. Pushing the control point density towards 0 or 1, we have:

[0207]

[0208] In the formula: For sharpness parameters, This is the threshold parameter.

[0209] To suppress the numerical singularity problem in low-density regions, the Ersatz parameter is introduced. To construct a material interpolation model for a functionally graded plate structure: using the volume interpolation function and the stiffness interpolation function, we have:

[0210]

[0211]

[0212] In the formula: , These are adjustable parameters.

[0213] To reduce the computational cost of subsequent material interpolation models, the above formula only calculates the material interpolation model at the Gaussian integration point located at the center point of the NURBS element, i.e. and .

[0214] Additionally, regarding design variables It characterizes the continuous and smooth gradient of functionally graded materials without the need to establish an additional material interpolation model.

[0215] In summary, to control point density and the volume fraction of strong materials on the unit Using the volumetric quantities of strong and weak phase materials as design variables and minimizing dynamic compliance as the optimization objective, a full-order time-domain dynamic multi-material topology optimization model for functionally graded plates is established as follows:

[0216]

[0217] In the formula: The dynamic compliance of a functionally graded plate is abbreviated as , The total number of isogeometric NURBS elements. , For design variables, , The volume constraints for strong and weak material phases are respectively abbreviated as... , , , These represent the upper volume boundaries of the strong and weak material phases, respectively. , They are plate units Upper-strength material volume fraction variable The upper and lower bounds.

[0218] S103: Seek the eigenmode matrix of the vibration of the functionally graded plate structure, and use it as a basis to establish a low-dimensional solution space that approximates the solution space of the above full-order model, thereby constructing a reduced-order finite element model of the functionally graded plate structure with equal geometric time-domain dynamics.

[0219] Step S103-1: Perform eigenvalue decomposition on the correlation matrix of the instantaneous image matrix of the displacement field to find the energy-optimal eigenmode matrix as the orthogonal basis of the low-dimensional solution space.

[0220] Solving the full-order finite element model Construct the instantaneous image matrix .

[0221] In order to construct a low-dimensional displacement solution space, we seek solutions that are similar to high-dimensional displacement solutions. Orthogonal basis satisfying the least squares best approximation The above optimization problem can be transformed into the following eigenvalue problem:

[0222]

[0223]

[0224] In the formula: for and The positive eigenvalues ​​are arranged in descending order. and These are the corresponding feature vectors, typically .

[0225] Retain the intrinsic modes that contribute significantly to structural vibration energy. Select a sufficiently small number This ensures that the ratio of the vibrational energy of the intrinsic modes preserved in the low-dimensional space to the vibrational energy of all intrinsic modes satisfies the following cutoff criterion:

[0226]

[0227] In the formula: This represents the optimal number of intrinsic modes.

[0228] Therefore, the truncated eigenmode matrix Then the displacement field approximation Represented as:

[0229]

[0230] In the formula: for The coefficient matrix, abbreviated as .

[0231] Step S103-2: Based on principal component analysis, establish a time-domain dynamic reduced-order model of the functionally graded plate structure to improve the computational efficiency of the instantaneous displacement field.

[0232] Based on principal component analysis, Substituting into the full-order finite element equation Thus, a reduced-order time-domain dynamic model of the functionally graded plate structure can be obtained, which yields:

[0233]

[0234]

[0235] The dimension of the solution space of the above time-domain dynamical reduced-order model This significantly reduces the solution size, and its initial conditions are:

[0236]

[0237] In the formula: These represent the initial displacements for the full-order finite element equations. Combining the initial conditions described above, applying the aforementioned HHT-α time integration scheme to the reduced-order model yields the displacements at each time step. , back to Thus, an approximate solution for the displacement field can be obtained.

[0238] S104: A low-order time-domain dynamic multi-material topology optimization model is proposed to minimize the dynamic compliance of functionally graded plate structures. A reduced-order sensitivity time-domain adjoint equation is constructed to efficiently calculate the sensitivity of dynamic compliance and volume constraints of functionally graded plate structures.

[0239] Based on the time-domain dynamics reduced-order model of the functionally graded plate structure in S103-2, a low-order time-domain dynamics multi-material topology optimization model for the functionally graded plate structure is proposed, which yields:

[0240]

[0241] In the formula: for An approximation of, denoted as .

[0242] A sensitivity analysis strategy of first discretizing and then differentiating is used to calculate the dual variables, thereby obtaining an approximate value of dynamic compliance for the design variables. , Sensitivity.

