A mechanical simulation analysis method for analyzing two-dimensional periodic inhomogeneous structures

Abstract

The present invention provides a mechanical simulation analysis method for analyzing a two-dimensional periodic non-homogeneous structure. The present invention considers using an isogeometric analysis method to simulate a multi-scale model, because the isogeometric analysis method uses a spline function to describe the geometric model, which is more accurate than the finite element method in describing certain smooth curve and surface structures, and the basis function used in the isogeometric analysis during the analysis and calculation is also a spline function, which realizes the unification of the physical model and the analysis model. Therefore, compared with the extended multi-scale finite element method, the extended multi-scale isogeometric analysis can solve the problem of heterogeneity between the analysis model and the physical model, and will also improve the calculation accuracy.

Description

Technical Field

[0001] The present invention relates to the technical field of mechanical analysis methods, and in particular to a mechanical simulation analysis method for analyzing two-dimensional periodic inhomogeneous structures. Background Art

[0002] Existing extended multiscale methods, primarily the extended multiscale finite element method (EM), construct numerical basis functions by solving local subproblems within a cell. By applying certain boundary conditions to the subdomain to solve the equilibrium equations, the displacements of each node within the subdomain are obtained, which are numerical basis functions, also called displacement basis functions. These basis functions accurately and effectively reflect the microscopic heterogeneity of the material, thus achieving accurate and effective solutions at the macroscopic level, thus avoiding the need to solve the problem at the microscopic level and significantly saving computational resources. The material properties within each cell can be heterogeneous.

[0003] Common microstructures such as Figure 1 As shown, it contains properties similar to holes or material inhomogeneities.

[0004] like Figure 2 As shown in (a), for a fine mesh model, we only need to select a representative set of fine meshes and use the obtained basis functions to convert the stiffness matrix of this set of fine meshes into an equivalent unit stiffness matrix with four nodes of coarse mesh. This set of equivalent unit stiffness matrices is assembled into an overall stiffness matrix. In this way, in actual calculation, the degrees of freedom of the analytical model calculation are as follows: Figure 2 (b), compared to Figure 2 (a) This significantly reduces the computational degrees of freedom. After obtaining the nodal displacements, simple matrix multiplication using basis functions can be performed to obtain the nodal displacement information within the fine mesh. Compared to models with fine meshes, this also reduces the number of equilibrium equations to solve, improving computational efficiency.

[0005] The most important step in extended multiscale finite elements is the calculation of basis functions. Different basis function construction methods can be used for different problems. For homogeneous structures, linear boundary conditions can be used, while for periodic structures, periodic boundary conditions can be used. Different basis function construction methods essentially impose different boundary conditions. By solving the equilibrium equations, the node displacements within the fine mesh are obtained as basis functions.

[0006] The extended multi-scale finite element method currently achieves good calculation results in multiple sets of examples. The displacement and stress distributions are in good agreement with the reference solution, and the calculation speed is significantly improved.

[0007] At the same time, when using the finite element method for multi-scale analysis, for certain structures, such as smooth surfaces or curves, the discretization of finite element units cannot accurately describe the model. The analytical model also uses the discretized model for calculation. This will cause inconsistencies between the actual model and the analytical model, which will have a certain impact on its accuracy. If you want to obtain accurate results, you often need to divide the model very finely, which increases the computational effort. At the same time, model meshing in finite element pre-processing is also a complex process.

[0008] The existing technology also includes isogeometric analysis, a new numerical calculation method. Compared to the discretization method used by finite element methods, isogeometric analysis uses spline functions to describe the model. Using spline functions can accurately express the model, for example, smooth surface and curve structures, isogeometric analysis is more accurate than finite element methods. It also avoids the tedious meshing process of finite element methods. During the analysis and calculation, the basis functions used in isogeometric analysis are also the spline functions used to express the model, so isogeometric analysis also has an advantage over finite element methods in terms of calculation accuracy.

