Isogeometric transient dynamics topological optimization method and system of functional gradient plate structure

Abstract

The invention discloses an isogeometric transient dynamics topological optimization method and system of a functional gradient plate structure, and belongs to the field of topological optimization design of functional gradient structures. According to the method, the problems of collaborative optimization of the gradient direction in the gradient material plane and the multi-material structure topology under the transient load effect and the like are solved. According to the technical key points, an isogeometric transient dynamics finite element analysis model of the functional gradient plate structure is constructed based on a reduced Gaussian integral method; constructing a smooth projection scheme and a material interpolation scheme of design variables of the functional gradient plate structure, and providing a transient dynamic topological optimization model of the functional gradient plate structure by taking the density of a control point and the volume fraction of a strong material attached to a plate unit as the design variables and the minimization of dynamic flexibility as a target; constructing a time domain adjoint equation of the topological optimization model, and obtaining sensitivity information of the target function; and finally, obtaining the optimal anti-vibration design of the functional gradient plate structure convenient for engineering application. The device is used for structural light-weight vibration reduction design of large engineering equipment under the strong vibration working condition.

Description

Technical Field

[0001] This invention relates to the field of topology optimization design of functionally graded structures, and more specifically, to a method and system for isogeometric transient dynamic topology optimization of functionally graded plate structures. Background Technology

[0002] With the continuous advancement of lightweight and high-performance equipment, aircraft and underwater vehicles often operate in high-vibration environments for extended periods, which can easily lead to vibration failure of airborne precision instruments and damage to the main structure. Functionally graded materials (FJTs) possess excellent specific stiffness and specific strength properties, and the smooth transition between material phases effectively avoids interfacial failures. Therefore, FJT structures are increasingly widely used in aerospace and marine equipment. Consequently, maximizing the optimal vibration resistance of FJT structures remains one of the most challenging research areas in mechanical engineering.

[0003] Topology optimization methods search for the optimal material distribution within the design space to achieve maximum structural performance with minimal material usage. Current research on topology optimization of functionally graded plate (FRP) structures mainly focuses on maximizing natural frequencies, minimizing harmonic responses, and maximizing band gaps, solving the topology optimization problem in the frequency domain dynamics. However, the extreme loads experienced by these devices are often transient, and the optimal solution in the frequency domain does not satisfy time-domain optimality. Furthermore, the aforementioned studies all assume that the gradient material is distributed gradually along a defined in-plane direction, classifying it as a topology optimization problem for single-material structures, only capable of optimizing the topology of FRP structures. To further expand the design space, the synergistic optimization of the in-plane gradient material direction and the structural topology should be considered to achieve multi-material structure dynamics topology optimization with smooth transitions between material phases. In particular, topology optimization results based on the finite element method often fail to form smooth and clear interface profiles in the transition regions between different material phases, which is detrimental to the subsequent fabrication of FRP structures.

[0004] In summary, from a design-manufacturing consistency perspective, this paper proposes a collaborative optimization of the in-plane gradient direction of graded materials and the structural topology under transient loads to achieve lightweight and high-performance equipment under strong vibration conditions, which has significant engineering implications. However, existing design technologies do not consider the impact of transient loads and the in-plane gradient direction of materials on the topology optimization design of multi-material structures, failing to maximize the optimal vibration resistance of functionally graded structures and making them unsuitable for engineering scenarios under the aforementioned strong vibration conditions. Therefore, a clear and effective integrated design-manufacturing solution is lacking. Summary of the Invention

[0005] The technical problem to be solved by this invention is:

[0006] Existing design techniques fail to consider the impact of transient loads and in-plane material gradient directions on the topology optimization design of multi-material structures. This prevents them from maximizing the optimal vibration resistance of functionally graded structures and makes them unsuitable for engineering scenarios under strong vibration conditions. Consequently, a clear and effective integrated design-manufacturing solution is lacking. Therefore, this paper presents an isogeometric transient dynamic topology optimization method and system for functionally graded plate structures.

[0007] To achieve the above objectives, in a first aspect, the present invention provides an isogeometric transient dynamic topology optimization method for functionally graded plate structures, comprising the following steps:

[0008] S101: A hybrid material model of two-phase functionally graded materials is established based on the Hashin-Shtrikman limit model, and a transient dynamic finite element model of functionally graded plate structure is constructed by the Gaussian reduced integral isogeometric analysis method.

[0009] S102: Construct a smooth projection scheme and material interpolation scheme for the structural design variables of functionally graded plates, and propose a transient dynamic topology optimization model for functionally graded plates with the goal of minimizing dynamic compliance.

[0010] S103: Construct time series of dual variables based on the sensitivity analysis strategy of first discretization and then differentiation, and calculate the sensitivity of dynamic compliance and volume constraint of functionally graded plate structure.

[0011] S104: 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 functionally graded plate structure with isogeometric transient dynamic topology optimization system includes:

[0013] The initialization module initializes the control point density of the functional gradient plate and the design variables such as the volume fraction of strong materials attached to the plate elements. It also pre-sets the property parameters of each phase material of the plate structure, the upper limit parameters of the volume constraint, the parameters of the HHT-α method time integration scheme, the parameters of the material interpolation model, the smoothing and projection parameters of the design variables, and the parameters of the MMA-GCMMA hybrid optimization algorithm.

