Piezoelectric particle composite material simulation method based on polygon-quadtree grid

Abstract

The invention relates to the technical field of piezoelectric particle composite materials, in particular to a piezoelectric particle composite material simulation method based on a polygon-quadtree grid, which comprises the following steps: S1, through an electromechanical coupling theory, determining a generalized equilibrium equation and a geometric equation, and defining boundary conditions; s2, constructing a constitutive equation and a complementary energy expression of electromechanical coupling; s3, deducing a functional expression by using a complementary energy principle; s4, constructing interpolation functions of a stress field and an electric displacement field by utilizing geometric characteristics of the polygon units; and S5, finely calculating a stiffness matrix of the polygon unit through a Gaussian integral method. The electromechanical coupling hybrid finite element method (EM-HFEM) is adopted, the continuity of displacement and electric displacement is effectively processed by introducing a Lagrange multiplier between units, free selection of boundaries and unit shapes is allowed, and dependence on grid quality is greatly reduced.

Description

Technical Field

[0001] The present invention relates to the technical field of piezoelectric particulate composites, and particularly relates to a simulation method of piezoelectric particulate composites based on polygon - quadtree meshes. Background Art

[0002] Piezoelectric material structures have unique piezoelectric - electro - mechanical coupling effects, and thus have excellent applications in sensors and actuators. They are continuously applied in fields such as manufacturing, automotive, aerospace engineering, medical instruments, information communication, etc. However, pure ceramic - based piezoelectric materials have some problems, such as high acoustic impedance, large weight, and low ductility. To overcome the inherent drawbacks of piezoelectric ceramics, piezoelectric particulate composites have been developed, which combine the excellent piezoelectric properties of piezoelectric ceramics with the mechanical properties of polymer matrices, have higher piezoelectric - electro - mechanical coupling characteristics compared with pure piezoelectric ceramics, and have higher flexibility and toughness. These composites are characterized by embedding particulate piezoelectric phases in polymer matrices.

[0003] With the rapid development of computer hardware and software, numerical calculation methods have become an essential means for solving piezoelectric structural mechanics problems. Among traditional numerical analysis methods, the finite element method (FEM) is widely used to solve electromechanical coupling problems. However, the calculation accuracy of the finite element method is greatly affected by the quality of model elements, so it is not suitable for large - deformation and mesh distortion situations. Especially for multiphase composites, it takes a lot of time to divide a large number of meshes at the interfaces of different materials to describe the interface discontinuities. In recent years, efforts have been made to construct new mesh - generation algorithms, but these methods still cannot solve problems related to composites.

[0004] In addition to the finite element method, new numerical analysis methods for electromechanical coupling are also being developed, such as the boundary element method and the meshless method. The BEM greatly simplifies the mesh generation while maintaining high accuracy, but when solving problems of heterogeneous materials, internal elements still need to be divided, which limits the application of the BEM. A variety of meshless methods have been developed and extended to the analysis of piezoelectric problems. Although there are many discretization methods, the calculation efficiency is still a problem.

[0005] For the case of randomly distributed large-scale inclusions, it is difficult to simulate using traditional finite element methods, and a large number of fine meshes need to be divided at the matrix and interface. To reduce the computational cost, the flexible performance of arbitrary polygons is considered to accurately represent the geometric features between the matrix and inclusions. The electromechanical coupling hybrid finite element method (EM-HFEM) proposed in the present invention is applicable to arbitrary polygon mesh models. In the model, the stress, electric displacement within each element domain, and the displacement and electric potential at the element boundary are all independently assumed. The equilibrium at the inter-element boundary is relaxed by introducing displacement and electric potential as Lagrange multipliers into the piezoelectric hybrid complementary energy functional, and a correction function is established. A function is constructed to relate the stress function, displacement variable, electric displacement function, and electric potential variable. The equation is transformed into a solvable form and presented in the form of the standard solution of FEM.

[0006] Quadtree meshes are very suitable for EM-HFEM because there is no need to impose additional restrictions on EM-HFEM due to hanging nodes. Each element can be modeled as a polygon with an arbitrary number of sides. Since quadtree meshes can finitely disperse geometric features of different scales, only arbitrary-shaped polygon elements are used at the matrix-inclusion interface, making it better fit the irregular curve at the interface. The mesh combining polygon and quadtree enables EM-HFEM to have higher accuracy with as few elements as possible. At the same time, the stiffness matrix of the finite element can be pre-stored and quickly read when needed, which greatly improves the computational efficiency, and only the polygon elements located at the boundary of two different materials are calculated.

