A functionally graded porous structure optimization method and system based on two-phase implicit field interpolation
By optimizing functionally graded porous structures through biphasic implicit field interpolation and adaptive multimodal cooperative strategies, the problems of stiffness-mass competition and spurious modes in dynamic environments are solved, achieving fundamental frequency maximization and stable convergence of porous structures, which are suitable for aerospace and precision equipment.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- HUAZHONG UNIV OF SCI & TECH
- Filing Date
- 2026-02-14
- Publication Date
- 2026-06-02
AI Technical Summary
Existing functionally graded porous structure optimization designs face problems such as non-monotonic competition between stiffness and mass, spurious modal disturbances, and modal exchange oscillations in dynamic environments, resulting in highly non-convex optimization problems that are difficult to converge.
A method based on biphase implicit field interpolation is adopted to optimize the fundamental frequency of functionally graded porous structures by constructing a unified biphase implicit geometric description function and a dual nonlinear penalty model, combined with an adaptive multimodal cooperative strategy and an asymmetric hybrid update strategy.
The fundamental frequency of the porous structure was maximized, spurious mode interference was eliminated, the numerical stability and efficient convergence of the structure were ensured, and a gradient distribution of "dense at the root and sparse at the tip" was obtained, which is suitable for aerospace and precision equipment fields.
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Figure CN122133389A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of structural dynamics topology optimization, and more specifically, relates to a functionally gradient porous structure optimization method and system based on biphasic implicit field interpolation. Background Technology
[0002] Functionally graded porous structures (FJPS) have significant application value in aerospace, precision equipment, and other fields due to their ability to achieve lightweight construction and customized mechanical property distribution. Existing FJPS optimization designs primarily focus on static load-bearing capacity (such as maximizing stiffness), but in practical engineering, structures often face complex dynamic environments. Increasing the structure's first natural frequency (fundamental frequency) is crucial for preventing resonance and improving dynamic stiffness.
[0003] Traditional topology optimization methods based on static stiffness (such as the level set method and the SIMP method) face significant challenges when applied to dynamic frequency optimization: 1. The competition mechanism between stiffness and mass: In dynamics, adding material increases stiffness but also introduces additional mass. This non-monotonic competition relationship makes the optimization problem highly non-convex and prone to getting trapped in local optima.
[0004] 2. Spurious mode interference: In low-density porous regions, stiffness decays faster than mass decays, which easily generates non-physical local low-frequency vibrations (spurious modes), leading to instability in numerical calculations.
[0005] 3. Modal exchange oscillation: During the optimization process, the fundamental frequency mode and higher-order modes may have similar or exchanged frequencies, causing the objective function to become non-differentiable and leading to oscillations and non-convergence of the algorithm.
[0006] While existing hybrid level set methods have solved the microstructure connectivity problem, their evolution mechanism based on the Hamilton-Jacobi equation is inadequate in dealing with the aforementioned dynamic-specific challenges and lacks effective update strategies for positive and negative sensitivity regions. Summary of the Invention
[0007] In view of the above-mentioned defects or improvement needs of the existing technology, the present invention provides a functionally graded porous structure optimization method and system based on biphase implicit field interpolation, the purpose of which is to achieve efficient and effective optimization of the fundamental frequency of functionally graded porous structures.
[0008] To achieve the above objectives, according to a first aspect of the present invention, a method for optimizing functionally graded porous structures based on biphasic implicit field interpolation is proposed, comprising the following steps: S1. Construct a parameterized physical geometry model, including: defining interpolation field variables as global design variables on the discretized finite element mesh nodes, and constructing a unified biphase implicit geometric description function for the entire field through spatial linear weighted interpolation; and then establishing a physical field mapping model that includes both stiffness and mass nonlinear penalties. S2. Establish a dynamic topology optimization mathematical model with the objective function of maximizing the structural fundamental frequency and the structural volume fraction as the constraint condition; S3. Perform multimodal finite element eigenvalue analysis on the functionally graded porous structure corresponding to the current global design variables to obtain the first N generalized eigenvalues of the structure. S4. Based on the parametric physical geometry model, calculate the sensitivity of the objective function with respect to the global design variables according to the first N generalized eigenvalues of the structure; S5. Update global design variables based on sensitivity; S6. Repeat steps S3 to S5 until the convergence condition is met to obtain the optimal functionally graded porous structure.
[0009] As a further preferred option, the interpolation field variable It is used to control the topology and porosity of local microstructures.
