Integrated design method for continuum structure and supporting assembly thereof

Abstract

The invention relates to an integrated design method for a continuum structure and a supporting assembly thereof, and the method comprises the following steps: (1) defining an explicit-implicit hybrid topology description function which is used for describing the updating and iteration of the geometric topology of the continuum structure and the supporting assembly; (2) constructing a weighted interpolation function based on the explicit topology description function, wherein the weighted interpolation function is used for applying a Dirichlet boundary condition; (3) deducing a corrected strain matrix and a stiffness matrix, and establishing a finite element analysis model; (4) establishing an integrated design model of the continuum structure and the supporting assembly thereof; (5) deriving and calculating the sensitivity of the topological optimization model; and (6) updating and iterating explicit and implicit design variables. By means of the method, customizable explicit description of the supporting assembly is achieved, frequent grid updating and introduction of a large number of grid related design variables are avoided, integrated design of a continuum structure and the supporting assembly is achieved, and the mechanical performance of a compliant mechanism and a general structure is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of mechanical and structural design, and more specifically, relates to an integrated design method for a continuum structure and its supporting components. Background Art

[0002] Topology optimization is a numerical method used to find the optimal configuration of a structure within a finite design domain. It plays a crucial role in multidisciplinary design problems and has become an important design approach for improving structural performance and reducing costs. In most cases, topology optimization of continuum structures uses the material distribution within a given design domain as a design variable. With the increasing demands of structural design, the interaction between continuum structures, their materials, and boundary conditions has begun to attract attention. Therefore, integrated design methods for continuum structures and these elements have become a research hotspot.

[0003] Support assemblies or Dirichlet boundaries play a decisive role in load transfer paths, and their influence on structural performance has long been recognized. For example, the stiffness and natural frequency of some plates or beams are significantly dependent on the orientation of the supports; the moment arms of compliant mechanisms are largely determined by the supports, thus affecting the output displacement or displacement amplification ratio.

[0004] Conventional topology optimization programs typically require that rigid supports be pre-defined and maintained throughout the iterations. This makes practical applications reliant on the designer's experience, making it difficult to optimally position supports. Therefore, how to adjust the position or orientation of supports while optimizing the structural topology becomes a crucial issue.

[0005] The key issue in the integrated design of continuum structures and supports is how to model the changing support components. Existing methods can be divided into two categories. The first is the background spring method, which was first proposed in the work of Buhl. This method adds a series of background springs consistent with the number and direction of degrees of freedom at the nodes of all finite element meshes to represent potential support components. This method uses the stiffness of these springs as design variables and penalizes the intermediate stiffness. It can be regarded as an extension of the solid isotropic material penalty method (SIMP). It has strong flexibility and can characterize complex support forms. It has been successfully applied to problems such as non-uniform deformation, maximum fundamental frequency, and compliant mechanisms. However, this method introduces a large number of design variables related to the node degrees of freedom, which increases the amount of calculation. At the same time, there is a grayscale transition area on the boundary of the obtained support component, and its boundary is not smooth, requiring secondary design.

[0006] In order to obtain smooth support component boundaries, some scholars have proposed a method to parameterize the support component boundaries using computational geometry techniques. Zhu and Zhang used movable components to characterize rigid supports and obtained sensitivity information through the difference method; Xia et al. used multiple level set functions to characterize free boundaries and supports respectively, and used Lagrangian functions and continuous adjoint methods to obtain the shape functions of the two boundaries. However, these methods require frequent body-fitting meshing of the support boundaries during iteration. To avoid this process, Zhang et al. used weighted B-spline functions defined by level set functions to penalize the displacement field to satisfy the Dirichlet boundary conditions, and obtained sensitivity information through the discrete adjoint method; Dapogny et al. proposed an approximate method in which the "exact" Laplace equation with mixed boundary conditions was replaced by a "smooth" version, so that it can be described as a Robin boundary condition distributed over a narrow band.

[0007] However, in practical applications, fixed-shape support components are more common, such as bolts in structures and anchor points in microelectromechanical systems (MEMS). Although some researchers have recently introduced explicit topological description methods to describe these engineering features, making some promising attempts, these efforts are still based on the background spring method and fail to address its inherent shortcomings. Summary of the Invention

[0008] In response to the defects of the existing technology and the need for improvement, the present invention provides an integrated design method for a continuum structure and its supporting components, the purpose of which is to achieve a customizable explicit description of the supporting components, avoid frequent mesh updates and the introduction of a large number of mesh-related design variables, realize the integrated design of the continuum structure and the supporting components, and improve the mechanical properties of the compliant mechanism and the general structure.