[0009] Isogeometric analysis such as Figure 3 As shown in the figure, for smooth curve structures, we first construct a parameter domain in two directions, divide the parameter domain by node vectors, and generate spline functions using control points and weights. The spline functions in two directions generate smooth surfaces through tensor products. By adjusting the control points and weights, different surfaces can be expressed. Spline functions are also used in analytical calculations, and numerical calculations are all performed on the parameter domain. The parameter domain and the physical domain are mapped through spline functions. Summary of the Invention

[0010] Based on the technical problems mentioned in the above-mentioned background technology, a mechanical simulation analysis method for analyzing two-dimensional periodic non-homogeneous structures is provided. The present invention considers using the isogeometric analysis method to expand multi-scale, because the isogeometric analysis method uses spline functions to describe the geometric model. Compared with the finite element method, it is more accurate in describing certain smooth curve and surface structures. Moreover, the basis functions used in the isogeometric analysis during analysis and calculation are also spline functions, which realizes the unification of the physical model and the analytical model. Therefore, compared with the extended multi-scale finite element method, the extended multi-scale isogeometric analysis can solve the problem of heterogeneity between the analytical model and the physical model, and will also improve the calculation accuracy.

[0011] The technical means adopted in the present invention are as follows:

[0012] A mechanical simulation analysis method for analyzing a two-dimensional periodic inhomogeneous structure, characterized by comprising the following steps:

[0013] Step 1: For a periodic inhomogeneous structure, extract its characteristic unit cell, where the domain of the extracted characteristic unit cell is a square, and construct basis functions by applying different boundary conditions;

[0014] Step 2: construct the geometric model of the characteristic unit cell and the isogeometric grid respectively through NURBS basis functions;

[0015] When the characteristic unit cell is purely homogeneous, it is expressed by a set of parameterized domains;

[0016] When the characteristic unit cell contains holes or heterogeneous material areas, the model is segmented according to the specific situation to form several parameterized domains. Then, during the calculation, the parameter domain models are combined according to the multi-piece splicing method to perform the overall calculation;

[0017] According to the accuracy requirements, different refinement times can be set to obtain geometric grids of different accuracies;

[0018] Step 3: Obtain the isogeometric analysis formula according to the potential energy method and calculate the stiffness matrix of the characteristic unit cell;

[0019] Step 4: Apply specific displacement boundary conditions to the characteristic unit cell. The boundary conditions are applied in the following way: Figure 9 As shown, constructing the basis function requires solving the equilibrium equations inside and at the boundaries of the unit:

[0020] Step 5: After applying the boundary conditions, solve the equilibrium equation inside the characteristic unit cell and obtain the displacement as the basis function;

[0021] Step 6: Obtain the equivalent stiffness matrix of the characteristic unit cell according to the displacement basis function obtained in step 4;

[0022] Step 7, calculating the equivalent global stiffness matrix of the macrostructure based on the equivalent unit stiffness matrix according to the mapping relationship from local to global;

[0023] Step 8: Perform a transformation on the load matrix to perform calculations on a macro scale.

[0024] Step 9, obtaining the macroscopic deformation displacement result of the entire structure by solving the linear equations, wherein the result is the macroscopic node displacement;

[0025] Step 10: After obtaining the macroscopic node displacement, the microscopic displacement and stress-strain information inside the characteristic unit cell are calculated by matrix multiplication.

[0026] Compared with the prior art, the present invention has the following advantages:

[0027] The extended multi-scale isogeometric analysis method involved in the invention of this article is compared and verified with the results of fine simulation. This method is effective for the analysis of periodic inhomogeneous structural materials. It can obtain more accurate microscopic displacement results on the basis of greatly reducing the amount of calculation. Moreover, compared with the traditional finite element method for extended multi-scale analysis, the results of the static and dynamic analysis of the extended multi-scale isogeometric method are also more accurate. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative labor.

[0029] Figure 1 Schematic diagram of a common microstructure in the prior art.

[0030] Figure 2 (a) is a schematic diagram of the model with fine mesh division; (b) is a schematic diagram of the degrees of freedom calculated for the analytical model.

[0031] Figure 3 Schematic diagram of isogeometric analysis.

[0032] Figure 4 Expanding the multi-scale isogeometric flow chart for the present invention.

[0033] Figure 5 It is a typical periodic heterogeneous structure of the present invention.

[0034] Figure 6 It is a schematic diagram of the constraint and force conditions of the present invention.

[0035] Figure 7 Schematic diagram of the characteristic unit cell of the present invention.

[0036] Figure 8 It is a schematic diagram of the slice form of the present invention.

[0037] Figure 9 The expression of the NURBS spline model of the present invention; wherein, (a) and (d) are schematic diagrams of the parameter space geometry model; (b) and (e) are schematic diagrams of the control point grid; (c) and (f) are schematic diagrams of the physical space geometry model.

[0038] Figure 10 There are two ways of applying boundary conditions; (a) is the linear boundary condition application method; (b) is the periodic boundary condition application method. DETAILED DESCRIPTION

[0039] In order to enable those skilled in the art to better understand the solutions of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the embodiments described are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts should fall within the scope of protection of the present invention.

