Topological optimization method and system for full-scale functional gradient porous hierarchical structure transient dynamics

Abstract

The invention discloses a full-scale functional gradient porous hierarchical structure transient dynamics topological optimization method and system, and belongs to the field of topological optimization design of functional gradient structures. The problem to be solved is how to construct a transient isogeometric analysis method compatible with a traditional finite element method and integrate the transient isogeometric analysis method into a full-scale functional gradient porous hierarchical structure topological optimization problem, and the engineering expandability of an optimization algorithm is improved. According to the method, a self-adaptive growth strategy of projection parameters is constructed, and continuous regulation and control of structure topology in the optimization process are realized. According to the technical key points, a structure isogeometric transient dynamics finite element model is constructed; constructing a functional gradient porous hierarchical constraint and a material interpolation model based on a sharpness parameter adaptive increase strategy, and proposing a transient dynamic topological optimization model of a full-scale functional gradient porous hierarchical structure; constructing a time domain adjoint equation of the topological optimization model; and the optimal anti-vibration design of the functional gradient porous hierarchical structure is obtained. The designed porous structure has good structural performance and functional characteristics, and can meet the requirements of complex engineering environments in the fields of aerospace, deep sea and the like.

Description

Technical Field

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

[0002] Functionally graded materials (FJCTs) introduce continuously varying material composition and microstructural features within the spatial domain, enabling targeted performance matching and optimization in different regions for service conditions. The gradient design concept is not only applicable to performance control at the material level but can also be extended to topology and configuration design at the structural level. In nature, the concept of functional gradient is widely applied to biological structures such as bones, teeth, horns, beaks, and wood, and the loads borne by these structures often exhibit significant dynamic characteristics. Furthermore, these biological structures not only display obvious gradient characteristics but also generally possess porous morphologies, enabling efficient material utilization for lightweight design while ensuring structural performance. Therefore, structural performance under dynamic loads can be significantly improved by introducing a gradient porosity distribution. Inspired by the efficient load-bearing mechanism of naturally occurring functionally graded porous structures under dynamic environments, this paper utilizes a full-scale dynamic topology optimization method to design a functionally graded porous hierarchical structure that meets vibration resistance requirements under transient excitation. This has significant theoretical and engineering value for solving the lightweighting problem of key load-bearing components in high-end equipment such as aerospace and deep-sea applications.

[0003] Existing research on vibration-resistant design is based on homogenization theory with the scale separation assumption, which cannot accurately characterize the mechanical properties of porous structures, leading to a mismatch between structural design performance and manufacturing performance. Using full-scale analysis instead of homogenization can avoid the over-averaging of microstructural mechanical behavior caused by the scale separation assumption, resulting in distortions in macroscopic response predictions and deviations in dynamic characteristics. Functionally graded porous structures often have complex structural boundaries. Existing research on isogeometric analysis methods based on NURBS elements can accurately characterize topology-optimized configurations. However, the cross-element characteristics of NURBS basis functions make them incompatible with existing finite element analysis and topology optimization frameworks, limiting the complexity of isogeometric analysis in topology optimization. Furthermore, to obtain a clear topological profile, Heaviside projection functions are often used to push design variables towards 0 or 1; however, the update strategy for the sharpness parameter lacks correlation with the optimization process, leading to discontinuities in the optimization process. In particular, key load-bearing components of high-end equipment in aerospace and deep-sea fields often operate in extreme environments with transient loads. Existing frequency-domain-based functionally graded porous structure vibration-resistant topology optimization design methods cannot meet the optimality requirements of time-domain problems.

[0004] In summary, integrating functionally graded materials (FJCTs) into gradient porous hierarchical structures and proposing a transient dynamic topology optimization method for full-scale FJCTs to achieve lightweight and high-performance equipment under strong vibration conditions has significant engineering implications. However, existing isogeometric analysis-based design frameworks lack consistent element structures and computational flows with classical finite element methods at the numerical implementation level, making it difficult to extend to the dynamic vibration-resistant design of complex engineering structures. Furthermore, in the topology optimization process, the growth strategy of the sharpness parameter in the Heaviside projection function is usually pre-assumed based on experience, severing the effective connection with the optimizer and failing to achieve continuous control of the 0-1 binary design optimization process. Therefore, existing technologies lack effective analysis and optimization methods and approaches in terms of isogeometric transient dynamic analysis, the engineering scalability of the topology optimization framework, and the adaptive control of the optimization process, making it difficult to adapt to the aforementioned engineering optimization design scenarios under strong vibration conditions. Summary of the Invention

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

[0006] Key load-bearing components of high-end equipment in aerospace and deep-sea fields frequently operate under extreme environments with transient loads. Existing frequency-domain-based functionally graded porous structures' vibration-resistant topology optimization design methods cannot meet the optimality requirements of time-domain problems. Furthermore, while design techniques based on isogeometric analysis offer significant advantages in representing complex structural profiles, they lack consistent element structures and computational flows with classical finite element methods, making it difficult to extend to the dynamic vibration-resistant design of complex engineering structures. Additionally, during topology optimization, the growth strategy of the sharpness parameter in the Heaviside projection function is typically based on empirical pre-assumptions, severing the effective connection with the optimizer and failing to achieve continuous control of the 0-1 binary design optimization process. Therefore, this paper addresses the lightweighting problem of these key load-bearing components in high-end equipment by providing a full-scale functionally graded porous hierarchical structure transient dynamic topology optimization method and system.

[0007] To achieve the above objectives, in a first aspect, the present invention provides a transient dynamic topology optimization method for full-scale functionally graded porous hierarchical structures, comprising the following steps:

[0008] S101: The mass / stiffness matrix of the NURBS element is expressed as a linear combination of the mass / stiffness matrix of the standard Bezier element and the Bezier extraction operator, which efficiently solves the isogeometric transient dynamic finite element equations of functionally graded porous hierarchical structures under the traditional finite element framework.

