CN115292953A

CN115292953A - Mechanical simulation analysis method for analyzing two-dimensional periodic heterogeneous structure

Info

Publication number: CN115292953A
Application number: CN202210995460.7A
Authority: CN
Inventors: 夏阳; 牛宏泽; 刘辉
Original assignee: Dalian University of Technology
Current assignee: Dalian University of Technology
Priority date: 2022-08-18
Filing date: 2022-08-18
Publication date: 2022-11-04

Abstract

The invention provides a mechanical simulation analysis method for analyzing a two-dimensional periodic heterogeneous structure, which considers the simulation of a multi-scale model by using an isogeometric analysis method. Compared with the expansion of multi-scale finite elements, the geometric analysis of expansion of multi-scale and the like can solve the problem of isomerism of an analysis model and a physical model, and the calculation precision is improved.

Description

Mechanical simulation analysis method for analyzing two-dimensional periodic heterogeneous structure

Technical Field

The invention relates to the technical field of mechanical analysis methods, in particular to a mechanical simulation analysis method for analyzing a two-dimensional periodic heterogeneous structure.

Background

The existing extended multi-scale method is mainly an extended multi-scale finite element method, and the main idea of the method is to construct a numerical basis function by solving a local subproblem inside a unit. The balance equation is solved by applying a certain boundary condition to the subdomain, and the displacement of each node in the subdomain is obtained, namely a numerical value basis function, which is also called a displacement basis function. The basis functions can accurately and effectively reflect the microscopic heterogeneity of the material, so that an accurate and effective solution can be obtained on a macroscopic level, the problem solving on the microscopic level is avoided, and the computing resources are greatly saved. The material properties inside each cell may be non-homogeneous.

A common microstructure is shown in fig. 1, and contains properties similar to pores, or non-homogeneity of the material.

As shown in fig. 2 (a), in the fine mesh partitioning model, only a representative group of fine meshes needs to be selected, and the group of fine mesh stiffness arrays is equivalent to an equivalent unit stiffness array having four-node coarse meshes through the obtained basis functions. The equivalent unit stiffness arrays are assembled into an integral stiffness array, so that the degree of freedom of analysis model calculation is as shown in fig. 2 (b) in actual calculation, compared with fig. 2 (a), the degree of freedom of calculation can be greatly reduced, and meanwhile, after node displacement is obtained, simple matrix multiplication operation is carried out through a basis function, and node displacement information in the fine grid can be obtained. Compared with a fine grid division model, the process of solving the balance equation is reduced, and the calculation efficiency is improved.

The most important step in expanding the multi-scale finite element is the calculation of the basis function, different basis function construction modes can be selected for different types of problems, a linear boundary condition construction mode can be adopted for a homogeneous structure, and a periodic boundary condition construction mode can be selected for a periodic structure. Different basis function construction modes are essentially to apply different boundary conditions, and the obtained fine grid internal node displacement is the basis function by solving a balance equation.

The method for expanding the multi-scale finite element has good calculation effect in multiple groups of calculation examples, the displacement and stress distribution and the reference solution are in good accordance, and the calculation speed is obviously improved.

Meanwhile, when the finite element method is adopted for multi-scale, for some structures such as smooth curved surfaces or curves, the model cannot be accurately described by using the method of dispersing the finite element units, and the analysis model is also calculated by using the discretized model, so that the problem that the actual model is inconsistent with the analysis model is caused, and the precision of the model is influenced to a certain extent. If accurate results are desired, the partitioning is often very fine, which increases the amount of computation. Meanwhile, the division of the model meshes in the finite element pretreatment is also a complex process.

In the prior art, an isogeometric analysis method exists, the isogeometric analysis method is a new numerical calculation method, compared with a method that a finite element adopts discretization, the isogeometric analysis adopts a spline function to describe a model, the spline function is used to accurately express the model, and the isogeometric analysis method is more accurate than the finite element expression, such as a smooth curved surface curve structure. Meanwhile, the process of dividing meshes by using finite elements is avoided. In the analysis and calculation, the basis functions adopted by the isogeometric analysis are spline functions used for expressing the model, so the isogeometric analysis is more advantageous than the finite element in the aspect of calculation precision.