[0007] The present invention develops EM-HFEM combined with polygon-quadtree meshes for piezoelectric particulate composites to calculate the stress and electric displacement of piezoelectric particulate composites. The outline of the rest of the present invention is as follows: First, the governing equations of the electromechanical coupling problem are introduced, and the state function of the electromechanical coupling problem is derived according to the constitutive equations. Next, the minimum complementary energy functional of the electromechanical coupling problem is established using the weighted residual method, and an arbitrary polygon model is constructed accordingly, and its correction function is established. Finally, by comparing five numerical examples with the traditional finite element software ABAQUS, the accuracy and efficiency of EM-HFEM combined with polygon-quadtree meshes are demonstrated. Summary of the Invention

[0008] The purpose of the present invention is to provide a simulation method for piezoelectric particulate composites based on polygon-quadtree meshes, using the electromechanical coupling hybrid finite element method (EM-HFEM). By introducing Lagrange multipliers between elements, the continuity of displacement and electric displacement is effectively processed, allowing free selection of boundaries and element shapes, greatly reducing the dependence on mesh quality. This method not only increases the adaptability of the model but also improves its computational stability when dealing with complex material systems, ensuring physical consistency between different elements.

[0009] To achieve the above technical objectives and effects, the present invention is realized through the following technical solutions:

[0010] A simulation method for piezoelectric particulate composites based on polygon - quadtree meshes, comprising the following steps:

[0011] S1: Through the electromechanical coupling theory, establish the generalized equilibrium equations and geometric equations, and clarify the boundary conditions, including natural conditions (such as free boundaries) and forced conditions (such as fixed boundaries or electric field application);

[0012] S2: Through the elastic constants, piezoelectric coefficients, and dielectric constants of piezoelectric materials, introduce material parameters to construct the constitutive equations of electromechanical coupling; through the increments of the generalized stress function and state function, derive the state function and complementary energy expressions for electromechanical coupling problems;

[0013] S3: Use the complementary energy principle to derive the functional expression, and apply the Gauss formula and boundary conditions to simplify the complementary energy functional into an expression containing stress, body force, electric displacement, and volume charge density.

[0014] S4: Utilize the geometric characteristics of polygon elements to construct interpolation functions for the stress field and electric displacement field, and represent the stress and electric displacement within each element in matrix form; introduce the Lagrange multiplier method to handle the continuity of displacement and electric displacement between elements, and combine the modified functional to ensure the satisfaction of natural boundary conditions and forced boundary conditions;

[0015] S5: Through the Gauss integration method, accurately calculate the stiffness matrix of polygon elements; by solving the stationary value conditions of the modified functional, form the stiffness matrix of elements and generate the final finite - element solution format; construct a polygon - quadtree mesh model and apply it to different numerical examples to verify the convergence and accuracy of the method.

[0016] Advantages of the present invention:

[0017] The present invention realizes the simulation of piezoelectric particulate composites by using polygon elements combined with quadtree meshes, greatly improving the accuracy and efficiency of calculations. Polygon elements allow for the geometric description of arbitrary shapes, enabling more accurate approximation of complex material boundaries and the morphology of embedded particles, while the layer - by - layer division characteristic of quadtree meshes enables intelligent adaptation of mesh generation to various regions of complex geometric structures, especially outstanding in the detailed representation of interfaces and heterogeneous regions. This not only improves the ability to capture the true physical characteristics of materials but also reduces the demand for computing resources, achieving higher accuracy and faster computing speed through an efficient finite - element solution format.

[0018] The traditional finite element method is prone to mesh distortion in the case of large deformations and complex geometric structures, leading to computational instability. In contrast, the electro-mechanical coupled hybrid finite element method (EM-HFEM) adopted in this method effectively processes the continuity of displacements and electric displacements by introducing Lagrange multipliers between elements, allowing free selection of boundaries and element shapes, and greatly reducing the dependence on mesh quality. This method not only increases the adaptability of the model but also improves its computational stability when dealing with complex material systems, ensuring physical consistency between different elements.

[0019] In piezoelectric particulate composites, the accurate simulation of the matrix and particle interface is crucial. The present invention effectively solves the high-density mesh requirement of traditional methods in interface division through a flexible combination of polygon-quadtree meshes. Polygonal elements can accurately describe the irregular shape and microscopic contact characteristics of the interface, reducing numerical errors caused by discontinuous and irregular interfaces and improving the simulation accuracy of the electro-mechanical coupling characteristics of the interface. This method is particularly suitable for material systems containing complex inclusions and heterogeneous characteristics, significantly improving the computational reliability of the interface region.