[0009] To achieve the above objectives, according to the present invention, a method for integrated design of a continuum structure and its supporting components is provided, the method comprising the following steps:

[0010] Define the explicit topological description function φ s , to describe the geometry of the support component, the function φ s By explicit design variable χ ex Control; define the implicit topological description function φ L , to describe the geometry of the continuum structure, the function φ L By implicit design variable χ im control.

[0011] A weighted interpolation function w is constructed to modify the displacement field U to obtain a local penalty displacement interpolation function u, which is used to impose the Dirichlet boundary condition.

[0012] The modified strain matrix is ​​derived from the local penalty displacement interpolation function u and the stiffness matrix The displacement field of the current continuum structure is calculated. According to the integrated design model of the continuum structure and its supporting components, the objective function value and sensitivity are calculated through the displacement field, and the explicit design variable χ is updated. ex and the implicit design variable χ im , until the objective function converges and the optimal continuum structure topology is obtained.

[0013] Further preferably, the local penalty displacement interpolation function u is preferably performed according to the following expression:

[0014] u=w(x)U+g(x)

[0015] Where x represents any point in the design domain, and g(x) represents the force applied on a given boundary Γ. D Dirichlet boundary conditions on .

[0016] Further preferably, the integrated design model of the continuum structure and its supporting components is preferably carried out according to the following expression:

[0017] Find:χ ex ,χ im

[0018]

[0019] Where J is the objective function and l is the vector describing the load. In the case of compliant mechanisms, l is 1 at the output degree of freedom and 0 at the rest. The objective function is displacement maximization. In general structural problems, l is the external force applied to the continuum and the objective function is flexibility minimization. a is the bilinear energy function and l is the linear load function. The linear elastic equilibrium equation is derived from a=l. u is the displacement field in the design domain, v is the virtual displacement field, and U is the displacement field in the design domain. ad is the Sobolev function space where the virtual displacement field is located; V H The material usage constraint for the continuum structure, V max is the maximum allowable value of the material, and H is the Heaviside function.

[0020] Further preferably, the weighted interpolation function w is performed according to the following expression:

[0021]

[0022] Where ψ is the implicit surface and δ is a small positive number.

[0023] Further preferably, the implicit surface ψ is defined as the explicit topological description function φ s :

[0024] ψ:=φ s

[0025] Further preferably, given the boundary Γ D Follow the following expression:

[0026]

[0027] Where, represents the explicit topological description function φ s The zero level set of .

[0028] Further preferably, the explicit topology description function φ s Follow the following expression:

[0029]

[0030] Where, Represents the composition of the explicit topological description function φ s A series of independently movable deformation components.

[0031] Further preferably, the movable deformable component Follow the following expression:

[0032]

[0033] Where, is the explicit design variable χ ex Components include geometric parameters such as the center coordinates and inclination angle of the movable deformable component.

[0034] A computer-readable storage medium includes a stored computer program, wherein when the computer program is executed by a processor, the device where the storage medium is located is controlled to execute the above-mentioned integrated design method for a continuum structure and its supporting components.

[0035] In general, the above technical solutions of the present invention can achieve the following beneficial effects:

[0036] 1. The method provided by the present invention constructs a weighted interpolation function based on an explicit topological description function, which can facilitate the construction of various types of supports commonly used in engineering, greatly reduce the number of design variables, and improve optimization efficiency.

[0037] 2. The method provided by the present invention is based on the extended finite element method, which can reconstruct the details of the support components without being affected by the quality of the background grid, ensuring that the theoretical optimization model is consistent with the analysis model, and improving the accuracy of finite element analysis and sensitivity analysis.

[0038] 3. The method provided by the present invention can realize the integrated design of continuum structures and support components. The optimization results do not require post-processing and can be directly transmitted to the computer-aided design (CAD) system. It is suitable for the study of general structure and flexible mechanism optimization design problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 A flow chart of an effective integrated design method for a continuum structure and its supporting components provided by an embodiment of the present invention;

[0040] Figure 2 The design domain and initial layout of the two-rod frame problem provided by an embodiment of the present invention, wherein (a) is the design domain of the two-rod frame problem, and (b) is the initial layout of the design domain of the two-rod frame problem;

[0041] Figure 3 Convergence history diagram of the two-rod frame problem provided by an embodiment of the present invention, wherein (a) is the iteration history diagram of the two-rod frame structure, and (b) is the iteration history diagram of the objective function and constraint function;

[0042] Figure 4 The design domain and initial layout of the compliant clamp problem provided by an embodiment of the present invention, wherein (a) is the design domain of the compliant clamp problem, and (b) is the initial layout of the design domain of the compliant clamp problem;

[0043] Figure 5 This is a convergence history diagram of the compliant clamp problem provided by an embodiment of the present invention, wherein (a) is an iteration history diagram of the compliant clamp mechanism, and (b) is an iteration history diagram of the objective function and constraint function. DETAILED DESCRIPTION

[0044] In order to make the objectives, technical solutions, and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are merely illustrative of the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

[0045] An integrated design approach for continuum structures and their supporting components, such as Figure 1 As shown, the following steps are included:

[0046] (1) Define an explicit and implicit hybrid topology description function to describe the update and iteration of the structural topology.

