Sheet shell meshless simulation analysis method and device

Abstract

The invention provides a meshless simulation analysis method and device for a thin-plate shell, and relates to the technical field of thin-plate shell structure analysis. The meshless simulation analysis method comprises the following steps: discretizing a middle surface of the thin shell through meshless nodes, establishing a discrete control equation by utilizing regenerated core approximate displacement and regenerated smooth gradient approximate strain, performing numerical integration, and finally solving displacement. The problems of shear locking, grid dependence, low-order continuity and the like of a traditional finite element method in thin shell analysis are solved, and particularly, the film self-locking phenomenon is avoided. Through the Hu-Washu variational principle, the method does not need manual parameters, simplifies the application process of essential boundary conditions, and improves the stability and efficiency of calculation. Compared with a traditional method, the method has remarkable advantages in calculation precision and efficiency, and is suitable for thin shell structure analysis of complex geometrical shapes and large deformation problems.

Description

Technical Field

[0001] The present invention relates to the technical field of thin plate and shell structure analysis, and particularly to a meshless simulation analysis method and device for thin plate and shell structures. Background Art

[0002] Currently, in the field of modern engineering, thin shell structures have attracted much attention due to their excellent mechanical properties and wide application scenarios. With the geometric feature that their thickness is much smaller than other dimensions, such structures exhibit excellent load-bearing capacity and material utilization efficiency, and are widely used in multiple industries such as architecture, aerospace, and mechanical manufacturing. However, the complex geometric shape and non-linear material properties of thin shell structures pose great challenges to their accurate analysis and optimal design.

[0003] However, the traditional finite element method (FEA) is one of the most commonly used numerical methods in current engineering analysis, but it has significant limitations when dealing with thin shell problems. The finite element method is based on the Mindlin-Reissner hypothesis and introduces rotational degrees of freedom to describe the deformation in the thickness direction. However, when the shell thickness decreases, this method is prone to shear locking, Poisson locking, and membrane locking phenomena, resulting in a significant decrease in calculation accuracy and being only applicable to relatively thick shell structures. In addition, the mesh generation of the finite element method has a significant impact on calculation accuracy and efficiency. The mesh generation for complex geometric shapes and large deformation problems is particularly difficult, and mesh distortion will further reduce the calculation accuracy.

[0004] In recent years, as an emerging numerical analysis method, the meshless method has received extensive attention due to its advantages in dealing with complex geometric shapes and large deformation problems. The meshless method constructs shape functions through nodal information and can construct high-order continuous and smooth shape functions over the entire domain, avoiding the complexity of mesh generation and mesh distortion problems in the traditional finite element method. However, the existing meshless methods still face some challenges in thin shell structure analysis. For example, although high-order shape functions are helpful in solving the membrane locking problem, they are highly complex in actual calculations and may lead to inaccurate numerical integration, affecting the overall calculation accuracy. In addition, it is difficult to apply essential boundary conditions in the meshless method because its shape functions lack the interpolation characteristics in the traditional finite element method and it is difficult to directly apply boundary conditions.

[0005] Specifically, in the prior art, although the improved methods of the finite element method (such as the mixed model and the hybrid model) alleviate the computational difficulties of the thin shell problem to a certain extent, there are still problems such as low-order continuity and shear locking. The collocation method lacks variational consistency, resulting in difficult-to-guarantee computational accuracy. In addition, boundary condition application methods such as the Nitsche method, the Lagrange multiplier method, and the penalty function method, although each has its own advantages, also have deficiencies such as low computational efficiency and dependence on artificial empirical parameters. In summary, there are still many deficiencies in the prior art in the analysis of thin shell structures, especially in terms of numerical integration accuracy, boundary condition application, and computational efficiency. Therefore, there is an urgent need for a new numerical analysis method that can improve computational efficiency, simplify the boundary condition application process, and reduce the dependence on artificial parameters while maintaining high accuracy, so as to provide more reliable technical support for the design and analysis of thin shell structures.

[0006] In view of this, the present application is proposed. Summary of the Invention

[0007] The present invention provides a meshless simulation analysis method and device for thin plate shells, which can at least partially improve the above problems.

[0008] To achieve the above object, the present invention adopts the following technical solutions: A meshless simulation analysis method for thin plate shells, which includes: Obtain the information of the thin plate shell to be analyzed, and based on the Hu-Washizu variational principle and the information of the thin plate shell, establish the Galerkin weak form of the elasticity problem; Obtain the preset meshless nodes, and discretize the middle surface of the thin plate shell to be analyzed through the meshless nodes to establish a reproducing kernel meshless approximation of the displacement; Obtain the preset background integration domain, and within the background integration domain, construct an approximation of the strain using polynomials and covariant bases of corresponding orders; Substitute the reproducing kernel meshless approximation of the displacement and the approximation of the strain into the Galerkin weak form to obtain a stiffness matrix dataset, and obtain a traditional force vector. Perform discrete control preprocessing according to the traditional force vector and the stiffness matrix dataset to obtain a discrete control equation; Perform numerical integration processing on the discrete control equation to obtain an integrated discrete control equation, and solve the integrated discrete control equation to obtain a simulation analysis result.