[0014] The preprocessing module uses the design domain of the functionally graded plate structure to be discretized using equal geometric elements. It combines material interpolation schemes and Gaussian reduced integration schemes to efficiently calculate the stiffness, mass, and damping matrices of the plate elements. It defines the initial and boundary conditions and applies transient loads in the time domain, ultimately forming an equal geometric transient dynamic finite element model of the functionally graded plate structure.

[0015] The design module includes variable smoothing and projection, and material interpolation. By combining the Shepard and NURBS functions, a compactly supported and smooth density distribution function is constructed. This function, combined with the Heaviside function, drives the density distribution function to approximate a 0-1 distribution. Subsequently, a very small but non-zero Ersatz number is introduced to construct a volumetric and stiffness interpolation model for multiphase materials.

[0016] The dynamic analysis module constructs a time integration scheme using the unconditionally stable HHT-α method to solve the isogeometric transient dynamic finite element model of the functionally graded plate structure, thereby calculating the dynamic compliance of the functionally graded plate structure and the volume constraint of the multiphase material.

[0017] The optimization solution module calculates the sensitivity information of the transient 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.

[0018] This invention relates to an isogeometric transient dynamic topology optimization method and system for functionally graded plate structures, which has the following beneficial technical effects:

[0019] This invention addresses the challenges of transient dynamic topology optimization methods for functionally graded plate structures: how to fully explore the design space, synergistically consider the in-plane gradient direction of the graded materials and the structural topology of multiple materials, and maximize the optimal vibration resistance performance of functionally graded structures. Based on this, an isogeometric transient dynamic topology optimization method for functionally graded plate structures is proposed to achieve optimal control of the vibration of functionally graded plate structures, thereby improving the operational reliability of functionally graded plate structures under strong vibration environments.

[0020] This invention establishes a transient dynamic topology optimization model for functionally graded plate (FGP) structures with the goal of minimizing dynamic compliance, using the control point density and the volume fraction of strong materials attached to the plate elements as design variables. Based on the isogeometric analysis framework of reduced Gaussian integrals, a dynamic co-optimization algorithm is constructed for the in-plane gradient direction of the gradient material and the topology of the multi-material structure. Therefore, an isogeometric transient dynamic topology optimization method for FGP structures is proposed, including: defining the initial design domain and algorithm parameters; constructing an isogeometric transient dynamic finite element analysis model of the FGP structure based on the reduced Gaussian integral method; constructing a smooth projection scheme and material interpolation scheme for the design variables of the FGP structure; proposing a transient dynamic topology optimization model for FGP structures with the goal of minimizing dynamic compliance; constructing the time-domain adjoint equation of the optimization model and obtaining the sensitivity information of the objective function; and updating the design variables such as the control point density and the volume fraction of strong materials attached to the plate elements based on the MMA-GCMMA hybrid optimization algorithm to obtain the optimal vibration-resistant design of the FGP structure that is convenient for engineering applications.

[0021] This invention proposes an isogeometric transient dynamic topology optimization method for functionally graded plate structures, which can be used to solve the problem of synergistic optimization of the in-plane gradient direction of the material and the topology of the functionally graded structure under transient excitation, and realize the optimal vibration-resistant design of functionally graded plate structures. Attached Figure Description

[0022] Figure 1 This is a flowchart of an isogeometric transient dynamic topology optimization method for a functionally graded plate structure according to the present invention.

[0023] Figure 2 This is a schematic diagram of the isogeometric transient dynamic topology optimization system of a functionally graded plate structure according to the present invention.

[0024] Figure 3 For the problem of a simply supported functionally graded square plate under a single-point load, (a) is the structural design domain in Embodiment 1 of the present invention;

[0025] (b) is the half-wave sinusoidal load in Embodiment 1 of the present invention.

[0026] Figure 4 shows a schematic diagram of the topology optimization results of a simply supported functionally graded square plate under a single-point load (using the finite element method).

[0027] (a) The load application time in Embodiment 1 of the present invention The topology optimization results;

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

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

[0030] Figure 5 This is a schematic diagram of the topology optimization results for a simply supported functionally graded square plate under a single-point load (using the method of this invention).

[0031] (a) The load application time in Embodiment 1 of the present invention The topology optimization results;

[0032] (b) The load application time in Embodiment 1 of the present invention The topology optimization results;

[0033] (c) The load application time in Embodiment 1 of the present invention The topology optimization results.

[0034] Figure 6 For a four-sided simply supported functional gradient square plate under multi-point load, (a) the structural design domain in Embodiment 2 of the present invention;

[0035] (b) Half-wave sinusoidal load in Embodiment 2 of the present invention.

[0036] Figure 7 This is a schematic diagram of the topology optimization results for a four-sided fixed functionally graded square plate under multi-point loading (using the finite element method).

[0037] (a) The load application time in Embodiment 2 of the present invention The topology optimization results;

[0038] (b) The load application time in Embodiment 2 of the present invention The topology optimization results;

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

[0040] Figure 8 This is a schematic diagram of the topology optimization results for a four-sided fixed functionally graded square plate under multi-point loading (using the method of this invention).