[0040] It should be noted that the terms "first", "second", etc. in the description and claims of the present invention and the above-mentioned drawings are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that the numbers used in this way can be interchanged where appropriate, so that the embodiments of the present invention described herein can be implemented in an order other than those illustrated or described herein. In addition, the terms "including" and "having" and any variations thereof are intended to cover non-exclusive inclusions. For example, a process, method, system, product or device that includes a series of steps or units is not necessarily limited to those steps or units clearly listed, but may include other steps or units that are not clearly listed or inherent to these processes, methods, products or devices.

[0041] like Figure 1-10 As shown, the present invention provides a mechanical simulation analysis method for analyzing a two-dimensional periodic inhomogeneous structure, comprising the following steps:

[0042] Step 1: For the periodic inhomogeneous structure, extract its characteristic unit cell. The domain of the extracted characteristic unit cell is a square. By applying different boundary conditions, construct the basis function.

[0043] Step 2: Construct the geometric model and isogeometric grid of the characteristic unit cell respectively through NURBS basis functions; when the characteristic unit cell is purely homogeneous, it is expressed through a set of parameterized domains; when the characteristic unit cell contains holes or heterogeneous material areas, the geometric model of the characteristic unit cell is divided according to the specific situation to form several subdomains described by the parameterized model, and the subdomains are spliced ​​together by overlapping adjacent control points to form the geometric model of the entire unit cell; the parameterized subdomain models formed by the division of the characteristic unit cell geometric model are combined, and the overall calculation is performed by superimposing the stiffness values ​​of the control points; when calculating the stiffness matrix of the characteristic unit cell, the number of control points of the unit cell geometric model can be adjusted to obtain isogeometric grids of different scales.

[0044] Step 3: According to the minimum potential energy principle, perform isogeometric analysis and calculate the stiffness matrix of the characteristic unit cell. The stiffness matrix of the characteristic unit cell is:

[0045]

[0046] Among them, U is the control point displacement matrix, K e is the control unit stiffness matrix, G is the displacement strain matrix, F e is the external force matrix.

[0047] Step 4: Apply linear or periodic displacement boundary conditions to the characteristic unit cell. Constructing the basis function requires solving the equilibrium equations inside and on the boundary of the unit cell. The equilibrium equations inside and on the boundary of the unit cell need to be solved as follows:

[0048]

[0049] Where L represents the elastic operator, satisfying:

[0050]

[0051] Among them, N i Represents the basis functions of each node of the macro unit. For two-dimensional plane problems, m=4; for two-dimensional vector field problems, basis functions need to be constructed in the x and y directions respectively.

[0052] Step 5: After applying the boundary conditions, solve the equilibrium equations inside the characteristic unit cell and obtain the displacement as the basis function. Since each node has basis functions in both the x and y directions, a total of 8 sets of equilibrium equations need to be solved to obtain 8 basis functions, which are in the following form:

[0053]

[0054] Among them, N is the displacement basis function, N ixx and N iyy is the basis function term. i=1,2,3,4,N ixy and N iyx is the additional coupling term due to the existence of Poisson's ratio;

[0055] For the basis function, the verification is performed through the properties of the basis function. The verification method for the properties of the basis function is:

[0056]

[0057] Since it is a displacement basis function, the results of the basis function are verified by importing the model data and result data files into the Hypermesh visualization software.

[0058] Step 6: Obtain the equivalent stiffness matrix of the characteristic unit cell according to the displacement basis function obtained in step 4. The calculation method of the equivalent stiffness matrix of the characteristic unit cell is:

[0059] K C =G T K S G;

[0060] Among them, G represents the transformation matrix, which is a matrix formed by arranging basis functions in a certain order, and its form is:

[0061]

[0062] K S represents the stiffness matrix of the characteristic unit cell, K C represents the four-node stiffness matrix of the macro unit, K C is an equivalent stiffness matrix of order 8×8.

[0063] Step 7: According to the mapping relationship from local to global, the equivalent unit stiffness matrix is ​​converted from the local coordinate system to the global coordinate system to calculate the equivalent global stiffness matrix of the macrostructure;

[0064] Step 8: For the load matrix, transform the local coordinate system to the global coordinate system so as to perform calculations on a macro scale. The transformation method for the load matrix is:

[0065] F C =G T F S ;

[0066] Among them, F C represents the macroscopic nodal force, F S Represents the element nodal force.

