A method for the integrated design of a continuum structure and its support assembly

Abstract

The application relates to a kind of integrated design method of continuum structure and its support assembly, method includes the following steps: (1) defining explicit topology description function, for describing the geometry topology of continuum structure and support assembly update and iteration;(2) based on explicit topology description function, construct weighted interpolation function, for applying Dirichlet boundary condition;(3) derive the modified strain matrix and stiffness matrix, establish finite element analysis model;(4) establish the integrated design model of continuum structure and its support assembly;(5) topology optimization model sensitivity derivation and calculation;(6) explicit and implicit design variable update and iteration.Through the application, the customizable explicit description of support assembly is realized, the frequent mesh update and the introduction of a large number of mesh-related design variables are avoided, the integrated design of continuum structure and support assembly is realized, and the mechanical properties of compliant mechanism and general structure are improved.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of mechanical and structural design, and more particularly relates to a method for integrated design of continuum structure and its support assembly. BACKGROUND

[0002] Topology optimization is a numerical method for finding 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 method for improving structural performance and reducing cost. In most cases, continuum structure topology optimization takes the material distribution in the given design domain as the design variable. With the increasing demand for structural design, the interaction between continuum structure and its own material and boundary conditions has begun to be valued, so the integrated design method of continuum structure and these elements has become a research hotspot.

[0003] The support assembly or Dirichlet boundary plays a decisive role in the load transmission path, and its influence on structural performance has been noticed early. For example, the stiffness and natural frequency of some plates or beams are largely dependent on the direction of support location; the force arm of a compliant mechanism is largely determined by the support, thereby affecting the output displacement or displacement amplification ratio.

[0004] Conventional topology optimization programs usually require the rigid support to be given manually in advance and remain unchanged during the iteration process. This makes it difficult to place the support at the best location in practical applications, which relies on the experience of designers. Therefore, how to adjust the position or direction of the support while optimizing the topology configuration of the structure has become an important issue.

[0005] The key problem of the integrated design of continuum structure and support is how to model the changing support assembly. Existing methods can be divided into two categories. The first is the background spring method, which was first proposed in Buhl's work. This method adds a series of background springs consistent with the number and direction of degrees of freedom at all finite element grid nodes to represent the potential support assembly. The stiffness of these springs is taken as a design variable, and the intermediate stiffness is penalized, which can be regarded as an extension of the solid isotropic material penalization (SIMP) method. It is flexible and can represent 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 node degrees of freedom, increasing the computational load. At the same time, the boundary of the support assembly obtained has a gray transition area, and the boundary is not smooth, which requires secondary design.

[0006] To get smooth support component boundary, some scholars proposed to use computational geometry techniques to parameterize the description method of support component boundary. Zhu and Zhang use movable components to represent rigid support, and obtain sensitivity information by difference method; Xia et al. adopt multiple level set functions to represent free boundary and support respectively, and use Lagrange function and continuous adjoint method to obtain shape function of two kinds of boundary. But these methods need to frequently divide support boundary in iteration. In order to avoid this process, Zhang et al. use weighted B-spline function defined by level set function to punish displacement field to meet Dirichlet boundary condition, and obtain sensitivity information by discrete adjoint method; Dapogny et al. proposed an approximate method, which replaces the "exact" Laplace equation with mixed boundary conditions into a "smooth" version, so that it can be described as Robin boundary condition distributed on a narrow band.

[0007] However, in practical applications, support components with fixed shape are more common, such as bolts in structures, anchor points in micro-electro-mechanical systems (MEMS). Although in recent years some scholars have begun to introduce explicit topological description method to describe these engineering features, some beneficial attempts have been made. But the related work is still based on background spring method, which cannot solve the inherent defects of background spring method. SUMMARY

[0008] In view of the defects of the prior art and the improvement needs, the present application provides an integrated design method of continuum structure and its support component, which aims to realize the customizable explicit description of the support component, avoid frequent mesh update and the introduction of a large number of mesh-related design variables, realize the integrated design of continuum structure and support component, and improve the mechanical properties of compliant mechanism and general structure.