[0009] S102: Construct a functionally graded porous hierarchical constraint and a material interpolation model based on an adaptive increase strategy of sharpness parameter, establish an optimization solution formula for the dynamic compliance minimization problem, and propose an efficient transient dynamic topology optimization method for full-scale functionally graded porous hierarchical structures;

[0010] S103: Construct a time-dependent sensitivity adjoint equation, solve it to obtain the time series of the dual variables, and then calculate the sensitivity of the dynamic compliance and volume constraint of the functionally graded porous hierarchical structure respectively.

[0011] S104: The optimal topology design of the functionally graded porous hierarchical structure is obtained by updating the Lagrange multipliers and design variables of the volume constraints in parallel using the generalized OC criterion method.

[0012] Secondly, a transient dynamic topology optimization system for a full-scale functionally graded porous hierarchical structure includes:

[0013] The initialization module initializes independent design variables within the design domain and pre-sets the property parameters, volume constraint upper limit parameters, Newmark method time integration scheme parameters, material interpolation model parameters, design variable smoothing and projection parameters, and generalized OC criterion optimization algorithm parameters for functionally graded materials within the porous hierarchical structure.

[0014] The preprocessing module uses isogeometric NURBS elements to discretize the design domain of the functional graded porous hierarchical structure, applies boundary conditions and transient dynamic loads; it represents the mass / stiffness matrix of the NURBS element as a linear combination of the mass / stiffness matrix of the standard Bezier element and the Bezier extraction operator, and then constructs an isogeometric transient dynamic finite element model of the functional graded porous hierarchical structure.

[0015] The dynamic analysis module constructs a time integration scheme using the unconditionally stable Newmark method to solve the isogeometric transient dynamic finite element model of the functionally graded porous hierarchical structure, thereby calculating the dynamic compliance and volume constraints of the functionally graded porous hierarchical structure.

[0016] The optimization solution module calculates the sensitivity information of the transient dynamic topology optimization model of the functionally graded porous hierarchical structure, updates the design variables based on the generalized OC criterion optimization algorithm, and determines whether the optimization iteration results meet the convergence criterion. If not,

[0017] Then continue to optimize and iterate until the convergence criterion is met.

[0018] This invention provides a transient dynamic topology optimization method and system for full-scale functionally graded porous hierarchical structures, which has the following beneficial technical effects:

[0019] Transient dynamic topology optimization of functionally graded porous hierarchical structures faces multiple coupling challenges: On the one hand, traditional homogenization methods based on scale separation struggle to accurately characterize the dynamic behavior of multi-scale porous structures, easily leading to a mismatch between design performance and actual manufacturing performance; on the other hand, while isogeometric analysis possesses the ability to accurately express complex boundaries and gradient configurations, its cross-element basis function characteristics lack compatibility with classical finite element and topology optimization frameworks, limiting its scalability in the transient vibration-resistant design of key components in engineering equipment; furthermore, existing Heaviside projection sharpness parameters rely on empirical updates and lack an adaptive coupling mechanism with the transient optimization process, making it difficult to achieve continuous and stable control of 0-1 topology evolution; simultaneously, current frequency-domain-based vibration-resistant optimization strategies fail to meet the optimality requirements under time-domain transient loads. These issues collectively constrain the design theory and engineering application of functionally graded porous hierarchical structures to achieve high performance and lightweight design under strong vibration environments.

[0020] This invention constructs a structural transient isogeometric analysis strategy based on Bezier extraction operators and functionally graded porous hierarchical constraints. Based on this, an optimization solution formula for minimizing dynamic compliance is established. A material interpolation model based on an adaptive increase strategy for sharpness parameters is constructed to achieve continuous control of the topology optimization process. Therefore, a transient dynamic topology optimization method for full-scale functionally graded porous hierarchical structures is proposed, including: defining the initial design domain and optimization algorithm parameters; utilizing the local preservation property of the Bezier basis function space to construct a structural isogeometric transient dynamic finite element model based on Bezier extraction operators; constructing a material interpolation model based on functionally graded porous hierarchical constraints and an adaptive increase strategy for sharpness parameters; using material volume usage as a constraint and minimizing dynamic compliance as the objective, proposing a transient dynamic topology optimization model for full-scale functionally graded porous hierarchical structures; constructing the time-domain adjoint equation of the topology optimization model to obtain sensitivity information on dynamic compliance and volume constraints; and updating the Lagrange multipliers of volume constraints and design variables in parallel based on the generalized OC criterion method to obtain the optimal vibration-resistant design of functionally graded porous hierarchical structures suitable for engineering applications.

[0021] This invention addresses how to construct a transient isogeometric analysis method compatible with the traditional finite element method and integrate it into the topology optimization problem of full-scale functionally graded porous hierarchical structures, thereby improving the engineering scalability of the optimization algorithm. It also addresses how to construct an adaptive growth strategy for projection parameters to achieve continuous control of the structural topology during the optimization process. The proposed transient dynamic topology optimization method for full-scale functionally graded porous hierarchical structures can meet the lightweight and vibration-resistant design requirements of key load-bearing components in high-end equipment under transient excitation. The designed porous structure possesses both excellent structural performance and functional characteristics, and can adapt to the complex engineering environments of aerospace and deep-sea fields. Attached Figure Description

[0022] Figure 1 This is a flowchart of a transient dynamic topology optimization method for a full-scale functionally graded porous hierarchical structure according to the present invention.

[0023] Figure 2 This is a schematic diagram illustrating the design principle of the functionally graded porous hierarchical structure of the present invention.

[0024] (a) is a functionally graded structure;

[0025] (b) is a functionally graded porous hierarchy constraint.