[0067] Step 9: Obtain the macroscopic deformation displacement of the entire structure by solving the linear equations. The result is the macroscopic node displacement.

[0068] Step 10: After obtaining the macroscopic node displacement, the microscopic displacement and stress-strain information inside the characteristic unit cell are calculated by matrix multiplication. The specific calculation formula for calculating the microscopic displacement and stress-strain information inside the characteristic unit cell by matrix multiplication is:

[0069] σ e =SU, S=D e B e G;

[0070] Where G represents the transformation matrix in step 6, and U represents the displacement of the microscopic nodes inside the characteristic unit cell. After obtaining the microscopic node displacement, the microscopic stress and strain are calculated. The specific formula is:

[0071] ε e =TU, T=B e G;

[0072] Among them, B e Denotes the displacement strain matrix of the characteristic unit cell, D e represents the elastic matrix of the characteristic unit cell.

[0073] A simple calculation example is performed for a beam structure. The right end is subjected to an upward force, and the displacement of the beam's centerline is measured and compared.

[0074] The calculated results are compared with the precise simulation results of Hypermesh, mainly showing the improvement of calculation accuracy under the same degree of freedom of the characteristic unit cell.

[0075] Calculation method Characteristic unit cell degrees of freedom Coarse grid Norm error Extended multi-scale geometry 526 20×4 0.15％ Extended Multiscale Finite Element 526 20×4 1.7％ Hypermesh 1049 20×4 /

[0076] Example 1

[0077] Periodic inclusion structures, such as Figure 6 As shown, the coarse grid size is 20×4, the characteristic subgrid is a square plate with an inclusion structure, the side length is 1, the inclusion model inside the subgrid is a circle with r = 0.15, and the material parameter is Young's modulus 10 6 , Poisson's ratio is 0.4, and the subgrid's external Young's modulus is 10 4 , with a Poisson's ratio of 0.3. When subjected to a concentrated upward force F = 4 at the right end and a clamped left end, the displacement of the centerline obtained using the extended multiscale isogeometric method is highly consistent with the Hypermesh simulation results. Compared to the extended multiscale finite element method with the same computational degrees of freedom, the second-norm error is significantly reduced (0.15% for the extended multiscale isogeometric method and 1.7% for the extended multiscale finite element method).

[0078] In Example 1, the characteristic cell is refined twice, and the obtained isogeometric mesh is as follows Figure 8 As shown, for Example 1, the macro model is a 20×4 grid.

[0079] Example 2

[0080] Considering the actual engineering application, the simplified flexible electronic model is calculated. The simplified flexible electronic model consists of two parts of materials, the substrate and the electronic device. The material properties of the two are different. The material parameter of the substrate is Young's modulus 10 6 , Poisson's ratio is 0.4, and Young's modulus of flexible electronic devices is 10 4 , with a Poisson's ratio of 0.3. The overall displacement and microscopic displacement distribution were calculated, and the results were highly consistent with the Hypermesh simulation results. Simultaneously, using the same computing resources, the results of the extended multi-scale isogeometric method and the full-scale finite element simulation were calculated. Under the two boundary conditions of extended multi-scale isogeometric, the calculation times were 18.146 seconds and 19.233 seconds, respectively, while the full-scale calculation time was 225.237 seconds. This shows that the extended multi-scale isogeometric method can significantly reduce calculation time while maintaining accuracy.

[0081] The serial numbers of the above embodiments of the present invention are for description only and do not represent the advantages or disadvantages of the embodiments. In the above embodiments of the present invention, the description of each embodiment has its own emphasis. For parts not described in detail in one embodiment, please refer to the relevant description of other embodiments.

