Kirchhoff sheet response adaptive analysis method based on virtual unit method

Abstract

The invention discloses a Kirchhoff sheet response adaptive analysis method based on a virtual unit method, and the method comprises the steps: carrying out the virtual unit discretization of a Kirchhoff sheet structure, constructing a virtual unit space, and calculating a unit stiffness matrix and a mass matrix; evaluating a spatial discrete error by adopting a layered grid method, and carrying out grid adaptive refinement on an initial grid; calculating the acceleration, evaluating the time discrete error of the virtual unit, establishing a time discrete error standard to adjust the time step length, and carrying out the adaptive adjustment of the time step length; and integrating evaluation results of the spatial discrete error and the time discrete error, controlling grid refinement and time step length adjustment, and realizing time-space adaptive analysis of the Kirchhoff thin plate response. According to the method, the flexibility and adaptability of the grid are effectively improved, the calculation efficiency is remarkably improved, and the method can be widely applied to thin plate structure analysis and design in the fields of machinery, buildings, bridges, aerospace and the like and has important engineering application value.

Description

Technical Field

[0001] The present invention belongs to the fields of computational mechanics and numerical analysis, and particularly relates to an adaptive analysis method for the response of thin plates in Kirchhoff engineering structures based on the virtual element method. Background Art

[0002] Thin plate structures are widely used in many engineering fields due to their light weight and high strength. Under the action of transverse loads, the bending and deflection of thin plates become key indicators for evaluating their deformation degree, while the buckling phenomenon that may be caused by external loads affects the stability of the structure. Although the finite element method (FEM) is widely used in thin plate analysis, it has limitations in dealing with complex geometries and boundary conditions, and the calculation accuracy and convergence speed are limited when the mesh quality is poor.

[0003] With the continuous development of engineering technology, thin plate structures, as important engineering components, are widely used in fields such as aerospace, machinery, architecture, and bridges. In order to improve the accuracy of response analysis of thin plate structures under static and dynamic loads, although the traditional finite element analysis method has been widely used, it still has certain limitations in dealing with complex geometries and boundary conditions.

[0004] Therefore, there is an urgent need to provide an efficient numerical simulation method that combines the virtual element method (VEM) and adaptive analysis to solve the limitations in dealing with complex geometries and boundary conditions. Summary of the Invention

[0005] To solve the above-mentioned defects in the prior art, the purpose of the present invention is to provide an adaptive analysis method for the response of Kirchhoff thin plates based on the virtual element method, which uses the virtual element method (Virtual Element Method, VEM) to numerically simulate the response of rectangular thin plates under static and dynamic loads, and optimizes the calculation process through an adaptive analysis system to be highly adaptable to the mesh and simplify complex geometric models. By using a projection operator mapping function, the isoparametric transformation is avoided, and the accuracy and calculation efficiency of thin plate response analysis are improved, accurately evaluating the response of thin plate structures under static and dynamic loads, and providing a scientific basis for engineering design and safety assessment.

[0006] The present invention is realized through the following technical solutions.

[0007] An adaptive analysis method for the response of Kirchhoff thin plates based on the virtual element method, comprising the following steps:

[0008] Discretize the Kirchhoff thin plate structure based on the virtual element method, construct a virtual element space, and calculate the element stiffness matrix and the element mass matrix;

[0009] Based on the element stiffness matrix, the hierarchical grid method is used to evaluate the spatial discretization error of the virtual element. By comparing the numerical solution differences between the fine grid and the initial grid, a spatial discretization error criterion is established, and the initial grid is refined and adjusted according to the spatial discretization error criterion;

[0010] Based on the element mass matrix, the Newmark integration method is used to calculate the acceleration. The time discretization error of the virtual element is evaluated by the reconstructed acceleration method. By comparing the differences between the acceleration calculated by the Newmark integration method and the acceleration evaluated by the reconstructed acceleration method, a time discretization error criterion is established. According to the time discretization error criterion, the time step is adjusted to achieve the adaptive adjustment of the time step;

[0011] Combining the adjustment results of the spatial discretization error and the time discretization error, the grid density and the time step are optimized in sequence until both the spatial discretization error and the time discretization error reach the preset error target, that is, the adaptive analysis of the Kirchhoff thin plate response is completed.