[0026] Figure 3 This is a schematic diagram of the transient dynamic topology optimization system of a full-scale functionally graded porous hierarchical structure according to the present invention.

[0027] Figure 4 This is a two-dimensional function-graded torsion ring problem.

[0028] (a) is the structural design domain in Embodiment 1 of the present invention;

[0029] (b) is the half-wave sinusoidal load in Embodiment 1 of the present invention.

[0030] Figure 5 This is a schematic diagram of the structural topology optimization results for a two-dimensional functionally graded torsion ring.

[0031] (a) The load application time in Embodiment 1 of the present invention The ratio of the elastic modulus of the outer ring to that of the inner ring topology

[0032] Optimization results;

[0033] (b) The load application time in Embodiment 1 of the present invention The ratio of the elastic modulus of the outer ring to that of the inner ring The expansion

[0034] Poaching optimization results;

[0035] (c) The load application time in Embodiment 1 of the present invention The ratio of the elastic modulus of the outer ring to that of the inner ring The topology optimization results;

[0036] (d) is the load application time in Embodiment 1 of the present invention. The ratio of the elastic modulus of the outer ring to that of the inner ring The topology optimization results.

[0037] Figure 6 For the three-dimensional functionally graded cantilever beam problem,

[0038] (a) is the structural design domain filled with equivalent homogeneous material in Embodiment 2 of the present invention;

[0039] (b) is the structural design domain for filling functionally graded materials in Embodiment 2 of the present invention.

[0040] Figure 7 This is a schematic diagram of the optimal topology of the three-dimensional equivalent homogeneous cantilever beam in Embodiment 2 of the present invention.

[0041] (a) The load application time in Embodiment 2 of the present invention The topology optimization results (optimal dynamic compliance is 5.6 Nm);

[0042] (b) The load application time in Embodiment 2 of the present invention The topology optimization results (optimal dynamic compliance is 2.5 Nm).

[0043] Figure 8 This is a schematic diagram of the optimal topology result of the three-dimensional functionally graded cantilever beam in Embodiment 2 of the present invention.

[0044] (a) The load application time in Embodiment 2 of the present invention The topology optimization results (optimal dynamic compliance is 4.0 Nm);

[0045] (b) The load application time in Embodiment 2 of the present invention The topology optimization results (optimal dynamic compliance is 1.9 Nm). Detailed Implementation

[0046] This paper addresses the challenges of transient dynamic topology optimization of full-scale functionally graded porous hierarchical structures. These challenges stem from multiple factors, including distorted dynamic response characterization due to the homogenization scale separation assumption, insufficient engineering compatibility between isogeometric analysis and traditional finite element topology optimization frameworks, lack of adaptive evolution mechanism for the Heaviside sharpness parameter, and the difficulty of guaranteeing temporal transient optimality using frequency domain methods. These factors severely restrict the theoretical development and engineering application of high-performance lightweight design under strong vibration conditions. This invention proposes a transient dynamic topology optimization method and system for full-scale functionally graded porous hierarchical structures, enabling optimal vibration-resistant design of lightweight functionally graded structures.

[0047] Combined with appendix Figures 1-8 This paper elaborates on the transient dynamic topology optimization method for full-scale functionally graded porous hierarchical 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 definite definition of the scope of protection of this invention.

[0048] In a first aspect, the present invention provides a transient dynamic topology optimization method for full-scale functionally graded porous hierarchical structures, the flowchart of which is shown below. Figure 1 As shown, the main steps include:

[0049] S101: The mass / stiffness matrix of the NURBS element is expressed as a linear combination of the mass / stiffness matrix of the standard Bezier element and the Bezier extraction operator, which efficiently solves the isogeometric transient dynamic finite element equations of functionally graded porous hierarchical structures within the traditional finite element framework.

[0050] Based on the isogeometric analysis of the Bezier extraction operator, the globally continuous NURBS basis functions are decomposed into C... 0 Continuous Bezier elements, while maintaining the high accuracy and geometric consistency of isogeometric analysis, enable isogeometric analysis to have an element structure and calculation process that is highly consistent with the classical finite element method at the numerical implementation level.

[0051] Bezier extraction operator It is possible Direction extraction operator and Direction extraction operator Obtained through direct product, we have:

[0052]

[0053] At the unit level, NURBS-shaped function arrays are... Decomposed into a column of Bernstein-shaped functions The linear combination of the Bezier extraction operator gives us:

[0054]

[0055] In the formula: , These are the Bezier extraction operator and weight function on the NURBS cell, respectively.

[0056] Based on the Bezier extraction operator, the stiffness matrix of NURBS solid elements With the mass matrix They are represented as follows:

[0057]

[0058]

[0059] In the formula: , These are the stiffness matrix and mass matrix of a standard Bezier element, respectively. Here is the strain matrix of the Bezier element. The constitutive matrix is , Functionally graded materials along The elastic modulus and mass density that vary in direction according to a power law are expressed as:

[0060]

[0061] In the formula: , For along The direction is related to the elastic modulus and mass density of the upper surface material. , For along The direction is related to the elastic modulus and mass density of the lower surface material. for Thickness in the direction, This is the power-law exponent.

[0062] At the control point ( , Introduce a density value on ) As a design variable, the mass matrix of the NURBS element Stiffness matrix With damping matrix They are respectively:

[0063]

[0064] In the formula: and These are the volume interpolation function and the material interpolation function, respectively. The Gaussian integration point is located at the center of the unit cell. and For Rayleigh damping parameters, , Hierarchical structure and The number of independent control points in a direction.

[0065] The mass, stiffness, and damping matrices of the assembly unit, and the isogeometric transient dynamic finite element equations for the functionally graded porous structure are as follows:

[0066]

[0067]

[0068] In the formula: , and These represent the total mass, stiffness, and mass matrix, respectively. , and For the first Incremental load The acceleration, velocity, and displacement matrix of the control point under action. This represents the total number of time sampling points during the load duration.