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

[0042] (b) The load application time in Embodiment 2 of the present invention The topology optimization results;

[0043] (c) The load application time in Embodiment 2 of the present invention The topology optimization results. Detailed Implementation

[0044] This invention addresses the challenge of optimizing the transient dynamic topology of functionally graded plate structures with uniform geometry: the collaborative optimization of the in-plane gradient direction of the graded material and the topology of the multi-material structure under transient loads. A functionally graded plate structure is proposed in this invention.

[0045] A geometric transient dynamic topology optimization method and system are proposed to achieve optimal vibration resistance design for functionally graded plate structures.

[0046] Combined with appendix Figures 1-8 This paper elaborates on the isogeometric transient dynamic topology optimization method for functionally graded plate structures 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.

[0047] In a first aspect, the present invention provides an isogeometric transient dynamic topology optimization method for functionally graded plate structures, the flowchart of which is shown below. Figure 1 As shown, the main steps include:

[0048] S101: A hybrid material model of two-phase functionally graded materials is established based on the Hashin-Shtrikman limit model, and a transient dynamic finite element model of functionally graded plate structure is constructed by the Gaussian reduced integral isogeometric analysis method.

[0049] In this invention, step S101 may further include:

[0050] Step S101-1: Based on the Hashin-Shtrikman limit model, calculate the equivalent elastic modulus, equivalent density and equivalent structural damping of the two-phase functionally graded materials respectively, and establish a material model describing the functionally graded plate structure.

[0051] 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:

[0052]

[0053] 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.

[0054] Assuming a homogeneous mixture of strong and weak material phases, the equivalent density at any point in the structure is... for:

[0055]

[0056] In the formula: , These are the physical densities of the strong and weak material phases, respectively.

[0057] The Rayleigh damping model is used to characterize the equivalent damping of a uniformly mixed functionally graded material.

[0058] Step S101-2: Construct a reduced Gaussian integral strategy for isogeometric analysis, and combine it with the material model of the functionally graded plate structure to efficiently construct the isogeometric transient dynamic finite element model of the functionally graded plate structure.

[0059] The isogeometric finite element method uses NURBS (non-uniform rational B-spline) functions describing the geometry as the shape functions for finite element analysis. Therefore, compared with the traditional finite element method, the isogeometric finite element method has advantages such as higher-order continuity and better numerical stability. In particular, topology optimization based on the isogeometric finite element method avoids numerical problems such as checkerboard patterns and mesh dependence, and can obtain optimal topologies with smooth boundaries.

[0060] Within the framework of isogeometric analysis, the isogeometric transient dynamic finite element equations for the functionally graded plate structure are:

[0061]

[0062]

[0063] In the formula: , and These represent the global stiffness matrix, mass matrix, and damping matrix of the functionally graded plate structure. , Units The element mass and stiffness matrix, , For Rayleigh damping parameters, , and The first Payload array at each time step The acceleration, velocity, and displacement matrix of the structural control points under action. This represents the total number of steps in the dynamic events.

[0064] To improve the efficiency of dynamic analysis, the reduced Gaussian integral method is used to calculate the element mass and stiffness matrices, resulting in:

[0065]

[0066]

[0067] In the formula: For the reduced Gaussian integral points, isogeometric unit Volume fraction of the medium-strength material phase. Let be the shape function matrix of the isogeometric unit. It is a volume interpolation function. Let Jacobian matrix be the determinant of the matrix that maps from the parameter space to the physical space. Let be the determinant of the Jacobian matrix that maps from the parent cell space to the parameter cell space. These are the weighting coefficients. For plate thickness, The strain matrix, For stiffness interpolation function, This is to reduce the number of Gaussian integration points.

[0068] By reducing the number of integration points to decrease computational complexity and thus improve computational efficiency, the rule for reduced Gaussian integrals on equal geometric units is as follows:

[0069]

[0070] In the formula: , Vector nodes , The degree of repetition Let be the order of the integrand polynomial.

[0071] Using the unconditionally stable HHT-α method as the time integration scheme, the isogeometric transient dynamic finite element equations of the above functionally graded plate structure are solved, resulting in:

[0072]

[0073]

[0074] In the formula: The residual matrix is... , and For HHT-α algorithm parameters, Increment the time step.

[0075] Update the displacement and velocity matrix using Newmark form:

[0076]

[0077] S102: Construct a smooth projection scheme and material interpolation scheme for the structural design variables of functionally graded plates, and propose a transient dynamic topology optimization model for functionally graded plates with the goal of minimizing dynamic compliance.

[0078] In this invention, step S102 may further include:

[0079] Step S102-1: Construct a smooth density distribution and projection function based on NURBS basis functions to establish a material interpolation scheme for the functionally graded plate structure.

[0080] At each structural control point Introduce a density value This refers to the control point density, which is used as a design variable. To improve the smoothness of the control point density, the Shepard function is used within the compact supports of the current control points. The density of all control points is weighted and used as the density value of the current control point. Then we have:

[0081]

[0082] In the formula: , These represent the current control points. The number of control points in two directions within the plane of the compact support.

[0083] NURBS function As a basis function, the control point density after linear combination smoothing Then, a smooth density distribution function is constructed. Then we have:

[0084]

[0085] In the formula: Let be the coordinates of any point within the parent element. , Two parameters respectively , The order of the NURBS function in the direction, and This represents the number of control points in the two directions within the plane.