[0009] The present invention also provides a meshless simulation analysis device for thin plate shells, which includes: A Galerkin weak form construction unit for obtaining the information of the thin plate shell to be analyzed, and based on the Hu-Washizu variational principle and the information of the thin plate shell, establishing the Galerkin weak form of the elasticity problem; A displacement approximation acquisition unit, configured to acquire preset meshless nodes, discretize the middle surface of the thin plate and shell to be analyzed through the meshless nodes, and establish a reproducing kernel meshless approximation of the displacement; A strain approximation acquisition unit, configured to acquire a preset background integration domain, and within the background integration domain, construct an approximation of the strain using polynomials and covariant bases of corresponding orders; A discrete control unit, configured to substitute the reproducing kernel meshless approximation of the displacement and the approximation of the strain into the Galerkin weak form to obtain a stiffness matrix data set, and acquire a traditional force vector, and perform discrete control preprocessing according to the traditional force vector and the stiffness matrix data set to obtain a discrete control equation; A result generation unit, configured to perform numerical integration processing on the discrete control equation to obtain an integrated discrete control equation, and solve the integrated discrete control equation to obtain a simulation analysis result.

[0010] In summary, the meshless simulation analysis method for thin plates and shells aims to solve problems such as low-order continuity, shear locking, mesh dependence, and insufficient calculation accuracy existing in the traditional finite element method when dealing with thin shell problems. This method develops a new meshless integration method by introducing the Hu-Washizu variational principle and combining the reproducing smooth gradient theory framework. Specifically, this method avoids the film self-locking phenomenon by mixing discrete film stress and bending stress, and simplifies the process of applying essential boundary conditions. In addition, this method does not require calculating high-order derivatives and does not depend on artificial parameters, significantly improving the calculation efficiency and accuracy. Through an optimized numerical integration scheme, the calculation stability and efficiency are further improved. The present invention provides an efficient and reliable numerical analysis tool for the engineering design and analysis of thin shell structures, and is particularly suitable for the analysis of complex geometric shapes and large deformation problems. Description of the Drawings

[0011] Figure 1 is a schematic flow chart of the meshless simulation analysis method for thin plates and shells provided by the first embodiment of the present invention; Figure 2 is a schematic diagram of the variational consistency weak form based on the Hu-Washizu variational principle provided by the embodiment of the present invention; Figure 3 is a schematic diagram of meshless discretization for patch testing provided by the embodiment of the present invention; Figure 4 is the M provided by the embodiment of the present invention for the bending shell patch test 12 schematic diagram of the exact solution; Figure 5 is the M provided by the embodiment of the present invention for the bending shell patch test 12 schematic diagram of the contour map of the Nitsche method; Figure 6 is the contour diagram of the penalty function method for the bending shell patch test provided by an embodiment of the present invention 12 ; Figure 7 is the contour diagram of the Hu-Washizu method for the bending shell patch test provided by an embodiment of the present invention 12 ; Figure 8 is the schematic diagram of the relationship between the calculation time and the number of nodes for the efficiency comparison of the cantilever beam problem provided by an embodiment of the present invention; Figure 9 is the schematic diagram of the efficiency analysis of the essential boundary condition application for the efficiency comparison of the cantilever beam problem provided by an embodiment of the present invention; Figure 10 is the module schematic diagram of the thin plate shell meshless simulation analysis device provided by the second embodiment of the present invention. Detailed implementation manners

[0012] In order to make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be further described in detail below with reference to embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0013] Refer to Figures 1 to 3 as shown, the first embodiment of the present invention discloses a thin plate shell meshless simulation analysis method, which can be executed by a thin plate shell meshless simulation analysis device (hereinafter referred to as the analysis device), and particularly, by one or more processors in the analysis device to implement the following method: S1. Obtain the thin plate shell information to be analyzed, and based on the Hu-Washizu variational principle and the thin plate shell information, establish the Galerkin weak form of the elasticity problem; Specifically, step S1 includes: The mathematical expression formula of the Galerkin weak form is: , where is the thickness of the shell of the thin plate shell, is the component of the fourth-order elastic tensor with respect to the covariant basis and the plane stress assumption, is the displacement of the shell of the thin plate shell, is the body force acting on the shell of the thin plate shell, is the integral sign, is the variational sign, , are both membrane strain components, , are both buckling strain components.