[0020] The present invention combines the efficient electro-mechanical coupled hybrid finite element method to construct a modified complementary energy functional, accurately expressing the stress and electric displacement fields within the element domain through linear interpolation, and optimizing the allocation of computational resources. Since the polygon-quadtree mesh can effectively reduce the number of elements, especially when dealing with systems with complex internal structures and multiple inclusions, the solution efficiency is significantly improved while maintaining high precision requirements. This optimization of resources not only reduces the computational complexity but also meets the needs of large-scale material simulations, providing an efficient solution for large-scale simulation problems in engineering applications.

[0021] Of course, it is not necessary for any product implementing the present invention to achieve all the above-mentioned advantages simultaneously. Brief Description of the Drawings

[0022] To more clearly illustrate the technical solutions of the embodiments of the present invention, the following will briefly introduce the drawings required for describing the embodiments. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0023] Figure 1 Schematic diagram of the mechanical and electrical boundaries of the piezoelectric structure;

[0024] Figure 2 Schematic diagram of a typical polygonal element;

[0025] Figure 3 Model diagram and result diagram;

[0026] Figure 4 are the schematic diagrams of mesh distortion and the relative error between different mesh models and ABAQUS results;

[0027] Figure 5 are the model diagram and result diagram of a model containing a circular inclusion;

[0028] Figure 6 are the model diagram and result diagram of a model containing sixteen regularly arranged circular inclusions;

[0029] Figure 7 is the diagram for selecting stress and electric displacement paths;

[0030] Figure 8 are the model diagram and result diagram of a model containing sixteen randomly distributed circular inclusions;

[0031] Figure 9 are the model diagram and result diagram of a model containing sixteen randomly distributed elliptical inclusions. Specific implementation manners

[0032] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without making creative efforts belong to the scope of protection of the present invention.

[0033] Embodiment 1

[0034] A simulation method for piezoelectric particulate composites based on polygon - quadtree meshes described in this embodiment includes the following steps:

[0035] S1: Through the electromechanical coupling theory, establish the generalized equilibrium equation and geometric equation, and clarify the boundary conditions, including natural conditions (such as free boundaries) and forced conditions (such as fixed boundaries or electric field application);

[0036] S2: Introduce material parameters through the elastic constants, piezoelectric coefficients, and dielectric constants of piezoelectric materials to construct the constitutive equation of electromechanical coupling; Derive the state function and complementary energy expression of the electromechanical coupling problem through the increments of the generalized stress function and state function;

[0037] S3: Use the complementary energy principle to derive the functional expression, and apply Gauss's formula and boundary conditions to simplify the complementary energy functional into an expression containing stress, body force, electric displacement, and volume charge density.

[0038] S4: Utilize the geometric properties of the polygonal elements to construct interpolation functions for the stress field and the electric displacement field, and represent the stress and electric displacement within each element in matrix form; introduce the Lagrange multiplier method to handle the continuity of displacements and electric displacements between elements, and combine with the modified functional to ensure the satisfaction of natural boundary conditions and essential boundary conditions;

[0039] S5: Through the Gaussian integration method, accurately calculate the stiffness matrix of the polygonal elements. This involves line integrals over the element boundaries and area integrals over the element domains. By solving the stationary value conditions of the modified functional, form the stiffness matrix of the elements and generate the final finite element solution format; construct a polygon-quadtree mesh model and apply it to different numerical examples to verify the convergence and accuracy of the method.

[0040] Example 2

[0041] Governing equations

[0042] The basic variables of the electromechanical coupling problem are divided into mechanical field variables and electric field variables. They mainly include generalized displacements, generalized stresses, and generalized strains. Divide the basic variables according to the physical fields, and their symbols in the mechanical field and the electric field are shown in Table 1.

[0043] Table 1 Basic variables in the force-electric coupling field

[0044]

[0045] Considering piezoelectric materials within the linear range, there is a finite closed region in the two-dimensional space Ω, whose boundary includes the element boundary S i , displacement boundary S u , traction boundary S t , electromotive force boundary S φ and charge density boundary S ω .

[0046] For two-dimensional problems, in the Cartesian coordinate system (x, y), the generalized equilibrium equations for the electromechanical coupling problem are

[0047]

[0048] D i,i - ρ = 0 (2)

[0049] where σ ij is the stress; is the body force; D i is the electric displacement; ρ is the volume charge density.