[0243] Dynamic compliance for design variables The sensitivity, expressed in components, is as follows:

[0244]

[0245] In the formula: To the residual of the reduced order The corresponding dual variable.

[0246] Constructing a reduced-order sensitivity time-domain adjoint equation to calculate the dual variable The formula is as follows:

[0247] Then:

[0248]

[0249]

[0250] Then:

[0251]

[0252] In the formula, , These are the dual variables corresponding to the reduced-order Newmark difference scheme.

[0253] Similarly, the approximate value of dynamic compliance For the volume fraction of strong material on the plate unit The sensitivity is:

[0254]

[0255] Finally, using the chain rule, the volume constraints of strong and weak material phases are applied to the design variables. , The sensitivity, expressed in component form, is as follows:

[0256]

[0257]

[0258] S105: The optimal topology design of the functionally graded plate structure is obtained by using a hybrid optimization algorithm of MMA (moving asymptote method) and GCMMA (globally convergent moving asymptote method).

[0259] The initial iteration phase of optimization uses the MMA algorithm, while the GCMMA algorithm is switched when the objective function approaches a local optimum. Once the objective function satisfies the following oscillation condition, MMA will switch to GCMMA, with the switching criterion being:

[0260]

[0261] In the formula: For the first The approximate dynamic compliance value obtained through step-by-step optimization iteration. For the middle A tiny positive number between.

[0262] The specific optimization results are as follows: Figures 5-7 As shown, the red area represents the region where strong materials are distributed, and the blue area represents the region where weak materials are distributed.

[0263] Figure 5 For the load application time Schematic diagrams of topology optimization results for a four-corner fixed functionally graded square plate under single-point load under three working conditions. In this invention, the optimized topology obtained by order reduction using principal component analysis is identical to the result of the full-priority topology optimization method. This is because the reduced-order model and the full-order model produce the same time-domain displacement response at the load application point (e.g., ...). Figure 6 As shown in the figure, the optimization accuracy of the reduced-order model is consistent with that of the full-order model. Figure 7 As shown, the load application time Under the three operating conditions, the optimal dynamic compliance of the reduced-order model and the full-order model are almost equal, which further verifies the consistency of the optimization accuracy between the reduced-order model and the full-order model. In particular, the optimization time of the reduced-order model is significantly lower than that of the full-order model, which shows the high efficiency of the reduced-order optimization method of the present invention.

[0264] The results of Embodiment 1 of this invention demonstrate that introducing principal component analysis (PCA) into time-domain dynamics analysis and optimization significantly improves computational efficiency, thereby efficiently achieving synergistic optimization of the functionally graded plate structure topology and the gradual gradient of multiphase materials along the in-plane direction of the plate, maximizing the temporal dynamic performance of the structure. Simultaneously, by combining the Bezier extraction operator, the mass, stiffness, and damping matrices of the NURBS elements can be obtained through offline single-step calculations, avoiding successive calculations in loops and further improving computational efficiency. Therefore, the algorithm of this invention effectively solves the computational efficiency problem of functionally graded plate structure topology optimization under time-domain loads and the problem of smooth gradient of multiphase graded materials along the in-plane direction of the plate, making it suitable for the engineering requirements of large-scale time-domain dynamics multi-material topology optimization of vibration equipment.

[0265] Example 2

[0266] Embodiment 2 of the present invention solves the transient dynamic topology optimization problem of a four-sided fixed functional gradient square plate under multi-point load, and further verifies the effectiveness of the method of the present invention.

[0267] Step S101: Based on the Bezier decomposition and extraction algorithm of NURBS function, calculate the stiffness, mass and damping matrices of NURBS element respectively to form a full-order finite element model of functionally graded plate structure in the time domain with equal geometry.

[0268] like Figure 8 As shown, four points in the central region of the four-sided fixed functional gradient square plate are subjected to half-wave sinusoidal concentrated loads, and other parameters are the same as in Example 1.

[0269] Step S102 is the same as in Example 1.

[0270] Step S103 is the same as in Example 1.

[0271] Step S104 is the same as in Example 1.

[0272] Step S105 is the same as in Example 1.

[0273] The specific optimization results are as follows: Figures 9-11 As shown, the red area represents the region where strong materials are distributed, and the blue area represents the region where weak materials are distributed.