[0012] Preferably, a virtual element space is constructed, and the element stiffness matrix and the element mass matrix are calculated, including:

[0013] Based on the equilibrium differential equation of the Kirchhoff thin plate, virtual element discretization is performed, and the analysis region is discretized into non-overlapping polygonal virtual elements; combining the element centroid and the geometric boundary, local degrees of freedom are defined to form a virtual element space;

[0014] Using the elliptic projection operator, the displacement field inside the element is mapped to the polynomial space, and the virtual element stiffness matrix is calculated. The virtual element stiffness matrix includes the consistent stiffness matrix and the stabilization stiffness matrix;

[0015] Based on the virtual element shape function, the element mass matrix is calculated. The element mass matrix includes the consistent mass matrix and the stabilization mass matrix.

[0016] Preferably, the hierarchical grid method is used to evaluate the spatial discretization error of the virtual element, including:

[0017] The Kirchhoff thin plate is meshed by the hierarchical grid method. By increasing the number of grids, the error between the numerical solution and the analytical solution is compared, and the fine solution obtained on the fine grid is calculated;

[0018] An error indicator is constructed. Taking the fine grid solution of the Kirchhoff thin plate bending as a reference, the energy error distribution between the initial grid and the fine grid is calculated, and a spatial discretization error criterion is established.

[0019] Preferably, the initial grid is refined and adjusted according to the spatial discretization error criterion, including:

[0020] Mark the grid elements whose energy error exceeds the preset target value;

[0021] Perform local refinement on the marked cells, update the grid and recalculate the error distribution until the errors of all cells meet the target requirements.

[0022] Preferably, evaluate the time discretization error of the virtual element by the reconstructed acceleration method, including:

[0023] Use the Newmark integration method to calculate the acceleration within each time step Δt, and integrate it to calculate the state variables of the next step. Reconstruct the acceleration based on the known displacement and velocity data.

[0024] Taking the reconstructed acceleration as the standard, calculate the acceleration error at a certain moment within the time step Δt, integrate it to obtain the velocity error at a certain moment and the displacement error within the time step Δt, and calculate the relative energy error of time discretization within the time step Δt according to the obtained displacement error.

[0025] Preferably, adjust the time step according to the time discretization error criterion, including:

[0026] Set the acceptable range of the time discretization error;

[0027] Change the length of the time step to make the time discretization error in each time step within the acceptable range. If the time discretization error does not meet the target, adjust the time step until the time discretization error in all time steps meets the requirements.

[0028] Preferably, comprehensively consider the adjustment results of the spatial discretization error and the time discretization error, and optimize the grid density and time step in sequence, including:

[0029] Perform grid adaptive refinement according to the obtained spatial discretization error criterion;

[0030] Perform time step adaptation according to the obtained time discretization error criterion. If the time discretization error does not meet the target, adjust the time step, calculate the time discretization error of the adjusted time step, and continue to compare until the time discretization error in all time steps meets the requirements.

[0031] Due to the adoption of the above technical solutions, the present invention has the following beneficial effects:

[0032] 1. The present invention discretizes Kirchhoff thin plates using the virtual element method and optimizes the mesh and time step through an adaptive analysis method, thereby achieving high-precision analysis of the response of thin plate structures. On the one hand, it can effectively overcome the limitations of traditional finite element methods in dealing with complex geometries and boundary conditions, avoiding subjective intervention in the mesh generation and time step selection processes. On the other hand, by dynamically adjusting the mesh density and time step, it significantly improves the computational efficiency while ensuring computational accuracy, reduces the consumption of computational resources, and is applicable to the analysis of thin plate structures in engineering design.

[0033] 2. By introducing the hierarchical mesh method and the reconstructed acceleration method, the spatial discretization error and the time discretization error are evaluated. The spatial discretization error is evaluated by the difference in numerical solutions between the fine mesh and the initial mesh, and this information is used to adaptively optimize the mesh density; the time discretization error is evaluated by the difference in acceleration between the reconstructed acceleration and the acceleration of the Newmark integration method, and the time step is adaptively adjusted according to the error criterion. This method provides high-precision error control in the discretization processes of space and time, making the results of thin plate response analysis more reliable.