[0010] As a further preferred embodiment, the biphasic implicit geometric description function It is expressed as follows:
[0011] In the formula, For design domain The spatial physical coordinates within; The shape function of the finite element mesh. The total number of nodes. For the first e Interpolation field variables for each node; For sparse phase basis functions, These are basis functions for the compact phase.
[0012] As a further preferred embodiment, the physical field mapping model that includes both stiffness and mass nonlinear penalties is represented as follows:
[0013]
[0014]
[0015] In the formula, For physical density field, tanh is a parameter used to control the steepness of the projection; For elastic modulus, Mass density; For the elastic modulus and mass density of solid materials, For porous phase materials, the elastic modulus and mass density are given. This is the stiffness penalty factor. This is a quality penalty factor.
[0016] As a further preferred option, the quality penalty factor Not less than the stiffness penalty factor .
[0017] As a further preferred embodiment, step S3 employs an adaptive multimodal collaborative strategy to dynamically adjust the objective function weights based on the eigenvalue spacing, including: Calculate the relative spacing between the first and second order eigenvalues; when the relative spacing is less than the preset spacing threshold, it is determined to be a modal proximity state, and bimodal averaging optimization is adopted, and the weights of the first and second order eigenvalues in the objective function are set accordingly; otherwise, it is determined to be a single-mode dominant state, and only the fundamental frequency is optimized, and the weights of the first and second order eigenvalues in the objective function are set accordingly.
[0018] As a further preferred option, the sensitivity calculation formula for the objective function with respect to the global design variables is as follows:
[0019] In the formula, Let be the objective function. For the first e Interpolation field variables for each node; For the first eigenvalues of order 1 For the first Weight coefficients of the first-order eigenvalues The corresponding feature vector; , These are the global stiffness matrix and the mass matrix, respectively.
[0020] As a further preferred step, step S5 involves updating the global design variables using an asymmetric hybrid update strategy: For elements with sensitivity less than 0, the global design variables are updated using the optimization criterion method to achieve compaction; for elements with sensitivity greater than or equal to 0, the global design variables are updated using the forced incremental step method to achieve sparsity.
[0021] According to a second aspect of the present invention, a functionally graded porous structure optimization system based on biphasic implicit field interpolation is provided, comprising a processor for executing the above-described functionally graded porous structure optimization method based on biphasic implicit field interpolation.
[0022] According to a third aspect of the present invention, a computer-readable storage medium is provided having a computer program stored thereon, which, when executed by a processor, implements the above-described functional gradient porous structure optimization method based on biphasic implicit field interpolation.
[0023] In summary, compared with the prior art, the above-described technical solutions conceived by this invention mainly possess the following technical advantages: 1. The biphase implicit field interpolation model (DP-IFI) proposed in this invention uses spatial linear weighted interpolation to construct a unified biphase implicit geometric description function for the entire field, abandoning the evolution of complex partial differential equations. Complex gradient microstructure design can be realized through simple parameterized interpolation, which is convenient to combine with various optimization algorithms.
[0024] 2. The dual-phase implicit geometric description function also realizes the smooth parameterized fusion of microstructure from sparse phase to dense phase, thereby solving the problem of multi-scale structural connectivity at the geometric level and ensuring that the skeleton between microstructures with different porosities is naturally connected without geometric breakage. This full-field continuous functional gradient distribution not only eliminates interface stress concentration, but also makes the structural boundary clear and smooth, and can be directly applied to additive manufacturing without complex smoothing post-processing.
[0025] 3. This invention constructs a dual nonlinear penalty model, which solves the spurious mode problem that has long existed in dynamic topology optimization from a physical perspective, and ensures the numerical stability of the low-density region.
[0026] 4. The adaptive multimodal collaborative and asymmetric hybrid update strategy proposed in this invention effectively overcomes the problems of mode exchange oscillation and local extrema in frequency optimization. It can automatically evolve into an efficient dynamic configuration with "dense roots and light ends", with fast convergence speed and strong robustness. Attached Figure Description
[0027] Figure 1 This is a flowchart of a method for optimizing the fundamental frequency of a functionally graded porous structure based on dual-phase implicit field interpolation and multimodal synergy, provided by an embodiment of the present invention. Figure 2 This is a schematic diagram of the geometric evolution of the biphase implicit field interpolation model according to an embodiment of the present invention; Figure 3 This is a schematic diagram illustrating the definition of two-dimensional design domain discrete mesh and structural topology optimization design variables in an embodiment of the present invention; Figure 4 This is a schematic diagram of a two-dimensional cantilever beam structure in an embodiment of the present invention; Figure 5 These are schematic diagrams of two initial unit cell structures defined in the embodiments of the present invention; Figure 6 This is a schematic diagram of the structural optimization results in an embodiment of the present invention. Detailed Implementation
[0028] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.