[0069] Using the unconditionally stable Newmark method as the time integration scheme, the isogeometric transient dynamic finite element equations of the functionally graded porous structure are solved, and the equations are transformed into the following residual form. :

[0070]

[0071] In the formula: Let t be the state variable for the t-th increment step. ~ These are the algorithm parameters.

[0072] Initial displacement With speed Given the initial acceleration The residuals at the following initial time can be solved. :

[0073]

[0074]

[0075] S102: Construct a functionally graded porous hierarchical constraint and a material interpolation model based on an adaptive increase strategy for sharpness parameters, establish an optimization solution formula for the dynamic compliance minimization problem, and propose an efficient transient dynamic topology optimization method for full-scale functionally graded porous hierarchical structures.

[0076] In this invention, step S102 may further include:

[0077] Step S102-1: Based on the principle of density consistency of hierarchical substructures with the same configuration and the control point coupling effect between hierarchical substructures under the same geometric analysis framework, construct functionally graded porous hierarchical constraints.

[0078] Functionally graded materials are incorporated into gradient porous hierarchical structures to construct functionally graded porous hierarchical constraints. For example... Figure 2 As shown, hierarchical structures are layered along the vertical direction of functionally graded material reinforcement. Hierarchical structure ① contains substructures (1,1) and (1,2), while hierarchical structure ② contains substructures (2,1) and (2,2). In the full-scale analysis of the hierarchical structure based on isogeometric analysis, a control point affects multiple isogeometric units, resulting in coupling control points between hierarchical structures ① and ②. The existence of these coupling control points influences the topology design of the hierarchical structure. Although... Figure 2Taking a finite hierarchical structure as an example, however, the above-mentioned coupling phenomenon exists in hierarchical structures with any number of layers.

[0079] To ensure that the hierarchical structures have the same topological configuration, the density values ​​of each hierarchy must be distributed consistently. Therefore, the corresponding control points of both hierarchical structures should have the same design variables. Thus:

[0080]

[0081] In the formula: The number of hierarchical structures, , This represents the total number of independent control points along the functional gradient change direction and its orthogonal direction in any hierarchical structure. The density value at coupled control points is a non-independent design variable, ensuring the geometric connection between hierarchical structures. Continuity, in which Let be the order of the NURBS basis functions perpendicular to the functional gradient direction.

[0082] Step S102-2: Construct a material interpolation model based on an adaptive increase strategy for sharpness parameters. With the amount of functionally graded materials as the volume constraint and the goal of minimizing dynamic compliance, a transient dynamic topology optimization model for full-scale functionally graded porous hierarchical structures is proposed.

[0083] NURBS function As a basis function, the density of all control points within the hierarchical structure is linearly combined to construct a smooth density distribution function. Then we have:

[0084]

[0085] In the formula: Let the coordinates 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 hierarchical control points in two directions within the plane.

[0086] To obtain clear functionally graded porous structure boundaries, the control point density is pushed towards 0 or 1 using the Heaviside threshold projection function, as shown below:

[0087]

[0088] In the formula: The density distribution function after threshold projection. Let be the sharpness parameter for the k-th optimization iteration step. This is the threshold parameter.

[0089] To ensure that the sharpness parameter grows adaptively according to the optimization process, its increment... Satisfy the following formula:

[0090]

[0091] In the formula: As a growth factor, Let be the dynamic compliance of the k-th optimization iteration step.

[0092] After the (k+1)th iteration, the sharpness parameter is updated as follows:

[0093]

[0094] In the formula: This is the sharpness value in the (k+1)th iteration. This represents the maximum growth rate.

[0095] The threshold projection function combined with the above density distribution Material interpolation schemes for constructing functionally graded porous hierarchical structures: volume interpolation and stiffness interpolation functions.

[0096] Volume interpolation With stiffness interpolation function Introducing the Ersatz parameter To suppress the numerical singularity problem in low-density regions, we have:

[0097]

[0098]

[0099] In the formula: For the order of the polynomial, This is a penalty factor.

[0100] Using the aforementioned volumetric and stiffness interpolation functions, the amount of functionally graded materials used is... (abbreviated as) ( ) as volume constraint, with dynamic compliance (abbreviated as) Let be the objective function, and its corresponding transient dynamic topology optimization model for the full-scale functionally graded porous hierarchical structure is:

[0101]

[0102] In the formula: To design the variable matrix, This represents the total number of geometrically equal units within the hierarchical structure. This represents the volume fraction of the material used.

[0103] S103: Construct a time-dependent sensitivity adjoint equation, solve it to obtain the time series of the dual variables, and then calculate the sensitivity of the dynamic compliance and volume constraint of the functionally graded porous hierarchical structure respectively.

[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 can be calculated using the following time-dependent sensitivity adjoint equation:

[0107]

[0108] Finally, using the chain rule, the volume constraint of functionally graded materials for design variables is... The sensitivity is expressed as:

[0109]

[0110] S104: The optimal topology design of the functionally graded porous hierarchical structure is obtained by updating the Lagrange multipliers and design variables of the volume constraints in parallel using the generalized OC criterion method.

[0111] Based on the Lagrange multiplier method, by simultaneously solving the objective function and the volume constraint function, the following unconstrained Lagrange function is constructed:

[0112]

[0113] In the formula: For Lagrange functions, For Lagrange multipliers, This is a stack variable.

[0114] Based on the generalized OC criterion, the parallel update format for Lagrange multipliers and design variables is as follows:

[0115]

[0116]

[0117] In the formula: To update parameters, , The first Second and third Lagrange multipliers in the next optimization iteration , The first Volume constraints and volume constraint increments in each optimization iteration. For the first The scaling factor for the next optimization iteration is expressed as:

[0118]

[0119] In summary, the method of this invention, under the isogeometric transient dynamics analysis framework based on Bezier extraction operators, constructs functionally graded porous hierarchical constraints and an adaptive increase strategy for sharpness parameters, thereby realizing continuous and controllable full-scale transient dynamics topology optimization of porous structures.