[0009] In order to achieve the above purpose, according to the present application, an integrated design method of continuum structure and its support component is provided, which comprises the following steps:

[0010] Defining an explicit topological description function φ s to describe the geometric shape of the support component, the function φ s is controlled by an explicit design variable χ ex ; Defining an implicit topological description function φ L to describe the geometric shape of the continuum structure, the function φ L is controlled by an implicit design variable χ im .

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

[0012] A modified strain matrix is derived from the locally penalized displacement interpolation function u and stiffness matrix The displacement field of the current continuum structure is calculated, and the objective function value and sensitivity are calculated through the displacement field according to the integrated design model of the continuum structure and its support assembly, and the explicit design variable χ ex and implicit design variable χ im are updated until the objective function converges, and the optimal continuum structure topology is obtained.

[0013] Further preferably, the locally penalized displacement interpolation function u is preferably 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 Dirichlet boundary condition applied on the given boundary Γ D .

[0016] Further preferably, the integrated design model of the continuum structure and its support assembly is preferably according to the following expression:

[0017] Find: χ ex , χ im

[0018]

[0019] where J is the objective function, and l is a vector describing the load. In the problem of compliant mechanism, l is 1 at the output degree of freedom and 0 at the rest, and the objective function is the maximization of displacement. In the general structure problem, l is the external force applied on the continuum, and the objective function is the minimization of compliance; a is the bilinear energy function, and 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, U ad is the Sobolev function space where the virtual displacement field v resides; V H is the material usage constraint of the continuum structure, V max is the maximum allowed value of the material, and H is the Heaviside function.

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

[0021]

[0022] where ψ is the implicit surface, and δ is a very small positive number.

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

[0024] Ψ := φ s

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

[0026]

[0027] In the formula, denotes the explicit topological description function φ s of the zero level set.

[0028] Further preferably, the explicit topological description function φ s According to the following expression:

[0029]

[0030] In the formula, denotes a series of independent movable deformation components constituting the explicit topological description function φ s .

[0031] Further preferably, the movable deformation component According to the following expression:

[0032]

[0033] In the formula, is a component of the explicit design variable χ ex , including geometric parameters such as movable deformation component center coordinates, inclination angles, etc.

[0034] A computer readable storage medium comprising a stored computer program, wherein the computer program, when executed by a processor, controls the device where the storage medium is located to perform an integrated design method of a continuum structure and its support component as described above.

[0035] Overall, the above technical solutions of the present application can achieve the following beneficial effects:

[0036] 1. The method provided by the present application is based on the construction of 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 the optimization efficiency.

[0037] 2. The method provided by the present application is based on the extended finite element method, which can reconstruct the details of the support component without being affected by the quality of the background mesh, ensure the consistency of the theoretical optimization model and the analysis model, and improve the accuracy of finite element analysis and sensitivity analysis.

[0038] 3、The method provided by the application can realize integrated design of continuum structure and support assembly, and the optimization result does not need post-processing and can be directly transmitted into a computer aided design (CAD) system, and is suitable for general structure and compliant mechanism optimization design problem research. BRIEF DESCRIPTION OF DRAWINGS

[0039] Figure 1 An effective integrated design method flow chart of continuum structure and its support assembly is provided for the embodiments of the application.

[0040] Figure 2 Design domain and initial layout of a two-bar frame problem are provided for the embodiments of the application, wherein (a) is the design domain of the two-bar frame problem, and (b) is the initial layout of the design domain of the two-bar frame problem.

[0041] Figure 3 Convergence history diagram of the two-bar frame problem is provided for the embodiments of the application, wherein (a) is the iteration history diagram of the two-bar frame structure, and (b) is the iteration history diagram of the objective function and the constraint function.

[0042] Figure 4 Design domain and initial layout of a compliant gripper problem are provided for the embodiments of the application, wherein (a) is the design domain of the compliant gripper problem, and (b) is the initial layout of the design domain of the compliant gripper problem.