[0029] This invention provides a method for optimizing functionally graded porous structures based on biphasic implicit field interpolation, such as... Figure 1 As shown, it includes the following steps: Step S1: Construct a parametric physical geometry model Based on microstructure unit cell configurations with two different volume fractions (sparse and dense phases), an implicit level set function is used for geometric description to construct a two-phase implicit field interpolation geometric model. In the discretized finite element mesh of the design domain, interpolation field variables are defined at each mesh node. As a global design variable, this variable is used to control the topology and porosity of local microstructures. Its physical meaning is: when At that time, the local microstructure tends to conform to a predefined sparse phase unit cell. ;when At that time, the local microstructure tends to form a predefined dense phase unit cell. .
[0030] A unified biphasic implicit geometric description function is constructed across the entire field using spatial linear weighted interpolation. Achieving smooth, parameterized fusion from sparse to dense phases:
[0031] In the formula, For design domain The spatial physical coordinates within; The shape function of the finite element mesh. The total number of nodes. For the first e Interpolation field variables for each node; The area represents physical materials. The region represents a void. This formula shows that any spatial location Microstructure geometry at the location It is formed by mixing two basis function fields, sparse phase and dense phase, according to local weights.
[0032] Subsequently, a dual nonlinear penalty physical field mapping model is established. First, the biphase implicit geometric description is represented by a hyperbolic tangent projection function. Mapped to physical density field :
[0033] In the formula, Parameters used to control the steepness of the projection.
[0034] Next, an asynchronous material property interpolation model is constructed:
[0035]
[0036] In the formula, For elastic modulus, Mass density; For physical material properties, Properties of porous phase materials; This is the stiffness penalty factor. This invention sets a quality penalty factor. With stiffness penalty factor Satisfying asynchronous penalty constraints This ensures that in low-density regions, the decay rate of mass properties is not lower than that of stiffness properties, thereby preventing the local Rayleigh quotient from approaching zero and ensuring that the stiffness-mass ratio of the material does not approach zero in low-density regions, thus eliminating spurious local modes in low-density regions.
[0037] Step S2: Establish a dynamic topology optimization mathematical model A dynamic topology optimization mathematical model is established with the maximization of the first natural frequency (fundamental frequency) of the structure as the objective function and the volume fraction of the porous structure as the constraint condition. The model uses the full-field interpolation field variables as design variables and aims to find the optimal material distribution to improve the overall dynamic stiffness of the structure.
[0038] The specific mathematical model for dynamic topology optimization is as follows:
[0039] In the formula, Design a global variable vector; The objective function is the cooperative dynamics. For the first Weighting coefficients of order eigenvalues; For the first eigenvalues of order 1; For the corresponding eigenvectors (mode shapes); and These are the global stiffness matrix and the mass matrix, respectively. To optimize the volume, For the total volume of the design domain, This represents the upper limit of the target volume fraction.
[0040] Step S3: Perform multimodal analysis and collaborative strategy Multimodal finite element eigenvalue analysis was performed on the functionally graded porous structure to obtain the first N order of the structure ( The generalized eigenvalues and their corresponding eigenvectors are used. The objective function weights are dynamically adjusted based on the eigenvalue intervals, and an adaptive multimodal collaborative strategy is implemented, as follows: Calculate the first-order eigenvalues With second-order eigenvalues relative spacing Preset spacing threshold ; When the relative spacing is less than the threshold When the fundamental frequency and the second-order frequency are too close, mode switching can easily occur during iteration, leading to algorithm oscillations. In this case, the judgment structure is in a "modal proximity state," and bimodal averaging optimization is enabled. Specifically, the first-order weights are forcibly set. With second-order weights By taking the arithmetic mean of the first two eigenvalues, a smooth cooperative objective function is constructed, which guides the optimization algorithm to simultaneously improve the stiffness of the first two modes, thereby passively widening the frequency spacing and suppressing numerical oscillations.
[0041] When the relative spacing is greater than or equal to the threshold When this condition is met, it indicates that the fundamental frequency mode is clear and independent, with no risk of mode aliasing. At this point, the structure is determined to be in a "single-mode dominant state," and optimization is performed only for the fundamental frequency. Specifically, the first-order weights are set. Second-order weights At this point, all sensitivity resources are concentrated to maximize the first-order natural frequency in order to obtain optimal dynamic performance and improve efficiency.