[0086] To obtain clear multiphase material interfaces and structural boundaries, the Heaviside threshold projection function is used to push the control point density toward 0 or 1, as shown below:

[0087]

[0088] In the formula: The density distribution function after threshold projection. For sharpness parameters, This is the threshold parameter.

[0089] The threshold projection function combined with the above density distribution Material interpolation schemes for constructing functionally graded plate structures: volume interpolation and stiffness interpolation functions.

[0090] Volume interpolation With stiffness interpolation function Introducing the Ersatz parameter To suppress the numerical singularity problem in low-density regions, we have:

[0091]

[0092]

[0093] In the formula: This is the uncorrected stiffness interpolation function.

[0094] Similarly, in order to suppress To address numerical instability issues arising in low-density regions, the RAMP function is used to penalize the threshold projection function. Then we have:

[0095]

[0096] In the formula: This is a penalty factor.

[0097] Additionally, regarding design variables It characterizes the continuous smooth gradient of functionally graded materials, thus eliminating the need for smoothing, projection, and penalty treatments.

[0098] Step S102-2: Considering the synergistic effect of functionally graded material ratio and structural topology, a transient dynamic topology optimization model for functionally graded plates is proposed, with the amount of two-phase materials as the volume constraint and the goal of minimizing dynamic compliance.

[0099] Considering a functionally graded material composed of strong and weak phases, the problem of minimizing the dynamic flexibility of a plate structure under the respective volume constraints is given by the following transient dynamic topology optimization model:

[0100]

[0101] In the formula: The dynamic compliance of a functionally graded plate is abbreviated as , The total number of equal geometric units. , 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.

[0102] S103: Based on the sensitivity analysis strategy of first discretization and then differentiation, construct the time series of dual variables and calculate the sensitivity of dynamic compliance and volume constraint of functionally graded plate structure.

[0103] To avoid consistency errors in sensitivity analysis of transient problems, a sensitivity analysis strategy of first discretizing and then differentiating is adopted to calculate the dual variables, thereby obtaining the dynamic compliance with respect to the design variables. , Sensitivity.

[0104] Dynamic compliance for design variables The sensitivity, expressed in components, is as follows:

[0105]

[0106] In the formula: To be with residuals The corresponding dual variable is calculated using the following formula:

[0107] Then:

[0108]

[0109]

[0110] Then:

[0111]

[0112] In the formula, , These are the dual variables corresponding to the Newmark difference scheme.

[0113] Similarly, dynamic compliance For the volume fraction of strong material on the plate unit The sensitivity is:

[0114]

[0115] 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:

[0116]

[0117]

[0118] S104: 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).

[0119] 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:

[0120]

[0121] In the formula: For the first Approximate transient heat dissipation efficiency obtained through step-by-step optimization iteration. For the middle A tiny positive number between.

[0122] In summary, the method of this invention combines isogeometric analysis with transient structural dynamics topology optimization, which can ensure that the functionally graded plate has clear and smooth multiphase material boundaries, facilitating subsequent processing and manufacturing, and achieving design-manufacturing consistency.

[0123] Secondly, the present invention provides an isogeometric transient dynamic topology optimization system for functionally graded plate structures, such as... Figure 2 As shown, it includes:

[0124] The initialization module initializes the control point density of the functional gradient plate and the design variables such as the volume fraction of strong materials attached to the plate elements. It also pre-sets the property parameters of each phase material of the plate structure, the upper limit parameters of the volume constraint, the parameters of the HHT-α method time integration scheme, the parameters of the material interpolation model, the smoothing and projection parameters of the design variables, and the parameters of the MMA-GCMMA hybrid optimization algorithm.

[0125] The preprocessing module uses the design domain of the functionally graded plate structure to be discretized using equal geometric elements. It combines material interpolation schemes and Gaussian reduced integration schemes to efficiently calculate the stiffness, mass, and damping matrices of the plate elements. It defines the initial and boundary conditions and applies transient loads in the time domain, ultimately forming an equal geometric transient dynamic finite element model of the functionally graded plate structure.

[0126] The design module includes variable smoothing and projection, and material interpolation. By combining the Shepard and NURBS functions, a compactly supported and smooth density distribution function is constructed. This function, combined with the Heaviside function, drives the density distribution function to approximate a 0-1 distribution. Subsequently, a very small but non-zero Ersatz number is introduced to construct a volumetric and stiffness interpolation model for multiphase materials.

[0127] The dynamic analysis module constructs a time integration scheme using the unconditionally stable HHT-α method to solve the isogeometric transient dynamic finite element model of the functionally graded plate structure, thereby calculating the dynamic compliance of the functionally graded plate structure and the volume constraint of the multiphase material.

[0128] The optimization solution module calculates the sensitivity information of the transient 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.

[0129] Example

[0130] Example 1

[0131] This invention addresses the transient dynamic topology optimization problem of a simply supported functionally graded square plate under a single-point load, further validating the effectiveness of the method.

[0132] Step S101: Establish a hybrid material model of two-phase functionally graded materials based on the Hashin-Shtrikman limit model, and construct a transient dynamic finite element model of the functionally graded plate structure through the isogeometric analysis method of Gaussian reduced integral.

[0133] like Figure 3 As shown, the functionally graded square plate structure has the following dimensions: length is... ,thickness The design domain is simply supported on all four sides, with a concentrated half-wave sinusoidal 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 , Let be the total volume of the functionally graded plate. The equivalent Rayleigh damping constant of the functionally graded plate is... , HHT-α algorithm parameters , , .