[0043] Figure 5 Convergence history diagram of the compliant gripper problem is provided for the embodiments of the application, wherein (a) is the iteration history diagram of the compliant gripper mechanism, and (b) is the iteration history diagram of the objective function and the constraint function. DETAILED DESCRIPTION

[0044] In order to make the objectives, technical solutions and advantages of the application clearer, the application is further described in detail below with reference to the drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the application and do not limit the application. In addition, the technical features involved in each embodiment of the application described below can be combined with each other as long as they do not conflict with each other.

[0045] An integrated design method of continuum structure and its support assembly, as shown in Figure 1 , includes the following steps:

[0046] (1) Defining an explicit-implicit hybrid topology description function for describing the update and iteration of the structure topology.

[0047] (1.1): Defining the topology description function of the movable deformable assembly Introducing an explicit design variable χ ex to control the center coordinates and inclination angle of the movable deformable assembly.

[0048] (1.2): Definition of explicit topology description function φ s :

[0049]

[0050] (1.3): Introduce C 2 compact support radial basis function κ i and implicit design variable χ im , construct implicit topology description function as follows:

[0051]

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

[0053]

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

[0055]

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

[0057] (2) Construct a weighted interpolation function based on the explicit topology description function, which is used to apply Dirichlet boundary conditions.

[0058] Introduce a weighted interpolation function w, modify the displacement field U to obtain a locally penalized displacement interpolation function u:

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

[0060] In the formula, x represents any point in the design domain, and g(x) represents the Dirichlet boundary condition applied on the given boundary Γ D .

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

[0062]

[0063] In the formula, ψ is an implicit surface, and δ is a very small positive number.

[0064] The implicit surface ψ is defined using the explicit topology description function φ s , which is given as follows:

[0065] ψ := φ s (7)

[0066] The boundary Γ in equation (4) is given D The explicit topological descriptor function φ s is defined by the zero level set of φ

[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 support components.

[0075] An integrated design model of the continuum structure and its support components is established with material usage as a constraint:

[0076]

[0077] where J is the objective function, and l is a vector describing the load. In the problem of compliant mechanism, l is 1 at the output degree of freedom and 0 at the remaining elements, and the objective function is the maximization of displacement. In the general structure problem, l is the external force applied to the continuum, and the objective function is the minimization of compliance; a is the bilinear energy function, and l is the linear load function, from which the linear elastic equilibrium equation is derived; u is the displacement field in the design domain, v is the virtual displacement field, and U ad is the Sobolev function space in which the virtual displacement field v resides; V H is the material usage constraint of the continuum structure, and V max is the maximum allowed value of the material.

[0078] (5) Sensitivity derivation and calculation of the topological optimization model.

[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 with respect to the implicit design variable and the explicit design variable 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] According to the calculated objective function, constraint function value, and their sensitivity, the explicit and implicit design variables are updated using the moving asymptotes method (MMA). When the convergence condition is reached, the calculation is ended and the optimized continuum structure is output.

[0087] In order to verify the effectiveness of the present application, the following example is given.

[0088] An effective integrated design method for continuum structure and its support assembly, the flowchart is as shown in Figure 1 .

[0089] Firstly, its effectiveness is verified by a two-bar frame problem. The design domain of the two-bar frame flexibility minimization problem is as shown in Figure 2 , with a size of 1.4x1.6x1, which is discretized into 70x80 bilinear elements. A unit force vertically downward is applied at point (0.6, 0.8). The rigid support is assumed to be a rectangle of 0.25x0.125, and the center coordinates of the four support assemblies are (0.1, 1.0), (0.1, 0.6), (1.3, 1.0), and (1.3, 0.6), respectively. In the iteration process, the center of the assembly is limited 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] The convergence history of continuum structure and support optimization is given in Figure 3 . It can be observed that under the limited volume, the structure finally only retains the structure closer to the force application point on one side, and forms two bars with an angle of ±45°. The final structure flexibility is 3.942.