[0050] The generalized geometric equations for the electromechanical coupling problem are

[0051]

[0052] E i + φ .i = 0 (4)

[0053] where ε ij is the strain; u i is the displacement; E i is the electric field strength; φ is the electric potential.

[0054] For the electromechanical coupling problem, the boundary conditions in the mechanical field and the electric field need to satisfy the natural boundary conditions and the forced boundary conditions as shown in Figure (1).

[0055] The generalized displacement boundary conditions satisfied in the electromechanical coupling problem are

[0056]

[0057] The generalized surface force boundary conditions that should be satisfied are

[0058]

[0059] Constitutive equations and state functions

[0060] For a quasi-static infinitesimal reversible process, the increment dH of the state function H of the electromechanical coupling problem is affected by mechanics and electricity, and its expression is

[0061] dH = σdε + DdE (9)

[0062] where:

[0063]

[0064] Taking the total differential of the generalized stress

[0065]

[0066] Obtained from Equation (12) and Equation (13)

[0067]

[0068] Introducing material parameters into Equations (14) and (15), we get

[0069]

[0070] Substituting Equations (16), (17), and (18) into (12) and (13), we obtain the constitutive equations of the electromechanical coupling problem

[0071]

[0072]

[0073] Where c, e, and d are the elastic constant, piezoelectric coefficient, and dielectric constant, respectively.

[0074] Assume that the piezoelectric material is two-dimensional isotropic in the x-y plane. Under the plane stress state, the constitutive relation equation can be obtained as follows:

[0075]

[0076] Substitute Equation (20) into Equation (9) to obtain the expression of the co-electroenthalpy of the state function for the electromechanical coupling problem

[0077]

[0078] Its tensor form is

[0079]

[0080] Derivation of the functional of the complementary energy principle

[0081] The weighted residual method is used to handle the geometric equations and boundary conditions. Let δG ij and δH i be arbitrary functions in the domain Ω, and δJ i and δK i be arbitrary functions on the boundary S u and the boundary S φ According to the requirement that the total complementary energy takes an extreme value to derive the generalized geometric equations (3)(4) and the generalized displacement boundary conditions (5)(6) based on the minimum complementary energy principle, we can obtain

[0082]

[0083] Equation (24) is similar to the virtual force principle. If we take δG ij = δσ ij , δH i = δD i , then we have

[0084] Apply Gauss's formula to obtain

[0085]

[0086] From the generalized equilibrium equations (1)(2) and the generalized stress boundary conditions (7)(8), we get

[0087] Substitute Equations (28)(29) into (25) to obtain

[0088]

[0089] Take δJ i = δσ ij n j, δK i = δD i n i , then there is

[0090]

[0091] Substitute the constitutive equation (20) into equation (31) to obtain

[0092]

[0093] Considering and being known quantities on the boundary, so equation (32) can be written as

[0094] Derivation of the electromechanical coupling hybrid finite element functional

[0095] Principle of constructing arbitrary polygon elements

[0096] The microstructure of the polygon mesh is as Figure 1 shown. Using σ i , D i to represent the stress field and the electric displacement field in Ω respectively, and using u i , φ to represent the displacement and electric potential on the boundary , and n e is the outer normal direction of the boundary.

[0097] The boundary satisfies:

[0098] The matrix form of equation (33) can be written as

[0099]

[0100] Considering the boundary conditions (5), (6) and the force continuity and electric displacement continuity between elements

[0101]

[0102] Apply the Lagrange multiplier method to introduce the above constraint conditions, and the modified functional is obtained as

[0103]

[0104] When applying the hybrid finite element method, the stress field and the electric displacement field in each element are represented as a complete polynomial in the Cartesian coordinate system

[0105]

[0106] On the boundary of the element, the displacement and the electric potential are interpolated by the generalized nodal displacements and the generalized nodal electric potentials, and are expressed as

[0107]

[0108] where L is the boundary interpolation function. For linear interpolation between two adjacent nodes i and i + 1, the function is expressed as

[0109]

[0110] Substituting equations (38) and (39) into equation (37), we get

[0111]

[0112] where

[0113]

[0114] According to the stationary condition of the modified complementary energy:

[0115] we can obtain

[0116]

[0117] Substituting formula (42) into (43), we get

[0118]

[0119] Equation (44) is the final finite element solution format, where K is the element stiffness matrix, expressed as

[0120]

[0121] Construct the stress function and the electric displacement function

[0122] In the finite element implementation, the element self - balancing stress function and the electric displacement function are introduced. For the convenience of subsequent differential and integral operations, it is assumed that the stress field and the electric displacement field are polynomials, represented by the interpolation matrix and the unknown interpolation coefficients. Since the order of the polynomial function is increased, the calculation accuracy is improved.