[0034] 3. When performing adaptive analysis, the present invention comprehensively considers the evaluation results of the spatial discretization error and the time discretization error, and optimizes the mesh density and time step in sequence to ensure high precision of the analysis. By this method, high-stress gradient regions can be accurately captured, and at the same time, the computational accuracy and efficiency are balanced in time integration, avoiding the problems of excessive computational cost and uneven accuracy caused by over-refined meshes or unreasonable time steps in traditional methods.

[0035] 4. While optimizing the computational accuracy, it can automatically adjust the mesh refinement and time step according to a preset error target, maximizing the computational efficiency of the analysis. This method is not only applicable to the response analysis of thin plates under static and dynamic loads, but also can be widely applied to the design and safety assessment of complex thin plate structures, and has important engineering application value.

[0036] The system of the present invention can be widely applied to the analysis of thin plate structures and the analysis of the strength, stability and vibration characteristics of thin plate structures in the design of mechanical engineering, civil engineering, bridge engineering, aerospace and electronic devices, etc., and has important engineering application value. Description of the Drawings

[0037] The drawings described herein are used to provide a further understanding of the present invention, form a part of this application, and do not constitute an improper limitation of the present invention. In the drawings:

[0038] Figure 1 is the flowchart of the adaptive analysis method for the response of Kirchhoff thin plates in the embodiments of the present invention;

[0039] Figure 2 It is a schematic diagram of the boundary conditions of the thin plate in the embodiment of the present invention;

[0040] Figure 3 It is a schematic diagram of the discrete polygon elements of the thin plate in the embodiment of the present invention;

[0041] Figure 4 It is a grid refinement strategy diagram in the embodiment of the present invention;

[0042] Figure 5 It is a schematic diagram of a single cycle process in the grid adaptive analysis in the embodiment of the present invention;

[0043] Figure 6 It is a basic flowchart of grid adaptivity in the embodiment of the present invention;

[0044] Figure 7 It is an analysis flowchart of the adaptive step size in the embodiment of the present invention;

[0045] Figure 8 It is a basic flowchart of the time - space adaptive analysis in the embodiment of the present invention;

[0046] Figure 9 It is a schematic diagram of the load conditions in the embodiment of the present invention;

[0047] Figures 10(a)-(b) are schematic diagrams of the Kirchhoff thin plate model and the initial grid in the embodiment of the present invention;

[0048] Figure 11 (a)-(d) are the grids of the adaptive virtual element method and the bending moment M of the Kirchhoff thin plate at some moments in the embodiment of the present invention x distribution diagram;

[0049] Figure 12 (a)-(c) are schematic diagrams of the coarse, medium, and fine grids in the embodiment of the present invention respectively;

[0050] Figure 13 It is the bending moment M of the thin plate at point B under the coarse, medium, and fine grids in the embodiment of the present invention x time - history response schematic diagram;

[0051] Figure 14 It is a schematic diagram of the distribution of the degrees of freedom at each time step of the proposed adaptive method in the embodiment of the present invention;

[0052] Figure 15 It is a schematic diagram of the comparison of the relative errors between each time step in the adaptive analysis and the analysis with a fixed time step length in the embodiment of the present invention. Detailed implementation manners

[0053] The present invention will be described in detail below in conjunction with the accompanying drawings and specific embodiments. The illustrative embodiments of the present invention and the description are used to explain the present invention, but do not limit the present invention.

[0054] An adaptive analysis method for the response of Kirchhoff thin plates based on the virtual element method provided by an embodiment of the present invention, as Figure 1 shown, includes the following steps:

[0055] Step 1, discretize the Kirchhoff thin plate structure based on the virtual element method, construct a virtual element space, and calculate the element stiffness matrix and the element mass matrix.

[0056] The specific steps are as follows:

[0057] 1a) According to the equilibrium differential equation of the Kirchhoff thin plate, perform virtual element discretization, discretize the analysis region into non-overlapping polygonal virtual elements; combine the element centroid and the geometric boundary, define local degrees of freedom, and form a virtual element space.

[0058] The equilibrium differential equation of the Kirchhoff thin plate is:

[0059]

[0060] Wherein, is the material elasticity matrix, Δ 2 is the bi-Laplacian operator, w is the deflection function, and f is the transverse load received by the Kirchhoff thin plate.