[0120] Secondly, this invention provides a transient dynamic topology optimization system for a full-scale functionally graded porous hierarchical structure, such as... Figure 3 As shown, it includes:

[0121] The initialization module initializes independent design variables within the design domain and pre-sets the property parameters, volume constraint upper limit parameters, Newmark method time integration scheme parameters, material interpolation model parameters, design variable smoothing and projection parameters, and generalized OC criterion optimization algorithm parameters for functionally graded materials within the porous hierarchical structure.

[0122] The preprocessing module uses isogeometric NURBS elements to discretize the design domain of the functional graded porous hierarchical structure, applies boundary conditions and transient dynamic loads; it represents the mass / stiffness matrix of the NURBS element as a linear combination of the mass / stiffness matrix of the standard Bezier element and the Bezier extraction operator, and then constructs an isogeometric transient dynamic finite element model of the functional graded porous hierarchical structure.

[0123] The dynamic analysis module constructs a time integration scheme using the unconditionally stable Newmark method to solve the isogeometric transient dynamic finite element model of the functionally graded porous hierarchical structure, thereby calculating the dynamic compliance and volume constraints of the functionally graded porous hierarchical structure.

[0124] The optimization solution module calculates the sensitivity information of the transient dynamic topology optimization model of the functionally graded porous hierarchical structure, updates the design variables based on the generalized OC criterion optimization algorithm, and determines whether the optimization iteration result meets the convergence criterion. If not, the optimization iteration continues until the convergence criterion is met.

[0125] Example

[0126] Example 1

[0127] This invention addresses the transient dynamic topology optimization problem of a two-dimensional functionally graded torsional circular ring under multi-point loads, further validating the effectiveness of the method.

[0128] Step S101: The mass / stiffness matrix of the NURBS element is expressed as a linear combination of the mass / stiffness matrix of the standard Bezier element and the Bezier extraction operator, so as to efficiently solve the isogeometric transient dynamic finite element equation of the functionally graded porous hierarchical structure under the traditional finite element framework.

[0129] like Figure 4 As shown, a fixed constraint is applied to the inner ring, and a half-wave sinusoidal load with an amplitude of 1 kN is applied to the outer ring. Design domain dimensions: outer ring radius. Inner radius ,thickness Number of subdomains Applying hierarchical constraints and material gradients along the radial direction, the elastic modulus of the inner ring material... mass density , Power Law Index Poisson's ratio is Rayleigh damping parameters: , Optimized operating conditions: Analysis of the elastic modulus ratio between the outer and inner rings. and Topology optimization configuration, volume fraction .

[0130] Based on the isogeometric analysis of the Bezier extraction operator, the globally continuous NURBS basis functions are decomposed into C... 0 Continuous Bezier elements, while maintaining the high accuracy and geometric consistency of isogeometric analysis, enable isogeometric analysis to have an element structure and calculation process that is highly consistent with the classical finite element method at the numerical implementation level.

[0131] Bezier extraction operator It is possible Direction extraction operator and Direction extraction operator Obtained through direct product, we have:

[0132]

[0133] At the unit level, NURBS-shaped function arrays are... Decomposed into a column of Bernstein-shaped functions The linear combination of the Bezier extraction operator gives us:

[0134]

[0135] In the formula: , These are the Bezier extraction operator and weight function on the NURBS cell, respectively.

[0136] Based on the Bezier extraction operator, the stiffness matrix of NURBS solid elements With the mass matrix They are represented as follows:

[0137]

[0138]

[0139] In the formula: , These are the stiffness matrix and mass matrix of a standard Bezier element, respectively. Here is the strain matrix of the Bezier element. The constitutive matrix is , Functionally graded materials along The elastic modulus and mass density that vary in direction according to a power law are expressed as:

[0140]

[0141] In the formula: , For along The direction is related to the elastic modulus and mass density of the upper surface material. , For along The direction is related to the elastic modulus and mass density of the lower surface material. for Thickness in the direction, This is the power-law exponent.

[0142] At the control point ( , Introduce a density value on ) As a design variable, the mass matrix of the NURBS element Stiffness matrix With damping matrix They are respectively:

[0143]

[0144] In the formula: and These are the volume interpolation function and the material interpolation function, respectively. The Gaussian integration point is located at the center of the unit cell. and For Rayleigh damping parameters, , Hierarchical structure and The number of independent control points in a direction.

[0145] The mass, stiffness, and damping matrices of the assembly unit, and the isogeometric transient dynamic finite element equations for the functionally graded porous structure are as follows:

[0146]

[0147]

[0148] In the formula: , and These represent the total mass, stiffness, and mass matrix, respectively. , and For the first Incremental load The acceleration, velocity, and displacement matrix of the control point under action. This represents the total number of response time sampling points during the load duration.

[0149] Using the unconditionally stable Newmark method as the time integration scheme, the isogeometric transient dynamic finite element equations of the functionally graded porous structure are solved, and the equations are transformed into the following residual form. :

[0150]

[0151] In the formula: Let t be the state variable for the t-th increment step. ~ These are the algorithm parameters.

[0152] Initial displacement With speed Given the initial acceleration The residuals at the following initial time can be solved. :

[0153]

[0154]

[0155] S102: Construct a functionally graded porous hierarchical constraint and a material interpolation model based on an adaptive increase strategy for sharpness parameters, establish an optimization solution formula for the dynamic compliance minimization problem, and propose an efficient transient dynamic topology optimization method for full-scale functionally graded porous hierarchical structures.