Isogeometric analysis As shown in FIG. 3, for a smooth curve structure, first, parameter domains in two directions are constructed, the parameter domains are divided by node vectors, spline functions are generated through control points and weight factors, the spline functions in the two directions generate smooth curved surfaces in a tensor product form, and curved surfaces in different forms can be expressed by adjusting the control points and the weight factors. The analytical calculation is also spline function, and the numerical calculation is carried out on parameter domain, and the parameter domain and physical domain are mapped by spline function.

Disclosure of Invention

In light of the above-mentioned technical problems in the background, a method for mechanical simulation analysis for analyzing a two-dimensional periodic heterogeneous structure is provided. The invention considers that the geometric analysis method is used for expanding multi-scale, because the geometric analysis method adopts the spline function to describe the geometric model, compared with finite elements, the description of certain smooth curve surface structures is more accurate, and the basis function adopted in the analysis and calculation of the geometric analysis is also the spline function, thereby realizing the unification of the physical model and the analysis model. Compared with the expansion of multi-scale finite elements, the geometric analysis of expansion of multi-scale and the like can solve the problem of isomerism of an analysis model and a physical model, and the calculation precision is improved.

The technical means adopted by the invention are as follows:

a mechanical simulation analysis method for analyzing a two-dimensional periodic heterogeneous structure is characterized by comprising the following steps:

step 1: for a periodic heterogeneous structure, extracting a characteristic unit cell of the periodic heterogeneous structure, wherein the domain of the extracted characteristic unit cell is a square, and constructing a basis function by applying different boundary conditions;

step 2: respectively constructing a geometric model and an isogeometric grid of the characteristic unit cell through NURBS basis functions;

when the signature unit cell is purely homogeneous, expressing through a set of parameterized domains;

when the characteristic unit cell contains holes or heterogeneous material areas, the model is divided according to specific conditions to form a plurality of parametric domains, and then all the parametric domain models are combined according to a multi-slice splicing method during calculation to perform overall calculation;

different thinning times can be set according to the precision requirement to obtain isogeometric grids with different precisions;

and step 3: obtaining an isogeometric analysis column according to a potential energy method, and calculating to obtain a stiffness array of the characteristic unit cell;

and 4, step 4: applying specific displacement boundary conditions to the characteristic unit cells, wherein the application mode of the boundary conditions is shown in FIG. 9, and the construction of the basis functions needs to solve the balance equations of the cell interior and the boundary:

step 5, after the boundary condition is applied, solving a balance equation in the characteristic unit cell to obtain displacement which is a basis function;

step 6, obtaining an equivalent stiffness array of the characteristic unit cell according to the displacement basis function obtained in the step 4;

step 7, calculating an equivalent integral stiffness array of the macrostructure according to the mapping relation from the local part to the whole and the equivalent unit stiffness array according to the mapping relation from the local part to the whole;

step 8, carrying out primary conversion on the load array so as to calculate on a macroscopic scale;

step 9, solving a linear equation set to obtain a result of macroscopic deformation displacement of the whole structure, wherein the result is macroscopic node displacement;

and step 10, after the macroscopic node displacement is obtained, calculating microscopic displacement and stress strain information inside the characteristic unit cell by matrix multiplication.

Compared with the prior art, the invention has the following advantages:

compared with a fine simulation result, the method for the extended multi-scale isogeometric analysis is effective in analysis of periodic heterogeneous structural materials, can obtain a more accurate micro-displacement result on the basis of greatly reducing calculated amount, and is more accurate in the result of the extended multi-scale isogeometric statics and dynamics analysis compared with the traditional finite element method.

Drawings

In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings needed to be used in the description of the embodiments or the prior art will be briefly introduced below, and it is obvious that the drawings in the following description are some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to these drawings without creative efforts.

Fig. 1 is a schematic view of a microstructure commonly used in the prior art.

FIG. 2 (a) is a schematic diagram of a model for fine meshing; and (b) is a degree of freedom diagram calculated by an analysis model.

FIG. 3 is a schematic diagram of isogeometric analysis.

FIG. 4 is a flow chart of the extended multi-scale isopeometric method of the present invention.

FIG. 5 illustrates an exemplary periodic heterogeneous structure according to the present invention.

FIG. 6 is a schematic diagram of the present invention showing the constraint and stress conditions.

FIG. 7 is a schematic diagram of a characteristic unit cell of the present invention.