[0061] Write the equilibrium differential equation of the Kirchhoff thin plate in variational form, that is, find the deflection function w:

[0062]

[0063] Wherein, a(u,v) and L(v) are the bilinear and linear forms of the two-dimensional linear elastic problem respectively, v and are the test function and the vector function space respectively;

[0064] For the rectangular thin plate OABC in Figure 2 , illustrate three boundary conditions, and the boundary conditions can be expressed as:

[0065]

[0066] Where: n is the outer normal direction, is the boundary of the Kirchhoff thin plate, and at the same time, it is assumed that the entire boundary of the thin plate is fixed and the boundary deflection w is 0.

[0067] Adopt the virtual element method to discretize the Kirchhoff thin plate into a finite number of non-overlapping polygonal elements K. Represents the edge of the element, x K and y K Represents the centroid coordinates of the element, h K Represents the diameter of the element, |K| represents the area of the element, and the polygonal element is as Figure 3 .

[0068] The virtual element space order for the lowest-order Kirchhoff thin plate bending problem is k = 2, and the corresponding local virtual element space is:

[0069]

[0070] Among them, Represents the second-order virtual element space, H 2 (K) is the Sobolev space, referring to the function space with a complete inner product, v| E ∈P3(E) means that on the element E, within the element, v| E Can be represented by a cubic polynomial, Represents on the boundary of the element E, v| E The normal derivative of, which means the normal derivative is a linear function.

[0071] 1b) According to the constructed virtual element space, using the elliptic projection operator, map the displacement field inside the element to the polynomial space and calculate the virtual element stiffness matrix, including the consistent stiffness matrix and the stabilization stiffness matrix.

[0072] Introduce the elliptic projection operator Discretize the variational form of the control equation, and thus obtain the polynomial projection of the function in the virtual element space:

[0073]

[0074] Among them, a K Represents the bilinear form on the polygonal element K, Represents the gradient calculation, Is the virtual element space of order k, P k (K) is the polynomial space of order k in the corresponding required element space, and p is a basis function in the polynomial space P k (K).

[0075] Element shape function The expression is:

[0076]

[0077] Among them, Is a constant, N dof Is the total number of degrees of freedom of the element, p α Is the polynomial space Pk The basis functions in (K), α represents the degree of the polynomial, n k is the dimension of the polynomial space P k (K), when k = 2, n k = 6;

[0078] The stiffness matrix of the Kirchhoff thin plate element in the virtual element form consists of two parts:

[0079]

[0080] Among them, is the consistent stiffness matrix, is the stabilization stiffness matrix. The consistent stiffness matrix is used to describe the accuracy of the physical field, and the stabilization stiffness matrix is used to ensure the stability of numerical calculations.

[0081] Consistent stiffness matrix The elements in it are calculated as follows:

[0082]

[0083] Among them, is the element in the i-th row and j-th column of the consistent stiffness matrix, and are constants used to represent the linear combination of shape functions, G is a matrix representing the inner product of shape functions, and are the element shape functions, p α and p β are the basis functions in the polynomial space P k (K), α and β represent the degrees of the polynomials, is the matrix form of the projection operator , N dof represents the total number of degrees of freedom of the element.

[0084] The stabilization stiffness matrix is calculated as follows:

[0085]

[0086] Among them, h K represents the element diameter, I represents the identity matrix, and the matrix D represents the degree-of-freedom matrix of the basis functions in the polynomial space:

[0087]

[0088] Among them, N v is the number of element vertices, dof i represents the i-th degree of freedom.

[0089] The elements in the degree-of-freedom matrix D can be obtained according to the selected degrees of freedom:

[0090]

[0091] where dof i (p) is the value of the basis function p at the node (x i , y i ). is the degree of freedom of the derivative of the basis function p with respect to the x - direction at node i, is the degree of freedom of the derivative of the basis function p with respect to the y - direction at node i, and h ξ is the characteristic length of the element vertex.

[0092] 1c) Calculate the element mass matrix according to the constructed virtual element space.