[0014] In this embodiment, relevant information of the thin plate and shell structure to be analyzed needs to be obtained first, including its geometric shape, material properties, boundary conditions, and external forces. These information are the basis for subsequent numerical analysis. Based on this thin plate and shell information, combined with the Hu-Washizu variational principle, the Galerkin weak form of the elasticity problem is established. By introducing the mixed formulas of displacement, strain, and stress, a complete theoretical framework is provided for subsequent numerical calculations.

[0015] Through this Galerkin weak form based on the Hu-Washizu variational principle, complex stress and deformation problems in the thin plate and shell structure can be effectively handled. This weak form not only provides a theoretical basis for subsequent numerical discretization, but also improves the accuracy and stability of the numerical solution through the introduction of mixed variables. Especially for the common membrane locking problem in the thin plate and shell structure, this method can significantly reduce the occurrence probability of the locking phenomenon through reasonable variable discretization and boundary condition treatment, thereby improving the reliability of the calculation results.

[0016] Specifically, in this embodiment, the meshless simulation analysis of the thin plate and shell uses the Hu-Washizu variational principle. In this method, the mixed formulas of displacement, strain, and stress are adopted. The displacement is discretized through meshless shape functions, while the strain and stress are expressed through smoothed gradients and covariant bases. By based on the Hu-Washizu variational principle, a more complete variational theory basis is established for the variational consistent Galerkin meshless method, so that accurate theoretical error estimation can be carried out. And, since there are naturally stable terms in the weak form of Hu-Washizu, no artificial parameters are required during the implementation process, which helps to automatically maintain the consistency of the stiffness matrix.

[0017] In addition, to improve the imposition of essential boundary conditions, this method adopts the way embedded in the Hellinger-Reissner Galerkin weak form, without the need to additionally impose essential boundary conditions, thus simplifying the imposition process and improving the stability of the calculation.

[0018] S2. Obtain preset meshless nodes, discretize the middle surface of the thin plate and shell to be analyzed through the meshless nodes, and establish a reproducing kernel meshless approximation of the displacement; Specifically, step S2 includes: The mathematical expression of the reproducing kernel meshless approximation of the displacement is: , where is the meshless shape function, is the displacement nodal coefficient, is the total number of nodes.

[0019] In this embodiment, it is first necessary to obtain the preset meshless node information. These meshless nodes are the basis for discretizing the mid-surface of the thin plate shell, and their distribution and quantity directly affect the accuracy and efficiency of numerical analysis. The selection of meshless nodes can be flexibly adjusted according to the geometric shape, boundary conditions, and expected analysis accuracy of the thin plate shell, which provides great flexibility for the present invention. Especially when dealing with complex geometric shapes and large deformation problems, compared with the mesh generation of the traditional finite element method, the arrangement of meshless nodes is more efficient and easier to implement.

[0020] After obtaining the meshless nodes, the mid-surface of the thin plate shell to be analyzed is discretized through these nodes. The core of this process is to establish a reproducing kernel meshless approximation of the displacement. The reproducing kernel meshless approximation is an advanced numerical method that can construct high-order continuous and smooth shape functions by using node information, so as to provide an accurate approximate expression for the displacement field of the thin plate shell structure. This method avoids the problem of accuracy degradation caused by mesh distortion in the traditional finite element method and can effectively handle the high-order continuity requirements in the thin plate shell structure. The meshless shape function is constructed by the reproducing kernel approximation method, which can ensure that the displacement value at each node can be accurately interpolated and maintain high-order continuity in the entire discrete domain.

[0021] Through this reproducing kernel meshless approximation method, this method can significantly improve the accuracy and efficiency of the thin plate shell structure analysis. Due to the high-order continuity of the meshless shape function, this method can effectively avoid the common shear locking and membrane locking phenomena in the traditional finite element method, thus providing strong support for the accurate analysis of the thin plate shell structure. In addition, the flexible arrangement of meshless nodes makes the present invention more advantageous when dealing with complex geometric shapes and large deformation problems, which can significantly reduce the calculation cost and improve the efficiency of numerical analysis.

[0022] S3. Obtain a preset background integration domain, and within the background integration domain, construct an approximation of the strain using polynomials and covariant bases of corresponding orders; Specifically, step S3 includes: The approximation of the strain includes membrane strain and buckling strain. The mathematical expression of the membrane strain is: , and the mathematical expression of the buckling strain is: , where, , , represents the approximate value of the membrane strain related to the I rd node, , represents the approximate value of the bending strain related to the I th node. The mathematical expressions of the membrane strain and buckling strain at the boundary conditions are: and . Where, , , , , , is a polynomial vector, , and , are integral constraint terms, is the moment matrix, , , are the covariant basis vectors of the mid-surface of the shell.