[0047] (1.1): Define the topological description function of the movable deformable component Introducing the explicit design variable χ ex , used to control the center coordinates and inclination angle of the movable deformation component.

[0048] (1.2): Define the explicit topological description function φ s :

[0049]

[0050] (1.3): Introducing C 2 Compactly supported radial basis function κ i and the implicit design variable χ im , construct the implicit topological description function as follows:

[0051]

[0052] (1.4): Based on explicit and implicit topological description functions, define an explicit-implicit hybrid topological description framework:

[0053]

[0054] The elasticity matrix of any point in the design domain is expressed as:

[0055]

[0056] Where D H is the elastic matrix of the continuum structure, and H is the Heaviside function.

[0057] (2) A weighted interpolation function is constructed based on the explicit topological description function to impose the Dirichlet boundary condition.

[0058] The weighted interpolation function w is introduced to modify the displacement field U to obtain the local penalty displacement interpolation function u:

[0059] u=w(x)U+g(x) (5)

[0060] Where x represents any point in the design domain, and g(x) represents the force applied on a given boundary Γ. D Dirichlet boundary conditions on .

[0061] The weighted interpolation function w is given as follows:

[0062]

[0063] Where ψ is the implicit surface and δ is a small positive number.

[0064] The implicit surface ψ is described by an explicit topological function φ s The definition is given as follows:

[0065] ψ:=φ s (7)

[0066] Given the boundary Γ in equation (4) D Using explicit topological description function φ s The zero level set of is defined as follows:

[0067]

[0068] (3) Derive the modified strain matrix and stiffness matrix and establish the finite element analysis model.

[0069] The modified strain matrix can be expressed as:

[0070]

[0071] Where N is the finite element shape function.

[0072] The modified stiffness matrix can be expressed as:

[0073]

[0074] (4) Establish an integrated design model of the continuum structure and its supporting components.

[0075] Establish an integrated design model of the continuum structure and its supporting components with material usage as a constraint:

[0076]

[0077] Where J is the objective function and l is the vector describing the load. In the case of compliant mechanisms, l is 1 at the output degree of freedom and 0 at the rest. The objective function is displacement maximization. In general structural problems, l is the external force applied to the continuum and the objective function is flexibility minimization. a is the bilinear energy function and l is the linear load function. The linear elastic equilibrium equation is derived from a=l. u is the displacement field in the design domain, v is the virtual displacement field, and U is the displacement field in the design domain. ad is the Sobolev function space where the virtual displacement field is located; V H The material usage constraint for the continuum structure, V max This is the maximum allowable value for the material.

[0078] (5) Derivation and calculation of topology optimization model sensitivity.

[0079] The first-order derivative of the objective function with respect to any design variable χ can be calculated using the adjoint method:

[0080]

[0081] According to formula (9), the sensitivity of each stiffness matrix component relative to the implicit design variables and explicit design variables is derived respectively:

[0082]

[0083] The sensitivity of the volume constraint to the explicit and implicit design variables is:

[0084]

[0085] (6) Explicit and implicit design variable updating and iteration.

[0086] Based on the calculated objective function, constraint function values, and their sensitivities, the moving asymptote method (MMA) is used to update the explicit and implicit design variables. Once convergence is achieved, the calculation ends and the optimized continuum structure is output.

[0087] In order to verify the effectiveness of the present invention, the following examples are given.

[0088] An effective integrated design method for continuum structures and their supporting components, the process is as follows Figure 1 shown.

[0089] First, the effectiveness of the proposed method is verified by using a two-bar frame problem. The design domain of the flexibility minimization problem of the two-bar frame is as follows: Figure 2 As shown, the dimensions are 1.4 × 1.6 × 1, discretized into 70 × 80 bilinear elements. A vertical downward unit force is applied at the point (0.6, 0.8). Assume that the rigid supports are 0.25 × 0.125 rectangles, and the center coordinates of the four support components are (0.1, 1.0), (0.1, 0.6), (1.3, 1.0), and (1.3, 0.6). During the iteration process, the center of the component is constrained to move within the gray rectangular area. The Young's modulus of the structural material is set to E = 1, the Poisson's ratio is set to v = 0.3, and the material volume ratio is 0.05.