[0093] The element consistent mass matrix M in the finite element method K can be calculated by integrating the shape function of element K and the material density ρ:

[0094]

[0095] Introduce the L2 projection operator Π 0 : Calculate the consistent mass matrix in the virtual element form, which satisfies the following orthogonality condition:

[0096] a K (Π 0 v - v, p) = 0, p ∈ P k (K)

[0097] According to the calculation method of the element consistent mass matrix in the finite element method, use the projected shape function and the stabilization part to replace the original shape function

[0098] The element consistent mass matrix M K can be expressed as:

[0099]

[0100] Similar to the element stiffness matrix, the element mass matrix consists of the consistent mass matrix and the stabilization mass matrix :

[0101]

[0102] where the elements in the consistent mass matrix can be calculated by the following formula:

[0103]

[0104] Among them, is the element of the \(i\)-th row and \(j\)-th column of the consistent stiffness matrix, is the matrix form of \(\Pi\) 0 and and are the shape functions of element \(K\), and \(H\) is a matrix representing the inner product of the shape functions.

[0105] Stable mass matrix can be obtained through approximate calculation:

[0106]

[0107] Among them, \(|K|\) is the element area, and the matrix \(\Pi\) 0 can be calculated through the matrix and the degree-of-freedom matrix \(D\):

[0108]

[0109] Through the decomposition design of the consistency and stability components, the accuracy and stability of the system in dynamic analysis are guaranteed.

[0110] Step 2: Based on the element stiffness matrix, use the hierarchical grid method to evaluate the spatial discretization error of the virtual element. By comparing the numerical solution differences between the fine grid and the initial grid, establish a spatial discretization error criterion, and refine and adjust the initial grid according to the spatial discretization error criterion to achieve grid adaptive refinement.

[0111] The specific steps are as follows:

[0112] A. Establish a spatial discretization error criterion based on the obtained element stiffness matrix.

[0113] 2a) Use the hierarchical grid method to divide the grid of the Kirchhoff thin plate. By increasing the number of grids, compare the error between the numerical solution and the analytical solution, and calculate the fine solution obtained on the fine grid.

[0114] Taking the analytical solution as the standard, continuously refine the grid to reduce the error between the numerical solution and the analytical solution, and calculate the fine solution \(u\) obtained on the fine grid.

[0115] 2b) Construct an error index. Taking the fine grid solution of the Kirchhoff thin plate bending problem as a reference, calculate the energy error distribution between the initial grid and the fine grid, and establish a spatial discretization error criterion.

[0116] Assume that for the numerical solution \(u\) h in the Kirchhoff thin plate bending problem, there is the following error \(e\) with the fine solution \(u\):

[0117] e = u - uh

[0118] Then the energy norm of the error e is:

[0119]

[0120] where Ω represents the domain where the Kirchhoff thin plate is located; m is the total number of discretized grids; κ and respectively represent the true solutions of the generalized strain and the generalized stress; κ h and respectively represent the numerical solutions of the generalized strain and the generalized stress; K is any one of the discretized elements.

[0121] For any element K, its energy norm of the error ||e|| K is:

[0122]

[0123] Adopt the hierarchical grid method to obtain the refined solution on the refined grid to replace the true solution and further calculate the energy error as:

[0124]

[0125] Then the total energy error ||e|| of the system at this time Ω is:

[0126]

[0127] Let the target value of the total error index be The target value of the element error is Its expression is:

[0128]

[0129] The global error η at this time Ω can be calculated through the total energy error ||e|| Ω and the system energy norm ||u|| as:

[0130]

[0131] B. Refine and adjust the initial grid according to the spatial discretization error criterion to achieve grid adaptive refinement.

[0132] 2c) Mark the grid elements whose energy error exceeds the preset target value;

[0133] 2d) Locally refine the marked elements, update the grid and recalculate the error distribution until the errors of all elements meet the target requirements.

[0134] Step 3: Based on the element mass matrix, use the Newmark integration method to calculate the acceleration. Evaluate the time discretization error of the virtual element through the reconstructed acceleration method. By comparing the difference between the acceleration calculated by the Newmark integration method and the acceleration evaluated by the reconstructed acceleration method, establish a time discretization error criterion. According to the time discretization error criterion, adjust the time step to achieve the adaptive adjustment of the time step and realize the adaptive adjustment of the time step.

[0135] The specific steps are as follows:

[0136] A. Establish a time discretization error criterion based on the obtained element mass matrix.

[0137] 3a) Adopt the Newmark integration method. Within each time step Δt, regard the acceleration and velocity as constants and integrate them to calculate the state variables of the next step. Reconstruct the acceleration based on the known displacement and velocity data.