[0023] In this embodiment, the goal of this stage is to construct approximate expressions of membrane strain and buckling strain in a preset background integral domain by using polynomials and covariant bases of corresponding orders. The setting of the background integral domain is an important part of the meshless method, which provides a calculation framework for numerical integration and ensures the accuracy and efficiency of the integration. The approximate expressions of membrane strain and buckling strain are constructed by combining the node information and the polynomial basis functions in the background integral domain.

[0024] Through this strain approximation method based on polynomials and covariant bases, the deformation behavior of thin plate and shell structures in the membrane and buckling directions can be effectively captured. A significant advantage of this method is that it can provide high-precision strain field approximation without increasing too much computational complexity. Compared with the traditional finite element method, the present invention avoids the problem of accuracy degradation caused by mesh distortion by combining the regenerating kernel meshless approximation and the background integral domain, and can better handle the high-order continuity requirements in thin plate and shell structures. In addition, another important benefit of this strain approximation method is its adaptability to complex geometric shapes and boundary conditions. Since the meshless method does not rely on fixed grid division, the setting of the background integral domain can be more flexibly adapted to the geometric characteristics of the thin plate and shell structure, thereby showing higher efficiency and accuracy in the analysis of thin plates and shells with complex shapes. At the same time, by constructing an approximate expression of strain in the background integral domain, the present invention can effectively reduce the amount of numerical integration calculations and further improve the computational efficiency.

[0025] S4, substituting the reproducing kernel meshless approximation of the displacement and the strain approximation into the Galerkin weak form to obtain a stiffness matrix data set, and obtaining a traditional force vector, performing discrete control preprocessing according to the traditional force vector and the stiffness matrix data set to obtain a discrete control equation; Specifically, step S4 includes: substituting the reproducing kernel meshless approximation of the displacement and the approximation of the strain into the Galerkin weak form to obtain the traditional stiffness matrix K , and the stability term stiffness matrix corresponding to the imposed essential boundary conditions and the consistency term stiffness matrix ; According to the traditional stiffness matrix K obtain the components of the traditional stiffness matrix ; According to the stabilization term stiffness matrix obtain the components of the stabilization term stiffness matrix , where is the displacement boundary, is the rotation boundary, is the displacement boundary at the corner point, where , , , is the component of the boundary outer normal direction vector; According to the consistent term stiffness matrix obtain the components of the consistent term stiffness matrix , where , , , , , , .

[0026] Obtain the traditional force vector f , and construct the components of the traditional force vector according to the external force boundary conditions and body force distribution , is the surface force, is the external normal bending moment, is the external concentrated load, is the stress boundary, is the bending moment boundary, is the concentrated load corner boundary; Perform boundary integral calculation to obtain the corresponding stabilization term force vector and consistent term force vector ; According to the stabilization term force vector obtain the components of the stabilization term force vector , where is the known boundary displacement, is the known boundary rotation; According to the consistent term force vector obtain the components of the consistent term force vector ; According to the traditional stiffness matrix K , stabilization term stiffness matrix , consistent term stiffness matrix , traditional force vector f , stabilization term force vector and consistent term force vector , assemble to obtain the discrete control equation: 。

[0027] In this embodiment, the approximate expressions of displacement and strain are substituted into the Galerkin weak form, thereby obtaining the complete discrete control equations. This process is the core link for realizing numerical analysis and directly determines the calculation accuracy and efficiency.

[0028] Specifically, the reproducing kernel meshless approximation of displacement and the approximation of strain are substituted into the Galerkin weak form. Through this substitution process, the traditional stiffness matrix, as well as the consistent term stiffness matrix and the stabilization term stiffness matrix corresponding to the imposed essential boundary conditions, can be obtained. Among them, the consistent term stiffness matrix is used to ensure the variational consistency of the overall solution, and the stabilization term stiffness matrix ensures the positive definiteness of the stiffness matrix.

[0029] The component calculations of these stiffness matrices cover the displacement boundaries, rotation boundaries, displacement boundaries at corner points, as well as the stress boundary terms, normal bending moments, and concentrated force terms related to the nodes, etc., thus comprehensively reflecting the mechanical behavior of the thin plate and shell structure. After obtaining the stiffness matrix dataset, the traditional force vector is further obtained through the influence of the load on the system; according to the external force boundary conditions and the body force distribution, the components of the traditional force vector are constructed. Through boundary integral calculations, the corresponding consistent term force vector and stabilization term force vector are obtained. Finally, based on the traditional stiffness matrix, consistent term stiffness matrix, stabilization term stiffness matrix, traditional force vector, consistent term force vector, and stabilization term force vector, the discrete control equations are assembled. These discrete control equations are linear equations and can be solved using a linear algebra library to obtain the approximate solution of the displacement field of the thin plate and shell structure.