[0134] Step S101-1: Based on the Hashin-Shtrikman limit model, calculate the equivalent elastic modulus, equivalent density and equivalent structural damping of the two-phase functionally graded materials respectively, and establish a material model describing the functionally graded plate structure.

[0135] 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:

[0136]

[0137] 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.

[0138] Assuming a homogeneous mixture of strong and weak material phases, the equivalent density at any point in the structure is... for:

[0139]

[0140] In the formula: , These are the physical densities of the strong and weak material phases, respectively.

[0141] The Rayleigh damping model is used to characterize the equivalent damping of a uniformly mixed functionally graded material.

[0142] Step S101-2: Construct a reduced Gaussian integral strategy for isogeometric analysis, and combine it with the material model of the functionally graded plate structure to efficiently construct the isogeometric transient dynamic finite element model of the functionally graded plate structure.

[0143] The isogeometric finite element method uses NURBS (non-uniform rational B-spline) functions describing the geometry as the shape functions for finite element analysis. Therefore, compared with the traditional finite element method, the isogeometric finite element method has advantages such as higher-order continuity and better numerical stability. In particular, topology optimization based on the isogeometric finite element method avoids numerical problems such as checkerboard patterns and mesh dependence, and can obtain optimal topologies with smooth boundaries.

[0144] Within the framework of isogeometric analysis, the isogeometric transient dynamic finite element equations for the functionally graded plate structure are:

[0145]

[0146]

[0147] In the formula: , and These represent the global stiffness matrix, mass matrix, and damping matrix of the functionally graded plate structure. , Units The element mass and stiffness matrix, , For Rayleigh damping parameters, , and The first Payload array at each time step The acceleration, velocity, and displacement matrix of the structural control points under action. This represents the total number of steps in the dynamic events.

[0148] To improve the efficiency of dynamic analysis, the reduced Gaussian integral method is used to calculate the element mass and stiffness matrices, resulting in:

[0149]

[0150]

[0151] In the formula: For the reduced Gaussian integral points, isogeometric unit Volume fraction of the medium-strength material phase. Let be the shape function matrix of the isogeometric unit. It is a volume interpolation function. Let Jacobian matrix be the determinant of the matrix that maps from the parameter space to the physical space. Let be the determinant of the Jacobian matrix that maps from the parent cell space to the parameter cell space. These are the weighting coefficients. For plate thickness, The strain matrix, For stiffness interpolation function, This is to reduce the number of Gaussian integration points.

[0152] By reducing the number of integration points to decrease computational complexity and thus improve computational efficiency, the rule for reduced Gaussian integrals on equal geometric units is as follows:

[0153]

[0154] In the formula: , Vector nodes , The degree of repetition Let be the order of the integrand polynomial.

[0155] Using the unconditionally stable HHT-α method as the time integration scheme, the isogeometric transient dynamic finite element equations of the above functionally graded plate structure are solved, resulting in:

[0156]

[0157]

[0158] In the formula: The residual matrix is... , and For HHT-α algorithm parameters, Increment the time step.

[0159] Update the displacement and velocity matrix using Newmark form:

[0160]

[0161] S102: Construct a smooth projection scheme and material interpolation scheme for the structural design variables of functionally graded plates, and propose a transient dynamic topology optimization model for functionally graded plates with the goal of minimizing dynamic compliance.

[0162] Step S102-1: Construct a smooth relative density distribution and projection function based on NURBS basis functions, and then combine this with the SIMP method to establish a material interpolation scheme for the functionally graded plate structure.

[0163] At each structural control point Introduce a density value This refers to the control point density, which is used as a design variable. To improve the smoothness of the control point density, the Shepard function is used within the compact supports of the current control points. Weight the density of all control points using the Shepard function. The density of all control points within the compact support of the current control point is weighted and used as the density value of the current control point. Then we have:

[0164]

[0165] In the formula: , These represent the current control points. The number of control points in two directions within the plane of the compact support.

[0166] NURBS function As a basis function, the control point density after linear combination smoothing Then, a smooth density distribution function is constructed. Then we have:

[0167]

[0168] In the formula: Let be the coordinates of any point within the parent element. , Two parameters respectively , The order of the NURBS function in the direction, and This represents the number of control points in the two directions within the plane.

[0169] To obtain clear multiphase material interfaces and structural boundaries, the Heaviside threshold projection function is used to push the control point density toward 0 or 1, as shown below:

[0170]

[0171] In the formula: The density distribution function after threshold projection. For sharpness parameters, This is the threshold parameter.

[0172] The threshold projection function combined with the above density distribution Material interpolation schemes for constructing functionally graded plate structures: volume interpolation and stiffness interpolation functions.

[0173] Volume interpolation With stiffness interpolation function Introducing the Ersatz parameter To suppress the numerical singularity problem in low-density regions, we have:

[0174]

[0175]

[0176] In the formula: This is the uncorrected stiffness interpolation function.

[0177] Similarly, in order to suppress To address numerical instability issues arising in low-density regions, the RAMP function is used to penalize the threshold projection function. Then we have:

[0178]

[0179] In the formula: This is a penalty factor.