[0123] For plane problems, the polynomial stress function and the electric displacement function can be expressed by Φ 1 , Φ 2 as follows

[0124]

[0125] By differentiating the stress function and the electric displacement function, the stress and the electric displacement at each point of the element can be obtained

[0126]

[0127] Taking the third - order polynomial stress function and electric displacement function as examples, the stress and electric displacement can be expressed as

[0128]

[0129] where

[0130]

[0131]

[0132] Numerical integration

[0133] It can be seen from the above equation (45) that the calculation of the element stiffness matrix is essentially the calculation of the H matrix and the G matrix. G e1 and G e2 are different because of their corresponding and matrix structures and constitutive relationships. However, the calculation methods are the same.

[0134] In this embodiment, the Gaussian integration method is used to perform numerical integration on the H matrix and the G matrix. For two - dimensional polygonal hybrid elements, the integration of G is a line integral on the element boundary as

[0135]

[0136] According to equation (51), its Gaussian integration formula is

[0137]

[0138] The integration of H is an area integral in the element domain, and the area integral form of Gaussian integration is

[0139]

[0140] According to formula (53), its Gaussian integration formula is

[0141]

[0142] Example 3

[0143] Numerical example

[0144] Comparing the calculation results with the traditional finite element is undoubtedly the most convenient and practical method to evaluate the performance and practicality of the previously derived elements. In this embodiment, five numerical examples will be used to illustrate the effectiveness and efficiency of the proposed method in analyzing piezoelectric materials. Table 1 shows the material parameters of the five models. In the calculation, for each 2×2mm 2 model, the electric potential of the lower boundary is set to 0. The displacements of the left boundary and the upper and lower boundaries are fixed, and a horizontal displacement of 0.0015mm is applied to the right boundary.

[0145] Table 2 Material Properties

[0146]

[0147] It should be noted that in the unit system, the orders of magnitude of C ij , d ij differ too much, resulting in unsatisfactory calculation results. To improve this problem, normalization is used

[0148]

[0149] When using a pure quadtree mesh for calculation, the element stiffness matrices of all elements can be directly read and called, greatly improving the calculation efficiency. However, all pure quadtree elements are regular squares, and the jagged areas at the matrix-inclusion boundary will make the calculation results inconsistent with the actual situation. To solve this problem, near the matrix-inclusion interface, arbitrary-shaped polygon elements are adopted in the model to better fit the irregular curves at the matrix-inclusion boundary and have higher accuracy. However, since they are irregular arbitrary-node elements and the number of element sides is not fixed, the element stiffness matrices, H matrices, and G matrices of polygon elements cannot be pre-calculated, which sacrifices a certain amount of calculation efficiency. Each of the two meshes has its own advantages and disadvantages, and can be selected according to the requirements of the solution accuracy and calculation efficiency for specific problems. Table 3 compares the number of meshes of ABAQUS mesh generation, polygon-quadtree mesh generation, and pure quadtree mesh generation.

[0150] Table 3 Number of Meshes

[0151]

[0152] Algorithm Verification

[0153] To verify the correctness of the algorithm, in this embodiment, first, for a homogeneous material model without inclusions, the finite element software ABAQUS and EM-HFEM are used for solution respectively. Generally, it is considered that when the mesh accuracy is high enough, the results are infinitely close to the real situation. The model working conditions are as described above, and the material parameters of the inclusions in Table 1 are selected.

[0154] Generally, it is considered that when the ABAQUS mesh accuracy is high enough, the calculation results are infinitely close to the real situation. Figure 3 (a) is the working condition diagram of the model, Figure 3 (b) is the ABAQUS 64×64 element model, Figure 3 (c) is the EM-HFEM 2×2 element model. Figure 3 (d)(e)(f) are the result comparison diagrams.

[0155] As described in the introduction, for irregular structures, the mesh quality is usually uneven, and computational errors inevitably occur in FEM. Based on Figure 3 the model in (c), two distortion schemes are proposed as shown in Figure 4 (b) and (c). The effectiveness and performance of EM-HFEM in harsh environments are demonstrated by calculating the relative error between the results and those of ABAQUS.