[0030] The essential boundary conditions have been considered in the weak form of the Hellinger-Reissner variational principle, thus realizing the automatic imposition of the essential boundary conditions without the need to impose additional essential boundary conditions. This method not only simplifies the process of imposing boundary conditions but also improves the calculation stability and efficiency.

[0031] S5. Perform numerical integration on the discrete control equations to obtain the integrated discrete control equations, and solve the integrated discrete control equations to obtain the simulation analysis results.

[0032] Specifically, step S5 includes: performing numerical integration on the components of the traditional stiffness matrix to obtain the integrated components of the traditional stiffness matrix, where is the total number of integration elements, is the number of integration points, is the integration point, is the weight; perform numerical integration on the components of the stabilization term stiffness matrix to obtain the integrated components of the stabilization term stiffness matrix ; Numerically integrate the components of the consistent term stiffness matrix to obtain the components of the integrated consistent term stiffness matrix .

[0033] Numerically integrate the components of the traditional force vector to obtain the components of the integrated traditional force vector ; Numerically integrate the components of the stabilizing term force vector to obtain the components of the integrated stabilizing term force vector ; Numerically integrate the components of the consistent term force vector to obtain the components of the integrated consistent term force vector ; According to the components of the integrated traditional stiffness matrix , the components of the integrated stabilizing term stiffness matrix , the components of the integrated consistent term stiffness matrix , the components of the integrated traditional force vector , the components of the integrated stabilizing term force vector , and the components of the integrated consistent term force vector , obtain the integrated discrete control equations; Use a preset linear algebra library to solve the integrated discrete control equations to obtain approximate displacement node coefficients and generate simulation analysis results.

[0034] In this embodiment, numerically integrate each matrix and vector in the discrete control equations to obtain the integrated discrete control equations, and obtain the simulation analysis results of the thin plate and shell structure by solving these equations. First, numerically integrate the components of the traditional stiffness matrix. According to the integration points and corresponding weights in the background integration domain, calculate the integral value of the traditional stiffness matrix. Through this numerical integration method, the integral value of the traditional stiffness matrix can be calculated efficiently, and at the same time, the complex analytical integration process in the traditional method is avoided.

[0035] Next, numerically integrate the components of the consistent term stiffness matrix to obtain the components of the integrated consistent term stiffness matrix. Similarly, numerically integrate the components of the stabilizing term stiffness matrix to obtain the components of the integrated stabilizing term stiffness matrix.

[0036] After completing the numerical integration of the stiffness matrix, the components of the traditional force vector are numerically integrated to obtain the integrated components of the traditional force vector. The components of the consistent term force vector and the stabilization term force vector are also numerically integrated respectively to obtain the integrated components of the consistent term force vector and the stabilization term force vector. Finally, based on the integrated traditional stiffness matrix, consistent term stiffness matrix, stabilization term stiffness matrix, traditional force vector, consistent term force vector, and stabilization term force vector, the integrated discrete control equations are assembled. Among them, optimized numerical integration points are selected within the background integration element to ensure integration accuracy and computational efficiency; according to the positions of the integration points and the geometric shape of the background integration element, the corresponding integration weights are determined to ensure the accuracy of numerical integration.

[0037] The integrated discrete control equations are linear equation systems, which are solved using a preset linear algebra library (such as LAPACK, PETSc, etc.) to obtain approximate displacement node coefficients. These displacement node coefficients are the simulation analysis results of the thin plate and shell structure, which can intuitively reflect the deformation of the structure after being stressed. Through this process, the efficient numerical analysis of the thin plate and shell structure is realized. The introduction of numerical integration not only simplifies the calculation process but also improves the calculation accuracy and efficiency. Due to the adoption of the reproducing kernel meshless approximation and the optimized numerical integration scheme, the present invention can effectively avoid the shear locking and membrane locking problems in the traditional finite element method, while simplifying the application process of the essential boundary conditions and reducing the dependence on artificial parameters. The comprehensive application of these technical means enables this method to show significant advantages in dealing with complex geometric shapes and large deformation problems, providing a reliable numerical tool for the engineering design and analysis of thin plate and shell structures.

[0038] Please refer to Figure 4 and Figure 9 , in this embodiment, Figure 8 shows the efficiency analysis in the overall solution process. The results show that the computational efficiency of the reproducing kernel gradient smoothing integration method (RKGSI) is significantly better than that of the Gaussian integration method (GI), and its CPU time consumption is only 2.5 times that of RKGSI. This advantage is attributed to the fact that RKGSI does not need to calculate the derivatives of the traditional meshless shape functions and adopts optimized numerical integration points, thus reducing the computational amount of the shape functions. Figure 9The efficiency of different methods for imposing essential boundary conditions was further compared. The results show that the penalty function method and the Lagrange multiplier method are comparable in terms of the efficiency of shape function calculation and are superior to other methods because they only rely on the meshless shape functions themselves. For the RKGSI-HR method based on the Hellinger-Reissner variational principle, although it does not require the calculation of the shape function gradient, it takes more time to construct a smooth gradient, about 1.6 times that of the former two methods. However, in terms of assembling the force vector, the RKGSI-HR method is comparable to the Nitsche method. Although the Lagrange multiplier method is the most efficient in some aspects, neither it nor the penalty function method can guarantee the theoretical error convergence rate. Overall, the RKGSI-HR method not only ensures the calculation accuracy and theoretical convergence, but also shows higher computational efficiency compared to the traditional Nitsche method.