[0156] In this invention, step S102 may further include:

[0157] Step S102-1: Based on the principle of density consistency of hierarchical substructures with the same configuration and the control point coupling effect between hierarchical substructures under the same geometric analysis framework, construct functionally graded porous hierarchical constraints.

[0158] To ensure that the hierarchical structures have the same topological configuration, the density values ​​of each hierarchy must be distributed consistently. Therefore, the corresponding control points of both hierarchical structures should have the same design variables. Thus:

[0159]

[0160] In the formula: The number of hierarchical structures, , This represents the total number of independent control points along the functional gradient change direction and its orthogonal direction in any hierarchical structure. The density value at coupled control points is a non-independent design variable, ensuring the geometric connection between hierarchical structures. Continuity, in which Let be the order of the NURBS basis functions perpendicular to the functional gradient direction.

[0161] Step S102-2: Construct a material interpolation model based on an adaptive increase strategy for sharpness parameters. With the amount of functionally graded materials as the volume constraint and the goal of minimizing dynamic compliance, a transient dynamic topology optimization model for full-scale functionally graded porous hierarchical structures is proposed.

[0162] NURBS function As a basis function, the density of all control points within the hierarchical structure is linearly combined to construct a smooth density distribution function. Then we have:

[0163]

[0164] In the formula: Let the coordinates 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 hierarchical control points in two directions within the plane.

[0165] To obtain clear functionally graded porous structure boundaries, the control point density is pushed towards 0 or 1 using the Heaviside threshold projection function, as shown below:

[0166]

[0167] In the formula: The density distribution function after threshold projection. Let be the sharpness parameter for the k-th optimization iteration step. This is the threshold parameter.

[0168] To ensure that the sharpness parameter grows adaptively according to the optimization process, its increment... Satisfy the following formula:

[0169]

[0170] In the formula: As a growth factor, Let be the dynamic compliance of the k-th optimization iteration step.

[0171] After the (k+1)th iteration, the sharpness parameter is updated as follows:

[0172]

[0173] In the formula: This is the sharpness value in the (k+1)th iteration. This represents the maximum growth rate.

[0174] The threshold projection function combined with the above density distribution Material interpolation schemes for constructing functionally graded porous hierarchical structures: volume interpolation and stiffness interpolation functions.

[0175] Volume interpolation With stiffness interpolation function Introducing the Ersatz parameter To suppress the numerical singularity problem in low-density regions, we have:

[0176]

[0177]

[0178] In the formula: For the order of the polynomial, This is a penalty factor.

[0179] Using the aforementioned volumetric and stiffness interpolation functions, the amount of functionally graded materials used is... (abbreviated as) ( ) as volume constraints, with structural dynamic flexibility (abbreviated as) Let be the objective function, and its corresponding transient dynamic topology optimization model for the full-scale functionally graded porous hierarchical structure is:

[0180]

[0181] In the formula: To design the variable matrix, This represents the total number of geometrically equal units within the hierarchical structure. This represents the volume fraction of the material used.

[0182] S103: Construct a time-dependent sensitivity adjoint equation, solve it to obtain the time series of the dual variables, and then calculate the sensitivity of the dynamic compliance and volume constraint of the functionally graded porous hierarchical structure respectively.

[0183] Dynamic compliance for design variables The sensitivity, expressed in components, is as follows:

[0184]

[0185] In the formula: To be with residuals The corresponding dual variable can be calculated using the following time-dependent sensitivity adjoint equation:

[0186]

[0187] Finally, using the chain rule, the volume constraint of functionally graded materials for design variables is... The sensitivity is expressed as:

[0188]

[0189] S104: The optimal topology design of the functionally graded porous hierarchical structure is obtained by updating the Lagrange multipliers and design variables of the volume constraints in parallel using the generalized OC criterion method.

[0190] Based on the Lagrange multiplier method, by simultaneously solving the objective function and the volume constraint function, the following unconstrained Lagrange function is constructed:

[0191]

[0192] In the formula: For Lagrange functions, For Lagrange multipliers, This is a stack variable.

[0193] Based on the generalized OC criterion, the parallel update format for Lagrange multipliers and design variables is as follows:

[0194]

[0195]

[0196] In the formula: To update parameters, , The first Second and third Lagrange multipliers in the next optimization iteration , The first Volume constraints and volume constraint increments in each optimization iteration. For the first The scaling factor for the next optimization iteration is expressed as:

[0197]

[0198] The specific optimization results are as follows: Figure 5 As shown, the area highlighted in red represents the hierarchical structure unit.

[0199] Figure 5 This represents the optimal topology for radially functionally graded toroids with different elastic modulus ratios. When the load application time is long ( As the ratio of the elastic modulus of the outer ring to that of the inner ring increases, the material within the layered subdomain gradually deposits towards the softening phase along the radial direction, i.e., the direction of the material gradient change, while forming a sandwich structure in the circumferential direction. When the load application time is relatively short ( The structural inertia effect causes the material within the layered subdomains to gradually aggregate towards the free end, transforming the interior into a spoke-like structure, replacing the previous structure. Gradient porous structure under working conditions.

[0200] The results of Embodiment 1 of this invention demonstrate that applying functionally graded materials along the hierarchical constraint direction can ensure the synergistic optimization of structural topology and material properties, maximizing the transient dynamic performance of the structure. Due to the coupling effect between hierarchical structures, the functionally graded porous hierarchical structure effectively ensures sufficiently smooth geometric connectivity between layers.

[0201] Example 2

[0202] Embodiment 2 of the present invention solves the transient dynamic topology optimization problem of the three-dimensional functionally graded cantilever beam problem, further verifying the effectiveness of the method of the present invention.