[0274] Figure 9 For the load application time Schematic diagrams of topology optimization results for a four-sided fixed functionally graded square plate under multi-point loads in three working conditions. The reduced-order topology optimization model proposed in this invention achieves the same optimized topology as the full-order topology optimization model. This is because the reduced-order model and the full-order model produce the same time-domain displacement response (e.g., ...). Figure 10 As shown in the figure), therefore the optimization accuracy of the two is consistent. Figure 11As shown, the load application time Under the three operating conditions, the optimal dynamic compliance of the reduced-order model and the full-order model are almost equal, which further verifies the consistency of the optimization accuracy between the reduced-order model and the full-order model. In particular, the optimization time of the reduced-order model is significantly lower than that of the full-order model, which shows the high efficiency of the reduced-order optimization method of the present invention.

[0275] The results of Embodiment 2 of the present invention show that the algorithm of the present invention can solve the multi-material dynamic topology optimization problem of functionally graded plate structures under complex time-domain load conditions, and is suitable for the engineering needs of large-scale time-domain dynamic multi-material topology optimization of vibration equipment.

[0276] 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 time-domain dynamic topology optimization of a functionally graded plate structure based on principal component analysis, characterized by, It comprises the following steps: S101: Based on the Bezier decomposition and extraction algorithm of NURBS function, the stiffness, mass and damping matrix of NURBS unit are calculated respectively to form the full-order finite element model of the functionally graded plate structure in the isogeometric time domain dynamics; S102: A tight and smooth functionally graded material interpolation model is established, and a full-order time domain dynamic multi-material topology optimization model of the functionally graded plate structure is proposed to minimize the dynamic compliance; S103: The eigenmode matrix of the functionally graded plate structure vibration is sought as the basis to establish a low-dimensional solution space approximating the solution space of the above full-order model, thereby constructing the reduced-order finite element model of the functionally graded plate structure in the isogeometric time domain dynamics; S104: A low-order time domain dynamic multi-material topology optimization model of the functionally graded plate structure is proposed to minimize the dynamic compliance, and a reduced-order sensitivity time domain adjoint equation is constructed to efficiently calculate the sensitivity of the dynamic compliance and volume constraint of the functionally graded plate structure; S105: A hybrid optimization algorithm of moving asymptote method MMA-global convergence moving asymptote method GCMMA is used to solve and obtain the optimal topology design of the functionally graded plate structure.

2. The method of claim 1, wherein, Step S101 comprises: Step S101-1, construct Bezier extraction operator and Bernstein basis function, and efficiently calculate the stiffness, mass and damping matrix of NURBS unit; Step S101-2, combined with the mixed material model of two-phase functionally graded material, construct the full-order finite element model of the functionally graded plate structure in the time domain dynamics based on Bezier isogeometric analysis.

3. The method of claim 2, wherein, The step S101-1 comprises: In Bezier decomposition of NURBS curve, each internal node in open node vector is repeatedly inserted in parameter space until its repetition number reaches the order of NURBS curve ; given initial open node vector , only and have repetition number , and the rest have repetition number 1; new node ( ) is inserted into it to form new node vector ; initial control points in physical space are updated to control points , then wherein: is the number of initial control points in the NURBS curve; By analogy, open knot vector All the nodes with the multiplicity of 1, each node is inserted in turn If N, then the NURBS curve can be decomposed into each segment The combination of continuous Bezier curves corresponds to the node insertion process in the parameter space, and the original NURBS control points are mapped into Bezier control points by constructing a global Bezier extraction operator in the physical space; the matrix expression of the global Bezier extraction operator is: In the formula: is the Bezier extraction operator for the th inserted knot, is the total number of inserted knots. Similarly, the parameter space can be obtained according to the above formula Global Bezier extraction operator in the direction At the unit level of the plate structure, a Bezier unit extraction operator is used Mapping the NURBS unit to the Bezier unit, we have: In the formulae: denotes the encoding of the Bezier element; Bezier curve can be represented as a linear combination of Bezier control points and Bernstein polynomial, and Bernstein polynomial is used as the basis function of Bezier curve, which is calculated according to the following recursive algorithm: wherein: is the Bernstein polynomial of the th Bezier control point; Similarly, the parameter space in the above recursive algorithm can be obtained Bernstein polynomials in the direction of order n); shape function matrix of plate structure Bezier element is represented as:​ Bezier extraction operator NURBS control points are mapped to Bezier control points with the following formula: wherein: is a diagonal matrix of NURBS basis function weights, is a diagonal matrix of Bezier basis function weights; For rectangular structured mesh, the stiffness of NURBS solid element and mass matrix can be obtained by linear transformation of the stiffness of Bezier solid element and mass matrix , i.e. wherein: is The diagonal matrix of the base function weight on the NURBS element, is the strain-displacement matrix of the Bezier element, , are the equivalent elastic modulus and equivalent physical density of the two-phase homogeneous functionally graded material, respectively, is the constitutive matrix of the functionally graded plate structure, is the volume fraction of the strong material phase in the NURBS element; The isogeometric analysis based on Bezier extraction operator helps to improve the numerical calculation efficiency; In the design domain, the mass matrix , the stiffness matrix and the damping matrix of the NURBS element are respectively given by wherein: and is the Rayleigh damping parameter, , are volume, stiffness interpolation functions, respectively, is the density at control point satisfying , , are physical coordinates , the total number of directional control points.