[0039] In summary, the meshless simulation analysis method for thin plates and shells aims to solve the problems faced by the traditional finite element method in dealing with thin shell problems, such as shear locking, mesh dependence, and insufficient calculation accuracy. By introducing the Hu-Washizu variational principle and the reproducing kernel meshless approximation technique, this method constructs an efficient and accurate numerical analysis framework, which can significantly improve the simulation analysis efficiency and reliability of thin shell structures. First, based on the geometric and mechanical information of the thin shell structure, the Galerkin weak form is established using the Hu-Washizu variational principle, providing a theoretical basis for subsequent numerical discretization. Subsequently, the mid-surface of the thin shell is discretized by preset meshless nodes, and a reproducing kernel meshless approximation of the displacement is established. This approximation method can construct high-order continuous and smooth shape functions using node information, effectively avoiding the problem of accuracy degradation caused by mesh distortion in the traditional finite element method and simplifying the discretization process of complex geometric shapes. Further, approximate expressions of the membrane strain and buckling strain are constructed using polynomials and covariant bases in the background integration domain and substituted into the Galerkin weak form to obtain the traditional stiffness matrix, the consistent term stiffness matrix, and the stabilization term stiffness matrix. This process not only ensures the variational consistency of the overall solution but also guarantees the positive definiteness of the stiffness matrix through the stabilization term stiffness matrix, providing a solid theoretical support for subsequent numerical solutions. In addition, the computational efficiency and accuracy are further improved through an optimized numerical integration scheme. After assembling the discrete control equations, an efficient linear algebra library is used to solve the integrated discrete control equations, and finally, an approximate solution of the displacement field of the thin shell structure is obtained. Through the comprehensive application of this series of technical means, the present invention not only effectively solves the problems of membrane self-locking and shear locking in the traditional method, but also significantly improves the calculation accuracy and efficiency, reduces the dependence on artificial parameters, and simplifies the process of imposing essential boundary conditions.

[0040] It mainly includes a meshless approximation through reproducing kernels and an optimized numerical integration scheme, which significantly improves the accuracy and efficiency of thin-shell structure analysis, especially suitable for the analysis of complex geometries and large deformation problems; a mixed formulation framework constructed using the Hu-Washizu variational principle, which can effectively avoid the membrane locking phenomenon and ensure the stability and reliability of the numerical solution; and the beneficial effects of simplifying the application process of essential boundary conditions, reducing the dependence on artificial parameters, and further improving the stability and efficiency of the calculation.

[0041] Briefly, the meshless thin-shell formulation is an innovative structural analysis method. Compared with traditional finite element methods, it does not rely on regular mesh generation, thus providing higher flexibility. This method is particularly suitable for thin-shell structure analysis because it can effectively handle the membrane locking problem, which is a common issue in thin-shell design. The membrane locking problem can lead to inaccurate stress distributions and affect the performance and safety of the structure. The application of the Hu-Washizu variational principle provides a mixed formulation framework for the meshless thin-shell formulation, including variables for displacement, strain, and stress. The advantage of this method is that it can naturally enforce the essential boundary conditions while maintaining numerical consistency. Combining the meshless thin-shell formulation and the Hu-Washizu variational principle can improve the accuracy and efficiency of the calculation. By using smoothed gradients and covariant bases to express strain and stress, this method avoids the complex calculations of high-order derivatives in traditional methods and improves the calculation efficiency. The natural stabilization of boundary conditions is another advantage of the meshless thin-shell formulation. The proposed formulation has naturally stable terms and does not require the addition of any artificial stabilization factors, which is different from traditional penalty function methods and eliminates the need for artificial parameters. Variational consistency is the key to maintaining the accuracy of the numerical solution. Through the use of smoothed gradients, the meshless method has quasi-variational consistency within each integration cell, which is crucial for ensuring the accuracy of the numerical solution. The effectiveness of the proposed method is verified through a series of classical standard thin-shell problems, including linear and nonlinear problems. In terms of implementing natural boundary conditions, this technique can evaluate all essential boundary conditions, including displacement and rotation, in the Hu-Washizu weak form. This method is similar to the Nitsche method but does not require any artificial parameters. Finally, the optimized integration scheme is another key advantage of the meshless method. To be consistent with the construction of smoothed gradients, this technique has an optimized integration scheme that reduces the computational cost of traditional meshless shape functions and their first derivatives.