[0156] Figure 4 (d), (e), and (f) represent the relative errors obtained by comparing the results of ABAQUS with those of different mesh models in 4(a), (b), and (c). As can be seen from Figure 4 it, for the selected points A and B, the relative errors between the results of ABAQUS and EM-HFEM in the ideal mesh and the two distorted meshes are both less than 0.003%. At the same time, the calculation results of the ideal mesh and the distorted mesh are highly consistent, and the error between them is only at the -9 order of magnitude of 10. This fully demonstrates that EM-HFEM is less sensitive to mesh distortion.

[0157] Model containing a central inclusion: G1

[0158] This numerical example is a model containing a circular inclusion. The ABAQUS mesh division diagram is as shown in Figure 5 a, the quadtree mesh division diagram is as shown in Figure 5 b, and the polygon-quadtree mesh division diagram is as shown in Figure 5 c. Figure 5 d, 5e, and 5f are the x-direction stress nephograms calculated by ABAQUS, the quadtree mesh of EM-HFEM, and the polygon-quadtree mesh of EM-HFEM, Figure 5 g, 5h, and 5i are the y-direction stress nephograms calculated by ABAQUS, the quadtree mesh of EM-HFEM, and the polygon-quadtree mesh of EM-HFEM, Figure 5 j, 5k, and 5l are the y-direction electric displacement nephograms calculated by ABAQUS, the quadtree mesh of EM-HFEM, and the polygon-quadtree mesh of EM-HFEM.

[0159] Model containing multiple regularly arranged circular inclusions: G2

[0160] This numerical example is a model containing sixteen regularly arranged circular inclusions. The ABAQUS mesh division diagram is as shown in Figure 6 a, the quadtree mesh division diagram is as shown in Figure 6 b, and the polygon-quadtree mesh division diagram is as shown in Figure 6 c. Figure 6d, 6e, and 6f are the stress nephograms in the x direction calculated by ABAQUS, the EM-HFEM quad-tree mesh, and the EM-HFEM polygon-quad-tree mesh, Figure 6 g, 6h, and 6i are the stress nephograms in the y direction calculated by ABAQUS, the EM-HFEM quad-tree mesh, and the EM-HFEM polygon-quad-tree mesh, Figure 6 j, 6k, and 6l are the electric displacement nephograms in the y direction calculated by ABAQUS, the EM-HFEM quad-tree mesh, and the EM-HFEM polygon-quad-tree mesh.

[0161] Extract the stress values and electric displacement values of 200 points along the path of x = 0.8 in the EM-HFEM results and compare them with ABAQUS, as Figure 5 shown. Select the midpoint on the path and compare the relative errors between the calculation results of ABAQUS and the polygon-quad-tree mesh and between ABAQUS and the quad-tree mesh, as shown in Table 3.

[0162] Table 4 Relative Errors between ABAQUS and Results of Different Mesh Models

[0163]

[0164]

[0165] Observation Figure 5 、 Figure 6 、 Figure 7 and Table 4 reveals that in the calculation results of EM-HFEM, since the quad-tree elements are regular squares and cannot fully conform to the irregular curves of the matrix and inclusions, the material boundaries are uneven. The stress concentration in these serrated regions does not match the actual situation, affecting the calculation accuracy. The polygon-quad-tree mesh model has higher accuracy compared to the calculation results of the quad-tree mesh model due to better fitting of geometric entities. Therefore, in the following models, the EM-HFEM model with polygon-quad-tree mesh division is selected for comparison with ABAQUS.

[0166] Model with Multiple Randomly Distributed Circular Inclusions: G3

[0167] This numerical example is a square model containing sixteen randomly distributed circular inclusions as Figure 8 shown in a. The ABAQUS mesh division diagram is as Figure 8 shown in b, and the EM-HFEM division diagram is as Figure 8 shown in c. Figure 8 d, 8e are the stress nephograms in the x direction calculated by ABAQUS and EM-HFEM, Figure 8 g, 8h are the stress nephograms in the y direction calculated by ABAQUS and EM-HFEM, Figure 8j, 8k are the electric displacement nephograms in the y direction obtained by ABAQUS and EM-HFEM. The stress values and electric displacement values of 200 points along the path of x = 1 in the EM-HFEM results are extracted and compared with ABAQUS as shown in Figure 8 f, 8i, and 8l. The relative errors of the upper and lower limits of the calculation results between ABAQUS and EM-HFEM are shown in Table 5.