FIG. 8 is a schematic view of the present invention in a slice format.

FIG. 9 is a representation of the NURBS spline model of the present invention; wherein, (a) and (d) are parameter space geometric model schematic diagrams; (b) and (e) are control point grid schematic diagrams; and (c) and (f) are schematic diagrams of physical space geometric models.

FIG. 10 illustrates two boundary condition applying methods; wherein, (a) is a linear boundary condition application mode; and (b) a periodic boundary condition application mode.

Detailed Description

In order to make those skilled in the art better understand the technical solutions of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.

It should be noted that the terms "first," "second," and the like in the description and claims of the present invention and in the drawings described above are used for distinguishing between similar elements and not necessarily for describing a particular sequential or chronological order. It is to be understood that the data so used is interchangeable under appropriate circumstances such that the embodiments of the invention described herein are capable of operation in other sequences than those illustrated or described herein. Furthermore, the terms "comprises," "comprising," and "having," and any variations thereof, are intended to cover a non-exclusive inclusion, such that a process, method, system, article, or apparatus that comprises a list of steps or elements is not necessarily limited to those steps or elements expressly listed, but may include other steps or elements not expressly listed or inherent to such process, method, article, or apparatus.

As shown in fig. 1-10, the present invention provides a mechanical simulation analysis method for analyzing a two-dimensional periodic heterogeneous structure, comprising the following steps:

step 1: for a periodic heterogeneous structure, extracting characteristic unit cells of the periodic heterogeneous structure, wherein the domain of the extracted characteristic unit cells is square, and constructing a basis function by applying different boundary conditions;

step 2: respectively constructing a geometric model and an isogeometric grid of the characteristic unit cell through the NURBS basis function; when the signature unit cell is purely homogeneous, expression is performed 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 specific conditions to form a plurality of subdomains described by the parameterized model, and the subdomains are spliced in a mode of overlapping adjacent control points to form the geometric model of the whole unit cell; combining parameterized subdomain models formed by dividing the characteristic unit cell geometric model, and performing overall calculation in a rigidity value superposition mode of control points; when the rigidity matrix of the characteristic unit cell is calculated, the number of control points of the unit cell geometric model can be adjusted to obtain the isogeometric grids with different scales.

And step 3: and (4) performing an isogeometric analysis formula according to a minimum potential energy principle, and calculating to obtain a stiffness array of the characteristic unit cell. The stiffness matrix of the characteristic unit cells is:

wherein U is a control point displacement array, K ^e For controlling the cell stiffness matrix, G is the displacement strain matrix, F ^e Is a matrix of external forces.

And 4, step 4: applying linear or periodic displacement boundary conditions to the characteristic unit cells, and constructing a basis function to solve a balance equation of the interior and the boundary of the unit; constructing the basis functions requires solving the equilibrium equations inside the cell and at the boundaries as:

wherein L represents an elastic operator, satisfying:

wherein, N _i Representing the basis functions of nodes of the macro unit, and for a two-dimensional plane problem, m =4; for the two-dimensional vector field problem, basis functions need to be constructed separately in both x and y directions.

Step 5, after the boundary condition is applied, solving a balance equation in the characteristic unit cell to obtain displacement which is a basis function; because each node has basis functions in x and y directions, 8 groups of balance equations need to be solved to obtain 8 basis functions, and the specific form is as follows:

wherein N is a shift basis function, N _ixx And N _iyy Is a basis function term i =1,2,3,4,N _ixy And N _iyx Is an additional coupling term due to the presence of the poisson's ratio;

for the basis functions, verification is performed by the nature of the basis functions. The verification method for verifying the properties of the basis functions comprises the following steps:

as the basic functions are displacement basic functions, the results of the basic functions are verified by importing model data and result data files into visualization software in Hypermesh.

Step 6, obtaining an equivalent stiffness array of the characteristic unit cell according to the displacement basis function obtained in the step 4; the calculation mode of the equivalent stiffness array of the characteristic unit cell is as follows:

K _C ＝G ^T K _S G；

wherein, G represents a transformation matrix, which is a matrix formed by arranging basis functions according to a certain sequence, and the form is as follows:

K _S stiffness array representing characteristic unit cells, K _C Representing a four-node stiffness array of macro-units, K _C Is an equivalent stiffness matrix of order 8 x 8.