[0042] Step S4: Calculate the cooperative dynamic sensitivity Based on eigenvalue sensitivity theory, the sensitivity formula of the cooperative dynamic objective function with respect to design variables is derived, and the strain energy and kinetic energy density of the element are calculated. According to the Rayleigh quotient property and eigenvalue sensitivity theory, the... The eigenvalues of order 2 with respect to the 1st order 3 Design variables for each unit The derivative is:
[0043] Using the chain rule and combining it with the double penalty model, it can be expanded as follows:
[0044] In the formula, the first term Related to element strain energy density (positive contribution), the second term It is related to the unit kinetic energy density (negative contribution). This reveals the competitive mechanism between stiffness enhancement and mass increase in dynamic optimization.
[0045] The sensitivity expression of the objective function with respect to the design variables is as follows:
[0046] This formula includes a stiffness term with positive contribution and a mass term with negative contribution. Through the difference in derivatives of the double penalty factors, it enhances the driving force for mass removal in low-density regions and accurately guides the evolution direction of material distribution.
[0047] Step S5: Global Design Variable Update Based on sensitivity information, an asymmetric hybrid update strategy is used to update global design variables.
[0048] Regarding sensitivity The elements (stiffness-dominant region) are updated using an optimization criterion (OC) formula based on Lagrange multipliers, driving the process. Reduce size to achieve material densification:
[0049] Regarding sensitivity The cells (quality-dominant regions) are updated using a forced incremental step method, driving the process. Increase the size to achieve material sparsity:
[0050] In the formula, The Lagrange multipliers obtained by the bisection method are: For damping operators, This is to force the incremental step size.
[0051] Step S6: Repeat steps S3 to S5 until the convergence condition is met, and output the optimal functionally graded porous structure; the convergence condition is:
[0052] In the formula, For the first The objective function value of each iteration. To reduce convergence tolerance, This represents the maximum number of iterations. If convergence is achieved, the optimal functionally graded porous structure geometric model is output.
[0053] The following are specific examples: The structure of this embodiment is as follows: Figure 4 As shown, the structure is a two-dimensional cantilever beam with length L and height H, where L = 2H. All degrees of freedom on the left boundary of the design domain are constrained, while the right end is free. The design domain is discretized as follows: A four-node planar quadrilateral element.
[0054] The design is performed using the optimization method based on biphasic implicit field interpolation and multimodal collaboration provided by this invention: Parameter settings: Select as follows Figure 5 The two microstructure unit cells shown are used as basis functions (sparse phase unit cell and dense phase unit cell); the target volume fraction constraint is set to 50%; and a stiffness penalty factor is set. With quality penalty factor To eliminate spurious modes in low-density regions; a multimodal cooperative spacing threshold is set. This is to prevent mode exchange oscillations during the iteration process.
[0055] Optimization process: Taking the maximization of the first-order natural frequency (fundamental frequency) of the structure as the objective function, the full-field interpolation variables are updated using an asymmetric hybrid update strategy. .
[0056] Optimization results: After approximately 120 iterations and convergence, the following results were obtained: Figure 6 The optimization results are shown.
[0057] The optimized results demonstrate that the method used in this embodiment has the following significant effects: Significant functional gradient characteristics: The optimized structure exhibits a gradient distribution feature of "dense at the root and sparse at the tip". This distribution conforms to the physical laws of cantilever beam dynamic design: the root enhances stiffness to resist bending moment, while the tip removes mass to reduce inertial forces.
[0058] The microstructure exhibits excellent connectivity: thanks to the two-phase implicit field interpolation model, the internal structure achieves a smooth transition from the dense phase to the sparse phase, without any obvious geometric fractures or scale separation.
[0059] No spurious modal interference: The results show that the structure exhibits clear global bending modes and no local spurious vibrations caused by numerical calculation errors, which verifies the effectiveness of the dual-penalty model.
[0060] Fundamental frequency maximization is achieved: the fundamental frequency value of the structure steadily increases during the optimization process, and the final convergence result is significantly improved compared with the initial configuration, which verifies the efficiency and solution stability of the present invention in dealing with the problem of maximizing dynamic stiffness.