[0138] Assume that within a time step [t, t + Δt], the recovered acceleration is linearly distributed. The reconstructed acceleration of the system at time τ is:

[0139]

[0140] where and are the accelerations at times t + Δt and t respectively; τ is a moment within the time step [t, t + Δt].

[0141] 3b) Taking the reconstructed acceleration as the standard, calculate the acceleration error at time τ within the time step Δt as:

[0142]

[0143] The acceleration error Integrating with respect to time, the velocity error at time τ can be obtained as:

[0144]

[0145] The error e(U t+Δt ) caused by displacement within the time step [t, t + Δt] can be obtained by integrating the velocity error with respect to time:

[0146]

[0147] The local time discretization error within the time [t, t + Δt] can be expressed in the form of energy norm as:

[0148]

[0149] Among them, K represents the system stiffness matrix; ||e(U t+Δt )|| is the energy norm of the displacement error within the time step [t, t+Δt].

[0150] The relative energy error η of time discretization within the time step [t, t+Δt] t can be expressed as:

[0151]

[0152] Among them, ||U|| max represents the energy norm corresponding to the maximum displacement of the system within the total calculation time.

[0153] B. Adjust the time step according to the time discretization error criterion.

[0154] 3c) Set the acceptable range of the time discretization error;

[0155] 3d) Change the length of the time step to make the time discretization error in each time step within the acceptable range. If the time discretization error does not meet the target, adjust the time step until the time discretization error in all time steps meets the requirements.

[0156] Step 4, comprehensively consider the adjustment results of the spatial discretization error and the time discretization error, and optimize the mesh density and the time step in turn until the spatial discretization error and the time discretization error reach the preset error target, and complete the adaptive analysis of the Kirchhoff thin plate response.

[0157] The specific steps are as follows:

[0158] 4a) Perform mesh adaptive refinement according to the obtained spatial discretization error criterion.

[0159] By comparing the numerical solutions obtained on the initial mesh and the fine mesh, mark the elements in the initial mesh that do not meet the criterion, refine the marked elements and then recalculate to obtain a higher-precision numerical solution. As Figure 4 shown, after completing the refinement of element ①, hanging nodes appear on the adjacent meshes. Use the shape function interpolation method to extrapolate the state variables at the new nodes with the state variables and element shape functions at the nodes obtained on the initial mesh and

[0160]

[0161] First, solve the numerical solution M on the initial Mesh0 h, based on the initial mesh, a refined mesh CompareMesh0 is obtained through refinement, and the refined solution on CompareMesh0 is calculated. Compare the error between the numerical solutions obtained from the initial mesh and the refined mesh, mark and refine the elements that do not meet the error requirements, and obtain the refined mesh Mesh1 after one cycle as Figure 5 shown. Repeat the above process until all meshes meet the error requirements, as Figure 6 shown.

[0162] 4b) According to the obtained time discretization error criterion, perform time-step adaptivity.

[0163] Change the length of the time step to keep the time discretization error η in each time step within an acceptable range:

[0164]

[0165] where is the target value of the time discretization error, and represent the upper and lower limits of the acceptable error respectively, and γ1, γ2 are the upper and lower limit parameters.

[0166] If the time discretization error η does not meet the target, the time step size needs to be adjusted, calculate the time discretization error η of the adjusted time step size, and continue the comparison until the time discretization error in all time steps meets the requirements. For the Newmark method, its local error should have a convergence rate of third order O(Δt 3 ), and when the time discretization error η does not meet the requirements, there is a new time step size The specific steps are as Figure 7 shown.

[0167] As Figure 8 shown, by integrating the results of time-step adaptivity refinement and mesh adaptivity refinement, repeat the process until all indicators meet the error criterion, and then realize the adaptive response analysis of the Kirchhoff thin plate.

[0168] The following further illustrates the present invention through specific embodiments.

[0169] Input the material properties of the Kirchhoff thin plate in Matlab, as shown in Table 1.

[0170] Table 1 Material properties of Kirchhoff thin plate

[0171]

[0172] Input the load on the surface of the Kirchhoff thin plate as Figure 9The uniformly distributed load shown is used to calculate the response of the structure within 0 - 2.4 s, with a fixed time step of Δt = 0.05 s. The Kirchhoff thin plate model and the initial mesh are shown in Figures 10(a) and (b). Among them, it is set that the target error for time step discretization is 1%, and the values of the calculation parameters γ1 and γ2 are 0.9 and 1.1 respectively, and the standard of the element spatial discretization error is 1%.