[0091] Secondly, its effectiveness is verified by a compliant gripper problem. As shown in Fig. 4, due to the symmetry of the design domain, only the lower half is considered. The design domain is a rectangle of 2x1x1, which is discretized into 80x40 bilinear elements. The upper left corner of the design domain is the input point, and a unit force F is applied. The upper right corner is the output point, and the desired output direction is consistent with the positive direction of the y-axis. Two prototype support assemblies are set in the design domain to simulate the screw holes commonly used in the assembly of compliant mechanisms. The support diameter is 0.15. The centers of the two assemblies are limited to the dark gray rectangle.

[0092] The convergence history of the compliant gripper is as shown in Figure 5As shown, the output displacement is 1.0754. It can be observed that the two support assemblies can evolve with the compliant mechanism. Finally, a result different from the classic compliant gripper benchmark problem is formed, and the output displacement is increased by 15% compared with the classic result, effectively increasing the output displacement of the compliant mechanism.

[0093] To sum up, the application provides a continuum structure and its support assembly integrated design method based on explicit-implicit hybrid topological description, uses an explicit topological description function to define a Dirichlet boundary condition, realizes customizable explicit description of the support assembly, avoids frequent mesh updates and introduction of a large number of mesh-related design variables, and improves the mechanical properties of the compliant mechanism and general structures.

[0094] The above embodiments of the application are merely examples for clearly illustrating the application, and are not intended to limit the implementation modes of the application. Based on the above description, other different forms of changes or variations can be made by those skilled in the art. Here, all the implementation modes are not required or can not be exhausted. Any modification, equivalent replacement and improvement made within the spirit and principle of the application shall be included in the protection scope of the claims of the application.

Claims

1. A method of integrated design of a continuum structure and its support assembly, characterized by, The method comprises the following steps: defining an explicit topology description function , to describe the geometry of the support assembly, the function controlled by explicit design variables defining an implicit topology description function , to describe the geometry of the continuum structure, the function controlled by implicit design variables controlled by Constructing a weighted interpolation function , for modifying the displacement field of standard finite elements , resulting in a locally penalized displacement interpolation function is ； wherein represents an arbitrary point in the design domain, represents a Dirichlet boundary condition applied on a given boundary of the design domain. Interpolation function with local penalty displacement Derive the corrected strain matrix and stiffness matrix Based on 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 then calculated using the displacement field, and the explicit design variables are updated accordingly. and implicit design variables This continues until the objective function converges, yielding the optimal continuum structure topology; The integrated design model of the continuum structure and its support assembly is: ； where, is the objective function, is the vector describing the load; in the compliant mechanism problem, is 1 on the output degree of freedom and 0 on the rest, the objective function is the displacement maximization; in the general structure problem, is the external force applied on the continuum, the objective function is the compliance minimization; is the bilinear energy function, is the linear load function, which is given by linear elastic equilibrium equation is derived from; is the virtual displacement field, is the Sobolev function space that the virtual displacement field resides in; is the material usage constraint of the continuum structure, is the maximum allowed value of the material, is the Heaviside function; The weighted interpolation function According to the following expression: ； wherein is an implicit surface, is a small positive number; The implicit surface defined as an explicit topological description function : 。 2. The method of integrating a continuum structure with its support assembly as recited in claim 1, wherein, the given boundary are defined according to the following expressions: ； wherein represents an explicit topological description function of zero level set.

3. The method of integrating a continuum structure with its support assembly of claim 1, wherein, the explicit topology description function is defined according to the following expression: ； In the formulae, represents a series of independent movable deformable components that make up the explicit topological description function .

4. The method of integrating a continuum structure with its support assembly of claim 3, wherein, The movable deforming assembly , is defined according to the following expression: ； wherein are explicit design variables components of the vector of design variables, including movable deformable assembly center coordinates, inclination geometric parameters.

5. A computer readable storage medium, characterized in that, The computer readable storage medium comprises a stored computer program, wherein when the computer program is run by a processor, the device where the storage medium is located performs an integrated design method of a continuum structure and its support assembly according to any one of claims 1 to 4.

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