[0180] Additionally, regarding design variables It characterizes the continuous smooth gradient of functionally graded materials, thus eliminating the need for smoothing, projection, and penalty treatments.

[0181] Step S102-2: Considering the synergistic effect of functionally graded material ratio and structural topology, a transient dynamic topology optimization model for functionally graded plates is proposed, with the amount of two-phase materials as the volume constraint and the goal of minimizing dynamic compliance.

[0182] Considering a functionally graded material composed of strong and weak phases, the problem of minimizing the dynamic flexibility of a plate structure under the respective volume constraints is given by the following transient dynamic topology optimization model:

[0183]

[0184] In the formula: The dynamic compliance of a functionally graded plate is abbreviated as , The total number of equal geometric units. , 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 The upper and lower bounds.

[0185] S103: Based on the sensitivity analysis strategy of first discretization and then integration, construct the time series of dual variables and calculate the sensitivity of dynamic compliance and volume constraint of functionally graded plate structure.

[0186] To avoid consistency errors in sensitivity analysis of transient problems, a sensitivity analysis strategy of first discretizing and then differentiating is adopted to calculate the dual variables, thereby obtaining the dynamic compliance with respect to the design variables. , Sensitivity.

[0187] Dynamic compliance for design variables The sensitivity, expressed in components, is as follows:

[0188]

[0189] In the formula: To be with residuals The corresponding dual variable is calculated using the following formula:

[0190] Then:

[0191]

[0192]

[0193] Then:

[0194]

[0195] In the formula, , These are the dual variables corresponding to the Newmark difference scheme.

[0196] Similarly, dynamic compliance For the volume fraction of strong material on the plate unit The sensitivity is:

[0197]

[0198] 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:

[0199]

[0200]

[0201] S104: 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).

[0202] 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:

[0203]

[0204] In the formula: For the first Approximate transient heat dissipation efficiency obtained through step-by-step optimization iteration. For the middle A tiny positive number between.

[0205] The specific optimization results are shown in Figures 4 and 5. The red area represents the strong material distribution area, and the blue area represents the weak material distribution area.

[0206] Figure 4 shows the structural topology optimization results using the traditional finite element method; Figure 5 The results of structural topology optimization using the equal-geometric finite element method are shown, which is the method of this invention. As the load application time decreases, to suppress the inertial effect of transient loads, the strong material phase concentrates in the transient load application region, while the weak material phase tends to move towards the simply supported ends. Compare Figure 4 with... Figure 5 Traditional finite element methods and isogeometric finite element methods yield nearly identical material distributions. However, the latter produces structural topologies with clearer and smoother boundaries, facilitating subsequent processing and manufacturing. This is because isogeometric analysis implements a representation method consistent with geometric shape and design optimization through NURBS basis functions. Its higher order ensures the smoothness and clarity of the optimized structural boundaries, achieving a unity between design and manufacturing.

[0207] The results of Embodiment 1 of this invention demonstrate that, unlike the gradient distribution of existing functionally graded materials along a fixed direction, the gradient distribution of each constituent phase material in this invention is controlled according to the optimal load-bearing path. This achieves smooth and gradual phase transitions in the functionally graded materials and synergistic optimization of the structural topology, thereby maximizing the transient dynamic performance of the structure. Simultaneously, the reduced Gaussian integral method effectively improves the computational efficiency of dynamic topology optimization. Therefore, the algorithm of this invention effectively solves the multi-material topology optimization problem of functionally graded plate structures under transient loads, and is suitable for the engineering requirements of lightweight and high-performance equipment under strong vibration environments.

[0208] Example 2

[0209] 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.

[0210] Step S101: Establish a hybrid material model of two-phase functionally graded materials based on the Hashin-Shtrikman limit model, and construct a transient dynamic finite element model of the functionally graded plate structure through the reduced integral isogeometric analysis method.

[0211] like Figure 6 As shown, four points in the central region of the four-sided fixed functional gradient square plate are subjected to half-wave cosine concentrated loads, and other parameters are the same as in Example 1.

[0212] Steps S102, S103, and S104 are the same as in Example 1.

[0213] The specific optimization results are as follows: Figure 7 ,8 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.

[0214] Figure 7 The results are from structural topology optimization using the traditional finite element method. Figure 8 The results of structural topology optimization using the isogeometric finite element method are presented in this invention. As the load application time decreases, the strong material phase is distributed in the neighborhood of the fixed end and the four concentrated load application points, especially as the load application time decreases. Under operating conditions, the weaker material phases interconnect at the load application point to form square frame elements, effectively suppressing the inertial effect of transient loads. (Comparison) Figure 7 and Figure 8 Traditional finite element method and isogeometric finite element method obtained almost the same material distribution. However, the latter obtained a structural topology with clearer and smoother boundaries, which is convenient for subsequent processing and manufacturing.

[0215] The results of Embodiment 2 of the present invention show that the algorithm of the present invention can solve the multi-material topology optimization problem of functionally graded plate structures under transient complex load conditions, and is suitable for the engineering requirements of lightweight and high-performance equipment under strong vibration environment.