[0168] Table 5 Relative Errors between the Upper and Lower Limits of ABAQUS and EM-HFEM Results

[0169]

[0170] Model with Multiple Randomly Distributed Elliptical Inclusions: G4

[0171] This numerical example is a square model containing sixteen randomly distributed elliptical inclusions as shown in Figure 9 a. The ABAQUS mesh diagram is shown in Figure 9 b, and the EM-HFEM mesh diagram is shown in Figure 9 c. Figure 9 d, 9e are the stress nephograms in the x direction obtained by ABAQUS and EM-HFEM, Figure 9 g, 9h are the stress nephograms in the y direction obtained by ABAQUS and EM-HFEM, Figure 9 j, 9k are the electric displacement nephograms in the y direction obtained by ABAQUS and EM-HFEM. The stress values and electric displacement values of 200 points along the path of x = 0.3 in the EM-HFEM results are extracted and compared with ABAQUS as shown in Figure 9 f, 9i, and 9l. The relative errors at the upper and lower limits of the calculation results between ABAQUS and EM-HFEM are shown in Table 6.

[0172] Table 6 Relative Errors between the Upper and Lower Limits of ABAQUS and EM-HFEM Results

[0173]

[0174] The present invention proposes an electromechanical coupling hybrid finite element model combined with a polygonal quadtree grid for calculating the stress and electric displacement of piezoelectric particle-reinforced composites. In the framework of this model, quadtree elements and polygonal elements with arbitrary sides are adopted. The present invention compares five numerical examples with ABAQUS, and the results show that the above method has good convergence and accuracy. The advantage of this calculation method is that the electromechanical coupling hybrid finite element is no longer limited to elements of a specific shape and has extremely low sensitivity to distorted grids, and arbitrary polygonal elements can be used for calculation. The numerical examples G1-4 are automatically divided into polygon-quadtree elements according to the inherent structural program of the model, which not only saves the cumbersome steps of hand-drawing grids but also greatly reduces the number of elements while ensuring the calculation accuracy. Compared with the traditional finite element method, the number of elements used by the electromechanical coupling hybrid element based on the polygon-quadtree grid is much smaller than that of ABAQUS, and the obtained results are very close. The above shows that when the stress field and the electric displacement field gradients are large and the number of inclusions is large, the method proposed by the present invention can obtain a solution that meets the accuracy requirements with as few elements as possible, improving the calculation efficiency and providing a new method for dealing with large-scale piezoelectric particle composite problems.

[0175] The preferred embodiments of the present invention disclosed above are only used to help illustrate the present invention. The preferred embodiments do not describe all the details in detail, nor do they limit the invention to the specific embodiments described. Obviously, many modifications and variations can be made according to the content of this specification. These embodiments are selected and specifically described in this specification to better explain the principles and practical applications of the present invention, so that those skilled in the art can well understand and utilize the present invention. The present invention is only limited by the claims and their full scope and equivalents.

Claims

1. A simulation method for piezoelectric particle composite materials based on polygon-quadtree grid, characterized in that: The following steps are involved: S1: Through electromechanical coupling theory, establish generalized equilibrium equations and geometric equations, and clarify boundary conditions, including natural conditions and forced conditions; S2: Through the elastic constant, piezoelectric coefficient and dielectric constant of piezoelectric materials, material parameters are introduced to construct the constitutive equation of electromechanical coupling; through the increment of generalized stress function and state function, the state function and complementary energy expression of electromechanical coupling problem are derived; S3: Use the complementary energy principle to derive the functional expression, apply Gauss's formula and boundary conditions, and simplify the complementary energy functional into an expression that includes stress, body force, electric displacement, and body charge density; S4: Using the geometric characteristics of polygonal units, the interpolation functions of stress field and electric displacement field are constructed, and the stress and electric displacement in each unit are expressed in matrix form; the Lagrange multiplier method is introduced to deal with the continuity of displacement and electric displacement between units, and combined with the modified functional, the natural boundary conditions and forced boundary conditions are ensured to be met; S5: The stiffness matrix of the polygonal unit is precisely calculated by the Gaussian integral method; the stiffness matrix of the unit is formed by solving the stationary value condition of the modified functional, and the final finite element solution format is generated; a polygon-quadtree mesh model is constructed and applied to different numerical examples to verify the convergence and accuracy of the method.