[0173] Figure 11 (a)-(d) show the meshes of the adaptive virtual element method and the bending moment M x distribution of the Kirchhoff thin plate at some moments. It can be seen from the regions with mesh refinement that the regions with high bending moment gradients are mainly concentrated near the edge on the side with fixed constraints, while the meshes on the side far from the fixed constraints and with gentle bending moment changes are not refined multiple times.

[0174] Calculate respectively Figure 12 the time - history responses of the bending moment M at point B of the thin plate under the three types of coarse, medium, and fine meshes shown in (a)-(c), x and compare them with the calculation results of the proposed method. The results are as Figure 13 shown. It can be seen that the bending moment response of point B obtained based on the space - time adaptive analysis is similar to the result obtained with the fine mesh, and is better than the results of the coarse mesh and the medium mesh.

[0175] Figure 14 Shows the change in the number of degrees of freedom at each time step of the adaptive method proposed in the present invention. It can be seen from the figure that in this example, the number of degrees of freedom remains at 700 - 800 for most time steps, and a relatively small number of degrees of freedom can be used to obtain a solution with high accuracy. Figure 15 Shows the comparison of the relative errors of each time step in the adaptive analysis and the fixed - time - step analysis. It can be seen that the time discretization errors of the adaptive variable - step - size analysis have reached the preset target error, and compared with the fixed - step - size analysis, the distribution of its time discretization errors is more uniform. In addition, when the fixed time step is 0.05 s, the total number of time steps is 50 steps, while the total number of steps using the variable - step - size analysis is 45 steps, which improves the calculation speed.

[0176] It can be seen from the above examples that the method proposed in the present invention can obtain better results with fewer degrees of freedom compared with the traditional virtual element method with uniformly refined meshes at a fixed time step, and can well balance the calculation accuracy and calculation efficiency.

Claims

1. A Kirchhoff thin plate response adaptive analysis method based on imaginary element method, characterized in that: The following steps are involved: The Kirchhoff thin plate structure is discretized based on the virtual element method, the virtual element space is constructed, and the element stiffness matrix and element mass matrix are calculated; Based on the element stiffness matrix, a hierarchical grid method is used to evaluate the spatial discretization error of the virtual element, and a spatial discretization error standard is established by comparing the difference in numerical solutions between the fine grid and the initial grid. The initial grid is refined and adjusted according to the spatial discretization error standard. Based on the unit mass matrix, the acceleration is calculated by the Newmark integral method, the time discretization error of the imaginary unit is evaluated by the reconstruction acceleration method, and the time discretization error standard is established by comparing the difference between the acceleration calculated by the Newmark integral method and the acceleration evaluated by the reconstruction acceleration method. According to the time discretization error standard, the time step is adjusted to achieve adaptive adjustment of the time step; Based on the adjustment results of spatial discretization error and temporal discretization error, the grid density and time step are optimized in turn until both spatial discretization error and temporal discretization error reach the preset error targets, thus completing the adaptive analysis of Kirchhoff thin plate response.

2. The Kirchhoff thin plate response adaptive analysis method based on the imaginary element method according to claim 1 is characterized in that: The construction of the virtual unit space and the calculation of the unit stiffness matrix and the unit mass matrix include: Based on the equilibrium differential equation of Kirchhoff thin plate, virtual unit discretization is performed to discretize the analysis area into non-overlapping polygonal virtual units; the local degree of freedom is defined by combining the unit centroid and geometric boundary to form a virtual unit space; Using the elliptic projection operator, the displacement field inside the unit is mapped to the polynomial space, and the virtual unit stiffness matrix is ​​calculated. The virtual unit stiffness matrix includes the consistent stiffness matrix and the stable stiffness matrix. Based on the imaginary element shape function, the element mass matrix is ​​calculated, and the element mass matrix includes the consistent mass matrix and the stable mass matrix.