[0203] Step S101: Represent the mass / stiffness matrix of the NURBS element as a linear combination of the mass / stiffness matrix of the standard Bezier element and the Bezier extraction operator, and efficiently solve the isogeometric transient dynamic finite element equation of the functionally graded porous hierarchical structure under the traditional finite element framework.

[0204] like Figure 6 As shown, a fixed constraint is applied to the left end of the three-dimensional cantilever beam, and a half-wave sinusoidal load with an amplitude of 10 kN is applied to the free edge of the right end. Design domain dimensions: length ,high ,width Softening phase elastic modulus mass density Enhance the phase elastic modulus Poisson's ratio , Power Law Index Rayleigh damping parameters: , Optimized operating condition: Comparing the optimization results of an equivalent homogeneous cantilever beam and a function-graded cantilever beam, the number of subdomains is... Hierarchical constraints along the height direction of the cantilever beam, volume fraction .

[0205] Step S102 is the same as in Example 1.

[0206] Step S103 is the same as in Example 1.

[0207] Step S104 is the same as in Example 1.

[0208] The specific optimization results are as follows: Figure 7 , 8 As shown, the area highlighted in red represents the hierarchical structure unit.

[0209] Figure 7 This is a schematic diagram of the optimal topology of the three-dimensional equivalent homogeneous cantilever beam in Embodiment 2 of the present invention. The vertical direction is the spatial gradient hierarchy constraint direction. Since the tensile deformation and compressive deformation of the longitudinal fibers of the cantilever beam are symmetrical about the neutral axis, the hierarchical substructure is symmetrical from top to bottom, and the cross-sectional profile is a symmetrical I-shape. As the load application time decreases, the mass is concentrated on the upper and lower free surfaces to resist the influence of transient effects. Figure 8 This diagram illustrates the optimal topology of the three-dimensional functionally graded cantilever beam in Embodiment 2 of the present invention. Unlike the equivalent homogeneous case, the optimal topology of the three-dimensional functionally graded cantilever beam consumes more softening phase material, the strengthening material gradient decreases along the beam height direction, and the cross-sectional profile is an asymmetrical I-shape. In particular, compared with the optimal dynamic compliance, the functionally graded cantilever beam is significantly lower than the corresponding equivalent homogeneous structure, meaning the former has superior dynamic stiffness performance.

[0210] The results of Embodiment 2 of the present invention show that the algorithm of the present invention can be extended from two-dimensional functionally graded porous hierarchical structures to the transient dynamics optimization problem of three-dimensional functionally graded porous hierarchical structures, and is suitable for the engineering requirements of vibration-resistant and lightweight design of engineering equipment under strong vibration environment.

[0211] 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 transient dynamic topology optimization method for full-scale functionally graded porous hierarchical structures, characterized in that, Includes the following steps: S101: The mass / stiffness matrix of the NURBS element is expressed as a linear combination of the mass / stiffness matrix of the standard Bezier element and the Bezier extraction operator, which enables efficient solution of the isogeometric transient dynamic finite element equations of functionally graded porous hierarchical structures within the finite element framework. S102: Construct a functionally graded porous hierarchical constraint and a material interpolation model based on an adaptive increase strategy of sharpness parameter, establish an optimization solution formula for the dynamic compliance minimization problem, and propose an efficient transient dynamic topology optimization method for full-scale functionally graded porous hierarchical structures; S103: Construct a time-dependent sensitivity adjoint equation, solve it to obtain the time series of the dual variables, and then calculate the sensitivity of the dynamic compliance and volume constraint of the functionally graded porous hierarchical structure respectively. S104: The optimal topology design of the functionally graded porous hierarchical structure is obtained by updating the Lagrange multipliers and design variables of the volume constraints in parallel using the generalized OC criterion method.

2. The transient dynamic topology optimization method for a full-scale functionally graded porous hierarchical structure according to claim 1, characterized in that, Step S101 includes: Based on the isogeometric analysis of the Bezier extraction operator, the globally continuous NURBS basis functions are decomposed into C... 0 Continuous Bezier elements, while maintaining the high accuracy and geometric consistency of isogeometric analysis, enable isogeometric analysis to have a unit structure and calculation process that is highly consistent with the classical finite element method at the numerical implementation level. Bezier extraction operator It is possible Direction extraction operator and Direction extraction operator Obtained through direct product, we have: At the unit level, NURBS-shaped function arrays are... Decomposed into a column of Bernstein-shaped functions The linear combination of the Bezier extraction operator gives us: In the formula: , These are the Bezier extraction operator and weight function on the NURBS cell, respectively. Based on the Bezier extraction operator, the stiffness matrix of NURBS solid elements With the mass matrix They are represented as follows: In the formula: , These are the stiffness matrix and mass matrix of a standard Bezier element, respectively. Here is the strain matrix of the Bezier element. The constitutive matrix is , Functionally graded materials along The elastic modulus and mass density that vary in direction according to a power law are expressed as: In the formula: , For along The direction is related to the elastic modulus and mass density of the upper surface material. , For along The direction is related to the elastic modulus and mass density of the lower surface material. for Thickness in the direction, The power-law exponent; At the control point ( , Introduce a density value on ) As a design variable, the mass matrix of the NURBS element Stiffness matrix With damping matrix They are respectively: In the formula: and These are the volume interpolation function and the material interpolation function, respectively. The Gaussian integration point is located at the center of the unit cell. and For Rayleigh damping parameters, , Hierarchical structure and The number of independent control points in the direction; The mass, stiffness, and damping matrices of the assembly unit, and the isogeometric transient dynamic finite element equations for the functionally graded porous structure are as follows: In the formula: , and These represent the total mass, stiffness, and mass matrix, respectively. , and For the first Incremental load The acceleration, velocity, and displacement matrix of the control point under action. This represents the total number of time sampling points during the load duration. Using the unconditionally stable Newmark method as the time integration scheme, the isogeometric transient dynamic finite element equations of the functionally graded porous structure are solved, and the equations are transformed into the following residual form. : In the formula: Let t be the state variable for the t-th increment step. ~ These are algorithm parameters; Initial displacement With speed Given the initial acceleration The residuals at the following initial time can be solved. : 。 3. The transient dynamic topology optimization method for a full-scale functionally graded porous hierarchical structure according to claim 2, characterized in that, Step S102 includes: Step S102-1: Based on the principle of density consistency of hierarchical substructures with the same configuration and the control point coupling effect between hierarchical substructures under the same geometric analysis framework, construct functionally graded porous hierarchical constraints. Step S102-2: Construct a material interpolation model based on an adaptive increase strategy for sharpness parameters. With the amount of functionally graded materials as the volume constraint and the goal of minimizing dynamic compliance, construct a transient dynamic topology optimization model for a full-scale functionally graded porous hierarchical structure.