4. The method according to claim 2 or 3, wherein, Step S101-2 comprises: The upper and lower bounds of the equivalent elastic modulus of the functionally graded material are calculated by Hashin-Shtrikman model, and the average value of the two bounds is taken as the approximate value of the equivalent elastic modulus of the functionally graded material. wherein: the upper bound of the equivalent elastic modulus of the corresponding strong material as the cladding phase, the lower bound of the equivalent elastic modulus of the corresponding weak material as the cladding phase, , the elastic modulus of the strong and weak material phases, respectively, the volume fraction of the strong material phase at any point in the structure; The equivalent density of any point in the homogeneous mixed two-phase functionally graded structure is where: wherein: , are the physical densities of the strong and weak material phases, respectively; Combined with the above material properties, the full-order finite element equation of the functionally graded plate structure in the isogeometric time domain dynamics is: In the formula: , and These represent the global stiffness matrix, mass matrix, and damping matrix of the functionally graded plate structure. , and The first Payload array at each time step The acceleration, velocity, and displacement matrices of the NURBS control points under the influence of the NURBS. This represents the total number of steps in the dynamic events; An unconditional stable HHT-alpha time integration scheme is used to solve the above full-order finite element equation of the functionally graded plate structure in the isogeometric transient dynamics.

5. The method of claim 1, wherein, The step S102 comprises: Shepard function is used to control the density of control points , and a control point density distribution function is constructed in combination with NURBS basis functions To greatly reduce the existence of intermediate density, a threshold projection Heaviside threshold projection function is introduced Push the control point density to 0 or 1, then In the formula: is a sharpness parameter, is a threshold parameter; An ersatz parameter is introduced to suppress the numerical singularity problem in low density regions The material interpolation model of functionally graded plate structure is constructed, including the volume interpolation function and the stiffness interpolation function. In the formulae: , are adjustable parameters; To reduce the computational cost of the material interpolation model in the subsequent material interpolation model, only the material interpolation model of the Gauss integration point located at the center point of the NURBS element is calculated in the above formula, that is and ; Design variables The present application does not need to establish the corresponding material interpolation model for characterizing the continuous and smooth gradient of the functionally graded material. with control point density and unit volume fraction of strong material The full-order model of the time-domain dynamics multi-material topology optimization of the functionally graded plate is established with the control point density, the volume fraction of the strong and weak materials as the design variables, the volume of the strong and weak materials as the constraints, and the minimum dynamic compliance as the optimization objective. where: is the dynamic compliance of the functionally graded plate, simply denoted as , is the total number of isogeometric NURBS elements, , is the design variable, , are the volume constraints of the strong and weak material phases, simply denoted as , , , are the upper volume bounds of the strong and weak material phases, , are the upper and lower bounds of the strong material volume fraction variable on the plate element .

6. The method of topology optimization of a functionally graded plate structure in time domain based on principal component analysis according to claim 1 or 5, characterized in that, The step S103 comprises: Step S103-1, perform eigenvalue decomposition on the correlation matrix of the displacement field instantaneous image matrix to seek the eigenmode matrix with optimal energy as the orthogonal basis of the low-dimensional solution space; Step S103-2, based on principal component analysis, establish the reduced-order model of the functionally graded plate structure in the time domain dynamics to improve the calculation efficiency of the instantaneous displacement field.