[0090] Convergence history of continuum structure and support optimization in Figure 3 Given in . It can be observed that within the limited volume, the structure ultimately retains only the structure closer to the force application point, forming two rods with an angle of ±45°. The final structural flexibility is 3.942.

[0091] Secondly, its effectiveness is verified by the compliant clamp problem. As shown in Figure 4, due to the symmetry of the design domain, only the lower half is considered. The design domain is a 2×1×1 rectangle, discretized into 80×40 bilinear elements. The upper left corner of the design domain is the input point, where a unit force F is applied, and the upper right corner is the output point, with the desired output direction aligned with the positive y-axis. Two prototype support components are set within the design domain to simulate the screw holes commonly used in compliant mechanism assembly. The support diameter is 0.15. The centers of the two components are confined within the dark gray rectangle.

[0092] The convergence history of the compliant clamp is as follows Figure 5As shown, the output displacement is 1.0754. It can be observed that the two support components can coevolve with the compliant mechanism. The resulting results are quite different from those of the classic compliant clamp benchmark problem, with the output displacement increasing by 15% compared to the classic results, effectively increasing the output displacement of the compliant mechanism.

[0093] In summary, the present invention proposes an integrated design method for continuum structures and their supporting components based on explicit and implicit hybrid topological description. Dirichlet boundary conditions are defined using explicit topological description functions to achieve customizable explicit descriptions of supporting components, avoid frequent mesh updates and the introduction of a large number of mesh-related design variables, and improve the mechanical properties of compliant mechanisms and general structures.

[0094] The above embodiments of the present invention are merely examples for the purpose of clearly illustrating the present invention and are not intended to limit the embodiments of the present invention. Those skilled in the art will appreciate that other variations or modifications may be made based on the above description. It is not necessary and impossible to enumerate all embodiments here. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention are intended to be included within the scope of protection of the claims of the present invention.

Claims

1. A method for integrated design of a continuum structure and its supporting components, characterized in that: The method comprises the following steps: Define the explicit topological description function φ S , to describe the geometry of the support component, the function φ S By explicit design variable χ ex Control; define the implicit topological description function φ L , to describe the geometry of the continuum structure, the function φ L By implicit design variable χ im control; Construct a weighted interpolation function w to modify the displacement field U of the standard finite element. The obtained local penalty displacement interpolation function u is: u=w(x)U+g(x) Where x represents any point in the design domain, and g(x) represents the force applied on a given boundary Γ. D Dirichlet boundary conditions on ; The modified strain matrix is ​​derived from the local penalty displacement interpolation function u and the stiffness matrix According to the integrated design model of the continuum structure and its supporting components, the displacement field of the current continuum structure is calculated, the objective function value and sensitivity are calculated through the displacement field, and the explicit design variable χ is updated. ex and the implicit design variable χ in , until the objective function converges and the optimal continuum structure topology is obtained; The integrated design model of the continuum structure and its supporting components is: Find:x ex ,x im Where J is the objective function, l is the vector describing the load; in the problem of compliant mechanism, l is 1 at the output degree of freedom and 0 for the other elements, and the objective function is displacement maximization; in general structural problems, l is the external force applied to the continuum, and the objective function is compliance minimization; a is the bilinear energy function, l is the linear load function, and the linear elastic equilibrium equation is derived from a=l; u is the displacement field in the design domain, v is the virtual displacement field, and U is ad is the Sobolev function space where the virtual displacement field is located; V H The material usage constraint for the continuum structure, V max is the maximum allowable value of the material, and H is the Heaviside function.

2. The integrated design method of a continuum structure and its supporting components according to claim 1, characterized in that: The weighted interpolation function w is performed according to the following expression: Where ψ is the implicit surface and δ is a small positive number.

3. The integrated design method of a continuum structure and its supporting components according to claim 2, characterized in that: The implicit surface ψ is defined as the explicit topological description function φ s : ψ:=φ s 。 4. The integrated design method of a continuum structure and its supporting components according to claim 1, characterized in that: The given boundary Γ D Defined by the following expression: Where, represents the explicit topological description function φ s The zero level set of .

5. The integrated design method of a continuum structure and its supporting components according to claim 1, characterized in that: The explicit topological description function φ s Defined by the following expression: Where, Represents the composition of the explicit topological description function φ s A series of independently movable deformation components.

6. The integrated design method of a continuum structure and its supporting components according to claim 5, characterized in that: The movable deformable component Defined by the following expression: Where, is the explicit design variable χ ex Components, including the center coordinates of the movable deformation component and the inclination geometric parameters.

7. A computer-readable storage medium, characterized in that The computer-readable storage medium includes a stored computer program, wherein when the computer program is executed by a processor, the device where the storage medium is located is controlled to execute the integrated design method of a continuum structure and its supporting components as described in any one of claims 1 to 6.

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