[0082] In the several embodiments provided in this application, it should be understood that the disclosed technical content can be implemented in other ways. Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A mechanical simulation analysis method for analyzing a two-dimensional periodic inhomogeneous structure, characterized in that: The following steps are involved: Step 1: For a periodic inhomogeneous structure, extract its characteristic unit cell, the domain of the extracted characteristic unit cell is a square, and construct a basis function by applying different boundary conditions; Step 2: constructing the geometric model and isogeometric grid of the characteristic unit cell respectively through NURBS basis functions; When the characteristic unit cell is purely homogeneous, it is expressed by a set of parameterized domains; When the characteristic unit cell contains holes or heterogeneous material areas, the geometric model of the characteristic unit cell is segmented according to the specific situation to form several subdomains described by parameterized models, and the subdomains are spliced ​​by overlapping adjacent control points to form the geometric model of the entire unit cell; the parameterized subdomain models formed by segmenting the characteristic unit cell geometric model are combined, and the overall calculation is performed by superimposing the stiffness values ​​of the control points; When calculating the stiffness matrix of the characteristic unit cell, the number of control points of the unit cell geometric model is adjusted to obtain equal geometric grids of different scales; Step 3: Perform isogeometric analysis according to the minimum potential energy principle to calculate the stiffness matrix of the characteristic unit cell; Step 4: applying linear or periodic displacement boundary conditions to the characteristic unit cell, and constructing the basis function requires solving the equilibrium equations inside and at the boundary of the unit cell; Step 5, after applying the boundary conditions, solving the equilibrium equation inside the characteristic unit cell, and obtaining the displacement as the basis function; Step 6, obtaining the equivalent stiffness matrix of the characteristic unit cell according to the displacement basis function obtained in step 5; Step 7, according to the mapping relationship from local to global, the equivalent unit stiffness matrix is ​​converted from the local coordinate system to the global coordinate system, and the equivalent global stiffness matrix of the macro structure is calculated; Step 8: For the load matrix, transform the local coordinate system to the global coordinate system so as to perform calculations on a macro scale; Step 9, obtaining the result of the macroscopic deformation displacement of the whole structure by solving the linear equation group, wherein the result is the macroscopic node displacement; Step 10: After obtaining the macroscopic node displacement, the microscopic displacement and stress-strain information inside the characteristic unit cell are calculated by matrix multiplication.

2. A mechanical simulation analysis method for analyzing a two-dimensional periodic inhomogeneous structure according to claim 1, characterized in that: The stiffness matrix of the characteristic unit cell is: Among them, U is the control point displacement matrix, K e is the control unit stiffness matrix, G is the displacement strain matrix, F e is the external force matrix.

3. A mechanical simulation analysis method for analyzing a two-dimensional periodic inhomogeneous structure according to claim 1, characterized in that: Constructing basis functions requires solving the equilibrium equations inside and at the boundaries of the unit: Where L represents the elastic operator, satisfying: Among them, N i Represents the basis function of each node of the macro unit. For two-dimensional plane problems, m=4; for two-dimensional vector field problems, it is necessary to construct basis functions in the x and y directions respectively.

4. The mechanical simulation analysis method for analyzing a two-dimensional periodic inhomogeneous structure according to claim 1, characterized in that: Since each node has basis functions in the x and y directions, a total of 8 sets of equilibrium equations need to be solved to obtain 8 basis functions, whose specific form is: Among them, N is the displacement basis function, N ixx and N iyy is the basis function term. i=1,2,3,4,N ixy and N iyx is the additional coupling term due to the existence of Poisson's ratio; The basis function is verified by the properties of the basis function.

5. The mechanical simulation analysis method for analyzing a two-dimensional periodic inhomogeneous structure according to claim 4, characterized in that: The verification method for verifying the properties of the basis function is: Since it is a displacement basis function, the results of the basis function are verified by importing the model data and result data files into the visualization software in Hypermesh.

6. The mechanical simulation analysis method for analyzing a two-dimensional periodic inhomogeneous structure according to claim 4, characterized in that: The calculation method of the equivalent stiffness matrix of the characteristic unit cell is: K C =G T K S G; Among them, G represents the transformation matrix, which is a matrix formed by arranging the basis functions in a certain order, and its form is: K S represents the stiffness matrix of the characteristic unit cell, K C represents the four-node stiffness matrix of the macro unit, K C is an equivalent stiffness matrix of order 8×8.

7. The mechanical simulation analysis method for analyzing a two-dimensional periodic inhomogeneous structure according to claim 1, characterized in that: The conversion method for the load matrix is: F C =G T F S ; Among them, F C represents the macroscopic nodal force, F S Represents the element nodal force.

8. The mechanical simulation analysis method for analyzing a two-dimensional periodic inhomogeneous structure according to claim 1, characterized in that: The specific calculation formula for calculating the microscopic displacement and stress-strain information inside the characteristic unit cell by matrix multiplication is: s e =SU,S=D e B e G; Wherein, G represents the conversion matrix in step 6, and U represents the displacement of the microscopic node inside the characteristic unit cell. After obtaining the microscopic node displacement, the microscopic stress and strain are calculated. The specific formula is: ε e =TU,T=B e G; Among them, B e represents the displacement strain matrix of the characteristic unit cell, D e Represents the elastic matrix of the characteristic unit cell.

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