[0042] The meshless simulation analysis method for thin plate shells provides an efficient and accurate new method for thin-shell structure analysis through its high flexibility, natural boundary condition enforcement, variational consistency, numerical verification, and optimized integration scheme. These advantages together address the challenges encountered by traditional finite element methods in dealing with the membrane locking problem.

[0043] Please refer to Figure 10 , the second embodiment of the present invention provides a meshless simulation analysis device for thin plate shells, which includes: The Galerkin weak form construction unit 201 is used to obtain the information of the thin plate shell to be analyzed, and based on the Hu-Washizu variational principle and the information of the thin plate shell, establish the Galerkin weak form of the elasticity problem; The displacement approximation acquisition unit 202 is used to obtain the preset meshless nodes, discretize the middle surface of the thin plate shell to be analyzed through the meshless nodes, and establish the reproducing kernel meshless approximation of the displacement; The strain approximation acquisition unit 203 is used to obtain the preset background integration domain, and within the background integration domain, construct the approximation of the strain using polynomials and covariant bases of corresponding orders; The discrete control unit 204 is used to substitute the reproducing kernel meshless approximation of the displacement and the approximation of the strain into the Galerkin weak form to obtain a stiffness matrix data set, and obtain the traditional force vector. According to the traditional force vector and the stiffness matrix data set, perform discrete control preprocessing to obtain a discrete control equation; The result generation unit 205 is used to perform numerical integration processing on the discrete control equation to obtain the integrated discrete control equation, and solve the integrated discrete control equation to obtain the simulation analysis result.

[0044] The above is the preferred embodiment of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the principle of the present invention, several improvements and refinements can be made, and these improvements and refinements are also regarded as the protection scope of the present invention.

Claims

1. A meshless simulation analysis method for thin plates and shells, characterized in that: include: Obtain the thin plate and shell information to be analyzed, and establish the Galerkin weak form of the elastic mechanics problem based on the Hu-Washizu variational principle and the thin plate and shell information; Obtaining preset meshless nodes, discretizing the mid-surface of the thin plate and shell to be analyzed through the meshless nodes, and establishing a reproducing core meshless approximation of displacement; Obtaining a preset background integral domain, and constructing an approximation of strain using a polynomial of a corresponding order and a covariant basis within the background integral domain; Substituting the reproducing kernel meshless approximation of the displacement and the approximation of the strain into the Galerkin weak form to obtain a stiffness matrix data set, and obtaining a traditional force vector, performing discrete control preprocessing according to the traditional force vector and the stiffness matrix data set to obtain a discrete control equation; The discrete control equation is numerically integrated to obtain an integrated discrete control equation, and the integrated discrete control equation is solved to obtain a simulation analysis result.

2. The thin plate and shell meshless simulation analysis method according to claim 1 is characterized in that: The mathematical expression formula of the Galerkin weak form is: ； in, is the thickness of the shell of the thin plate shell, are the components of the fourth-order elasticity tensor with respect to the covariant basis and plane stress assumption, is the displacement of the shell of the thin plate shell, is the body force acting on the shell of the thin plate shell, is the integral number, For variable semicolon, , are the film strain components, , are all buckling strain components.

3. The thin plate and shell meshless simulation analysis method according to claim 2 is characterized in that: The mathematical expression of the reproducing kernel meshless approximation of the displacement is: ； in, is the mesh-free shape function, is the displacement node coefficient, is the total number of nodes.

4. The thin plate and shell meshless simulation analysis method according to claim 3 is characterized in that: The strain approximation includes film strain and buckling strain. The mathematical expression of the film strain is: ； The mathematical expression of buckling strain is: ； in, ; ； Indicates I Approximate film strain associated with each node; ； Indicates I The approximate bending strain associated with each node is is the transpose of the linear polynomial vector used to describe the background integration unit; The mathematical expressions of membrane strain and buckling strain at the boundary conditions are: and ; in, ; ； ； ； ； is a polynomial vector, , , , , and are integral constraint terms, is the moment matrix, , , are the covariant basis vectors of the mid-surface of the shell.