Step 7, converting the equivalent unit stiffness array from a local coordinate system to a global coordinate system according to the mapping relation from local to global, and calculating the equivalent global stiffness array of the macrostructure;

step 8, for the load array, converting a local coordinate system into a whole coordinate system so as to calculate on a macroscopic scale; the conversion mode for load matrix conversion is as follows:

F _C ＝G ^T F _S ；

wherein, F _C Represents the macroscopic nodal force, F _S Representing the unit node force.

And 9, solving a linear equation set to obtain a result of macroscopic deformation displacement of the whole structure, wherein the result is macroscopic node displacement.

And step 10, after the macroscopic node displacement is obtained, calculating microscopic displacement and stress strain information inside the characteristic unit cell 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 as follows:

σ _e ＝SU，S＝D _e B _e G；

wherein G represents the transformation matrix in the step 6, U represents the displacement of the microcosmic nodes in the characteristic unit cell, and after the displacement of the microcosmic nodes is obtained, the microcosmic stress strain is calculated by the following specific formula:

ε _e ＝TU，T＝B _e G；

wherein, B _e Representing the displacement-strain matrix of the characteristic unit cell, D _e Representing an elastic matrix of characteristic cells.

The calculation is carried out for a simple calculation example, the right end of a beam structure is subjected to upward force, and the displacement of the central axis of the beam is measured and compared.

And comparing the calculated result with the Hypermesh accurate simulation result, and mainly improving the calculation precision under the condition of the same degree of freedom of the characteristic unit cells.

Calculation method	Characteristic unit cell degree of freedom	Coarse mesh	Norm error
				Expanding multi-scale iso-geometry	526	20×4	0.15％
Expanding multi-scale finite elements	526	20×4	1.7％
				Hypermesh	1049	20×4	/

Example 1

Periodic inclusion structure, as shown in FIG. 6, coarse meshThe grid size is 20 multiplied by 4, the characteristic sub-grid is a square plate containing an inclusion structure, the side length is 1, the inclusion model in the sub-grid is a circle with r =0.15, and the material parameter is Young modulus 10 ⁶ Poisson's ratio of 0.4 and Young's modulus of the exterior of the subgrid of 10 ⁴ And the Poisson ratio is 0.3. When the left end is fixed and the right end upwards concentrates force F =4, the result obtained by using the geometric method of expanding multi-scale and the like is highly consistent with the Hypermesh simulation result for the displacement of the central axis. Compared with the method for expanding the multi-scale finite element with the same calculation degree of freedom, the error of the two norms is obviously reduced (the error of the geometry such as the expanded multi-scale is 0.15 percent, and the error of the expanded multi-scale finite element is 1.7 percent).

In example 1, the characteristic cells were refined twice and the resulting isogeometric grid is shown in fig. 8, while for example 1 the macroscopic model is a 20 x 4 grid.

Example 2

Considering engineering practice, calculations were performed on a simplified flexible electronic model comprising two parts of material, a substrate part and an electronics part. The two materials have different properties, and the material parameter of the substrate part is Young modulus 10 ⁶ Poisson's ratio of 0.4 and Young's modulus of flexible electronic device portion of 10 ⁴ The Poisson's ratio was 0.3. And calculating the whole displacement and the microscopic displacement distribution of the displacement, wherein the microscopic displacement distribution result is highly consistent with the Hypermesh simulation result. Meanwhile, under the same computing resource, the results of the geometric method such as the expansion multi-scale and the like and the finite element full-scale simulation are computed, under two boundary conditions of the geometric method such as the expansion multi-scale and the like, the computing time is respectively 18.146 seconds and 19.233 seconds, and the computing time under the full scale is 225.237 seconds, so that the computing time can be greatly shortened while the precision is kept by the geometric method such as the expansion multi-scale and the like.

The above-mentioned serial numbers of the embodiments of the present invention are only for description, and do not represent the advantages and disadvantages of the embodiments. In the above embodiments of the present invention, the descriptions of the respective embodiments have respective emphasis, and for parts that are not described in detail in a certain embodiment, reference may be made to related descriptions of other embodiments.