[0216] This invention addresses the problem of co-optimizing the in-plane gradient direction of graded materials and the topology of multi-material structures under transient loads from a design-manufacturing consistency perspective. It achieves optimal vibration-resistant design for functionally graded plate structures under strong vibration conditions, meeting the technical requirements of lightweight and high-performance engineering equipment. This invention constructs an isogeometric transient dynamic finite element analysis model of the functionally graded plate structure based on the reduced Gaussian integral method; it constructs a smooth projection scheme and material interpolation scheme for the design variables of the functionally graded plate structure, using control point density and the volume fraction of strong materials attached to the plate elements as design variables, with the goal of minimizing dynamic compliance, and proposes a transient dynamic topology optimization model for the functionally graded plate structure; it constructs the time-domain adjoint equation of the topology optimization model to obtain the sensitivity information of the objective function; and based on the MMA-GCMMA hybrid optimization algorithm, it updates the design variables such as the control point density and the volume fraction of strong materials attached to the plate elements of the functionally graded plate, thereby obtaining an optimal vibration-resistant design for the functionally graded plate structure that is convenient for engineering applications. This invention maximizes the optimal vibration-resistant performance of functionally graded structures and is particularly suitable for the lightweight vibration reduction design of large-scale engineering equipment under strong vibration conditions.

[0217] 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 isogeometric transient dynamic topology optimization of a functionally graded plate structure, characterized in that, Includes the following steps: S101: A hybrid material model of two-phase functionally graded materials is established based on the Hashin-Shtrikman limit model, and a transient dynamic finite element model of functionally graded plate structure is constructed by the Gaussian reduced integral isogeometric analysis method. S102: Construct a smooth projection scheme and material interpolation scheme for the design variables of functionally graded plate structures, and propose a transient dynamic topology optimization model for functionally graded plate structures with the goal of minimizing dynamic compliance. S103: Construct time series of dual variables based on the sensitivity analysis strategy of first discretization and then differentiation, and calculate the sensitivity of dynamic compliance and volume constraint of functionally graded plate structure. S104: The optimal topology design of the functionally graded plate structure is obtained by using the moving asymptote method (MMA) and the globally convergent moving asymptote method (GCMMA) hybrid optimization algorithm.

2. The isogeometric transient dynamic topology optimization method for a functionally graded plate structure according to claim 1, characterized in that, Step S101 includes: Step S101-1: Based on the Hashin-Shtrikman limit model, calculate the equivalent elastic modulus, equivalent density and equivalent structural damping of the two-phase functionally graded material respectively, and establish a material model describing the functionally graded plate structure. Step S101-2: Construct a reduced Gaussian integral strategy for isogeometric analysis, and combine it with the material model of the functionally graded plate structure to efficiently construct the isogeometric transient dynamic finite element model of the functionally graded plate structure.

3. The isogeometric transient dynamic topology optimization method for a functionally graded plate structure according to claim 2, characterized in that, Step S101-1 includes: 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: 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. The volume fraction of the strong material phase at any point in the structure; Assuming a homogeneous mixture of strong and weak material phases, the equivalent density at any point in the structure is... for: In the formula: , These are the physical densities of the strong and weak material phases, respectively. The Rayleigh damping model is used to characterize the equivalent damping of uniformly mixed functionally graded materials.

4. The isogeometric transient dynamic topology optimization method for a functionally graded plate structure according to claim 2, characterized in that, Step S101-2 includes: The isogeometric finite element method uses the non-uniform rational B-spline NURBS function describing the geometry as the shape function of the finite element analysis; it takes advantage of the advantages of the isogeometric finite element method such as high-order continuity and good numerical stability; the topology optimization based on the isogeometric finite element method will not have checkerboard pattern and mesh-dependent numerical problems, and can obtain the optimal topology structure with smooth boundaries. Within the framework of isogeometric analysis, the isogeometric transient dynamic finite element equations for the functionally graded plate structure are: In the formula: , and These represent the global stiffness matrix, mass matrix, and damping matrix of the functionally graded plate structure. , Units The element mass and stiffness matrix, , For Rayleigh damping parameters, , and The first Payload array at each time step The acceleration, velocity, and displacement matrix of the structural control points under action. This represents the total number of steps in the dynamic events; To improve the efficiency of dynamic analysis, the reduced Gaussian integral method is used to calculate the element mass and stiffness matrices, resulting in: In the formula: For the reduced Gaussian integral points, isogeometric unit Volume fraction of the medium-strength material phase. Let be the shape function matrix of the isogeometric unit. It is a volume interpolation function. Let Jacobian matrix be the determinant of the matrix that maps from the parameter space to the physical space. Let be the determinant of the Jacobian matrix that maps from the parent cell space to the parameter cell space. These are the weighting coefficients. For plate thickness, The strain matrix, For stiffness interpolation function, To reduce the number of Gaussian integration points; By reducing the number of integration points to decrease computational complexity and thus improve computational efficiency, the rule for reduced Gaussian integrals on equal geometric units is as follows: In the formula: , Vector nodes , The degree of repetition Let the order of the integrand be the polynomial order. Using the unconditionally stable HHT-α method as the time integration scheme, the isogeometric transient dynamic finite element equations of the above functionally graded plate structure are solved, resulting in: In the formula: The residual matrix is... , and For HHT-α algorithm parameters, Increment the time step; Update the displacement and velocity matrix using Newmark form: 。 5. The isogeometric transient dynamic topology optimization method for a functionally graded plate structure according to claim 1, characterized in that, Step S102 includes: Step S102-1: Construct a smooth density distribution and projection function based on NURBS basis functions to establish a material interpolation scheme for the functionally graded plate structure. Step S102-2: Considering the synergistic effect of functionally graded material ratio and structural topology, a transient dynamic topology optimization model for functionally graded plates is proposed, with the amount of two-phase materials as the volume constraint and the goal of minimizing dynamic compliance.