7. The method of claim 6, wherein, The step S103-1 comprises: solving the full order finite element model , constructing the snapshot matrix ; To construct a low-dimensional displacement solution space, we seek orthogonal bases that satisfy the best approximation in the least square sense orthogonal bases that satisfy the best approximation in the least square sense The above optimal problem can be transformed into the following eigenvalue problem: wherein: is and the positive eigenvalues in descending order, and are the corresponding eigenvectors, respectively, and typically ; retaining the eigenmodes with larger contribution to the structural vibration energy , selecting a sufficient small number of , so that the ratio of the vibration energy of the retained eigenmodes to the vibration energy of all eigenmodes in the low-dimensional space satisfies the following truncation criterion: In the formula: is the optimal number of eigenmodes; Thus, the truncated eigenmodal matrix The displacement field is approximated by is approximated by wherein: is a coefficient matrix of .​ 8. The method according to claim 6 or 7, wherein, The step S103-2 comprises: Based on the principal component analysis method, we have Substituting the full-order finite element equation The time-domain dynamic reduced-order model of the functionally graded plate structure is obtained, that is: The solution space dimension of the above time-domain dynamic reduced-order model , for reducing the scale of solving, the initial conditions are: In the formula: is the initial displacement of the full-order finite element equation; combining the above initial conditions, the aforementioned HHT-α time integration scheme is used for the reduced-order model, and the displacement at each time is obtained , and the approximate solution of the displacement field is obtained. , and the approximate solution of the displacement field is obtained.

9. The method of claim 1, wherein, The step S104 comprises: Based on the reduced-order model of the functionally graded plate structure in the time domain dynamics in S103-2, a low-order time domain dynamic multi-material topology optimization model of the functionally graded plate structure is proposed, that is: wherein: is an approximation of ; and ; The sensitivity analysis strategy of first-discrete and then-differential is adopted to calculate the dual variables, so as to obtain the sensitivity of the dynamic compliance approximation value to the design variables , ​ Dynamic compliance with respect to the sensitivity of the design variable is expressed in components as: In the formulae: is the reduced residual corresponding dual variable; Constructing a reduced order sensitivity time domain adjoint equation to compute dual variables which is given by the formula Then, there is: If, then, wherein , are the dual variables corresponding to the reduced Newmark difference scheme, respectively. By analogy, the dynamic compliance approximation For a plate unit with a high volume fraction of stiff material The sensitivity is: Finally, by the chain rule of differentiation, the sensitivities of the volume constraints of the strong and weak material phases with respect to the design variables , are expressed in component form as 。 10. The method of claim 1, wherein the method is a time domain dynamic topology optimization method for a functionally graded plate structure based on a principal component analysis method. The step S105 comprises: The optimization initial iteration stage adopts the MMA optimization algorithm, and when approaching the local optimal solution, the GCMMA optimization algorithm is adopted; once the target function meets the following oscillation condition, MMA will switch to GCMMA, and the switching criterion is as follows: wherein: is the first step optimization iteration of the dynamic compliance approximation, is a small positive number between and 11. A system for topological optimization of time-domain dynamics of a functionally graded plate structure based on principal component analysis, comprising: The system is realized based on the method in any one of claims 1 to 10, and comprises the following program modules: An initialization module, which constructs a design domain of the functionally graded plate structure, initializes design variables such as control point density of the functionally graded plate and volume fraction of strong material attached to the plate unit, pre-sets attribute parameters of the strong and weak phase materials of the plate structure, upper limit parameters of volume constraints, algorithm parameters of the HHT-alpha time integration scheme, material interpolation model parameters and parameters of the MMA-GCMMA hybrid optimization algorithm; A preprocessing module, which discretizes the design domain of the functionally graded plate structure by using the isogeometric NURBS unit, defines Dirichlet and Neumann boundary conditions, initial conditions and applied time-domain dynamic loads, calculates extraction operators, basis functions and derivatives of the Bezier unit, and then combines the material interpolation scheme to calculate stiffness, mass and damping matrices of the isogeometric NURBS unit, thereby forming an isogeometric time-domain dynamic full-order finite element model of the functionally graded plate structure; A dynamic analysis module, which solves the isogeometric time-domain dynamic full-order finite element model of the functionally graded plate structure by using the unconditionally stable HHT-alpha time integration scheme, and obtains an instantaneous matrix; An eigenmode matrix is constructed, and an isogeometric time-domain dynamic reduced-order finite element model is established based on the principal component analysis method, so as to calculate an approximate value of dynamic compliance of the functionally graded plate structure and volume constraints of the multi-phase material; An optimization solution module, which calculates sensitivity information of the time-domain dynamic topology optimization reduced-order model of the functionally graded plate structure, updates design variables such as control point density of the functionally graded plate and volume fraction of strong material attached to the plate unit based on the MMA-GCMMA hybrid optimization algorithm, judges whether the optimization iteration result meets the convergence criterion, and if not, continues the optimization iteration until the convergence criterion is met.

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