3. The Kirchhoff thin plate response adaptive analysis method based on the imaginary unit method according to claim 2 is characterized in that: Consistent stiffness matrix The elements in are calculated as follows: in, is the element in the i-th row and j-th column of the consistent stiffness matrix; a K represents the bilinear form on polygonal element K, and is the unit shape function, and is a constant used to represent the linear combination of shape functions, p α and p β The polynomial space P k (K) is the basis function, α and β represent the degree of the polynomial, n k is the polynomial space dimension, is the material elastic matrix, G is a matrix representing the inner product of shape functions, N dof represents the total number of degrees of freedom of the element, represents the gradient operation, is the elliptic projection operator, It is the projection operator The matrix form of Stable stiffness matrix The calculation is as follows: Among them, h K is the unit diameter, I is the unit matrix, and D is the degree of freedom matrix.

4. The Kirchhoff thin plate response adaptive analysis method based on the imaginary unit method according to claim 3 is characterized in that: Consistent mass matrix Calculated by the following formula: in, is the element in the i-th row and j-th column of the consistent mass matrix, ρ is the material density, and is the shape function of unit K, Π 0 is the L2 projection operator, Yes 0 In matrix form, H is a matrix representing the inner product of shape functions, N dof Represents the total number of degrees of freedom of the unit; Stable mass matrix Calculated by the following formula: Where |K| is the unit area.

5. The Kirchhoff thin plate response adaptive analysis method based on imaginary element method according to claim 1 is characterized in that: A hierarchical grid method is used to evaluate the spatial discretization error of the imaginary element, including: The Kirchhoff thin plate is meshed by using the hierarchical mesh method. By increasing the number of meshes, the error between the numerical solution and the analytical solution is compared, and the refined solution on the fine mesh is calculated. The error index is constructed, and the energy error distribution between the initial grid and the fine grid is calculated using the fine grid solution of Kirchhoff thin plate bending as a reference, and the spatial discretization error standard is established.

6. The Kirchhoff thin plate response adaptive analysis method based on imaginary element method according to claim 1 is characterized in that: The initial grid is refined according to the spatial discretization error criteria, including: Mark the grid cells whose energy errors exceed the preset target value; The marked cells are locally refined, the mesh is updated and the error distribution is recalculated until the errors of all cells meet the target requirements.

7. The Kirchhoff thin plate response adaptive analysis method based on imaginary element method according to claim 1 is characterized in that: The time discretization error of imaginary units is evaluated by reconstructing the acceleration method, including: The Newmark integration method is used to calculate the acceleration in each time step Δt, and the state quantity of the next step is calculated by integrating it, and the acceleration is reconstructed based on the known displacement and velocity data; Taking the reconstructed acceleration as the standard, the acceleration error at a certain moment within the time step Δt is calculated, and the velocity error at a certain moment and the displacement error within the time step Δt are obtained by integrating it. The relative energy error of the time discretization within the time step Δt is calculated based on the obtained displacement error.

8. The Kirchhoff thin plate response adaptive analysis method based on the imaginary element method according to claim 7 is characterized in that: Calculate the acceleration error at a certain moment within the time step Δt for: in, and are the accelerations at time t+Δt and time t respectively; τ is a moment within the time step [t, t+Δt]; is the reconstructed acceleration of the system at time τ; Speed ​​error at time τ for: in, is the acceleration error at time τ; The displacement error e(U t+Δt ): in, is the speed error at time τ; The relative energy error η of the time discretization within Δt t for: Among them, ||U|| max is the energy norm of the system corresponding to the maximum displacement in the total calculation time, e(U t+Δt ) is the energy norm of the displacement error in the time step [t, t+Δt].

9. The Kirchhoff thin plate response adaptive analysis method based on imaginary element method according to claim 1 is characterized in that: Adjust the time step size based on the time discretization error criteria, including: Set the acceptable range of time discretization error; Change the length of the time step to make the time discretization error in each time step within an acceptable range. If the time discretization error does not meet the target, adjust the time step until the time discretization error in all time steps meets the requirements.

10. The Kirchhoff thin plate response adaptive analysis method based on imaginary element method according to claim 1 is characterized in that: Based on the adjustment results of spatial discretization error and temporal discretization error, the grid density and time step are optimized in turn, including: According to the obtained spatial discretization error standard, the grid is adaptively refined; According to the obtained time discretization error standard, the time step is adapted. If the time discretization error does not meet the target, the time step is adjusted, the time discretization error of the adjusted time step is calculated, and the comparison is continued until the time discretization error in all time steps meets the requirements.

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