6. The isogeometric transient dynamic topology optimization method for a functionally graded plate structure according to claim 5, characterized in that, Step S102-1 includes: At each structural control point Introduce a density value The control point density, as a design variable, is improved by using the Shepard function within the compact supports of the current control points to enhance its smoothness. The density of all control points is weighted and used as the density value of the current control point. Then we have: In the formula: , These represent the current control points. The number of control points in the plane along two directions within a compact support; NURBS function As a basis function, the control point density after linear combination smoothing Then, a smooth density distribution function is constructed. Then we have: In the formula: Let be the coordinates of any point within the parent element. , Two parameters respectively , The order of the NURBS function in the direction, and This refers to the number of control points in two directions within the plane; To obtain clear multiphase material interfaces and structural boundaries, the control point density is pushed towards 0 or 1 using the Heaviside threshold projection function, as shown below: In the formula: The density distribution function after threshold projection. For sharpness parameters, For threshold parameters; The threshold projection function combined with the above density distribution Material interpolation schemes for constructing functionally graded plate structures: volume interpolation and stiffness interpolation functions; Volume interpolation With stiffness interpolation function Introducing the Ersatz parameter To suppress the numerical singularity problem in low-density regions, we have: In the formula: This is the uncorrected stiffness interpolation function; Similarly, to suppress To address numerical instability issues arising in low-density regions, the RAMP function is used to penalize the threshold projection function. Then we have: In the formula: As a penalty factor; Design variables It characterizes the continuous smooth gradient of functionally graded materials without requiring smoothing, projection, and penalty processing.

7. The isogeometric transient dynamic topology optimization method for a functionally graded plate structure according to claim 5, characterized in that, Step S102-2 includes: Considering a functionally graded material composed of strong and weak phases, the problem of minimizing the dynamic flexibility of a plate structure under the respective volume constraints is given by the following transient dynamic topology optimization model: In the formula: The dynamic compliance of a functionally graded plate is abbreviated as , The total number of equal geometric units. , 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 The upper and lower bounds.

8. The isogeometric transient dynamic topology optimization method for a functionally graded plate structure according to claim 1, characterized in that, Step S103 includes: To avoid consistency errors in sensitivity analysis of transient problems, a sensitivity analysis strategy of first discretizing and then differentiating is adopted to calculate the dual variables, thereby obtaining the dynamic compliance with respect to the design variables. , Sensitivity; Dynamic compliance for design variables The sensitivity, expressed in components, is as follows: In the formula: To be with residuals The corresponding dual variable is calculated using the following formula: Then: Then: In the formula, , These are the dual variables corresponding to the Newmark difference scheme; Similarly, dynamic compliance For the volume fraction of strong material on the plate unit The sensitivity is: 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: 。 9. The isogeometric transient dynamic topology optimization method for a functionally graded plate structure according to claim 8, characterized in that, Step S104 includes: The initial iteration phase of optimization uses the MMA optimization algorithm, while the GCMMA optimization 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: In the formula: For the first Approximate transient heat dissipation efficiency obtained through step-by-step optimization iteration. For the middle The smallest positive number between; By combining isogeometric analysis methods with transient structural dynamics topology optimization methods, functionally graded plates can be guaranteed to have clear and smooth multiphase material boundaries, which facilitates subsequent processing and manufacturing and achieves design-manufacturing consistency.

10. A transient dynamic topology optimization system for a functionally graded plate structure with isogeometric properties, characterized in that, The system is implemented based on the method of any one of claims 1 to 9, and includes the following program modules: The initialization module initializes the control point density of the functional gradient plate and the design variables such as the volume fraction of strong materials attached to the plate elements. It also pre-sets the property parameters of each phase material of the plate structure, the upper limit parameters of the volume constraint, the parameters of the HHT-α method time integration scheme, the parameters of the material interpolation model, the smoothing and projection parameters of the design variables, and the parameters of the MMA-GCMMA hybrid optimization algorithm. The preprocessing module uses the design domain of the functionally graded plate structure to be discretized using equal geometric elements. It combines material interpolation schemes and Gaussian reduced integration schemes to efficiently calculate the stiffness, mass, and damping matrices of the plate elements. It defines the initial and boundary conditions and applies transient loads in the time domain, ultimately forming an equal geometric transient dynamic finite element model of the functionally graded plate structure. The design module includes variable smoothing and projection, and material interpolation. By combining the Shepard and NURBS functions, a compactly supported and smooth density distribution function is constructed. This function, combined with the Heaviside function, drives the density distribution function to approximate a 0-1 distribution. Subsequently, a very small but non-zero Ersatz number is introduced to construct a volumetric and stiffness interpolation model for multiphase materials. The dynamic analysis module constructs a time integration scheme using the unconditionally stable HHT-α method to solve the isogeometric transient dynamic finite element model of the functionally graded plate structure, thereby calculating the dynamic compliance of the functionally graded plate structure and the volume constraint of the multiphase material. The optimization solution module calculates the sensitivity information of the transient 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.

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