2. The method for simulating piezoelectric particle composite materials based on polygon-quadtree grid according to claim 1, characterized in that: The step S1 specifically includes: basic variables include generalized displacement, generalized stress, and generalized strain; boundaries include unit boundaries S i , displacement boundary S u , traction boundary S t , electromotive force boundary S φ and the charge density boundary S ω ; In the Cartesian coordinate system (x, y), the generalized equilibrium equation for the electromechanical coupling problem is D i,i -ρ=0 (2) In the formula, σ ij for stress; For physical strength; D i is the electric displacement; ρ is the body charge density; The generalized geometric equations for the electromechanical coupling problem are: (3) E i +φ .i =0 (4) In the formula, ε ij is strain; u i is the displacement; E i is the electric field strength; φ is the electric potential; The generalized displacement boundary condition satisfied in the electromechanical coupling problem is At the border S u Previous (5) At the border S φ Previous (6) The generalized traction boundary condition that should be satisfied is At the border S t Previous (7) At the border S ω Previous (8).

3. The method for simulating piezoelectric particle composite materials based on polygon-quadtree grid according to claim 1, characterized in that: The step S2 specifically includes: For quasi-static infinitesimal reversible processes, the increment dH of the state function H of the electromechanical coupling problem is affected by mechanical and electrical factors, and its expression is: dH=σdε+DdE (9) Where: Total differential of generalized stress From equations (12) and (13), we get Introducing material parameters into equations (14) and (15), we obtain Substituting equations (16), (17), and (18) into (12) and (13), we obtain the constitutive equations for the electromechanical coupling problem: In the formula, c, e, and d are elastic constant, piezoelectric coefficient, and dielectric constant, respectively; Assuming that the piezoelectric material is two-dimensionally isotropic in the xy plane, under the plane stress state, its constitutive relationship equation can be obtained: Substituting equation (20) into equation (9) yields the residual enthalpy expression of the state function for the electromechanical coupling problem: Its tensor form is 4. The method for simulating piezoelectric particle composite materials based on polygon-quadtree grid according to claim 1, characterized in that: The step S3 specifically includes: The residual method is used to deal with the geometric equations and boundary conditions; let δG ij and δH i is any function in the Ω domain, δJ i and δK i For the boundary S u and boundary S φ Any function on the surface; According to the principle of minimum complementary energy, the total complementary energy takes the extreme value to derive the generalized geometric equations (3)(4) and the generalized displacement boundary conditions (5)(6), which can be obtained. Equation (24) is similar to the virtual force principle. If δG ij =δσ ij , δH i =δD i , then Applying Gauss's formula we get From the generalized equilibrium equations (1)(2) and the generalized stress boundary conditions (7)(8), we get Substituting equations (28) and (29) into (25), we obtain Take δJ i = δσ ij n j , δK i = δD i n i Then there is Substituting the constitutive equation (20) into equation (31), we obtain consider and is a known quantity on the boundary, so equation (32) can be written as Using σ i , D i denote the stress field and electric displacement field in Ω respectively, and u i ,φ represents the boundary The displacement and potential on the e is the normal direction outside the boundary; Boundaries meet: The matrix form of equation (33) can be written as Considering the boundary conditions (5), (6) and the force continuity and electric displacement continuity between elements Applying the Lagrange multiplier method to introduce the above constraints, the modified functional is obtained as follows: When applying the hybrid finite element method, the stress field and electric displacement field within each element are expressed as a complete polynomial in the rectangular coordinate system At the boundaries of the element, the displacements and potentials are interpolated from the generalized nodal displacements and generalized nodal potentials, expressed as Where L is the boundary interpolation function; to perform linear interpolation between two adjacent nodes i and i+1, the function is expressed as Substituting equations (38) and (39) into equation (37), we obtain in According to the modified residual energy stationary value condition: Can get Substituting formula (42) into (43) we can obtain Formula (44) is the final finite element solution format, where K is the element stiffness matrix, expressed as 5. The method for simulating piezoelectric particle composite materials based on polygon-quadtree grid according to claim 1, characterized in that: The step S4 specifically includes: introducing a unit self-equilibrium stress function and an electric displacement function in the finite element implementation; The polynomial stress function and electric displacement function can be expressed by Φ1 and Φ2. By differentiating the stress function and the electric displacement function, the stress and electric displacement at each unit can be obtained. The stress and electric displacement are expressed as in 6. The method for simulating piezoelectric particle composite materials based on polygon-quadtree grid according to claim 1, characterized in that: The step S5 specifically includes: Gaussian integration method is used to numerically integrate H matrix and G matrix; for two-dimensional polygonal hybrid cells, the integral of G is the line integral of the cell boundary: According to equation (51), the Gaussian integral formula is The H integral is the area integral in the unit domain. The area integral form of the Gaussian integral is According to formula (53), the Gaussian integral formula is:

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