4. The transient dynamic topology optimization method for a full-scale functionally graded porous hierarchical structure according to claim 3, characterized in that, Step S102-1 includes: To ensure that the hierarchical structures have the same topological configuration, the density distribution of each hierarchy must be consistent. Therefore, the corresponding control points of both should have the same design variables, which leads to: In the formula: The number of hierarchical structures, , This represents the total number of independent control points along the functional gradient change direction and in its orthogonal direction in any hierarchical structure; the density value at coupled control points is a non-independent design variable, ensuring the geometric connection between hierarchical structures has... Continuity, in which Let be the order of the NURBS basis functions perpendicular to the functional gradient direction.

5. A transient dynamic topology optimization method for a full-scale functionally graded porous hierarchical structure according to claim 3 or 4, characterized in that, Step S102-2 includes: NURBS function As a basis function, the density of all control points within the hierarchical structure is linearly combined to construct a smooth density distribution function. Then we have: In the formula: Let the coordinates 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 hierarchical control points in two directions within the plane. To obtain clear functionally graded porous structure 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. Let be the sharpness parameter for the k-th optimization iteration step. For threshold parameters; To ensure that the sharpness parameter grows adaptively according to the optimization process, its increment... Satisfy the following formula: In the formula: As a growth factor, The dynamic compliance of the k-th optimization iteration step; After the (k+1)th iteration, the sharpness parameter is updated as follows: In the formula: This is the sharpness value in the (k+1)th iteration. The maximum growth rate; The threshold projection function combined with the above density distribution Material interpolation schemes for constructing functionally graded porous hierarchical 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: For the order of the polynomial, As a penalty factor; Using the aforementioned volumetric and stiffness interpolation functions, the amount of functionally graded materials used is... (abbreviated as) ( ) as volume constraint, with dynamic compliance abbreviated as Let be the objective function, and its corresponding transient dynamic topology optimization model for the full-scale functionally graded porous hierarchical structure is: In the formula: To design the variable matrix, This represents the total number of geometrically equal units within the hierarchical structure. This represents the volume fraction of the material used.

6. The transient dynamic topology optimization method for a full-scale functionally graded porous hierarchical structure according to claim 1, characterized in that, Step S103 includes: Dynamic compliance for design variables The sensitivity, expressed in components, is as follows: In the formula: To be with residuals The corresponding dual variable can be calculated using the following time-dependent sensitivity adjoint equation: Finally, using the chain rule, the volume constraint of functionally graded materials for design variables is... The sensitivity is expressed as: 。 7. The transient dynamic topology optimization method for a full-scale functionally graded porous hierarchical structure according to claim 1, characterized in that, Step S104 includes: Based on the Lagrange multiplier method, by simultaneously solving the objective function and the volume constraint function, the following unconstrained Lagrange function is constructed: In the formula: For Lagrange functions, For Lagrange multipliers, For stack variables; Based on the generalized OC criterion, the parallel update format for Lagrange multipliers and design variables is as follows: In the formula: To update parameters, , The first Second and third Lagrange multipliers in the next optimization iteration , The first Volume constraints and volume constraint increments in each optimization iteration. For the first The scaling factor for the next optimization iteration is expressed as: Within the framework of isogeometric transient dynamics analysis based on Bezier extraction operators, a functionally graded porous hierarchical constraint and an adaptive increase strategy for sharpness parameters are constructed to achieve continuous and controllable full-scale transient dynamics topology optimization of porous structures.

8. A transient dynamic topology optimization system for a functionally graded porous hierarchical structure with full-scale work, characterized in that, The system is implemented based on the method of any one of claims 1 to 7, and includes the following program modules: The initialization module initializes independent design variables within the design domain and pre-sets the property parameters, volume constraint upper limit parameters, Newmark method time integration scheme parameters, material interpolation model parameters, design variable smoothing and projection parameters, and generalized OC criterion optimization algorithm parameters for functionally graded materials within the porous hierarchical structure. The preprocessing module uses isogeometric NURBS elements to discretize the design domain of the functional graded porous hierarchical structure, applies boundary conditions and transient dynamic loads; it represents the mass / stiffness matrix of the NURBS element as a linear combination of the mass / stiffness matrix of the standard Bezier element and the Bezier extraction operator, and then constructs an isogeometric transient dynamic finite element model of the functional graded porous hierarchical structure. The dynamic analysis module constructs a time integration scheme using the unconditionally stable Newmark method to solve the isogeometric transient dynamic finite element model of the functionally graded porous hierarchical structure, thereby calculating the dynamic compliance and volume constraints of the functionally graded porous hierarchical structure. The optimization solution module calculates the sensitivity information of the transient dynamic topology optimization model of the functionally graded porous hierarchical structure, updates the design variables based on the generalized OC criterion optimization algorithm, and determines whether the optimization iteration result meets the convergence criterion. If not, the optimization iteration continues until the convergence criterion is met.

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