5. The thin plate and shell meshless simulation analysis method according to claim 4 is characterized in that: Substituting the reproducing kernel meshless approximation of the displacement and the strain approximation into the Galerkin weak form, a stiffness matrix data set is obtained, specifically: Substituting the reproducing kernel meshless approximation of the displacement and the strain approximation into the Galerkin weak form, the traditional stiffness matrix is ​​obtained K , and the stability term stiffness matrix corresponding to the imposed essential boundary conditions and the consistency term stiffness matrix ; According to the traditional stiffness matrix K Get the components of the traditional stiffness matrix: ； in, , indicating that J Approximate film strain associated with each node; ； ； , are the covariant basis vectors of the mid-surface of the thin shell; Indicates J Approximate bending strains associated with each node; ； , , All of them are integral constraint terms; According to the stability term stiffness matrix Get the components of the stabilizing term stiffness matrix: ； in, is the displacement boundary, is the corner boundary, is the displacement boundary at the corner point, is a meshless shape function based on the reproducing kernel approximation, Represents mesh-free shape function The partial derivative with respect to the coordinates, It means integrating the physical quantity distributed on the boundary; in, ; ； ； ； ； ； ； ； ； ； , , are integral constraint terms, is the buckling strain component, is the component of the external normal vector in the covariant basis, is the tangent vector Covariant basis vectors in the midplane The amount of is the component of the normal direction vector outside the boundary According to the consistent terms stiffness matrix Get the components of the consistent term stiffness matrix: ； in, is a meshless shape function based on the reproducing kernel approximation, Represents mesh-free shape function Partial derivatives with respect to coordinates; ； ； ； ； ； ； ； is the tangent vector Covariant basis vectors in the midplane The amount of , are the components of the external normal vector in the covariant basis.

6. The thin plate and shell meshless simulation analysis method according to claim 5 is characterized in that: The traditional force vector is obtained, and discrete control preprocessing is performed according to the traditional force vector and the stiffness matrix data set to obtain the discrete control equation, which is specifically: Get the traditional force vector f , and construct the components of the traditional force vector according to the external boundary conditions and body force distribution: ； in, For surface force, is the external normal bending moment, is the external concentrated load, is the stress boundary, is the bending moment boundary, is the concentrated load angle boundary; Perform boundary integral calculation to obtain the corresponding stable force vector and the consistent force vector ; According to the stability term force vector Get the components of the stabilizing force vector: ； in, is the known boundary displacement, is a known boundary corner; According to the consistent term force vector Get the components of the consistent force vector: ； According to the traditional stiffness matrix K , Stability stiffness matrix , consistent term stiffness matrix , Traditional force vector f , stabilizing force vector and the consistent force vector , assemble to get the discrete control equation: 。 7. The thin plate and shell meshless simulation analysis method according to claim 6 is characterized in that: The discrete control equation is numerically integrated to obtain the integrated discrete control equation, and the integrated discrete control equation is solved to obtain the simulation analysis results, which are specifically: The components of the traditional stiffness matrix Perform numerical integration to obtain the components of the integrated traditional stiffness matrix: ； in, is the total number of integration units, is the number of integral points, is the integral point, is the weight; Components of the stiffness matrix for the stability term Perform numerical integration to obtain the components of the integrated stability term stiffness matrix: ； Components of the stiffness matrix for the consistent terms Perform numerical integration to obtain the components of the integral consistent stiffness matrix: 。 8. The thin plate and shell meshless simulation analysis method according to claim 7 is characterized in that: Also includes: Components of the traditional force vector Perform numerical integration to obtain the components of the integrated traditional force vector: ； Components of the stabilizing force vector Perform numerical integration to obtain the components of the integrated stabilizing force vector: ； Components of the force vector for the consistent term Perform numerical integration to obtain the components of the integrated consistent force vector: ； According to the components of the integrated traditional stiffness matrix , the components of the integrated stability term stiffness matrix , the components of the integral consistent term stiffness matrix , the components of the traditional force vector after integration , the components of the integrated stabilizing force vector , the components of the force vector after integration , and obtain the integrated discrete control equation; The preset linear algebra library is used to solve the integrated discrete control equations to obtain approximate displacement node coefficients and generate simulation analysis results.

9. A thin plate and shell meshless simulation analysis device, characterized in that: include: Galerkin weak form construction unit is used to obtain the thin plate and shell information to be analyzed. Based on the Hu-Washizu variational principle and the thin plate and shell information, the Galerkin weak form of the elastic mechanics problem is established; A displacement approximation acquisition unit, used for acquiring preset meshless nodes, discretizing the mid-surface of the thin plate and shell to be analyzed through the meshless nodes, and establishing a reproducing core meshless approximation of the displacement; A strain approximation acquisition unit, used for acquiring a preset background integral domain, and constructing an approximation of the strain using a polynomial of a corresponding order and a covariant basis in the background integral domain; A discrete control unit, used for substituting the reproducing kernel meshless approximation of the displacement and the approximation of the strain into the Galerkin weak form to obtain a stiffness matrix data set, and obtaining a traditional force vector, and performing discrete control preprocessing according to the traditional force vector and the stiffness matrix data set to obtain a discrete control equation; The result generating unit is used to perform numerical integration processing on the discrete control equation to obtain the integrated discrete control equation, and solve the integrated discrete control equation to obtain the simulation analysis result.

Images