In the embodiments provided in the present application, it should be understood that the disclosed technology can be implemented in other ways. Finally, it should be noted that: the above embodiments are only used to illustrate the technical solution of the present invention, and not to limit the same; while the invention has been described in detail and with reference to the foregoing embodiments, it will be understood by those skilled in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some or all of the technical features may be equivalently replaced; and the modifications or the substitutions do not make the essence of the corresponding technical solutions depart 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 heterogeneous structure is characterized by comprising the following steps:

and 2, step: respectively constructing a geometric model and an isogeometric grid of the characteristic unit cells through NURBS basis functions;

when the characteristic unit cell contains holes or heterogeneous material regions, the geometric model of the characteristic unit cell is divided according to specific conditions to form a plurality of subdomains described by the parameterized model, and the subdomains are spliced in a mode that adjacent control points are overlapped to form the geometric model of the whole unit cell; combining parametric subdomain models formed by dividing the characteristic unit cell geometric model, and performing overall calculation in a mode of overlapping rigidity values of control points;

when the rigidity matrix of the characteristic unit cell is calculated, the number of control points of the unit cell geometric model can be adjusted to obtain equal geometric grids of different scales;

and step 3: performing an isogeometric analysis column according to a minimum potential energy principle, and calculating to obtain a stiffness array of the characteristic unit cell;

and 4, step 4: applying linear or periodic displacement boundary conditions to the characteristic unit cells, and constructing a basis function to solve balance equations of the interior and the boundary of the unit;

step 6, obtaining an equivalent stiffness array of the characteristic unit cell according to the displacement basis function obtained in the step 5;

step 8, for the load array, converting a local coordinate system into a whole coordinate system so as to calculate on a macroscopic scale;

and step 10, after the macroscopic node displacement is obtained, calculating microscopic displacement and stress strain information inside the characteristic unit cell through matrix multiplication.

2. The mechanical simulation analysis method for analyzing a two-dimensional periodic heterogeneous structure according to claim 1, wherein the stiffness matrix of the feature unit cells is:

wherein U is a control point displacement array, K ^e For controlling the unit stiffness matrix, G is the displacement strain matrix, F ^e Is a matrix of external forces.

3. A mechanical simulation analysis method for analyzing a two-dimensional periodic heterogeneous structure according to claim 1, wherein the balance equations of the cell interior and the boundary needed to be solved for constructing the basis functions are:

wherein L represents an elasticity operator, satisfying:

wherein N is _i Representing the basis functions of each node of the macro unit, and for a two-dimensional plane problem, m =4; for the two-dimensional vector field problem, basis functions need to be constructed separately in both x and y directions.

4. The mechanical simulation analysis method for analyzing the two-dimensional periodic heterogeneous structure according to claim 1, wherein:

because each node has basis functions in x and y directions, 8 groups of balance equations need to be solved to obtain 8 basis functions, and the specific form is as follows:

and verifying the base functions through the properties of the base functions.

5. The mechanical simulation analysis method for analyzing the two-dimensional periodic heterogeneous structure according to claim 4, wherein: the verification method for verifying the property of the basis function comprises the following steps:

6. The mechanical simulation analysis method for analyzing the two-dimensional periodic heterogeneous structure according to claim 4, wherein: the calculation mode of the equivalent stiffness array of the characteristic unit cells is as follows:

K _C ＝G ^T K _S G；

K _S stiffness matrix representing characteristic cells, K _C Representing a four-node stiffness array of macro-units, K _C Is an equivalent stiffness matrix of order 8 x 8.

7. The mechanical simulation analysis method for analyzing the two-dimensional periodic heterogeneous structure according to claim 1, wherein: the conversion mode for the load array conversion is as follows:

F _C ＝G ^T F _S ；

wherein, F _C Represents macroscopic nodal force, F _S Representing the unit node force.

8. The mechanical simulation analysis method for analyzing the two-dimensional periodic heterogeneous structure according to claim 1, wherein: the specific calculation formula for calculating the microscopic displacement and stress strain information inside the characteristic unit cell through matrix multiplication is as follows:

σ _e ＝SU,S＝D _e B _e G；

wherein G represents the transformation matrix in the step 6, U represents the displacement of the microscopic node inside the characteristic unit cell, and after the displacement of the microscopic node is obtained, the microscopic stress strain is calculated by the following specific formula:

ε _e ＝TU,T＝B _e G；