[0061] In summary, this invention uses a two-phase implicit field interpolation model for geometric description based on two unit cell configurations: sparse phase and dense phase, and constructs a unified parameterized geometric description function across the entire field. It establishes a physical field mapping model that includes dual nonlinear penalties for stiffness and mass, with the goal of maximizing the fundamental frequency. By introducing an adaptive multimodal cooperative strategy and an asymmetric hybrid update strategy, it solves for the dynamically optimal porous structure in which geometry and function are distributed in a gradient in space.
[0062] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
Claims
1. A method for optimizing functionally graded porous structures based on biphasic implicit field interpolation, characterized in that, Includes the following steps: S1. Construct a parameterized physical geometry model, including: defining interpolation field variables as global design variables on the discretized finite element mesh nodes, and constructing a unified biphase implicit geometric description function for the entire field through spatial linear weighted interpolation; and then establishing a physical field mapping model that includes both stiffness and mass nonlinear penalties. S2. Establish a dynamic topology optimization mathematical model with the objective function of maximizing the structural fundamental frequency and the structural volume fraction as the constraint condition; S3. Perform multimodal finite element eigenvalue analysis on the functionally graded porous structure corresponding to the current global design variables to obtain the first N generalized eigenvalues of the structure. S4. Based on the parametric physical geometry model, calculate the sensitivity of the objective function with respect to the global design variables according to the first N generalized eigenvalues of the structure; S5. Update global design variables based on sensitivity; S6. Repeat steps S3 to S5 until the convergence condition is met to obtain the optimal functionally graded porous structure.
2. The functionally graded porous structure optimization method based on biphasic implicit field interpolation as described in claim 1, characterized in that, The interpolation field variable It is used to control the topology and porosity of local microstructures.
3. The functionally graded porous structure optimization method based on biphasic implicit field interpolation as described in claim 2, characterized in that, The biphasic implicit geometric description function It is expressed as follows: In the formula, For design domain The spatial physical coordinates within; The shape function of the finite element mesh. The total number of nodes. For the first e Interpolation field variables for each node; For sparse phase basis functions, These are basis functions for the compact phase.
4. The functionally graded porous structure optimization method based on biphasic implicit field interpolation as described in claim 3, characterized in that, The physical field mapping model, which includes dual nonlinear penalties for stiffness and mass, is represented as follows: In the formula, For physical density field, tanh is a parameter used to control the steepness of the projection; For elastic modulus, Mass density; For the elastic modulus and mass density of solid materials, For porous phase materials, the elastic modulus and mass density are given. This is the stiffness penalty factor. This is a quality penalty factor.
5. The functionally graded porous structure optimization method based on biphasic implicit field interpolation as described in claim 4, characterized in that, Quality penalty factor Not less than the stiffness penalty factor .
6. The functionally graded porous structure optimization method based on biphasic implicit field interpolation as described in claim 1, characterized in that, Step S3 involves employing an adaptive multimodal collaborative strategy to dynamically adjust the objective function weights based on the eigenvalue spacing, including: Calculate the relative spacing between the first and second order eigenvalues; when the relative spacing is less than the preset spacing threshold, it is determined to be a modal proximity state, and bimodal averaging optimization is adopted, and the weights of the first and second order eigenvalues in the objective function are set accordingly; otherwise, it is determined to be a single-mode dominant state, and only the fundamental frequency is optimized, and the weights of the first and second order eigenvalues in the objective function are set accordingly.
7. The functionally graded porous structure optimization method based on biphasic implicit field interpolation as described in claim 1, characterized in that, The sensitivity calculation formula for the objective function with respect to the global design variables is as follows: In the formula, Let be the objective function. For the first e Interpolation field variables for each node; For the first eigenvalues of order 1 For the first Weight coefficients of the first-order eigenvalues The corresponding feature vector; , These are the global stiffness matrix and the mass matrix, respectively.
8. The method for optimizing functionally graded porous structures based on biphasic implicit field interpolation as described in any one of claims 1-7, characterized in that, Step S5: Update the global design variables using an asymmetric hybrid update strategy: For elements with sensitivity less than 0, the global design variables are updated using the optimization criterion method to achieve compaction; for elements with sensitivity greater than or equal to 0, the global design variables are updated using the forced incremental step method to achieve sparsity.
9. A functionally graded porous structure optimization system based on two-phase implicit field interpolation, characterized in that, Includes a processor for executing the functional gradient porous structure optimization method based on biphasic implicit field interpolation as described in any one of claims 1-8.
10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the functional gradient porous structure optimization method based on biphasic implicit field interpolation as described in any one of claims 1-8.